Existence and Convergence of Ground State Solutions for a ( p,q )-Laplacian System on Weighted Graphs

Abstract

We investigate the existence of ground state solutions for a $(p, q)$ -Laplacian system with $p, q > 1$ and potential wells on a weighted locally finite graph $G = (V, E)$ . By making use of the method of Nehari manifold and the Lagrange multiplier rule, we prove that if the nonlinear term $F$ takes on the super- $(p, q)$ -linear growth and the potential functions $a (x)$ and $b (x)$ satisfy some suitable conditions, then for any fixed parameter $λ \geq 1$ , the system is provided with a ground state solution $(u_{λ}, v_{λ})$ . Additionally, we set up the convergence property of the solutions set ${(u_{λ}, v_{λ})}$ when $λ \to + \infty$ .

Keywords

Ground state solution locally finite graph Nehari manifold

1. Introduction

In recent years, the analysis on graphs has attracted some attentions. For example, Bauer et al. (2015) proved a discrete version of Li–Yau inequality valid for solutions to the heat equation on graphs. Horn et al. (2019) proved a Gaussian estimate for the heat kernel, along with Poincaré and Harnack inequalities. It is remarkable that some analysis on graphs has been applied to the investigation of machine learning, data analysis, neural network, image processing, etc. (e.g., see Alkama et al., 2014; Arnaboldi et al., 2015; Elmoataz & Toutain M, 2015; Ta et al., 2010, 2008).

Next, we would like to address some investigations of the partial differential equations on graphs. For example, the heat equations (Huang, 2012), the Liouville type equations (Ge et al., 2018), the Fokker-Planck equation (Chow et al., 2017), and the Schrödinger equation (Chow et al., 2019). Especially, Grigoryan et al. (2017) studied the following nonlinear equation on a locally finite graph $G = (V, E)$ :

- Δ u + h u = f (x, u) in V,

(1.1)

where

Δ

is a graph Laplacian defined (1) below,

V

is the vertex set, and

E

is the edge set,

h : V \to R

satisfies some suitable assumptions and

f

satisfies the superquadratic growth condition. They established some existence results via mountain pass theorem. Grigor’yan et al. (2016) established some existence results to the Yamabe type equation

\begin{aligned} {\begin{cases} - Δ u - α u = | u |^{p - 2} u, in Ω^{\circ}, \\ u = 0, on \partial Ω, \end{cases} \end{aligned}

by the mountain pass theorem, where

p \geq 2, α < λ_{1} (Ω)

which is the first eigenvalue of

- Δ

. They also considered a

p

-Laplacian equation with Dirichlet boundary value and a generalized poly-Laplacian equation with Dirichlet boundary value on weighted locally finite graph, and a

p

-Laplacian equation and a generalized poly-Laplacian equation on finite graph by the same method. Inspired by Grigor’yan et al. (2016), and Pinamonti & Stefani (2022) studied a poly-Laplacian equation and proved existence of weak solutions under different conditions. Besides, they obtained a uniqueness result for a

p

-Laplacian equation by the Brezis-Strauss theorem.

Besides, the existence of ground state solutions on graphs have also attracted some academics via the Nehari manifold method which was developed by Nehari (1959) firstly. Zhang & Zhao (2018) studied the convergence of ground state solutions for the following nonlinear Schrödinger equation on a locally finite graph $G = (V, E)$ ,

- Δ u + (λ a + 1) u = | u |^{p - 1} u in V,

where

λ > 1

and

a (x)

satisfies the following assumptions:

(a_1)

$a (x) \geq 0$ and the potential well $Ω = {x \in V : a (x) = 0}$ is a non-empty, connected, and bounded domain in $V$ ;

(a_2)

there exists a vertex $x_{0} \in V$ such that $a (x) \to 0$ as $d (x, x_{0}) \to + \infty$ .

And, as

λ \to + \infty

, the solutions family

{u_{λ}}

converges to a ground state solution of the following Dirichlet problem:

\begin{aligned} {\begin{cases} - Δ u + u = | u |^{p - 1} u in Ω, \\ u = 0, on \partial Ω . \end{cases} \end{aligned}

Han et al. (2020) considered the following biharmonic equation on locally finite graph

G = (V, E)

Δ^{2} u - Δ u + (λ a + 1) u = | u |^{p - 2} u in V,

where

λ > 1

a (x)

satisfied

(a_{1})

and

(a_{2})

and they investigated the existence of ground state solution

u_{λ}

on graphs by the Nehari manifold method. Moreover, as

λ \to + \infty

, they also obtained that the solution

u_{λ}

converges to a ground state solution of the following Dirichlet problem:

\begin{aligned} {\begin{cases} Δ^{2} u - Δ u + u = | u |^{p - 1} u in Ω, \\ u = 0, on \partial Ω . \end{cases} \end{aligned}

Han & Shao (2021) investigated the following

p

-Laplacian equation with

p \geq 2

on locally finite graph

G = (V, E)

- Δ_{p} u + (λ a + 1) u^{p - 2} u = f (x, u) in V,

where

λ > 1

. They got that the equation admits a ground state solution

u_{λ}

via the Nehari manifold method and the deformation lemma. Besides, they also proved the convergence of ground state solution. They assumed that the nonlinear term

f

satisfies the following conditions:

(f_1)

for any $x \in V$ , $f (x, s)$ is continuous in $s \in R$ , $f (x, 0) = 0$ , and for any fixed $M > 0$ there exists a constant $A_{M}$ such that $max_{| s | < M} f (x, s) \leq A_{M}$ for all $x \in V$ ;

(f_2)

there exists some $α > p$ such that for any $s \in R ∖ {0}$ and $x \in V$ there holds

0 < α F (x, s) := α \int_{0}^{s} f (x, t) d t \leq s f (x, s);

(f_3)

for any $x \in V$ , there holds

\underset{| s | \to 0}{lim sup} \frac{| f (x, s) |}{| s |^{p - 1}} < λ_{p} := inf_{u ≢ 0} \frac{\int_{V} (| \nabla u |^{p} + (λ a + 1) | u |^{p}) d μ}{\int_{V} | u |^{p} d μ};

(f_4)

there exist some $q > p$ and $C > 0$ such that

| f (x, s) | \leq C (1 + | s |^{q - 1}) uniformly in x \in V;

(f_5)

$s \mapsto \frac{f (x, s)}{| s |^{p - 1}}$ is strictly increasing on $(- \infty, 0)$ and $(0, + \infty)$ for all $x \in V$ .

The assumptions on the potential $a (x)$ satisfies $(a_{1})$ and

(a_2)'

$(a (x) + 1)^{- 1} \in L^{\frac{1}{p - 1}} (V)$ .

Shao (2023) studied the following

p

-Laplacian systems with

p \geq 2

on locally finite graph

G = (V, E)

\begin{aligned} {\begin{cases} - Δ_{p} u + (λ a + 1) | u |^{p - 2} u = \frac{1}{κ} F_{u} (u, v), in V, \\ - Δ_{p} v + (λ b + 1) | v |^{p - 2} v = \frac{1}{κ} F_{v} (u, v), in V, \end{cases} \end{aligned}

where

λ > 1, 2 \leq p < κ < \infty

and

F

satisfies the following conditions:

(F)

$F \in C^{1} (R^{2}, R^{+})$ and $F (t u, t v) = t^{κ} F (u, v)$ holds for all $(u, v) \in R^{2}$ , where $t > 0$ . The assumptions on the potential $a (x)$ and $b (x)$ satisfies

(a_3)

$a (x) \geq 0, b (x) \geq 0$ . The potential well $Ω_{a} := {x \in V : a (x) = 0}$ , $Ω_{b} := {x \in V : b (x) = 0}$ , and $Ω_{a}, Ω_{b}$ , and $Ω_{a} \cap Ω_{b}$ are all non-empty, connected, and bounded domains in $V$ .

(a_4)

there exists a vertex $x_{0} \in V$ such that $a (x) \to 0$ and $b (x) \to 0$ as $d (x, x_{0}) \to + \infty$ .

Shao established the existence of ground state solutions by the Nehari manifold method. When $λ \to + \infty$ , the ground state solutions family converges to ground state solutions of the corresponding Dirichlet equation.

The ground state solutions for partial differential equations in the Euclidean setting have been studied extensively. We mainly refer to the work by Papageorgiou et al. (2020) which inspired our work. Papageorgiou et al. (2020) established the existence result of the ground state solution by the Nehari manifold method and the Lagrange multiplier rule for the following double phase problems:

\begin{aligned} {\begin{cases} - Δ_{p} u - div (a (z) | D u |^{q - 2} D u) = f (x, u), in Ω, \\ u = 0, in \partial Ω, 1 < q < p, \end{cases} \end{aligned}

where

Ω

is a bounded domain in

R^{N} (N \geq 2)

with a smooth boundary,

p^{*}

is the critical Sobolev exponent corresponding to

p

, which is defined by the following equation:

\begin{aligned} p^{*} = {\begin{cases} \frac{N p}{N - p}, & if p < N, \\ + \infty, & if N \leq p, \end{cases} \end{aligned}

f

satisfies the following assumptions:

H (f)

f : Ω \times R \to R

is a measurable function such that for a.e.

z \in Ω, f (z, 0) = 0, f (z, \cdot) \in C^{1} (R ∖ {0})

and

(i)

$| f_{x}^{'} (z, x) | \leq a_{0} (z) (1 + | x |^{r - 2})$ for a.e. $z \in Ω$ and all $x \in R$ , with $a_{0} \in L^{\infty} (Ω), 1 < q < p < r < p^{*};$

(ii)

Let $F (z, x) = \int_{0}^{x} f (z, x) d s$ . Then $lim_{x \pm \infty} F (z, x) / | x |^{p} = + \infty$ uniformly for a.e. $z \in Ω$ and there exist constants $τ \in ((r - p) max {1, \frac{N}{p}}, p^{*})$ and $β_{0} > 0$ such that

β_{0} \leq \underset{x \to \pm \infty}{lim inf} \frac{f (z, x) x - p F (z, x)}{| x |^{τ_{1}}} for all (x, t, s) \in V \times R^{2};

(iii)

$lim_{x \to 0} \frac{f (z, x)}{| x |^{q - 2} x} = 0$ for all $x \in Ω$ ;

(iv)

$0 < (p - 1) f (z, x) x \leq f_{x}^{'} (z, x) x^{2}$ for a.e. $x \in Ω$ and all $x \neq 0$ .

Motivated by Han & Shao (2021), Grigor’yan et al. (2016), Papageorgiou et al. (2020), and Shao (2023), we consider the following

(p, q)

-Laplacian systems

\begin{aligned} {\begin{cases} - Δ_{p} u + (λ a + 1) | u |^{p - 2} u = F_{u} (x, u, v), in V, \\ - Δ_{q} v + (λ b + 1) | v |^{q - 2} v = F_{v} (x, u, v), in V, \end{cases} \end{aligned}

(1.2)

where

V

is the vertex set,

p, q > 1

λ \geq 1

, and

a (x)

and

b (x)

are potential functions. The nonlinear term

F : V \times R \times R \to R

and

Δ_{s}, s = p, q

is the discrete

s

-Laplacian on graphs. We prove that for every given

λ > 1

, the system admits a ground state solution

(u_{λ}, v_{λ})

via the method of Nehari manifold and the Lagrange multiplier rule. As

λ \to + \infty

, we also obtained that the solutions family

{(u_{λ}, v_{λ})}

converges to the solution of the limit problem defined on the potential wells

Ω_{a}

and

Ω_{b}

\begin{aligned} {\begin{cases} - Δ_{p} u + | u |^{p - 2} u = F_{u} (x, u, v) in Ω_{a}, \\ - Δ_{q} v + | v |^{q - 2} v = F_{v} (x, u, v) in Ω_{b}, \\ u = 0 on \partial Ω_{a}, \\ v = 0 on \partial Ω_{b}, \end{cases} \end{aligned}

(1.3)

where

Ω_{a}, Ω_{b} \subset V

are bounded domains,

Ω_{a} = {x \in V : a (x) = 0}

and

Ω_{b} = {x \in V : b (x) = 0}

\partial Ω_{a}

denotes the boundary of

Ω_{a}

and

\partial Ω_{b}

denotes the boundary of

Ω_{b}

which is defined by (1.4) below.

To describe our problems and results more clearly, we review some concepts and assumptions (see Han & Shao, 2021; Grigor’yan et al., 2016; Grigoryan et al., 2017). Let $G = (V, E)$ be a graph, where $V$ denotes the vertex set and $E$ denotes the edge set. For any $x \in V$ , if there are only finite $y \in V$ such that $x y \in E$ , then $G$ is called a locally finite graph. $G$ is connected if any two vertices $x$ and $y$ can be connected via finitely many edges. Let a measure $μ : V \to R^{+}$ be finite. If there exists a constant $μ_{min} > 0$ such that $μ (x) \geq μ_{min}$ for all $x \in V$ , we call that $μ$ is a uniformly positive measure. $G$ is said to be symmetric, namely $ω_{x y} = ω_{y x}$ . For any edge $x y \in E$ with two vertexes of $x, y \in V$ , assume that its weight $ω_{x y} > 0$ and for any $x \in V$ , $\sum_{y \sim x} ω_{x y} < C$ , where $C$ is a positive constant. Here and throughout this article, $y \sim x$ stands for any vertex $y$ connected with $x$ by an edge $x y \in E$ . The distance $d (x, y)$ of two vertices $x, y \in V$ is defined by the minimal number of edges which connect these two vertices. $Ω \subset V$ is a bounded domain in $V$ , if the distance $d (x, y)$ is uniformly bounded from above for any $x, y \in Ω$ . Obviously, a bounded domain of a locally finite graph contains only finite vertices. Denote that the boundary of $Ω$ by

\partial Ω := {y \in V ∖ Ω : x \in Ω such that x y \in E}

(1.4)

and the interior of

Ω

Ω \circ

. Obviously,

Ω \circ = Ω

For any $x \in V$ , we define

Δ ψ (x) = \frac{1}{μ (x)} \sum_{y \sim x} w_{x y} (ψ (y) - ψ (x)),

where

ψ : V \to R

. The corresponding gradient form is

Γ (ψ_{1}, ψ_{2}) (x) = \frac{1}{2 μ (x)} \sum_{y \sim x} w_{x y} (ψ_{1} (y) - ψ_{1} (x)) (ψ_{2} (y) - ψ_{2} (x)) .

Write

Γ (ψ) = Γ (ψ, ψ)

. The length of the gradient is defined by the following equation:

| \nabla ψ | (x) = \sqrt{Γ (ψ) (x)} = {(\frac{1}{2 μ (x)} \sum_{y \sim x} w_{x y} (ψ (y) - ψ (x))^{2})}^{\frac{1}{2}} .

For any function

ψ : V \to R

, we denote

\int_{V} ψ (x) d μ = \sum_{x \in V} μ (x) ψ (x) .

When

p \geq 2

, we define the

p

-Laplacian operator by

Δ_{p} ψ

with

Δ_{p} ψ (x) = \frac{1}{2 μ (x)} \sum_{y \sim x} (| \nabla ψ |^{p - 2} (y) + | \nabla ψ |^{p - 2} (x)) ω_{x y} (ψ (y) - ψ (x)) .

In the distributional sense,

Δ_{p} ψ

can be written as follows. For any

ϕ \in C_{c} (V)

\int_{V} (Δ_{p} ψ) ϕ d μ = - \int_{V} | \nabla ψ |^{p - 2} Γ (ψ, ϕ) d μ,

where

C_{c} (V) := {u : V \to R : {x \in V : u (x) \neq 0} is of finite cardinality}

When $1 \leq θ < + \infty$ , we define

L^{θ} (V) = {ψ : V \to R | \int_{V} | ψ (x) |^{θ} d μ < \infty}

and the norm by

‖ ψ ‖_{θ, V} = {(\int_{V} | ψ (x) |^{θ} d μ)}^{\frac{1}{θ}} .

When

θ = + \infty

, we define

L^{\infty} (V) = {ψ : V \to R | sup_{x \in V} | ψ (x) | < \infty}

with the norm

‖ ψ ‖_{\infty, V} = sup_{x \in V} | ψ (x) | < \infty .

We define

‖ (ψ, ϕ) ‖_{L (V)} = ‖ ψ ‖_{θ_{1}} + ‖ ϕ ‖_{θ_{2}}

, where

1 \leq θ_{i} \leq + \infty, i = 1, 2.

Next, we introduce the following assumptions on $F : V \times R \times R \to R$ and (F_1)

$F (x, 0, 0) = 0$ and $F (x, t, s)$ is twice continuously differentiable in $(t, s) \in R^{2}$ for all $x \in V$ ;

(F_2)

there exist constants $α > β := max {p, q}$ and $0 < ϱ < min {\frac{α - p}{p}, \frac{α - q}{q}}$ such that

α F (x, t, s) \leq F_{t} (x, t, s) t + F_{s} (x, t, s) s + ϱ (| t |^{p} + | s |^{q}) for all (x, t, s) \in V \times R^{2},

where

F_{t} (x, t, s) = \partial F (x, t, s) / \partial t

and

F_{s} (x, t, s) = \partial F (x, t, s) / \partial s

;

(F_3)

there exist constants $r_{1}, r_{2} > α$ , two functions $C_{i} : V \to R, i = 1, 2$ such that $C_{1} \in L^{1} (V) \cap L^{\infty} (V)$ with $0 < C_{1} (x) \leq \frac{1}{1 + β}$ for all $x \in V$ , $C_{2} \in L^{1} (V) \cap L^{\infty} (V)$ with $C_{2} (x) > 0$ for all $x \in V$ and

\begin{aligned} | F_{t} (x, t, s) | & \leq C_{1} (x) (| t |^{p - 1} + | s |^{\frac{q (p - 1)}{p}}) + C_{2} (x) (| t |^{r_{1} - 1} + | s |^{\frac{r_{2} (r_{1} - 1)}{r_{1}}}), for all x \in V and \forall (s, t) \in R^{2}; \end{aligned}

\begin{aligned} | F_{s} (x, t, s) | & \leq C_{1} (x) (| t |^{\frac{p (q - 1)}{q}} + | s |^{q - 1}) + C_{2} (x) (| t |^{\frac{r_{1} (r_{2} - 1)}{r_{2}}} + | s |^{r_{2} - 1}), for all x \in V and \forall (s, t) \in R^{2}; \end{aligned}

(F_4)

$F_{t s} (x, t, s) t s \geq 0$ and $0 < (β - 1) (F_{t} (x, t, s) t + F_{s} (x, t, s) s) < F_{t t} (x, t, s) t^{2} + F_{s s} (x, t, s) s^{2}$ for all $x \in V$ and $\forall (t, s) \in R^{2} ∖ {(0, 0)}$ .

The assumptions on the potential functions

a (x)

and

b (x)

are:

(A_1)

$a (x) \geq 0, b (x) \geq 0$ for all $x \in V$ . The potential wells $Ω_{a} := {x \in V : a (x) = 0}$ , $Ω_{b} := {x \in V : b (x) = 0}$ , and $Ω_{a}, Ω_{b}$ and $Ω_{a} \cap Ω_{b}$ are all non-empty, connected and bounded domains in $V$ .

(A_2)

$(a (x) + 1)^{- 1} \in L^{\frac{1}{p - 1}} (V)$ and $(b (x) + 1)^{- 1} \in L^{\frac{1}{q - 1}} (V)$ .

We define

W^{1, s} (V) = {u : V \to R | \int_{V} (| \nabla u |^{s} + | u |^{s}) d μ < \infty}

which is provided with the norm

‖ u ‖_{W^{1, s} (V)} = {(\int_{V} (| \nabla u |^{s} + | u |^{s}) d μ)}^{\frac{1}{s}},

where

s > 1

. Let us consider the space

W := W^{1, p} (V) \times W^{1, q} (V)

and

‖ (u, v) ‖_{W} = ‖ u ‖_{W^{1, p} (V)} + ‖ v ‖_{W^{1, q} (V)}

Define

W_{λ} (a) := {u \in W^{1, p} (V) | \int_{V} λ a | u |^{p} d μ < \infty}

and

W_{λ} (b) := {v \in W^{1, q} (V) | \int_{V} λ b | v |^{q} d μ < \infty} .

To study problem (1.2), it is natural to consider the function space

W_{λ} := {(u, v) \in W | \int_{V} (λ a | u |^{p} + λ b | v |^{q}) d μ < \infty} .

It is easy to see that the space

W_{λ} := W_{λ} (a) \times W_{λ} (b)

. Define the norm

‖ (u, v) ‖_{W_{λ}} = ‖ u ‖_{W_{λ} (a)} + ‖ v ‖_{W_{λ} (b)}

, where

‖ u ‖_{W_{λ} (a)} = {(\int_{V} (| \nabla u |^{p} + (λ a + 1) | u |^{p}) d μ)}^{\frac{1}{p}}

and

‖ v ‖_{W_{λ} (b)} = {(\int_{V} (| \nabla v |^{q} + (λ b + 1) | v |^{q}) d μ)}^{\frac{1}{q}} .

Then,

(W_{λ}, ‖ \cdot ‖_{W_{λ}})

is a reflexive Banach space (see Shao et al., 2023).

The functional related to (1.2) is defined by the following equation:

J_{λ} (u, v) = \frac{1}{p} \int_{V} (| \nabla u |^{p} + (λ a + 1) | u |^{p}) d μ + \frac{1}{q} \int_{V} (| \nabla v |^{q} + (λ b + 1) | v |^{q}) d μ - \int_{V} F (x, u, v) d μ .

Under assumptions

(F_{1})

and

(F_{3})

, a standard procedure can show that

J_{λ} \in C^{1} (W_{λ}, R)

and

\begin{aligned} ⟨ J_{λ}^{'} (u, v), (ϕ_{1}, ϕ_{2}) ⟩ = & \int_{V} (| \nabla u |^{p - 2} Γ (u, ϕ_{1}) + (λ a + 1) | u |^{p - 2} u ϕ_{1}) d μ + \int_{V} (| \nabla v |^{q - 2} Γ (v, ϕ_{2}) + (λ b + 1) | v |^{q - 2} v ϕ_{2}) d μ \\ - \int_{V} (F_{u} (x, u, v) ϕ_{1} + F_{v} (x, u, v) ϕ_{2}) d μ \end{aligned}

for all

(u, v), (ϕ_{1}, ϕ_{2}) \in W_{λ}

. By Lemma 3.1 below, the critical point of

J

is the point-wise solution of system (1.2).

Define the Nehari manifold

N_{λ} := {(u, v) \in W_{λ} ∖ {(0, 0)} : ⟨ J_{λ}^{'} (u, v), (u, v) ⟩ = 0}

and the least energy level

m_{λ} := inf_{(u, v) \in N_{λ}} J_{λ} (u, v) .

It is well-known that if

m_{λ}

can be achieved by some function

(u_{λ}, v_{λ}) \in N_{λ}

and

(u_{λ}, v_{λ})

is a critical point of the functional

J_{λ}

, then

(u_{λ}, v_{λ})

is a ground state solution of (1.2). Next, we provide our main results.

Theorem 1.1

Let $G = (V, E)$ be a locally finite graph and it is connected, symmetric and there exists $μ_{min} > 0$ such that $μ (x) \geq μ_{min}$ for all $x \in V$ . For any $x \in V$ , $\sum_{y \sim x} ω_{x y} < C$ , where $C$ is a positive constant. Assume $\frac{1}{γ} < β + \frac{1}{β} - 1$ , where $γ = min {p, q}$ , and $(A_{1}), (A_{2})$ , and $(F_{1}) - (F_{4})$ hold. Then for every $λ \geq 1$ , there exists a ground state solution $(u_{λ}, v_{λ})$ of system (1.2) and $‖ (u_{λ}, v_{λ}) ‖_{W_{λ}}$ satisfies (3.21) below.

Remark 1.1

There exists examples satisfying Theorem 1.1. For example, let $α > β := max {p, q}$ and

\begin{aligned} C_{1} (x) & = \frac{1}{(1 + β) (1 + | x |)^{2}}, C_{2} (x) = \frac{2 β}{(1 + | x |)^{2}}, \\ F (x, t, s) & = \frac{| t |^{α} + | s |^{α}}{2 α (1 + β) (1 + | x |)^{2}} (1 - \frac{1}{\ln (e^{2} + | t |^{α} + | s |^{α})}) . \end{aligned}

In order to investigate the asymptotic behavior of $(u_{λ}, v_{λ})$ to the solution of (1.3), we define $W_{Ω_{a}}$ by the completion of $C_{c} (Ω_{a})$ under the norm

‖ u ‖_{W_{Ω_{a}}} = {(\int_{Ω_{a} \cup \partial Ω_{a}} | \nabla u |^{p} d μ + \int_{Ω_{a}} | u |^{p} d μ)}^{\frac{1}{p}},

and

W_{Ω_{b}}

by the completion of

C_{c} (Ω_{b})

under the norm

‖ v ‖_{W_{Ω_{b}}} = {(\int_{Ω_{b} \cup \partial Ω_{b}} | \nabla v |^{q} d μ + \int_{Ω_{b}} | v |^{q} d μ)}^{\frac{1}{q}},

where

C_{c} (Ω)

denotes the set of all functions

u : Ω \to R

satisfying

s u p p u \subset Ω

and

u = 0

V ∖ A

for some subset

Ω \subset V

. It is suitable to study (1.3) in the space

W_{Ω} := W_{Ω_{a}} \times W_{Ω_{b}}

and

W_{Ω}

is a reflexive Banach space with finite dimensional. Define the norm

‖ (u, v) ‖_{W_{Ω}} = ‖ u ‖_{W_{Ω_{a}}} + ‖ v ‖_{W_{Ω_{b}}}

We define when $1 \leq θ_{1} < + \infty$ ,

L^{θ_{1}} (Ω_{a}) = {ψ : Ω_{a} \to R | \int_{Ω_{a}} | ψ (x) |^{θ_{1}} d μ < \infty}

and the norm by

‖ ψ ‖_{θ_{1}, Ω_{a}} = {(\int_{Ω_{a}} | ψ (x) |^{θ_{1}} d μ)}^{\frac{1}{θ_{1}}} .

When

θ_{1} = + \infty

, we define

L^{\infty} (Ω_{a}) = {ψ : Ω_{a} \to R | sup_{x \in Ω_{a}} | ψ (x) | < \infty}

with the norm

‖ ψ ‖_{\infty, Ω_{a}} = sup_{x \in Ω_{a}} | ψ (x) | < \infty .

Similarly, when

1 \leq θ_{2} < + \infty

L^{θ_{2}} (Ω_{b}) = {ϕ : Ω_{b} \to R | \int_{Ω_{b}} | ϕ (x) |^{θ_{2}} d μ < \infty}

and the norm by

‖ ϕ ‖_{θ_{2}, Ω_{b}} = {(\int_{Ω_{b}} | ϕ (x) |^{θ_{2}} d μ)}^{\frac{1}{θ_{2}}} .

When

θ_{2} = + \infty

, we define

L^{\infty} (Ω_{b}) = {ϕ : Ω_{b} \to R | sup_{x \in Ω_{b}} | ϕ (x) | < \infty}

with the norm

‖ ϕ ‖_{\infty, Ω_{b}} = sup_{x \in Ω_{b}} | ϕ (x) | < \infty .

We also define

‖ (ψ, ϕ) ‖_{θ, Ω} = ‖ ψ ‖_{θ_{1}, Ω_{a}} + ‖ ϕ ‖_{θ_{2}, Ω_{b}}

The functional related to (1.3) is

\begin{aligned} J_{Ω} (u, v) & = \frac{1}{p} (\int_{{\bar{Ω}}_{a}} | \nabla u |^{p} d μ + \int_{Ω_{a}} | u |^{p} d μ) + \frac{1}{q} (\int_{{\bar{Ω}}_{b}} | \nabla v |^{q} d μ + \int_{Ω_{b}} | v |^{q} d μ) - \int_{Ω_{a} \cup Ω_{b}} F (x, u, v) d μ, \end{aligned}

where

{\bar{Ω}}_{a} := Ω_{a} \cup \partial Ω_{a}

and

{\bar{Ω}}_{b} := Ω_{b} \cup \partial Ω_{b}

. By a standard argument, we can verify that

J_{Ω} \in C^{1} (W_{Ω}, R)

and

\begin{aligned} ⟨ J_{Ω}^{'} (u, v), (ϕ_{1}, ϕ_{2}) ⟩ = & \int_{{\bar{Ω}}_{a}} | \nabla u |^{p - 2} Γ (u, ϕ_{1}) d μ + \int_{Ω_{a}} | u |^{p - 2} u ϕ_{1} d μ + \int_{{\bar{Ω}}_{b}} | \nabla v |^{q - 2} Γ (v, ϕ_{2}) d μ + \int_{Ω_{b}} | v |^{q - 2} v ϕ_{2} d μ \\ - \int_{Ω_{a} \cup Ω_{b}} F_{u} (x, u, v) ϕ_{1} d μ - \int_{Ω_{a} \cup Ω_{b}} F_{v} (x, u, v) ϕ_{2} d μ \end{aligned}

for any

(u, v), (ϕ_{1}, ϕ_{2}) \in W_{Ω}

. The corresponding Nehari manifold is

N_{Ω} := {(u, v) \in W_{Ω} ∖ {(0, 0)} : ⟨ J_{Ω}^{'} (u, v), (u, v) ⟩ = 0} .

Let

m_{Ω} := inf_{(u, v) \in N_{Ω}} J_{Ω} (u, v) .

Similar to Theorem 1.1, system (1.3) also has a ground state solution and the ground state solutions

{(u_{λ}, v_{λ})}

of system (1.2) converge to a ground state solution of system (1.3).

Theorem 1.2

Let $G = (V, E)$ be a locally finite graph and it is connected, symmetric, and there exists $μ_{min} > 0$ such that $μ (x) \geq μ_{min}$ for all $x \in Ω$ . For any $x \in V$ , $\sum_{y \sim x} ω_{x y} < C$ , where $C$ is a positive constant. $Ω_{a}, Ω_{b}$ , and $Ω_{a} \cap Ω_{b}$ are non-empty, connected, and bounded domains in $V$ . Assume $\frac{1}{γ} < β + \frac{1}{β} - 1$ , $(A_{1})$ , $(A_{2})$ , and $F : V \times R \times R \to R$ satisfies $(F_{1}) - (F_{4})$ . Then system (1.3) has a ground state solution $(u_{0}, v_{0}) \in W_{Ω}$ . Besides for any sequence $λ_{k} \to + \infty$ , up to a sequence, the ground state solutions family ${(u_{λ_{k}}, v_{λ_{k}})}$ of (1.2) converge to a ground state solution of (1.3).

Remark 1.2

Our works generalize those results by Han & Shao (2021) in some sense, and $(F_{2})$ corresponding to the scalar case is weaker than $(f_{2})$ . Moreover, Shao (2023) studied system (1.2) with $p = q$ and $p \geq 2$ and the nonlinear term $F$ is independent on $x$ (see system (1.1) for details). However, our system (1.2) allowed the fact that $p, q > 1$ and $p, q$ are allowed to be unequal, and the nonlinear term $F$ is allowed to depend on $x$ . It is easy to verify that the example in Remark 1.1 does not satisfy the condition $(F)$ . Hence our result is different from that by Shao (2023).

2. Preliminaries

In this section, we present some Sobolev embedding theorems on the locally finite graph and their proofs.

Lemma 2.1
Let $G = (V, E)$ be a locally finite graph and assume $p, q > 1$ . Then for any $λ \geq 1$ and all $(ψ, ϕ) \in W_{λ}$ , there is
$‖ ψ ‖_{\infty, V} \leq d_{1} ‖ ψ ‖_{W_{λ} (a)}, ‖ ϕ ‖_{\infty, V} \leq d_{2} ‖ ϕ ‖_{W_{λ} (b)}, ‖ (ψ, ϕ) ‖_{\infty, V} \leq d ‖ (ψ, ϕ) ‖_{W_{λ}},$
where $d_{1} = (\frac{1}{μ_{min}})^{\frac{1}{p}}, d_{2} = (\frac{1}{μ_{min}})^{\frac{1}{q}}$ and $d = max {d_{1}, d_{2}}$ .
Proof.
It follows from Lemma 2.6 by Han & Shao (2021) that $‖ ψ ‖_{\infty, V} \leq d_{1} ‖ ψ ‖_{W_{λ} (a)}$ and $‖ ϕ ‖_{\infty, V} \leq d_{2} ‖ ϕ ‖_{W_{λ} (b)}$ . Furthermore,
$‖ (ψ, ϕ) ‖_{\infty, V} = max_{x \in V} | (ψ, ϕ) | \leq ‖ ψ ‖_{\infty, V} + ‖ ϕ ‖_{\infty, V} \leq d_{1} ‖ ψ ‖_{W_{λ} (a)} + d_{2} ‖ ϕ ‖_{W_{λ} (b)} \leq max {d_{1}, d_{2}} ‖ (ψ, ϕ) ‖_{W_{λ}} .$
The proof is complete. □
Lemma 2.2
Let $G = (V, E)$ be a locally finite graph and assume that $(A_{1})$ and $(A_{2})$ hold. Then for any $λ \geq 1$ , $W_{λ} (a)$ and $W_{λ} (b)$ are continuously embedded into $L^{θ_{1}} (V)$ and $L^{θ_{2}} (V)$ and for any $ψ \in W_{λ} (a)$ and $ϕ \in W_{λ} (b)$ ,
$‖ ψ ‖_{θ_{1}} \leq K_{(p, θ_{1})} ‖ ψ ‖_{W_{λ} (a)}, ‖ ϕ ‖_{θ_{2}} \leq K_{(q, θ_{2})} ‖ ϕ ‖_{W_{λ} (b)},$
(2.1)
where $θ_{1}, θ_{2} \in [1, + \infty)$ ,
$\begin{aligned} K_{(p, θ_{1})} & = {\begin{cases} d_{1}^{\frac{θ_{1} - p}{θ_{1}}}, & θ_{1} \geq p, \\ min {d_{1}^{\frac{θ_{1} - 1}{θ_{1}}} ‖ (a + 1)^{- 1} ‖_{\frac{1}{p - 1}}^{\frac{1}{p θ_{1}}}, d_{1}^{\frac{p (θ_{1} - 1)}{θ_{1}}} ‖ (a + 1)^{- 1} ‖_{\frac{1}{p - 1}}^{\frac{1}{p}}}, & 1 \leq θ_{1} < p, \end{cases} \end{aligned}$

$\begin{aligned} K_{(q, θ_{2})} & = {\begin{cases} d_{2}^{\frac{θ_{2} - q}{θ_{2}}}, & θ_{2} \geq q, \\ min {d_{2}^{\frac{θ_{2} - 1}{θ_{2}}} ‖ (b + 1)^{- 1} ‖_{\frac{1}{q - 1}}^{\frac{1}{q θ_{2}}}, d_{2}^{\frac{q (θ_{2} - 1)}{θ_{2}}} ‖ (b + 1)^{- 1} ‖_{\frac{1}{q - 1}}^{\frac{1}{q}}}, & 1 \leq θ_{2} < q . \end{cases} \end{aligned}$
Moreover, $W_{λ}$ is continuously embedded into $L (V) := L^{θ_{1}} (V) \times L^{θ_{2}} (V)$ for all $θ_{1}, θ_{2} \in [1, + \infty)$ and
$‖ (ψ, ϕ) ‖_{L (V)} \leq max {K_{(p, θ_{1})}, K_{(q, θ_{2})}} ‖ (ψ, ϕ) ‖_{W_{λ}},$
(2.2)
Proof.
The proof is similar to Lemma 2.6 by Han & Shao (2021) with some slight modifications. According to Lemma 2.1, when $θ_{1} \geq p$ , we have
$‖ ψ ‖_{θ_{1}}^{θ_{1}} = \int_{V} | ψ |^{θ_{1} - p} \cdot | ψ |^{p} d μ \leq ‖ ψ ‖_{\infty, V}^{θ_{1} - p} \int_{V} (| \nabla ψ |^{p} + (λ a + 1) | ψ (x) |^{p}) d μ \leq d_{1}^{θ_{1} - p} ‖ ψ ‖_{W_{λ} (a)}^{θ_{1} - p} \cdot ‖ ψ ‖_{W_{λ} (a)}^{p} = d_{1}^{θ_{1} - p} ‖ ψ ‖_{W_{λ} (a)}^{θ_{1}} .$
By $(A_{2})$ and noticing that $λ \geq 1$ , we have
$\begin{aligned} \int_{V} | ψ | d μ = & \int_{V} (λ a + 1)^{- \frac{1}{p}} (λ a + 1)^{\frac{1}{p}} | ψ | d μ \\ \leq {(\int_{V} (λ a + 1)^{- \frac{1}{p} \times \frac{p}{p - 1}} d μ)}^{\frac{p - 1}{p}} {(\int_{V} (λ a + 1)^{\frac{1}{p} \times p} | ψ |^{p} d μ)}^{\frac{1}{p}} \\ \leq ‖ (a + 1)^{- 1} ‖_{\frac{1}{p - 1}}^{\frac{1}{p}} ‖ ψ ‖_{W_{λ} (a)} . \end{aligned}$
(2.3)
On the one hand, when $1 \leq θ_{1} < p$ , from (2.3), we have
$‖ ψ ‖_{θ_{1}}^{θ_{1}} = \int_{V} | ψ |^{θ_{1} - 1} \cdot | ψ | d μ \leq ‖ ψ ‖_{\infty, V}^{θ_{1} - 1} \int_{V} | ψ | d μ \leq d_{1}^{θ_{1} - 1} ‖ (a + 1)^{- 1} ‖_{\frac{1}{p - 1}}^{\frac{1}{p}} ‖ ψ ‖_{W_{λ} (a)}^{θ_{1}} .$
(2.4)
On the other hand, by Lemma 2.6 by Han & Shao (2021), $‖ ψ ‖_{θ_{1}} \leq d_{1}^{\frac{p (θ_{1} - 1)}{θ_{1}}} ‖ (a + 1)^{- 1} ‖_{\frac{1}{p - 1}}^{\frac{1}{p}} ‖ ψ ‖_{W_{λ} (a)}$ . So combining (2.4), there is
$‖ ψ ‖_{θ_{1}} \leq min {d_{1}^{\frac{θ_{1} - 1}{θ_{1}}} ‖ (a + 1)^{- 1} ‖_{\frac{1}{p - 1}}^{\frac{1}{p θ_{1}}}, d_{1}^{\frac{p (θ_{1} - 1)}{θ_{1}}} ‖ (a + 1)^{- 1} ‖_{\frac{1}{p - 1}}^{\frac{1}{p}}} ‖ ψ ‖_{W_{λ} (a)} .$
Similarly, it is easy to prove that the second inequality also holds in (2.1).Thus,
$‖ (ψ, ϕ) ‖_{L (V)} = ‖ ψ ‖_{θ_{1}} + ‖ ϕ ‖_{θ_{2}} \leq K_{(p, θ_{1})} ‖ ψ ‖_{W_{λ} (a)} + K_{(q, θ_{2})} ‖ ϕ ‖_{W_{λ} (b)} \leq max {K_{(p, θ_{1})}, K_{(q, θ_{2})}} ‖ (ψ, ϕ) ‖_{W_{λ}} .$
The proof is complete. □
Lemma 2.3
Let $G = (V, E)$ be a locally finite graph and assume that $(A_{1})$ and $(A_{2})$ hold. Then for any given $λ \geq 1$ and any bounded sequence $(ψ_{k}, ϕ_{k}) \in W_{λ}$ , there exists $(ψ, ϕ) \in W_{λ}$ such that, up to subsequence,
$\begin{aligned} {\begin{cases} (ψ_{k}, ϕ_{k}) ⇀ (ψ, ϕ), & i n W_{λ}, a s k \to \infty, \\ ψ_{k} (x) \to ψ (x), ϕ_{k} (x) \to ϕ (x), & \forall x \in V, a s k \to \infty, \\ ψ_{k} \to ψ, & i n L^{θ_{1}} (V), \forall θ_{1} \in [1, + \infty], a s k \to \infty, \\ ϕ_{k} \to ϕ, & i n L^{θ_{2}} (V), \forall θ_{2} \in [1, + \infty], a s k \to \infty . \end{cases} \end{aligned}$

Proof.
Since $W_{λ}$ is a reflexive Banach space. Thus for any bounded sequence $(ψ_{k}, ϕ_{k}) \in W_{λ}$ , we get that, up to a subsequence, $(ψ_{k}, ϕ_{k}) ⇀ (ψ, ϕ)$ in $W_{λ}$ . In addition, by Lemma 2.2, we can get $ψ_{k} ⇀ ψ$ in $L^{θ_{1}} (V)$ and $ϕ_{k} ⇀ ϕ$ in $L^{θ_{2}} (V)$ . The remaining proof is similar to Lemma 2.6 by Han & Shao (2021) with substituting $ψ_{k}$ and $ϕ_{k}$ for $u_{k}$ by Han & Shao (2021), respectively. We omit it here. □
Lemma 2.4
$W_{Ω_{a}}$ and $W_{Ω_{b}}$ are compactly embedded into $L^{θ_{1}} (Ω_{a})$ and $L^{θ_{2}} (Ω_{b})$ for any $θ_{1}, θ_{2} \in [1, + \infty]$ , respectively, and for any $ψ \in W_{Ω}$ ,
$‖ ψ ‖_{θ_{1}, Ω_{a}} \leq K_{(p, θ_{1})}^{} ‖ ψ ‖_{W_{Ω_{a}}}, ‖ ϕ ‖_{θ_{2}, Ω_{b}} \leq K_{(q, θ_{2})}^{} ‖ ϕ ‖_{W_{Ω_{b}}}, ‖ (ψ, ϕ) ‖_{θ, Ω} \leq K^{} ‖ (ψ, ϕ) ‖_{W_{Ω}},$
(2.5)
where
$K_{(p, θ_{1})}^{} = {(\sum_{x \in Ω_{a}} μ (x))}^{\frac{1}{θ_{1}}} {(\frac{1}{μ_{min}})}^{\frac{1}{p}}, K_{(q, θ_{2})}^{} = {(\sum_{x \in Ω_{b}} μ (x))}^{\frac{1}{θ_{2}}} {(\frac{1}{μ_{min}})}^{\frac{1}{q}}, K^{} = max {K_{(p, θ_{1})}^{}, K_{(q, θ_{2})}^{}} .$
Moreover, $W_{Ω_{a}} (W_{Ω_{b}})$ is pre-compact, namely, if $ψ_{k} (ϕ_{k})$ is bounded in $W_{Ω_{a}} (W_{Ω_{b}})$ , then up to a subsequence, there exists some $ψ \in W_{Ω_{a}} (ϕ \in W_{Ω_{b}})$ such that $ψ_{k} \to ψ$ in $W_{Ω_{a}} (ϕ_{k} \to ϕ$ in $W_{Ω_{b}})$ .
Proof.
The proof of (2.5) is similar to Lemmas 2.1 and 2.2. In fact,
$‖ ψ ‖_{W_{Ω_{a}}}^{p} = \int_{{\bar{Ω}}_{a}} | \nabla ψ |^{p} d μ + \int_{Ω_{a}} | ψ (x) |^{p} d μ \geq \sum_{x \in Ω_{a}} μ (x) | ψ (x) |^{p} \geq μ_{min} \sum_{x \in Ω_{a}} | ψ (x) |^{p} \geq μ_{min} ‖ ψ ‖_{\infty, Ω_{a}}^{p} .$
(2.6)
From (2.6), we have
$‖ ψ ‖_{θ_{1}, Ω_{a}}^{θ_{1}} = \sum_{x \in Ω_{a}} μ (x) | ψ |^{θ_{1}} \leq \sum_{x \in Ω_{a}} μ (x) ‖ ψ ‖_{\infty, Ω_{a}}^{θ_{1}} \leq \sum_{x \in Ω_{a}} μ (x) {(\frac{1}{μ_{min}})}^{\frac{θ_{1}}{p}} ‖ ψ ‖_{W_{Ω_{a}}}^{θ_{1}} .$
The proof of the first inequality in (2.5) is completed. Similarly, we can also easily complete the proofs of the other two inequalities in (2.5). Since $Ω_{a} (Ω_{b})$ is a finite set in $V$ , $W_{Ω_{a}} (W_{Ω_{b}})$ and $L^{θ_{1}} (Ω_{a}) (L^{θ_{2}} (Ω_{b}))$ are finite dimensional spaces. Hence, $W_{Ω_{a}} (W_{Ω_{b}})$ is pre-compact. Then for any bounded sequence ${(ψ_{k}, ϕ_{k})} \in W_{Ω}$ , there is subsequence, still denoted by ${(ψ_{k}, ϕ_{k})} \in W_{Ω}$ , such that $‖ ψ_{k} - ψ ‖_{W_{Ω_{a}}} \to 0$ and $‖ ϕ_{k} - ϕ ‖_{W_{Ω_{b}}} \to 0$ . Furthermore, by (2.5), we can get $‖ ψ_{k} - ψ ‖_{θ_{1}, Ω_{a}} \to 0$ and $‖ ϕ_{k} - ϕ ‖_{θ_{2}, Ω_{b}} \to 0$ . So $W_{Ω_{a}} (W_{Ω_{b}})$ is compactly embedded into $L^{θ_{1}} (Ω_{a}) (L^{θ_{2}} (Ω_{b}))$ for any $θ_{1}, θ_{2} \in [1, + \infty]$ .
3. The Existence of Ground State Solutions

In this section, we prove Theorem 1.1 by the Nehari manifold method and the Lagrange multiplier rule.

Definition 3.1
Suppose that $(u, v) \in W_{λ}$ . If for any $(ϕ_{1}, ϕ_{2}) \in W_{λ}$ , there holds
$\begin{aligned} \int_{V} (| \nabla u |^{p - 2} Γ (u, ϕ_{1}) + (λ a + 1) | u |^{p - 2} u ϕ_{1}) d μ + \int_{V} (| \nabla v |^{q - 2} Γ (v, ϕ_{2}) + (λ b + 1) | v |^{q - 2} v ϕ_{2}) d μ \\ = \int_{V} F_{u} (x, u, v) ϕ_{1} d μ + \int_{V} F_{v} (x, u, v) ϕ_{2} d μ, \end{aligned}$
then $(u, v)$ is called a weak solution of (1.2).
Definition 3.2
Suppose that $(u, v) \in W_{Ω}$ . If for any $(ϕ_{1}, ϕ_{2}) \in W_{Ω}$ , there holds
$\begin{aligned} \int_{{\bar{Ω}}_{a}} | \nabla u |^{p - 2} Γ (u, ϕ_{1}) d μ + \int_{Ω_{a}} | u |^{p - 2} u ϕ_{1} d μ + \int_{{\bar{Ω}}_{b}} | \nabla v |^{q - 2} Γ (v, ϕ_{2}) d μ + \int_{Ω_{b}} | v |^{q - 2} v ϕ_{2} d μ \\ = \int_{Ω_{a} \cup Ω_{b}} F_{u} (x, u, v) ϕ_{1} d μ + \int_{Ω_{a} \cup Ω_{b}} F_{v} (x, u, v) ϕ_{2} d μ, \end{aligned}$
then $(u, v)$ is called a weak solution of (1.3).
Lemma 3.1
If $(u, v) \in W_{λ}$ is a weak solution of (1.2), $(u, v)$ is also a point-wise solution of (1.2).
Proof.
The proof is standard (e.g. see Yang & Zhang, 2024). For completeness, we also present it here. Since $(u, v) \in W_{λ}$ is a weak solution of (1.2), for any $(ϕ, ψ) \in W_{λ}$ , there holds
$\begin{aligned} \int_{V} (| \nabla u |^{p - 2} Γ (u, ϕ) + (λ a + 1) | u |^{p - 2} u ϕ) d μ + \int_{V} (| \nabla v |^{q - 2} Γ (v, ψ) + (λ b + 1) | v |^{q - 2} v ψ) d μ \\ = \int_{V} F_{u} (x, u, v) ϕ d μ + \int_{V} F_{v} (x, u, v) ψ d μ . \end{aligned}$
From Lemma 2.1 by Han & Shao (2021), we have
$\int_{V} | \nabla u |^{p - 2} Γ (u, ϕ) d μ = \int_{V} (- Δ_{p} u) ϕ d μ, \int_{V} | \nabla v |^{q - 2} Γ (v, ψ) d μ = \int_{V} (- Δ_{q} v) ψ d μ .$
(3.1)
Then we get
$\begin{aligned} \int_{V} ((- Δ_{p} u) ϕ + (λ a + 1) | u |^{p - 2} u ϕ) d μ + \int_{V} ((- Δ_{q} v) ψ + (λ b + 1) | v |^{q - 2} v ψ) d μ \\ = \int_{V} F_{u} (x, u, v) ϕ d μ + \int_{V} F_{v} (x, u, v) ψ d μ . \end{aligned}$
(3.2)
If for any $x_{0} \in V$ , we take the test function $(ϕ (x), ψ (x)) : V \to R$ in (3.2) with
$(ϕ (x), ψ (x)) = {\begin{cases} (1, 0) & x = x_{0}, \\ (0, 0) & x \neq x_{0}, \end{cases} and (ϕ (x), ψ (x)) = {\begin{cases} (0, 1) & x = x_{0}, \\ (0, 0) & x \neq x_{0}, \end{cases}$
respectively, then
$\begin{aligned} {\begin{cases} - Δ_{p} u (x_{0}) + (λ a + 1) | u (x_{0}) |^{p - 2} u (x_{0}) = F_{u} (x, u (x_{0}), v (x_{0})), & x_{0} \in V, \\ - Δ_{q} v (x_{0}) + (λ b + 1) | v (x_{0}) |^{q - 2} v (x_{0}) = F_{v} (x, u (x_{0}), v (x_{0})), & x_{0} \in V . \end{cases} \end{aligned}$
Since $x_{0}$ is arbitrary, we conclude that $(u, v)$ is a point-wise solution of (1.2). □
Lemma 3.2
If $(u, v) \in W_{Ω}$ is a weak solution of (1.3), then $(u, v)$ is also a point-wise solution of (1.3).
Proof.
The proof is similar to Lemma 3.1. Since $(u, v) \in W_{Ω}$ is a weak solution of (1.3), for any $(ϕ, ψ) \in C_{c} (Ω_{a}) \times C_{c} (Ω_{b})$ , there holds
$\begin{aligned} \int_{{\bar{Ω}}_{a}} | \nabla u |^{p - 2} Γ (u, ϕ) d μ + \int_{Ω_{a}} | u |^{p - 2} u ϕ d μ + \int_{{\bar{Ω}}_{b}} | \nabla v |^{q - 2} Γ (v, ψ) d μ + \int_{Ω_{b}} | v |^{q - 2} v ψ d μ \\ = \int_{Ω_{a} \cup Ω_{b}} F_{u} (x, u, v) ϕ d μ + \int_{Ω_{a} \cup Ω_{b}} F_{v} (x, u, v) ψ d μ . \end{aligned}$
Similar to (3.1), we also have
$\int_{{\bar{Ω}}_{a}} | \nabla u |^{p - 2} Γ (u, ϕ) d μ = \int_{{\bar{Ω}}_{a}} (- Δ_{p} u) ϕ d μ, \int_{{\bar{Ω}}_{b}} | \nabla v |^{q - 2} Γ (v, ψ) d μ = \int_{{\bar{Ω}}_{b}} (- Δ_{q} v) ψ d μ,$
and then
$\begin{aligned} \int_{{\bar{Ω}}_{a}} (- Δ_{p} u) ϕ d μ + \int_{Ω_{a}} | u |^{p - 2} u ϕ d μ + \int_{{\bar{Ω}}_{b}} (- Δ_{q} v) ψ d μ + \int_{Ω_{b}} | v |^{q - 2} v ψ d μ \\ = \int_{Ω_{a} \cup Ω_{b}} F_{u} (x, u, v) ϕ d μ + \int_{Ω_{a} \cup Ω_{b}} F_{v} (x, u, v) ψ d μ . \end{aligned}$
(3.3)
If for any given $x_{1} \in Ω_{a}$ , we take the test function $(ϕ (x), ψ (x)) : Ω_{a} \times Ω_{b} \to R \times R$ in (3.3) with
$\begin{aligned} (ϕ (x), ψ (x)) = {\begin{cases} (1, 0) & x = x_{1}, \\ (0, 0) & x \neq x_{1}, \end{cases} \end{aligned}$
and for any given $x_{2} \in Ω_{b}$ , we take the test function $(ϕ (x), ψ (x)) : Ω_{a} \times Ω_{b} \to R \times R$ in (3.3) with
$\begin{aligned} (ϕ (x), ψ (x)) = {\begin{cases} (0, 1) & x = x_{2}, \\ (0, 0) & x \neq x_{2}, \end{cases} \end{aligned}$
respectively, then
$\begin{aligned} {\begin{cases} - Δ_{p} u (x_{1}) + | u (x_{1}) |^{p - 2} u (x_{1}) = F_{u} (x, u (x_{1}), v (x_{1})), \\ - Δ_{q} v (x_{2}) + | v (x_{2}) |^{q - 2} v (x_{2}) = F_{v} (x, u (x_{2}), v (x_{2})) . \end{cases} \end{aligned}$
Since $W_{Ω_{a}} (W_{Ω_{b}})$ is the completion of $C_{c} (Ω_{a}) (C_{c} (Ω_{b}))$ , $u (x) = 0$ on $\partial Ω_{a}$ and $v (x) = 0$ on $\partial Ω_{b}$ . Finally, by the arbitrary of $x_{1}$ and $x_{2}$ , we complete the proof.
Lemma 3.3
If $(F_{1})$ and $(F_{4})$ hold and $(u, v) \in W_{λ} ∖ {(0, 0)}$ , then there exists a unique $t_{0} > 0$ such that $(t_{0} u, t_{0} v) \in N_{λ}$ and then $N_{λ}$ is non-empty.
Proof.
We define a fibering map $g (t) := J_{λ} (t u, t v)$ on $R$ for all $t > 0$ . Clearly, we have
$g^{'} (t) = 0 ⟺ ⟨ J_{λ}^{'} (t u, t v), (u, v) ⟩ = 0 ⟺ ⟨ J_{λ}^{'} (t u, t v), (t u, t v) ⟩ = 0 ⟺ (t u, t v) \in N_{λ} .$
Since $t > 0$ and $β = max {p, q}$ , then
$\begin{aligned} \frac{g^{'} (t)}{t^{β - 1}} = & 0 ⟺ g^{'} (t) = 0 ⟺ (t u, t v) \in N_{λ} ⟺ t^{p - β} ‖ u ‖_{W_{λ} (a)}^{p} \\ + t^{q - β} ‖ v ‖_{W_{λ} (b)}^{q} = t^{- β} \int_{V} [F_{t u} (x, t u, t v) t u + F_{t v} (x, t u, t v) t v] d μ . \end{aligned}$
(3.4)
Without loss of generality, we let $β = p$ . Then we infer that
$‖ u ‖_{W_{λ} (a)}^{p} = t^{1 - p} \int_{V} [F_{t u} (x, t u, t v) u + F_{t v} (x, t u, t v) v] d μ - t^{q - p} ‖ v ‖_{W_{λ} (b)}^{q} .$
(3.5)
Let $H_{1} (t) = t^{1 - p} \int_{V} [F_{t u} (x, t u, t v) u + F_{t v} (x, t u, t v) v] d μ - t^{q - p} ‖ v ‖_{W_{λ} (b)}^{q}$ . Then from $(F_{4})$ , we have
$\begin{aligned} H_{1}^{'} (t) \\ = (1 - p) t^{- p} \int_{V} [F_{t u} (x, t u, t v) u + F_{t v} (x, t u, t v) v] d μ \\ + t^{1 - p} \int_{V} [F_{t u, t u} (x, t u, t v) u^{2} + F_{t u, t v} (x, t u, t v) u v + F_{t v, t v} (x, t u, t v) v^{2} \\ + F_{t v, t u} (x, t u, t v) v u] d μ - (q - p) t^{q - p - 1} ‖ v ‖_{W_{λ} (b)}^{q} \\ = (1 - p) t^{- p} \int_{V} [F_{t u} (x, t u, t v) u + F_{t v} (x, t u, t v) v] d μ \\ + t^{- p} \int_{V} [F_{t u, t u} (x, t u, t v) t u^{2} + F_{t v, t v} (x, t u, t v) t v^{2}] d μ \\ + t^{1 - p} \int_{V} [F_{t u, t v} (x, t u, t v) u v + F_{t v, t u} (x, t u, t v) v u] d μ - (q - p) t^{q - p - 1} ‖ v ‖_{W_{λ} (b)}^{q} \\ > (1 - p) t^{- p} \int_{V} [F_{t u} (x, t u, t v) u + F_{t v} (x, t u, t v) v] d μ \\ + (p - 1) t^{- p} \int_{V} [F_{t u} (x, t u, t v) u + F_{t v} (x, t u, t v) v] d μ \\ + t^{1 - p} \int_{V} [F_{t u, t v} (x, t u, t v) u v + F_{t v, t u} (x, t u, t v) v u] d μ - (q - p) t^{q - p - 1} ‖ v ‖_{W_{λ} (b)}^{q} \\ \geq 0. \end{aligned}$
Thus $H_{1} (t)$ is strictly increasing in $(0, + \infty)$ . Hence by observing (3.5), there exists a unique $t_{0} > 0$ such that (3.4) holds and $(t_{0} u, t_{0} v) \in N_{λ}$ . Therefore, $N_{λ}$ is non-empty.
Lemma 3.4
If $(F_{1})$ and $(F_{4})$ hold, then $J_{λ} (t u, t v) \leq J_{λ} (u, v)$ for any given $(u, v) \in N_{λ}$ and all $t > 0$ .
Proof.
Since $(u, v) \in N_{λ}$ , we have $g^{'} (1) = 0$ , where $g (\cdot)$ is introduced in the proof of Lemma 3.3, and 1 is the unique critical point of $g (\cdot) .$ Then
$g (t) = J_{λ} (t u, t v) = \frac{t^{p}}{p} ‖ u ‖_{W_{λ} (a)}^{p} + \frac{t^{q}}{q} ‖ v ‖_{W_{λ} (b)}^{q} - \int_{V} F (x, t u, t v) d μ .$
Then by $(F_{4})$ , we get
$\begin{array}{rcl} g^{'} (t) & = & t^{p - 1} ‖ u ‖_{W_{λ} (a)}^{p} + t^{q - 1} ‖ v ‖_{W_{λ} (b)}^{q} - \int_{V} [F_{t u} (x, t u, t v) u + F_{t v} (x, t u, t v) v] d μ . \\ g^{″} (t) & = & (p - 1) t^{p - 2} ‖ u ‖_{W_{λ} (a)}^{p} + (q - 1) t^{q - 2} ‖ v ‖_{W_{λ} (b)}^{q} - \int_{V} [F_{t u, t u} (x, t u, t v) u^{2} \\ + F_{t v, t v} (x, t u, t v) v^{2} + F_{t u, t v} (x, t u, t v) u v + F_{t v, t u} (x, t u, t v) v u] d μ \\ < & (p - 1) t^{p - 2} ‖ u ‖_{W_{λ} (a)}^{p} + (q - 1) t^{q - 2} ‖ v ‖_{W_{λ} (b)}^{q} - \int_{V} [(β - 1) \frac{1}{t} F_{t u} (x, t u, t v) u \\ + (β - 1) \frac{1}{t} F_{t v} (x, t u, t v) v + F_{t u, t v} (x, t u, t v) u v + F_{t v, t u} (x, t u, t v) v u] d μ . \end{array}$
Let $t = 1$ . By $(F_{4})$ , we have
$\begin{aligned} g^{″} (1) & < (p - 1) ‖ u ‖_{W_{λ} (a)}^{p} + (q - 1) ‖ v ‖_{W_{λ} (b)}^{q} - \int_{V} [(β - 1) F_{u} (x, u, v) u \\ + (β - 1) F_{v} (x, u, v) v + F_{u v} (x, u, v) u v + F_{v u} (x, u, v) v u] d μ \\ \leq (β - 1) (‖ u ‖_{W_{λ} (a)}^{p} + ‖ v ‖_{W_{λ} (b)}^{q}) - \int_{V} [(β - 1) F_{u} (x, u, v) u \\ + (β - 1) F_{v} (x, u, v) v + F_{u v} (x, u, v) u v + F_{v u} (x, u, v) v u] d μ \\ = (β - 1) \int_{V} [F_{u} (x, u, v) u + F_{v} (x, u, v) v] d μ - \int_{V} [(β - 1) F_{u} (x, u, v) u \\ + (β - 1) F_{v} (x, u, v) v + F_{u v} (x, u, v) u v + F_{v u} (x, u, v) v u] d μ \\ \leq 0. \end{aligned}$
(3.6)
From (3.6) and the fact that 1 is the unique critical point of $g (t)$ , we obtain that $t = 1$ is the maximizer of $g (\cdot)$ . Hence $J_{λ} (t u, t v) \leq J_{λ} (u, v)$ for all $t > 0$ .
Lemma 3.5
If $(F_{3})$ holds, there exists a constant $η > 0$ such that $m_{λ} = inf_{(u, v) \in N_{λ}} J_{λ} (u, v) \geq η > 0$ .
Proof.
By $(F_{3})$ and Appendix A.2, we can get
$| F (x, s, t) | \leq \frac{2 C_{1} (x)}{p} | s |^{p} + (\frac{C_{1} (x) (p - 1)}{p} + \frac{C_{1} (x)}{q}) | t |^{q} + \frac{2 C_{2} (x)}{r_{1}} | s |^{r_{1}} + (\frac{C_{2} (x) (r_{1} - 1)}{r_{1}} + \frac{C_{2} (x)}{r_{2}}) | t |^{r_{2}},$
(3.7)
for all $x \in V$ . For $(u, v) \in W_{λ} ∖ {(0, 0)}$ with $‖ (u, v) ‖_{W_{λ}} < 1$ , from (3.7) and Lemma 2.2, we have
$\begin{aligned} J_{λ} (u, v) \\ \geq \frac{1}{p} ‖ u ‖_{W_{λ} (a)}^{p} + \frac{1}{q} ‖ v ‖_{W_{λ} (b)}^{q} - \int_{V} \frac{2 C_{1} (x)}{p} | u |^{p} d μ - \int_{V} (\frac{C_{1} (x) (p - 1)}{p} + \frac{C_{1} (x)}{q}) | v |^{q} d μ \\ - \int_{V} \frac{2 C_{2} (x)}{r_{1}} | u |^{r_{1}} d μ - \int_{V} (\frac{C_{2} (x) (r_{1} - 1)}{r_{1}} + \frac{C_{2} (x)}{r_{2}}) | v |^{r_{2}} d μ \\ \geq \frac{1}{p} ‖ u ‖_{W_{λ} (a)}^{p} + \frac{1}{q} ‖ v ‖_{W_{λ} (b)}^{q} - \frac{2 ‖ C_{1} ‖_{\infty, V}}{p} ‖ u ‖_{p}^{p} - (\frac{‖ C_{1} ‖_{\infty, V} (p - 1)}{p} + \frac{‖ C_{1} ‖_{\infty, V}}{q}) ‖ v ‖_{q}^{q} \\ - \frac{2 ‖ C_{2} ‖_{\infty, V}}{r_{1}} ‖ u ‖_{r_{1}}^{r_{1}} - (\frac{‖ C_{2} ‖_{\infty, V} (r_{1} - 1)}{r_{1}} + \frac{‖ C_{2} ‖_{\infty, V}}{r_{2}}) ‖ v ‖_{r_{2}}^{r_{2}} \\ \geq (\frac{1}{p} - \frac{2 ‖ C_{1} ‖_{\infty, V}}{p} - \frac{2 ‖ C_{2} ‖_{\infty, V}}{r_{1}} K_{(p, r_{1})}^{r_{1}} ‖ u ‖_{W_{λ} (a)}^{r_{1} - p}) ‖ u ‖_{W_{λ} (a)}^{p} \\ + [\frac{1}{q} - (\frac{‖ C_{1} ‖_{\infty, V} (p - 1)}{p} + \frac{‖ C_{1} ‖_{\infty, V}}{q}) - (\frac{‖ C_{2} ‖_{\infty, V} (r_{1} - 1)}{r_{1}} + \frac{‖ C_{2} ‖_{\infty, V}}{r_{2}}) K_{(q, r_{2})}^{r_{2}} ‖ v ‖_{W_{λ} (b)}^{r_{2} - q}] ‖ v ‖_{W_{λ} (b)}^{q} \\ \geq \frac{1}{2^{β - 1}} ‖ (u, v) ‖_{W_{λ}}^{β} min {\frac{1}{p} - \frac{2 ‖ C_{1} ‖_{\infty, V}}{p} - \frac{2 ‖ C_{2} ‖_{\infty, V}}{r_{1}} K_{(p, r_{1})}^{r_{1}} ‖ (u, v) ‖_{W_{λ}}^{r_{1} - p}, \\ \frac{1}{q} - (\frac{‖ C_{1} ‖_{\infty, V} (p - 1)}{p} + \frac{‖ C_{1} ‖_{\infty, V}}{q}) - (\frac{‖ C_{2} ‖_{\infty, V} (r_{1} - 1)}{r_{1}} + \frac{‖ C_{2} ‖_{\infty, V}}{r_{2}}) K_{(q, r_{2})}^{r_{2}} ‖ (u, v) ‖_{W_{λ}}^{r_{2} - q}} . \end{aligned}$
Let
$ρ \in (0, min {1, {(\frac{\frac{1}{p} - \frac{2 ‖ C_{1} ‖_{\infty, V}}{p}}{\frac{2 ‖ C_{2} ‖_{\infty, V}}{r_{1}} K_{(p, r_{1})}^{r_{1}}})}^{\frac{1}{r_{1} - p}}, {(\frac{\frac{1}{q} - \frac{‖ C_{1} ‖_{\infty, V} (p - 1)}{p} - \frac{‖ C_{1} ‖_{\infty, V}}{q}}{(\frac{‖ C_{2} ‖_{\infty, V} (r_{1} - 1)}{r_{1}} + \frac{‖ C_{2} ‖_{\infty, V}}{r_{2}}) K_{(q, r_{2})}^{r_{2}}})}^{\frac{1}{r_{2} - q}}}),$
where $p < r_{1}$ , $q < r_{2}$ , and $C_{1} (x) \leq \frac{1}{1 + β} < \frac{1}{2}$ , and
$\begin{aligned} η & = \frac{1}{2^{β - 1}} ρ^{β} min {\frac{1}{p} - \frac{2 ‖ C_{1} ‖_{\infty, V}}{p} - \frac{2 ‖ C_{2} ‖_{\infty, V}}{r_{1}} K_{(p, r_{1})}^{r_{1}} ρ^{r_{1} - p}, \\ \frac{1}{q} - (\frac{‖ C_{1} ‖_{\infty, V} (p - 1)}{p} + \frac{‖ C_{1} ‖_{\infty, V}}{q}) - (\frac{‖ C_{2} ‖_{\infty, V} (r_{1} - 1)}{r_{1}} + \frac{‖ C_{2} ‖_{\infty, V}}{r_{2}}) K_{(q, r_{2})}^{r_{2}} ρ^{r_{2} - q}} . \end{aligned}$
Then
$J_{λ} (u, v) \geq η > 0 for\;all ‖ (u, v) ‖_{W_{λ}} = ρ .$
For any given $(u, v) \in N_{λ} \subset W_{λ}$ , there exists $τ_{λ, u, v} > 0$ such that $τ_{λ, u, v} ‖ (u, v) ‖_{W_{λ}} = ρ$ . Then by Lemma 3.4, we have
$\begin{aligned} J_{λ} (u, v) & \geq J_{λ} (τ_{λ, u, v} u, τ_{λ, u, v} v) \geq η > 0 \\ \Rightarrow m_{λ} = inf_{(u, v) \in N_{λ}} J_{λ} (u, v) > 0. \end{aligned}$
Thus we finish the proof.
Lemma 3.6
If $(F_{2})$ holds, then $J_{λ} |_{N_{λ}}$ is coercive.
Proof.
We prove the inverse negative proposition of coercive. That is, if ${(u_{k}, v_{k})} \subseteq N_{λ}$ and $J_{λ} (u_{k}, v_{k}) \leq M$ for some $M > 0$ and all $k \in N$ , then ${(u_{k}, v_{k})} \subseteq N_{λ}$ is bounded.

Indeed, from $(F_{2})$ , we have
$\begin{aligned} M & \geq \frac{1}{p} ‖ u_{k} ‖_{W_{λ} (a)}^{p} + \frac{1}{q} ‖ v_{k} ‖_{W_{λ} (b)}^{q} - \int_{V} F (x, u_{k}, v_{k}) d μ \\ \geq \frac{1}{p} ‖ u_{k} ‖_{W_{λ} (a)}^{p} + \frac{1}{q} ‖ v_{k} ‖_{W_{λ} (b)}^{q} - \frac{1}{α} \int_{V} (F_{u_{k}} (x, u_{k}, v_{k}) u_{k} + F_{v_{k}} (x, u_{k}, v_{k}) v_{k} + ϱ (| u_{k} |^{p} + | v_{k} |^{q})) d μ \\ = (\frac{1}{p} - \frac{1}{α}) ‖ u_{k} ‖_{W_{λ} (a)}^{p} + (\frac{1}{q} - \frac{1}{α}) ‖ v_{k} ‖_{W_{λ} (b)}^{q} - \frac{ϱ}{α} ‖ u_{k} ‖_{p}^{p} - \frac{ϱ}{α} ‖ v_{k} ‖_{q}^{q} \\ \geq (\frac{1}{p} - \frac{1}{α} - \frac{ϱ}{α}) ‖ u_{k} ‖_{W_{λ} (a)}^{p} + (\frac{1}{q} - \frac{1}{α} - \frac{ϱ}{α}) ‖ v_{k} ‖_{W_{λ} (b)}^{q} . \end{aligned}$
Since $0 < ϱ < min {\frac{α - p}{p}, \frac{α - q}{q}}$ , we infer that ${(u_{k}, v_{k})} \subseteq N_{λ}$ is bounded. Therefore, $J_{λ} |_{N_{λ}}$ is coercive.
Lemma 3.7
If $(F_{1})$ , $(F_{3})$ , and $(F_{4})$ hold, then there exists some $(u_{λ}, v_{λ}) \in N_{λ}$ such that $m_{λ}$ can be achieved.
Proof.
Let ${(u_{k}, v_{k})} \subseteq N_{λ}$ be a minimizing sequence, that is,
$lim_{k \to + \infty} J_{λ} (u_{k}, v_{k}) = m_{λ} .$
From Lemma 3.6, we have ${(u_{k}, v_{k})} \subseteq N_{λ}$ is bounded, that is, there exists a positive constant $T$ such that $‖ (u_{k}, v_{k}) ‖_{W_{λ}} \leq T$ . Then
$T \geq ‖ u_{k} ‖_{W_{λ} (a)} \geq {(\int_{V} | u |^{p} d μ)}^{\frac{1}{p}} \geq ‖ u_{k} ‖_{\infty, V} .$
(3.8)
Similarly, $‖ v_{k} ‖_{\infty, V} \leq T$ . Lemma 2.3 tells us that there exists some $(u_{λ}, v_{λ}) \in W_{λ}$ such that, up to a subsequence,
$\begin{aligned} {\begin{cases} (u_{k}, v_{k}) ⇀ (u_{λ}, v_{λ}) & in W_{λ}, \\ u_{k} (x) \to u_{λ} (x), v_{k} (x) \to v_{λ} (x) & \forall x \in V, \\ u_{k} \to u_{λ} & in L^{θ_{1}} (V), \forall θ_{1} \in [1, + \infty], \\ v_{k} \to v_{λ} & in L^{θ_{2}} (V), \forall θ_{2} \in [1, + \infty], \end{cases} \end{aligned}$
(3.9)
as $k \to + \infty$ . By $(F_{3})$ , (3.8), and Appendix A.3, there exists positive functions $C_{3} (x) = max {1 + \frac{1}{p} + \frac{q - 1}{q}, 1 + \frac{1}{q} + \frac{p - 1}{p}} \cdot C_{1} (x), C_{4} (x) = max {1 + \frac{1}{r_{1}} + \frac{r_{2} - 1}{r_{2}}, 1 + \frac{1}{r_{2}} + \frac{r_{1} - 1}{r_{1}}} \cdot C_{2} (x)$ such that
$\begin{aligned} | F_{u_{k}} (x, u_{k}, v_{k}) u_{k} + F_{v_{k}} (x, u_{k}, v_{k}) v_{k} | \\ \leq C_{3} (x) (| u_{k} |^{p} + | v_{k} |^{q}) + C_{4} (x) (| u_{k} |^{r_{1}} + | v_{k} |^{r_{2}}) \\ \leq C_{3} (x) (‖ u_{k} ‖_{\infty, V}^{p} + ‖ v_{k} ‖_{\infty, V}^{q}) + C_{4} (x) (‖ u_{k} ‖_{\infty, V}^{r_{1}} + ‖ v_{k} ‖_{\infty, V}^{r_{2}}) \\ \leq C_{3} (x) (T^{p} + T^{q}) + C_{4} (x) (T^{r_{1}} + T^{r_{2}}) \\ \leq max {T^{p} + T^{q}, N^{r_{1}} + T^{r_{2}}} (C_{3} (x) + C_{4} (x)) \\ := g_{1} (x) . \end{aligned}$
Note that $C_{1}, C_{2} \in L^{1} (V)$ . Then we can obtain $g_{1} (x) \in L^{1} (V)$ . Thus from Lebesgue dominated convergence theorem and (3.9), we have
$lim_{k \to + \infty} \int_{V} [F_{u_{k}} (x, u_{k}, v_{k}) u_{k} + F_{v_{k}} (x, u_{k}, v_{k}) v_{k}] d μ = \int_{V} [F_{u_{λ}} (x, u_{λ}, v_{λ}) u_{λ} + F_{v_{λ}} (x, u_{λ}, v_{λ}) v_{λ}] d μ .$
(3.10)
Simiarly, by $(F_{3})$ , (3.8), (3.9), Appendix A.2, and Lebesgue dominated convergence theorem, we also have
$lim_{k \to + \infty} \int_{V} F (x, u_{k}, v_{k}) d μ = \int_{V} F (x, u_{λ}, v_{λ}) d μ .$
(3.11)
Since $(u_{k}, v_{k}) \in N_{λ}$ , we have
$‖ u_{k} ‖_{W_{λ} (a)}^{p} + ‖ v_{k} ‖_{W_{λ} (b)}^{q} = \int_{V} [F_{u_{k}} (x, u_{k}, v_{k}) u_{k} + F_{v_{k}} (x, u_{k}, v_{k}) v_{k}] d μ .$
(3.12)
By the weak lower continuity of the norm, (3.10) and (3.12), we have
$\begin{aligned} ‖ u_{λ} ‖_{W_{λ} (a)}^{p} + ‖ v_{λ} ‖_{W_{λ} (b)}^{q} & \leq \underset{k \to + \infty}{lim inf} (‖ u_{k} ‖_{W_{λ} (a)}^{p} + ‖ v_{k} ‖_{W_{λ} (b)}^{q}) \\ = \underset{k \to + \infty}{lim inf} (\int_{V} F_{u_{k}} (x, u_{k}, v_{k}) u_{k} d μ + \int_{V} F_{v_{k}} (x, u_{k}, v_{k}) v_{k} d μ) \\ = \int_{V} F_{u_{λ}} (x, u_{λ}, v_{λ}) u_{λ} d μ + \int_{V} F_{v_{λ}} (x, u_{λ}, v_{λ}) v_{λ} d μ . \end{aligned}$
(3.13)
Next, we show that $(u_{λ}, v_{λ}) \in N_{λ}$ and $(u_{λ}, v_{λ}) \neq (0, 0)$ .

If $(u_{λ}, v_{λ}) = (0, 0)$ , then from (3.10) and (3.12), we have
$\begin{aligned} \underset{k \to + \infty}{lim inf} (‖ u_{k} ‖_{W_{λ} (a)}^{p} + ‖ v_{k} ‖_{W_{λ} (b)}^{q}) & = \underset{k \to + \infty}{lim inf} \int_{V} [F_{u_{k}} (x, u_{k}, v_{k}) u_{k} + F_{v_{k}} (x, u_{k}, v_{k}) v_{k}] d μ \\ = \int_{V} [F_{u_{λ}} (x, u_{λ}, v_{λ}) u_{λ} + F_{v_{λ}} (x, u_{λ}, v_{λ}) v_{λ}] d μ \\ = 0. \end{aligned}$
Thus, $(u_{k}, v_{k}) \to (0, 0)$ in $W_{λ}$ and then by the continuity of $J_{λ}$ , (3.11) and $(F_{1})$ we get $J_{λ} (u_{k}, v_{k}) \to m_{λ} = 0.$ It is a contradiction to Lemma 3.5. Therefore, $(u_{λ}, v_{λ}) \neq (0, 0)$ .

Moreover, from (3.13), we suppose that
$‖ u_{λ} ‖_{W_{λ} (a)}^{p} + ‖ v_{λ} ‖_{W_{λ} (b)}^{q} < \int_{V} [F_{u_{λ}} (x, u_{λ}, v_{λ}) u_{λ} + F_{v_{λ}} (x, u_{λ}, v_{λ}) v_{λ}] d μ,$
(3.14)
which implies that $g^{'} (1) < 0$ . By Lemma 3.3, we know that for $(u_{λ}, v_{λ}) \in W_{λ} ∖ {(0, 0)}$ , there is a unique $t_{λ}$ such that $(t_{λ} u_{λ}, t_{λ} v_{λ}) \in N_{λ}$ and $g^{'} (t_{λ}) = 0$ . By Lemma 3.4 and $(F_{4})$ , we get
$\begin{aligned} g^{'} (t_{λ}) = t_{λ}^{p - 1} ‖ u_{λ} ‖_{W_{λ} (a)}^{p} + t_{λ}^{q - 1} ‖ v_{λ} ‖_{W_{λ} (b)}^{q} - \int_{V} [F_{t_{λ} u_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) u_{λ} + F_{t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) v_{λ}] d μ = 0 \\ \Rightarrow t_{λ}^{p - 1} ‖ u_{λ} ‖_{W_{λ} (a)}^{p} + t_{λ}^{q - 1} ‖ v_{λ} ‖_{W_{λ} (b)}^{q} = \int_{V} [F_{t_{λ} u_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) u_{λ} + F_{t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) v_{λ}] d μ, \end{aligned}$
and it follows from (3.14), $(F_{1})$ and $(F_{4})$ that
$\begin{aligned} g^{″} (t_{λ}) = & (p - 1) t_{λ}^{p - 2} ‖ u_{λ} ‖_{W_{λ} (a)}^{p} + (q - 1) t_{λ}^{q - 2} ‖ v_{λ} ‖_{W_{λ} (b)}^{q} \\ - \int_{V} [F_{t_{λ} u_{λ}, t_{λ} u_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) u_{λ}^{2} + F_{t_{λ} v_{λ}, t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) v_{λ}^{2} \\ - \int_{V} [F_{t_{λ} u_{λ}, t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) u_{λ} v_{λ} + F_{t_{λ} v_{λ}, t_{λ} u_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) v_{λ} u_{λ}] d μ \\ < & (p - 1) t_{λ}^{p - 2} ‖ u_{λ} ‖_{W_{λ} (a)}^{p} + (q - 1) t_{λ}^{q - 2} ‖ v_{λ} ‖_{W_{λ} (b)}^{q} \\ - (β - 1) \int_{V} [\frac{1}{t_{λ}} F_{t_{λ} u_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) u_{λ} + \frac{1}{t_{λ}} F_{t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) v_{λ}] d μ \\ - \int_{V} [F_{t_{λ} u_{λ}, t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) u_{λ} v_{λ} + F_{t_{λ} v_{λ}, t_{λ} u_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) u_{λ} v_{λ}] d μ . \\ < & 0. \end{aligned}$
Thus we obtain that $t_{λ}$ is the unique maximizer of $g (t)$ . Then together with $g^{'} (1) < 0$ and $g (0) = 0$ we can infer that there exists $t_{λ} \in (0, 1)$ such that $(t_{λ} u_{λ}, t_{λ} v_{λ}) \in N_{λ}$ .

When $β = p$ , by Appendix A.4, (3.10), (3.11), and the weak lower continuity of the norm, we get
$\begin{aligned} m_{λ} \\ \leq J_{λ} (t_{λ} u_{λ}, t_{λ} v_{λ}) \\ = \frac{1}{p} ‖ t_{λ} u_{λ} ‖_{W_{λ} (a)}^{p} + \frac{1}{q} ‖ t_{λ} v_{λ} ‖_{W_{λ} (b)}^{q} - \int_{V} F (x, t_{λ} u_{λ}, t_{λ} v_{λ}) d μ \\ = \frac{1}{p} [\int_{V} F_{t_{λ} u_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) t_{λ} u_{λ} d μ + \int_{V} F_{t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) t_{λ} v_{λ} d μ - ‖ t_{λ} v_{λ} ‖_{W_{λ} (b)}^{q}] \\ + \frac{1}{q} ‖ t_{λ} v_{λ} ‖_{W_{λ} (b)}^{q} - \int_{V} F (x, t_{λ} u_{λ}, t_{λ} v_{λ}) d μ \\ = \int_{V} [\frac{1}{p} F_{t_{λ} u_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) t_{λ} u_{λ} + \frac{1}{p} F_{t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) t_{λ} v_{λ} - F (x, t_{λ} u_{λ}, t_{λ} v_{λ})] d μ \\ + (\frac{1}{q} - \frac{1}{p}) ‖ t_{λ} v_{λ} ‖_{W_{λ} (b)}^{q} \\ < \int_{V} [\frac{1}{p} F_{u_{λ}} (x, u_{λ}, v_{λ}) u_{λ} + \frac{1}{p} F_{v_{λ}} (x, u_{λ}, v_{λ}) v_{λ} - F (x, u_{λ}, v_{λ})] d μ \\ + (\frac{1}{q} - \frac{1}{p}) ‖ v_{λ} ‖_{W_{λ} (b)}^{q} \\ \leq \underset{k \to + \infty}{lim inf} [\int_{V} (\frac{1}{p} F_{u_{k}} (x, u_{k}, v_{k}) u_{k} + \frac{1}{p} F_{v_{k}} (x, u_{k}, v_{k}) v_{k} - F (x, u_{k}, v_{k})) d μ \\ + (\frac{1}{q} - \frac{1}{p}) ‖ v_{k} ‖_{W_{λ} (b)}^{q}] \\ = \underset{k \to + \infty}{lim inf} [\frac{1}{p} ‖ u_{k} ‖_{W_{λ} (a)}^{p} + \frac{1}{p} ‖ v_{k} ‖_{W_{λ} (b)}^{q} + (\frac{1}{q} - \frac{1}{p}) ‖ v_{k} ‖_{W_{λ} (b)}^{q} - \int_{V} F (x, u_{k}, v_{k}) d μ] \\ = m_{λ} \end{aligned}$
(3.15)
which is a contradiction. When $β = q$ , we also have the similar contradiction. Hence (3.14) does not hold.

Therefore,
$\int_{V} F_{u_{λ}} (x, u_{λ}, v_{λ}) u_{λ} d μ + \int_{V} F_{v_{λ}} (x, u_{λ}, v_{λ}) v_{λ} d μ = ‖ u_{λ} ‖_{W_{λ} (a)}^{p} + ‖ v_{λ} ‖_{W_{λ} (b)}^{q},$
and then $(u_{λ}, v_{λ}) \in N_{λ}$ . Thus $m_{λ}$ is achieved by $(u_{λ}, v_{λ})$ .
Lemma 3.8
If $(F_{3})$ holds, there exists a constant $ξ > 0$ which is independent of $λ$ , such that $‖ (u, v) ‖_{W_{λ}} \geq ξ$ for all $(u, v) \in N_{λ}$ .
Proof.
Since $(u, v) \in N_{λ}$ , by Lemma 2.2 and Appendix A.3, we have
$\begin{aligned} 0 & = ⟨ J_{λ}^{'} (u, v), (u, v) ⟩ \\ = ‖ u ‖_{W_{λ} (a)}^{p} + ‖ v ‖_{W_{λ} (b)}^{q} - \int_{V} F_{u} (x, u, v) u d μ - \int_{V} F_{v} (x, u, v) u d μ \\ \geq ‖ u ‖_{W_{λ} (a)}^{p} + ‖ v ‖_{W_{λ} (b)}^{q} - \int_{V} [C_{3} (x) (| u |^{p} + | v |^{q}) + C_{4} (x) (| u |^{r_{1}} + | v |^{r_{2}})] d μ \\ = ‖ u ‖_{W_{λ} (a)}^{p} + ‖ v ‖_{W_{λ} (b)}^{q} - ‖ C_{3} ‖_{\infty, V} ‖ u ‖_{p}^{p} - ‖ C_{3} ‖_{\infty, V} ‖ v ‖_{q}^{q} - ‖ C_{4} ‖_{\infty, V} ‖ u ‖_{r_{1}}^{r_{1}} - ‖ C_{4} ‖_{\infty, V} ‖ v ‖_{r_{2}}^{r_{2}} \\ \geq ‖ u ‖_{W_{λ} (a)}^{p} + ‖ v ‖_{W_{λ} (b)}^{q} \\ - ‖ C_{3} ‖_{\infty, V} ‖ u ‖_{W_{λ} (a)}^{p} - ‖ C_{3} ‖_{\infty, V} ‖ v ‖_{W_{λ} (b)}^{q} - ‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}} ‖ u ‖_{W_{λ} (a)}^{r_{1}} - ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}} ‖ v ‖_{W_{λ} (b)}^{r_{2}} \\ = (1 - ‖ C_{3} ‖_{\infty, V}) ‖ u ‖_{W_{λ} (a)}^{p} + (1 - ‖ C_{3} ‖_{\infty, V}) ‖ v ‖_{W_{λ} (b)}^{q} \\ - ‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}} ‖ u ‖_{W_{λ} (a)}^{r_{1}} - ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}} ‖ v ‖_{W_{λ} (b)}^{r_{2}} . \end{aligned}$
(3.16)
We will discuss the above formula in categories.

Case $(1) :$ Assume that $p > q$ , $‖ u ‖_{W_{λ} (a)} \geq 1$ and $‖ v ‖_{W_{λ} (b)} \geq 1$ . Then $‖ (u, v) ‖_{W_{λ}} \geq 1$ and by (3.16),
$\begin{aligned} 0 \\ \geq (1 - ‖ C_{3} ‖_{\infty, V}) ‖ u ‖_{W_{λ} (a)}^{p} + (1 - ‖ C_{3} ‖_{\infty, V}) ‖ v ‖_{W_{λ} (b)}^{q} - ‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}} ‖ u ‖_{W_{λ} (a)}^{r_{1}} - ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}} ‖ v ‖_{W_{λ} (b)}^{r_{2}} \\ \geq (1 - ‖ C_{3} ‖_{\infty, V}) (‖ u ‖_{W_{λ} (a)}^{q} + ‖ v ‖_{W_{λ} (b)}^{q}) - max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}} (‖ u ‖_{W_{λ} (a)}^{r_{1}} + ‖ v ‖_{W_{λ} (b)}^{r_{2}}) \\ \geq \frac{1 - ‖ C_{3} ‖_{\infty, V}}{2^{q - 1}} (‖ u ‖_{W_{λ} (a)} + ‖ v ‖_{W_{λ} (b)})^{q} - max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}} (‖ (u, v) ‖_{W_{λ}}^{r_{1}} + ‖ (u, v) ‖_{W_{λ}}^{r_{2}}) \\ \geq \frac{1 - ‖ C_{3} ‖_{\infty, V}}{2^{q - 1}} ‖ (u, v) ‖_{W_{λ}}^{q} - 2 max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}} ‖ (u, v) ‖_{W_{λ}}^{max {r_{1}, r_{2}}} . \end{aligned}$
Thus we have
$‖ (u, v) ‖_{W_{λ}} \geq {(\frac{1 - ‖ C_{3} ‖_{\infty, V}}{2^{q} max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}}})}^{\frac{1}{max {r_{1}, r_{2}} - q}} .$
Case $(2) :$ Assume that $p > q$ , $‖ u ‖_{W_{λ} (a)} \geq 1$ and $‖ v ‖_{W_{λ} (b)} < 1$ . Then $‖ (u, v) ‖_{W_{λ}} \geq 1$ and by (3.16),
$\begin{aligned} 0 \\ \geq (1 - ‖ C_{3} ‖_{\infty, V}) ‖ u ‖_{W_{λ} (a)}^{p} + (1 - ‖ C_{3} ‖_{\infty, V}) ‖ v ‖_{W_{λ} (b)}^{q} - ‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}} ‖ u ‖_{W_{λ} (a)}^{r_{1}} - ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}} ‖ v ‖_{W_{λ} (b)}^{r_{2}} \\ \geq (1 - ‖ C_{3} ‖_{\infty, V}) ‖ u ‖_{W_{λ} (a)}^{p} - ‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}} ‖ u ‖_{W_{λ} (a)}^{r_{1}} - ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}} ‖ v ‖_{W_{λ} (b)}^{r_{2}} . \end{aligned}$
Thus we have
$\begin{aligned} 1 - ‖ C_{3} ‖_{\infty, V} & \leq ‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}} ‖ u ‖_{W_{λ} (a)}^{r_{1} - p} + ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}} \frac{‖ v ‖_{W_{λ} (b)}^{r_{2}}}{‖ u ‖_{W_{λ} (a)}^{p}} \\ \leq ‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}} ‖ u ‖_{W_{λ} (a)}^{r_{1} - p} + ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}} ‖ v ‖_{W_{λ} (b)}^{r_{2}} \\ \leq ‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}} ‖ (u, v) ‖_{W_{λ}}^{r_{1} - p} + ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}} ‖ (u, v) ‖_{W_{λ}}^{r_{2}} \\ \leq 2 max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}} ‖ (u, v) ‖_{W_{λ}}^{max {r_{1} - p, r_{2}}} \\ \Rightarrow ‖ (u, v) ‖_{W_{λ}} & \geq {(\frac{1 - ‖ C_{3} ‖_{\infty, V}}{2 max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}}})}^{\frac{1}{max {r_{1} - p, r_{2}}}} . \end{aligned}$
Case $(3) :$ Assume that $p > q$ , $‖ u ‖_{W_{λ} (a)} < 1$ and $‖ v ‖_{W_{λ} (b)} \geq 1$ . Then $‖ (u, v) ‖_{W_{λ}} \geq 1$ and similar to case $(2)$ , we have
$‖ (u, v) ‖_{W_{λ}} \geq {(\frac{1 - ‖ C_{3} ‖_{\infty, V}}{2 max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}}})}^{\frac{1}{max {r_{1}, r_{2} - q}}} .$
Case $(4) :$ Assume that $p > q$ , $‖ u ‖_{W_{λ} (a)} < 1$ and $‖ v ‖_{W_{λ} (b)} < 1$ , by (3.16),
$\begin{aligned} 0 \\ \geq (1 - ‖ C_{3} ‖_{\infty, V}) (‖ u ‖_{W_{λ} (a)}^{p} + ‖ v ‖_{W_{λ} (b)}^{p}) - max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}} (‖ u ‖_{W_{λ} (a)}^{r_{1}} + ‖ v ‖_{W_{λ} (b)}^{r_{2}}) \\ \geq \frac{1 - ‖ C_{3} ‖_{\infty, V}}{2^{p - 1}} ‖ (u, v) ‖_{W_{λ}}^{p} - max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}} (‖ (u, v) ‖_{W_{λ}}^{r_{1}} + ‖ (u, v) ‖_{W_{λ}}^{r_{2}}) . \end{aligned}$
Thus we have
$max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}} (‖ (u, v) ‖_{W_{λ}}^{r_{1} - p} + ‖ (u, v) ‖_{W_{λ}}^{r_{2} - p}) \geq \frac{1 - ‖ C_{3} ‖_{\infty, V}}{2^{p - 1}} .$
(3.17)
When $‖ (u, v) ‖_{W_{λ}} \geq 1$ , from (3.17), we have
$\begin{aligned} 2 max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}} ‖ (u, v) ‖^{max {r_{1} - p, r_{2} - p}} \geq \frac{1 - ‖ C_{3} ‖_{\infty, V}}{2^{p - 1}} \\ \Rightarrow ‖ (u, v) ‖_{W_{λ}} \geq {(\frac{1 - ‖ C_{3} ‖_{\infty, V}}{2^{p} max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}}})}^{\frac{1}{max {r_{1}, r_{2}} - p}} . \end{aligned}$
When $‖ (u, v) ‖_{W_{λ}} < 1$ , from (3.17), we have
$\begin{aligned} 2 max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}} ‖ (u, v) ‖^{min {r_{1} - p, r_{2} - p}} \geq \frac{1 - ‖ C_{3} ‖_{\infty, V}}{2^{p - 1}} \\ \Rightarrow ‖ (u, v) ‖_{W_{λ}} \geq {(\frac{1 - ‖ C_{3} ‖_{\infty, V}}{2^{p} max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}}})}^{\frac{1}{min {r_{1}, r_{2}} - p}} . \end{aligned}$
For the case $p < q$ , using the same method, we can obtain that

Case $(5) :$ Assume that $‖ u ‖_{W_{λ} (a)} \geq 1$ and $‖ v ‖_{W_{λ} (b)} \geq 1$ ,
$‖ (u, v) ‖_{W_{λ}} \geq {(\frac{1 - ‖ C_{3} ‖_{\infty, V}}{2^{p} max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}}})}^{\frac{1}{max {r_{1}, r_{2}} - p}};$
Case $(6) :$ Assume that $‖ u ‖_{W_{λ} (a)} \geq 1$ and $‖ v ‖_{W_{λ} (b)} < 1$ ,
$‖ (u, v) ‖_{W_{λ}} \geq {(\frac{1 - ‖ C_{3} ‖_{\infty, V}}{2 max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}}})}^{\frac{1}{max {r_{1} - p, r_{2}}}};$
Case $(7) :$ Assume that $‖ u ‖_{W_{λ} (a)} < 1$ and $‖ v ‖_{W_{λ} (b)} \geq 1$ ,
$‖ (u, v) ‖_{W_{λ}} \geq {(\frac{1 - ‖ C_{3} ‖_{\infty, V}}{2 max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}}})}^{\frac{1}{max {r_{1}, r_{2} - q}}};$
Case $(8) :$ Assume that $‖ u ‖_{W_{λ} (a)} < 1$ and $‖ v ‖_{W_{λ} (b)} < 1$ ,
$\begin{aligned} ‖ (u, v) ‖_{W_{λ}} & \geq {(\frac{1 - ‖ C_{3} ‖_{\infty, V}}{2^{q} max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}}})}^{\frac{1}{max {r_{1}, r_{2}} - q}}, for ‖ (u, v) ‖_{W_{λ}} \geq 1. \end{aligned}$

$\begin{aligned} ‖ (u, v) ‖_{W_{λ}} & \geq {(\frac{1 - ‖ C_{3} ‖_{\infty, V}}{2^{q} max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}}})}^{\frac{1}{min {r_{1}, r_{2}} - q}}, for ‖ (u, v) ‖_{W_{λ}} < 1. \end{aligned}$
Case $(9) :$ Assume that $p = q$ , from (3.16), we have
$\begin{aligned} 0 \\ \geq (1 - ‖ C_{3} ‖_{\infty, V}) ‖ u ‖_{W_{λ} (a)}^{p} + (1 - ‖ C_{3} ‖_{\infty, V}) ‖ v ‖_{W_{λ} (b)}^{q} - ‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}} ‖ u ‖_{W_{λ} (a)}^{r_{1}} - ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}} ‖ v ‖_{W_{λ} (b)}^{r_{2}} \\ \geq \frac{1}{2^{p - 1}} (1 - ‖ C_{3} ‖_{\infty, V}) ‖ (u, v) ‖^{p} - max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}} (‖ (u, v) ‖_{W_{λ}}^{r_{1}} + ‖ (u, v) ‖_{W_{λ}}^{r_{2}}) . \end{aligned}$
Thus we get
$max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}} (‖ (u, v) ‖_{W_{λ}}^{r_{1} - p} + ‖ (u, v) ‖_{W_{λ}}^{r_{2} - p}) \geq \frac{1}{2^{p - 1}} (1 - ‖ C_{3} ‖_{\infty, V}) .$
(3.18)
When $‖ (u, v) ‖_{W_{λ}} \geq 1$ , from (3.18), there is
$\begin{aligned} 2 max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}} ‖ (u, v) ‖^{max {r_{1} - p, r_{2} - p}} \geq \frac{1}{2^{p - 1}} (1 - ‖ C_{3} ‖_{\infty, V}) \\ \Rightarrow ‖ (u, v) ‖_{W_{λ}} \geq {(\frac{1 - ‖ C_{3} ‖_{\infty, V}}{2^{p} max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}}})}^{\frac{1}{max {r_{1}, r_{2}} - p}} . \end{aligned}$
When $‖ (u, v) ‖_{W_{λ}} < 1$ , from (3.18), there is
$\begin{aligned} 2 max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}} ‖ (u, v) ‖^{min {r_{1} - p, r_{2} - p}} \geq \frac{1}{2^{p - 1}} (1 - ‖ C_{3} ‖_{\infty, V}) \\ \Rightarrow ‖ (u, v) ‖_{W_{λ}} \geq {(\frac{1 - ‖ C_{3} ‖_{\infty, V}}{2^{p} max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}}})}^{\frac{1}{min {r_{1}, r_{2}} - p}} . \end{aligned}$
Let
$\begin{aligned} ξ_{1} & = {(\frac{1 - ‖ C_{3} ‖_{\infty, V}}{2^{min {p, q}} max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}}})}^{\frac{1}{max {r_{1}, r_{2}} - min {p, q}}}, \\ ξ_{2} & = {(\frac{1 - ‖ C_{3} ‖_{\infty, V}}{2 max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}}})}^{\frac{1}{max {r_{1} - p, r_{2}}}}, \\ ξ_{3} & = {(\frac{1 - ‖ C_{3} ‖_{\infty, V}}{2 max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}}})}^{\frac{1}{max {r_{1}, r_{2} - q}}}, \\ ξ_{4} & = {(\frac{1 - ‖ C_{3} ‖_{\infty, V}}{2^{max {p, q}} max {‖ C_{4} ‖_{\infty, V} K_{(p, r_{1})}^{r_{1}}, ‖ C_{4} ‖_{\infty, V} K_{(q, r_{2})}^{r_{2}}}})}^{\frac{1}{max {r_{1}, r_{2}} - max {p, q}}} . \end{aligned}$
All in all, we can obtain that $‖ (u, v) ‖_{W_{λ}} \geq ξ$ , where $ξ = min {ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}}$ . By Appendix A.1 and noting that $C_{1} (x) \leq \frac{1}{1 + β}$ , we obtain that $‖ C_{3} ‖_{\infty, V} < 1$ and then $ξ > 0$ . Thus the lemma is proved.
Lemma 3.9
If $(F_{2})$ holds, then for a minimizing sequence ${(u_{k}, v_{k})} \subseteq N_{λ}$ , there holds
$lim_{k \to + \infty} ‖ (u_{k}, v_{k}) ‖_{W_{λ}} \leq {(max {2^{p - 1}, 2^{q - 1}} (\frac{m_{λ}}{\frac{1}{β} - \frac{1 + ϱ}{α}} + 1))}^{\frac{1}{min {p, q}}} .$

Proof.
Since ${(u_{k}, v_{k})} \subseteq N_{λ}$ is a minimizing sequence, we have
$0 = ‖ u_{k} ‖_{W_{λ} (a)}^{p} + ‖ v_{k} ‖_{W_{λ} (b)}^{q} - \int_{V} F_{u_{k}} (x, u_{k}, v_{k}) u_{k} d μ - \int_{V} F_{v_{k}} (x, u_{k}, v_{k}) u_{k} d μ .$
(3.19)
Then by $(F_{2})$ and (3.19),
$\begin{aligned} m_{λ} & = lim_{k \to + \infty} (\frac{1}{p} ‖ u_{k} ‖_{W_{λ} (a)}^{p} + \frac{1}{q} ‖ v_{k} ‖_{W_{λ} (b)}^{q} - \int_{V} F (x, u_{k}, v_{k}) d μ) \\ \geq lim_{k \to + \infty} [\frac{1}{p} ‖ u_{k} ‖_{W_{λ} (a)}^{p} + \frac{1}{q} ‖ v_{k} ‖_{W_{λ} (b)}^{q} \\ - \frac{1}{α} \int_{V} (F_{u_{k}} (x, u_{k}, v_{k}) u_{k} + F_{v_{k}} (x, u_{k}, v_{k}) v_{k} + ϱ (| u_{k} |^{p} + | v_{k} |^{q})) d μ] \\ = lim_{k \to + \infty} (\frac{1}{p} ‖ u_{k} ‖_{W_{λ} (a)}^{p} + \frac{1}{q} ‖ v_{k} ‖_{W_{λ} (b)}^{q} \\ - \frac{1}{α} ‖ u_{k} ‖_{W_{λ} (a)}^{p} - \frac{1}{α} ‖ v_{k} ‖_{W_{λ} (b)}^{q} - \frac{ϱ}{α} ‖ u_{k} ‖_{p}^{p} - \frac{ϱ}{α} ‖ v_{k} ‖_{q}^{q}) \\ \geq lim_{k \to + \infty} [(\frac{1}{p} - \frac{1}{α} - \frac{ϱ}{α}) ‖ u_{k} ‖_{W_{λ} (a)}^{p} + (\frac{1}{q} - \frac{1}{α} - \frac{ϱ}{α}) ‖ v_{k} ‖_{W_{λ} (b)}^{q}] \\ \geq lim_{k \to + \infty} [min {\frac{1}{p} - \frac{1 + ϱ}{α}, \frac{1}{q} - \frac{1 + ϱ}{α}} (‖ u_{k} ‖_{W_{λ} (a)}^{p} + ‖ v_{k} ‖_{W_{λ} (b)}^{q})] \\ \geq lim_{k \to + \infty} [min {\frac{1}{p} - \frac{1 + ϱ}{α}, \frac{1}{q} - \frac{1 + ϱ}{α}} (\frac{‖ (u_{k}, v_{k}) ‖_{W_{λ}}^{min {p, q}}}{max {2^{p - 1}, 2^{q - 1}}} - 1)] (see Xie \& Zhang, 2018, p.66) . \end{aligned}$
Then we can obtain that
$lim_{k \to + \infty} ‖ (u_{k}, v_{k}) ‖_{W_{λ}} \leq {(max {2^{p - 1}, 2^{q - 1}} (\frac{m_{λ}}{\frac{1}{β} - \frac{1 + ϱ}{α}} + 1))}^{\frac{1}{min {p, q}}} .$
The proof is complete.

Proof of Theorem 1.1. We denote the critical set
$K_{J_{λ}} = {(u, v) \in W_{λ} : J_{λ}^{'} (u, v) = 0} .$
Let $k (u, v) := ⟨ J_{λ}^{'} (u, v), (u, v) ⟩$ , that is,
$k (u, v) = ‖ u ‖_{W_{λ} (a)}^{p} + ‖ v ‖_{W_{λ} (b)}^{q} - \int_{V} F_{u} (x, u, v) u d μ - \int_{V} F_{v} (x, u, v) v d μ .$
Then by $(F_{1})$ , $k (u, v) \in C^{1} (W_{λ}, R)$ and
$\begin{aligned} ⟨ k^{'} (u, v), (h_{1}, h_{2}) ⟩ \\ = p \int_{V} (| \nabla u |^{p - 2} Γ (u, h_{1}) + (λ a + 1) | u |^{p - 2} u h_{1}) d μ + q \int_{V} (| \nabla v |^{q - 2} Γ (v, h_{2}) + (λ b + 1) | v |^{q - 2} v h_{2}) d μ \\ - \int_{V} [F_{u u} (x, u, v) u h_{1} + F_{u v} (x, u, v) u h_{2} + F_{u} (x, u, v) h_{1}] d μ \\ - \int_{V} [F_{v v} (x, u, v) v h_{2} + F_{v u} (x, u, v) v h_{1} + F_{v} (x, u, v) h_{2}] d μ, \end{aligned}$
for all $(h_{1}, h_{2}) \in W_{λ}$ . From Lemma 3.7, we know that
$J_{λ} (u_{λ}, v_{λ}) = m_{λ} = inf {J_{λ} (u, v) : k (u, v) = 0, (u, v) \in W_{λ} ∖ (0, 0)} .$
Then by the Lagrange multiplier rule, we can find $ϑ \in R$ such that for $(u_{λ}, v_{λ}) \in N_{λ}$ , we have
$\begin{aligned} J_{λ}^{'} (u_{λ}, v_{λ}) & + ϑ k^{'} (u_{λ}, v_{λ}) = 0 \\ \Rightarrow ⟨ J_{λ}^{'} (u_{λ}, v_{λ}), (u_{λ}, v_{λ}) ⟩ + ϑ ⟨ k^{'} (u_{λ}, v_{λ}), (u_{λ}, v_{λ}) ⟩ = 0 \\ \Rightarrow ϑ ⟨ k^{'} (u_{λ}, v_{λ}), (u_{λ}, v_{λ}) ⟩ = 0. \end{aligned}$
(3.20)
Suppose $ϑ \neq 0$ . Then $⟨ k^{'} (u_{λ}, v_{λ}), (u_{λ}, v_{λ}) ⟩ = 0$ . When $β = p$ , by $(F_{4})$ , we get
$\begin{aligned} ⟨ k^{'} (u_{λ}, v_{λ}), (u_{λ}, v_{λ}) ⟩ \\ = p ‖ u_{λ} ‖_{W_{λ} (a)}^{p} + q ‖ v_{λ} ‖_{W_{λ} (b)}^{q} - \int_{V} [F_{u_{λ}} (x, u_{λ}, v_{λ}) u_{λ} + F_{v_{λ}} (x, u_{λ}, v_{λ}) v_{λ}] d μ \\ - \int_{V} [F_{u_{λ} u_{λ}} (x, u_{λ}, v_{λ}) u_{λ}^{2} + F_{v_{λ} v_{λ}} (x, u_{λ}, v_{λ}) v_{λ}^{2} + 2 F_{u_{λ} v_{λ}} (x, u_{λ}, v_{λ}) u_{λ} v_{λ}] d μ \\ = p [‖ u_{λ} ‖_{W_{λ} (a)}^{p} + ‖ v_{λ} ‖_{W_{λ} (b)}^{q} - \int_{V} [F_{u_{λ}} (x, u_{λ}, v_{λ}) u_{λ} + F_{v_{λ}} (x, u_{λ}, v_{λ}) v_{λ}] d μ] \\ - \int_{V} [F_{u_{λ} u_{λ}} (x, u_{λ}, v_{λ}) u_{λ}^{2} + F_{v_{λ} v_{λ}} (x, u_{λ}, v_{λ}) v_{λ}^{2} + 2 F_{u_{λ} v_{λ}} (x, u_{λ}, v_{λ}) u_{λ} v_{λ}] d μ \\ + (q - p) ‖ v_{λ} ‖_{W_{λ} (b)}^{q} + (p - 1) \int_{V} [F_{u_{λ}} (x, u_{λ}, v_{λ}) u_{λ} + F_{v_{λ}} (x, u_{λ}, v_{λ}) v_{λ}] d μ \\ = (q - p) ‖ v_{λ} ‖_{W_{λ} (b)}^{q} - \int_{V} [F_{u_{λ} u_{λ}} (x, u_{λ}, v_{λ}) u_{λ}^{2} + F_{v_{λ} v_{λ}} (x, u_{λ}, v_{λ}) v_{λ}^{2} \\ + 2 F_{u_{λ} v_{λ}} (x, u_{λ}, v_{λ}) u_{λ} v_{λ}] d μ + (p - 1) \int_{V} [F_{u_{λ}} (x, u_{λ}, v_{λ}) u_{λ} + F_{v_{λ}} (x, u_{λ}, v_{λ}) v_{λ}] d μ \\ < 0. \end{aligned}$
It is a contradiction. Similarly, when $β = q$ , we can also have the same contradiction. So $ϑ = 0$ . From (3.20), we get $J_{λ}^{'} (u_{λ}, v_{λ}) = 0$ and so $(u_{λ}, v_{λ})$ is a nontrivial solution of (1.2). Moreover, by Lemma 3.8, the weak lower semi-continuity of the norm $‖ \cdot ‖_{W_{λ}}$ and Lemma 3.9, it is easy to obtain that
$ξ \leq ‖ (u_{λ}, v_{λ}) ‖_{W_{λ}} \leq {(max {2^{p - 1}, 2^{q - 1}} (\frac{m_{λ}}{\frac{1}{β} - \frac{1 + ϱ}{α}} + 1))}^{\frac{1}{min {p, q}}} .$
(3.21)
Thus the proof is completed.
4. Convergence of the Ground State Solutions Family

In this section, we prove that the ground state solutions family ${(u_{λ}, v_{λ})}$ of (1.2) converge to a ground state solution of (1.3) as $λ \to + \infty$ , which imply Theorem 1.2.

Lemma 4.1

If $(F_{1})$ and $(F_{2})$ hold, then $m_{λ} \to m_{Ω}$ as $λ \to \infty$ , where $m_{Ω}$ is defined by (1).

Proof.

Since $N_{Ω} \subset N_{λ}$ (see Appendix A.5), we obviously have that $m_{λ} \leq m_{Ω}$ for any $λ > 0$ . Take a sequence $λ_{k} \to \infty$ as $k \to \infty$ such that

lim_{k \to \infty} m_{λ_{k}} = M \leq m_{Ω} < + \infty,

(4.1)

where

m_{λ_{k}}

is the ground state of (1.2) with

λ = λ_{k}

and

(u_{λ_{k}}, v_{λ_{k}}) \in N_{λ_{k}}

is the corresponding ground state solution. Similar to the proof of Lemma 3.3, it can be obtain that

N_{Ω}

is non-empty, that is, there exists a

(u_{*}, v_{*}) \in N_{Ω}

. Then by the definition of

J_{Ω}

, we obtain that

m_{Ω} = inf_{(u, v) \in N_{Ω}} J_{Ω} (u, v) \leq J_{Ω} (u_{*}, v_{*}) < + \infty

. Lemma 3.5 tells us that

m_{Ω} \geq M \geq η > 0

From (3.21) and (4.1), we get

‖ (u_{λ_{k}}, v_{λ_{k}}) ‖_{W_{λ_{k}}} \leq {(max {2^{p - 1}, 2^{q - 1}} (\frac{m_{Ω}}{\frac{1}{β} - \frac{1 + ϱ}{α}} + 1))}^{\frac{1}{min {p, q}}} := L .

(4.2)

Besides by the definitions of

W

and

W_{λ}

, it is easy to see that

{(u_{λ_{k}}, v_{λ_{k}})} \subset W

and so

{(u_{λ_{k}}, v_{λ_{k}})}

is uniformly bounded in

W

. So there exists

(u_{0}, v_{0}) \subset W

such that

\begin{aligned} {\begin{cases} (u_{λ_{k}}, v_{λ_{k}}) ⇀ (u_{0}, v_{0}), & in W, \\ u_{λ_{k}} (x) \to u_{0} (x), v_{λ_{k}} (x) \to v_{0} (x) & \forall x \in V . \end{cases} \end{aligned}

(4.3)

Here we have used the proof of Lemma 2.3 with replacing

W_{λ} (a), W_{λ} (b)

, and

W_{λ}

with

W^{1, p} (V), W^{1, q} (V)

, and

W

, respectively, which is easily verified.

We claim that $u_{0} |_{Ω_{a}^{c}} \equiv 0$ and $v_{0} |_{Ω_{b}^{c}} \equiv 0$ . Otherwise, without loss of generality, we assume that there exists a vertex $x_{0} \in Ω_{a}^{c}$ such that $u_{0} (x_{0}) \neq 0$ . Since $(u_{λ_{k}}, v_{λ_{k}}) \in N_{λ_{k}}$ , we have

\begin{aligned} J_{λ_{k}} (u_{λ_{k}}, v_{λ_{k}}) \\ = J_{λ_{k}} (u_{λ_{k}}, v_{λ_{k}}) - \frac{1}{α} ⟨ J_{λ_{k}}^{'} (u_{λ_{k}}, v_{λ_{k}}), (u_{λ_{k}}, v_{λ_{k}}) ⟩ \\ = (\frac{1}{p} - \frac{1}{α}) ‖ u_{λ_{k}} ‖_{W_{λ} (a)}^{p} + (\frac{1}{q} - \frac{1}{α}) ‖ v_{λ_{k}} ‖_{W_{λ} (b)}^{q} \\ + \frac{1}{α} \int_{V} F_{u_{λ_{k}}} (x, u_{λ_{k}}, v_{λ_{k}}) u_{λ_{k}} d μ + \frac{1}{α} \int_{V} F_{v_{λ_{k}}} (x, u_{λ_{k}}, v_{λ_{k}}) v_{λ_{k}} d μ - \int_{V} F (x, u_{λ_{k}}, v_{λ_{k}}) d μ \\ \geq (\frac{1}{p} - \frac{1}{α}) ‖ u_{λ_{k}} ‖_{W_{λ} (a)}^{p} + (\frac{1}{q} - \frac{1}{α}) ‖ v_{λ_{k}} ‖_{W_{λ} (b)}^{q} - \frac{ϱ}{α} \int_{V} (| u_{λ_{k}} |^{p} + | v_{λ_{k}} |^{q}) d μ \\ \geq (\frac{1}{β} - \frac{1}{α} - \frac{ϱ}{α}) (‖ u_{λ_{k}} ‖_{W_{λ} (a)}^{p} + ‖ v_{λ_{k}} ‖_{W_{λ} (b)}^{q}) \\ \geq (\frac{1}{β} - \frac{1}{α} - \frac{ϱ}{α}) \int_{V} (| \nabla u_{λ_{k}} |^{p} + (λ_{k} a (x) + 1) | u_{λ_{k}} |^{p}) d μ \\ \geq (\frac{1}{β} - \frac{1}{α} - \frac{ϱ}{α}) λ_{k} \int_{V} a (x) | u_{λ_{k}} |^{p} d μ \\ \geq (\frac{1}{β} - \frac{1}{α} - \frac{ϱ}{α}) λ_{k} a (x_{0}) | u_{λ_{k}} (x_{0}) |^{p} μ (x_{0}) . \end{aligned}

Since

x_{0} \notin Ω_{a}, a (x_{0}) > 0

μ (x_{0}) \geq μ_{min} > 0

, and

u_{λ_{k}} (x_{0}) \to u_{0} (x_{0}) \neq 0

, then by

λ_{k} \to + \infty

, we get that

lim_{k \to \infty} J_{λ_{k}} (u_{λ_{k}}, v_{λ_{k}}) = + \infty

, which is a contraction to the fact that

m_{λ_{k}} \leq m_{Ω}

. So

u_{0} |_{Ω_{a}^{c}} \equiv 0

. Similarly, we can also obtain that

v_{0} |_{Ω_{b}^{c}} \equiv 0

Similar to (3.11), by (4.3), Appendices A.2, A.3, and Lebesgue dominated convergence theorem, for all $t > 0$ , we have

\begin{aligned} lim_{k \to \infty} \int_{V} F (x, t u_{λ_{k}}, t v_{λ_{k}}) d μ & = \int_{V} F (x, t u_{0}, t v_{0}) d μ \end{aligned}

(4.4)

\begin{aligned} lim_{k \to \infty} \int_{V} (F_{t u_{λ_{k}}} (x, t u_{λ_{k}}, t v_{λ_{k}}) t u_{λ_{k}} + F_{t v_{λ_{k}}} (x, t u_{λ_{k}}, t v_{λ_{k}}) t v_{λ_{k}}) d μ & = \int_{V} (F_{t u_{0}} (x, t u_{0}, t v_{0}) t u_{0} + F_{t v_{0}} (x, t u_{0}, t v_{0}) t v_{0}) d μ . \end{aligned}

(4.5)

Since

(u_{0}, v_{0}) \in W

and

u_{0} |_{Ω_{a}^{c}} \equiv 0

and

v_{0} |_{Ω_{b}^{c}} \equiv 0

, it follows from

(A_{1})

that

\int_{V} (λ a | u_{0} |^{p} + λ b | v_{0} |^{q}) d μ = \int_{Ω_{a}} λ a | u_{0} |^{p} + \int_{V ∖ Ω_{a}} λ a | u_{0} |^{p} + \int_{Ω_{b}} λ b | v_{0} |^{q} + \int_{V ∖ Ω_{b}} λ b | v_{0} |^{q} d μ = 0

which shows that

(u_{0}, v_{0}) \in W_{λ}

for all

λ > 0

. Next, we claim that

(u_{0}, v_{0}) \neq (0, 0)

. In fact, if

(u_{0}, v_{0}) = (0, 0)

, without loss of generality, setting

p \geq q

, then by the fact that

(u_{λ_{k}}, v_{λ_{k}}) \in N_{λ_{k}}

and

F (x, 0, 0) = 0

for all

x \in V

, (4.4) and (4.5), we have

\begin{aligned} 0 < M & = lim_{k \to \infty} J_{λ_{k}} (u_{λ_{k}}, v_{λ_{k}}) \\ = lim_{k \to \infty} [\frac{1}{p} ‖ u_{λ_{k}} ‖_{W_{λ} (a)}^{p} + \frac{1}{q} ‖ v_{λ_{k}} ‖_{W_{λ} (b)}^{q} - \int_{V} F (x, u_{λ_{k}}, v_{λ_{k}}) d μ] \\ = lim_{k \to \infty} [\frac{1}{q} ‖ u_{λ_{k}} ‖_{W_{λ} (a)}^{p} + \frac{1}{q} ‖ v_{λ_{k}} ‖_{W_{λ} (b)}^{q} + (\frac{1}{p} - \frac{1}{q}) ‖ u_{λ_{k}} ‖_{W_{λ} (a)}^{p} - \int_{V} F (x, u_{λ_{k}}, v_{λ_{k}}) d μ] \\ = lim_{k \to \infty} [\frac{1}{q} \int_{V} F_{u_{λ_{k}}} (x, u_{λ_{k}}, v_{λ_{k}}) u_{λ_{k}} d μ + \frac{1}{q} \int_{V} F_{v_{λ_{k}}} (x, u_{λ_{k}}, v_{λ_{k}}) v_{λ_{k}} d μ \\ + (\frac{1}{p} - \frac{1}{q}) ‖ u_{λ_{k}} ‖_{W_{λ} (a)}^{p} - \int_{V} F (x, u_{λ_{k}}, v_{λ_{k}}) d μ] \\ = \frac{1}{q} \int_{V} (F_{u_{0}} (x, u_{0}, v_{0}) u_{0} + F_{v_{0}} (x, u_{0}, v_{0}) v_{0}) d μ + lim_{k \to \infty} (\frac{1}{p} - \frac{1}{q}) ‖ u_{λ_{k}} ‖_{W_{λ} (a)}^{p} - \int_{V} F (x, u_{0}, v_{0}) d μ \\ = lim_{k \to \infty} (\frac{1}{p} - \frac{1}{q}) ‖ u_{λ_{k}} ‖_{W_{λ} (a)}^{p} \\ \leq 0. \end{aligned}

It is a contradiction. Hence

(u_{0}, v_{0}) \neq (0, 0)

Combining Lemma 3.3, it is easy to verify that there exists a constant $t^{'} > 0$ such that $(t^{'} u_{0}, t^{'} v_{0}) \in N_{λ}$ and further by $u_{0} |_{Ω_{a}^{c}} \equiv 0$ and $v_{0} |_{Ω_{b}^{c}} \equiv 0$ , we can obtain $(t^{'} u_{0}, t^{'} v_{0}) \in N_{Ω}$ . Then by the weak lower semi-continuity of norm, (4.3) and (4.4), we can infer that

\begin{aligned} m_{Ω} & \leq J_{Ω} (t^{'} u_{0}, t^{'} v_{0}) \\ = \frac{1}{p} (\int_{{\bar{Ω}}_{a}} | \nabla (t^{'} u_{0}) |^{p} d μ + \int_{Ω_{a}} | t^{'} u_{0} |^{p} d μ) + \frac{1}{q} (\int_{{\bar{Ω}}_{b}} | \nabla (t^{'} v_{0}) |^{q} d μ + \int_{Ω_{b}} | t^{'} v_{0} |^{q} d μ) \\ - \int_{Ω_{a} \cup Ω_{b}} F (x, t^{'} u_{0}, t^{'} v_{0}) d μ \\ = \frac{1}{p} \int_{V} (t^{'})^{p} (| \nabla u_{0} |^{p} + | u_{0} |^{p}) d μ + \frac{1}{q} \int_{V} (t^{'})^{q} (| \nabla v_{0} |^{q} + | v_{0} |^{q}) d μ - \int_{V} F (x, t^{'} u_{0}, t^{'} v_{0}) d μ \\ \leq \underset{k \to \infty}{lim inf} [\frac{1}{p} \int_{V} (t^{'})^{p} (| \nabla u_{λ_{k}} |^{p} + | u_{λ_{k}} |^{p}) d μ + \frac{1}{q} \int_{V} (t^{'})^{q} (| \nabla v_{λ_{k}} |^{q} + | v_{λ_{k}} |^{q}) d μ - \int_{V} F (x, t^{'} u_{λ_{k}}, t^{'} v_{λ_{k}}) d μ] \\ \leq \underset{k \to \infty}{lim inf} [\frac{1}{p} \int_{V} (t^{'})^{p} (| \nabla u_{λ_{k}} |^{p} + (λ_{k} a + 1) | u_{λ_{k}} |^{p}) d μ + \frac{1}{q} \int_{V} (t^{'})^{q} (| \nabla v_{λ_{k}} |^{q} + (λ_{k} b + 1) | v_{λ_{k}} |^{q}) d μ \\ - \int_{V} F (x, t^{'} u_{λ_{k}}, t^{'} v_{λ_{k}}) d μ] \\ = \underset{k \to \infty}{lim inf} J_{λ_{k}} (t^{'} u_{λ_{k}}, t^{'} v_{λ_{k}}) \\ \leq \underset{k \to \infty}{lim inf} J_{λ_{k}} (u_{λ_{k}}, v_{λ_{k}}) (by Lemma 3.4) \\ = M . \end{aligned}

Then combining with (4.1), we get

lim_{λ \to \infty} m_{λ} = m_{Ω} .

The proof is complete.

Proof of Theorem 1.2. From Lemma 4.1, $(t^{'} u_{0}, t^{'} v_{0}) \in N_{λ}$ and $t^{'} > 0$ , we obtain that $(u_{0}, v_{0}) ≢ (0, 0)$ . We have proved in Lemma 4.1 that $u_{0} |_{Ω_{a}^{c}} = 0$ and $v_{0} |_{Ω_{b}^{c}} = 0$ . Then there exists some $t > 0$ such that $(t^{'} u_{0}, t^{'} v_{0}) \in N_{Ω}$ . First we claim that as $k \to + \infty$ , there hold

λ_{k} \int_{V} a | u_{λ_{k}} |^{p} d μ \to 0, λ_{k} \int_{V} b | v_{λ_{k}} |^{q} d μ \to 0

(4.6)

and

\int_{V} (| \nabla u_{λ_{k}} |^{p} + | u_{λ_{k}} |^{p}) d μ \to \int_{V} (| \nabla u_{0} |^{p} + | u_{0} |^{p}) d μ, \int_{V} (| \nabla v_{λ_{k}} |^{q} + | v_{λ_{k}} |^{p}) d μ \to \int_{V} (| \nabla v_{0} |^{q} + | v_{0} |^{q}) d μ .

(4.7)

Otherwise, by (4.3) and the weak lower semi-continuity of the norm, for some

η_{1} > 0

and

η_{2} > 0

, there holds

lim_{k \to \infty} (λ_{k} \int_{V} a | u_{λ_{k}} |^{p} d μ) = η_{1} or lim_{k \to \infty} (λ_{k} \int_{V} b | v_{λ_{k}} |^{q} d μ) = η_{2}

\underset{k \to \infty}{lim inf} (\int_{V} (| \nabla u_{λ_{k}} |^{p} + | u_{λ_{k}} |^{p}) d μ) > \int_{V} (| \nabla u_{0} |^{p} + | u_{0} |^{p}) d μ

\underset{k \to \infty}{lim inf} (\int_{V} (| \nabla v_{λ_{k}} |^{q} + | v_{λ_{k}} |^{q}) d μ) > \int_{V} (| \nabla v_{0} |^{q} + | v_{0} |^{q}) d μ .

If any of the above four formulas hold, we get

\begin{aligned} J_{Ω} (t^{'} u_{0}, t^{'} v_{0}) \\ = \frac{1}{p} (\int_{{\bar{Ω}}_{a}} | \nabla (t^{'} u_{0}) |^{p} d μ + \int_{Ω_{a}} | t^{'} u_{0} |^{p} d μ) + \frac{1}{q} (\int_{{\bar{Ω}}_{b}} | \nabla (t^{'} v_{0}) |^{q} d μ + \int_{Ω_{b}} | t^{'} v_{0} |^{q} d μ) \\ - \int_{Ω_{a} \cup Ω_{b}} F (x, t^{'} u_{0}, t^{'} v_{0}) d μ \\ = \frac{1}{p} \int_{V} (t^{'})^{p} (| \nabla u_{0} |^{p} + | u_{0} |^{p}) d μ + \frac{1}{q} \int_{V} (t^{'})^{q} (| \nabla v_{0} |^{q} + | v_{0} |^{q}) d μ - \int_{V} F (x, t^{'} u_{0}, t^{'} v_{0}) d μ \\ < \underset{k \to \infty}{lim inf} [\frac{1}{p} \int_{V} (t^{'})^{p} (| \nabla u_{λ_{k}} |^{p} + (λ_{k} a + 1) | u_{λ_{k}} |^{p}) d μ + \frac{1}{q} \int_{V} (t^{'})^{q} (| \nabla v_{λ_{k}} |^{q} + (λ_{k} b + 1) | v_{λ_{k}} |^{q}) d μ \\ - \int_{V} F (x, t^{'} u_{λ_{k}}, t^{'} v_{λ_{k}}) d μ] \\ = \underset{k \to \infty}{lim inf} J_{λ_{k}} (t^{'} u_{λ_{k}}, t^{'} v_{λ_{k}}) \\ \leq \underset{k \to \infty}{lim inf} J_{λ_{k}} (u_{λ_{k}}, v_{λ_{k}}) (by Lemma 3.3) \\ = m_{Ω}, \end{aligned}

which contradicts to the definition of

m_{Ω}

and then the claim is proved.

Now we can prove that $(u_{0}, v_{0})$ is a ground state solution of (1.3). Firstly, we shall prove $(u_{0}, v_{0})$ is a weak solution of (1.3). Since $(u_{λ_{k}}, v_{λ_{k}})$ is the ground state solution, then $J_{λ_{k}}^{'} (u_{λ_{k}}, v_{λ_{k}}) = 0$ . Thus for any $(ϕ_{1}, ϕ_{2}) \in W_{Ω_{a}} \times W_{Ω_{b}} \subset W_{λ}$ , we have

\begin{aligned} \int_{V} (| \nabla u_{λ_{k}} |^{p - 2} Γ (u_{λ_{k}}, ϕ_{1}) + (λ_{k} a + 1) | u_{λ_{k}} |^{p - 2} u_{λ_{k}} ϕ_{1}) d μ + \int_{V} (| \nabla v_{λ_{k}} |^{q - 2} Γ (v_{λ_{k}}, ϕ_{2}) + (λ_{k} b + 1) | v_{λ_{k}} |^{q - 2} v_{λ_{k}} ϕ_{2}) d μ \\ = \int_{V} (F_{u_{λ_{k}}} (x, u_{λ_{k}}, v_{λ_{k}}) ϕ_{1} + F_{v_{λ_{k}}} (x, u_{λ_{k}}, v_{λ_{k}}) ϕ_{2}) d μ . \end{aligned}

(4.8)

(A_{1})

and noting that

Ω_{a} = {x \in V, a (x) = 0}

and

Ω_{b} = {x \in V, b (x) = 0}

, it is easy to obtain that

a (x) ϕ_{1} (x) \equiv 0

and

b (x) ϕ_{2} (x) \equiv 0

for any

x \in V

. Then (4.8) reduces to

\begin{aligned} \int_{{\bar{Ω}}_{a}} | \nabla u_{λ_{k}} |^{p - 2} Γ (u_{λ_{k}}, ϕ_{1}) d μ + \int_{Ω_{a}} | u_{λ_{k}} |^{p - 2} u_{λ_{k}} ϕ_{1} d μ + \int_{{\bar{Ω}}_{b}} | \nabla v_{λ_{k}} |^{q - 2} Γ (v_{λ_{k}}, ϕ_{2}) d μ + \int_{Ω_{b}} | v_{λ_{k}} |^{q - 2} v_{λ_{k}} ϕ_{2} d μ \\ = \int_{Ω_{a} \cup Ω_{b}} F_{u_{λ_{k}}} (x, u_{λ_{k}}, v_{λ_{k}}) ϕ_{1} d μ + \int_{Ω_{a} \cup Ω_{b}} F_{v_{λ_{k}}} (x, u_{λ_{k}}, v_{λ_{k}}) ϕ_{2} d μ . \end{aligned}

(4.9)

Note that

\begin{aligned} \int_{{\bar{Ω}}_{a}} | \nabla u_{λ_{k}} |^{p - 2} Γ (u_{λ_{k}}, ϕ_{1}) d μ - \int_{{\bar{Ω}}_{a}} | \nabla u_{0} |^{p - 2} Γ (u_{0}, ϕ_{1}) d μ \\ = \int_{{\bar{Ω}}_{a}} (| \nabla u_{λ_{k}} |^{p - 2} Γ (u_{λ_{k}}, ϕ_{1}) d μ - | \nabla u_{λ_{k}} |^{p - 2} Γ (u_{0}, ϕ_{1})) d μ + \int_{{\bar{Ω}}_{a}} (| \nabla u_{λ_{k}} |^{p - 2} Γ (u_{0}, ϕ_{1}) - | \nabla u_{0} |^{p - 2} Γ (u_{0}, ϕ_{1})) d μ \\ = \int_{{\bar{Ω}}_{a}} (| \nabla u_{λ_{k}} |^{p - 2} Γ (u_{λ_{k}} - u_{0}, ϕ_{1}) d μ + \int_{{\bar{Ω}}_{a}} (| \nabla u_{λ_{k}} |^{p - 2} - | \nabla u_{0} |^{p - 2}) Γ (u_{0}, ϕ_{1}) d μ . \end{aligned}

(4.10)

By (4.1),

‖ (u_{λ_{k}}, v_{λ_{k}}) ‖_{W_{Ω}} \leq L

. According to the definition of the local finite graph, we obtain that

Ω_{a}, Ω_{b}, \partial Ω_{a}

and

\partial Ω_{b}

are finite sets. So

W_{Ω}

is a finite dimensional space. Then, up to a subsequence,

(u_{λ_{k}}, v_{λ_{k}}) \to (u_{0}, v_{0})

W_{Ω}

. Thus we can get

\int_{{\bar{Ω}}_{a}} | \nabla (u_{λ_{k}} - u_{0}) |^{p} d μ \to 0

and

\int_{{\bar{Ω}}_{b}} | \nabla (v_{λ_{k}} - v_{0}) |^{q} d μ \to 0

. Similar to (7.10) and (7.11) by Yang & Zhang (2024), we can get

\int_{{\bar{Ω}}_{a}} | \nabla u_{λ_{k}} |^{p - 2} Γ (u_{λ_{k}} - u_{0}, ϕ_{1}) d μ \to 0

and

\int_{{\bar{Ω}}_{a}} (| \nabla u_{λ_{k}} |^{p - 2} - | \nabla u_{0} |^{p - 2}) Γ (u_{0}, ϕ_{1}) d μ \to 0

. Thus (4.10) shows that

lim_{k \to \infty} \int_{{\bar{Ω}}_{a}} | \nabla u_{λ_{k}} |^{p - 2} Γ (u_{λ_{k}}, ϕ_{1}) d μ = \int_{{\bar{Ω}}_{a}} | \nabla u_{0} |^{p - 2} Γ (u_{0}, ϕ_{1}) d μ,

and similarly, we also have

lim_{k \to \infty} \int_{{\bar{Ω}}_{b}} | \nabla v_{λ_{k}} |^{q - 2} Γ (v_{λ_{k}}, ϕ_{2}) d μ = \int_{{\bar{Ω}}_{b}} | \nabla v_{0} |^{q - 2} Γ (v_{0}, ϕ_{2}) d μ .

Using (4.1) again, we have

L \geq ‖ u_{λ_{k}} ‖_{W_{Ω_{a}}} \geq {(\int_{Ω_{a}} | u_{λ_{k}} |^{p} d μ)}^{\frac{1}{p}} \geq | Ω_{a} | \cdot ‖ u_{λ_{k}} ‖_{\infty, Ω_{a}}^{p} .

| u_{λ_{k}} | \leq (\frac{L}{| Ω_{a} |})^{\frac{1}{p}}

for all

x \in Ω_{a}

. Similarly, we can also get

| v_{λ_{k}} | \leq (\frac{L}{| Ω_{b} |})^{\frac{1}{q}}

for all

x \in Ω_{b}

. Note that

Ω_{a}, Ω_{b}

and

Ω_{a} \cup Ω_{b}

are finite sets. Then by (4.3) and

(F_{1})

, we have

\begin{aligned} lim_{k \to \infty} \int_{Ω_{a}} | u_{λ_{k}} |^{p - 2} u_{λ_{k}} ϕ_{1} d μ = \int_{Ω_{a}} | u_{0} |^{p - 2} u_{0} ϕ_{1} d μ, \\ lim_{k \to \infty} \int_{Ω_{b}} | v_{λ_{k}} |^{q - 2} v_{λ_{k}} ϕ_{2} d μ = \int_{Ω_{b}} | v_{0} |^{q - 2} v_{0} ϕ_{2} d μ, \\ lim_{k \to \infty} \int_{Ω_{a} \cup Ω_{b}} F_{u_{λ_{k}}} (x, u_{λ_{k}}, v_{λ_{k}}) ϕ_{1} d μ = \int_{Ω_{a} \cup Ω_{b}} F_{u_{0}} (x, u_{0}, v_{0}) ϕ_{1} d μ, \\ lim_{k \to \infty} \int_{Ω_{a} \cup Ω_{b}} F_{v_{λ_{k}}} (x, u_{λ_{k}}, v_{λ_{k}}) ϕ_{2} d μ = \int_{Ω_{a} \cup Ω_{b}} F_{v_{0}} (x, u_{0}, v_{0}) ϕ_{2} d μ . \end{aligned}

Then as

k \to \infty

, (4.9) becomes

\begin{aligned} \int_{{\bar{Ω}}_{a}} | \nabla u_{0} |^{p - 2} Γ (u_{0}, ϕ_{1}) d μ + \int_{Ω_{a}} | u_{0} |^{p - 2} u_{0} ϕ_{1} d μ + \int_{{\bar{Ω}}_{b}} | \nabla v_{0} |^{q - 2} Γ (v_{0}, ϕ_{2}) d μ + \int_{Ω_{b}} | v_{0} |^{q - 2} v_{0} ϕ_{2} d μ \\ = \int_{Ω_{a} \cup Ω_{b}} F_{u_{0}} (x, u_{0}, v_{0}) ϕ_{1} d μ + \int_{Ω_{a} \cup Ω_{b}} F_{v_{0}} (x, u_{0}, v_{0}) ϕ_{2} d μ, \end{aligned}

which tells us that

J_{Ω}^{'} (u_{0}, v_{0}) = 0

. Hence

(u_{0}, v_{0}) \in N_{Ω}

is a solution of (1.3).

Next, we shall prove $(u_{0}, v_{0})$ is a ground state solution of (1.3). By $u_{0} |_{Ω_{a}^{c}} = 0, v_{0} |_{Ω_{b}^{c}} = 0$ , (4.3), (4.4), (4.6), and (4.7), we have

\begin{aligned} lim_{k \to \infty} J_{λ_{k}} (u_{λ_{k}}, v_{λ_{k}}) \\ = lim_{k \to \infty} [\frac{1}{p} \int_{V} (| \nabla u_{λ_{k}} |^{p} + (λ_{k} a + 1) | u_{λ_{k}} |^{p}) d μ + \frac{1}{q} \int_{V} (| \nabla v_{λ_{k}} |^{q} + (λ_{k} b + 1) | v_{λ_{k}} |^{q}) d μ - \int_{V} F (x, u_{λ_{k}}, v_{λ_{k}}) d μ] \\ = \frac{1}{p} \int_{V} (| \nabla u_{0} |^{p} + | u_{0} |^{p}) d μ + \frac{1}{q} \int_{V} (| \nabla v_{0} |^{q} + | v_{0} |^{q}) d μ - \int_{V} F (x, u_{0}, v_{0}) d μ + o_{k} (1) \\ = \frac{1}{p} (\int_{{\bar{Ω}}_{a}} | \nabla u_{0} |^{p} d μ + \int_{Ω_{a}} | u_{0} |^{p} d μ) + \frac{1}{q} (\int_{{\bar{Ω}}_{b}} | \nabla v_{0} |^{q} d μ + \int_{Ω_{b}} | v_{0} |^{q} d μ) - \int_{Ω_{a} \cup Ω_{b}} F (x, u_{0}, v_{0}) d μ + o_{k} (1) \\ = J_{Ω} (u_{0}, v_{0}) + o_{k} (1) . \end{aligned}

Since

lim_{k \to \infty} J_{λ_{k}} (u_{λ_{k}}, v_{λ_{k}}) = lim_{k \to \infty} m_{λ_{k}}

and Lemma 4.1 tells us that

lim_{k \to \infty} m_{λ_{k}} = m_{Ω}

, then

J_{Ω} (u_{0}, v_{0}) = m_{Ω}

. Thus we get that

(u_{0}, v_{0})

is a solution of (1.3) which achieves the ground state. Finally, by Lemma 4.1 and the above proofs, we can conclude that for any sequence

λ_{k} \to + \infty

, up to a subsequence, the corresponding ground state solutions

(u_{λ_{k}}, v_{λ_{k}}) \in N_{λ_{k}}

of (1.2) satisfying

J_{λ_{k}} (u_{λ_{k}}, v_{λ_{k}}) = m_{λ_{k}}

converge to a ground state solution of (1.3) with ground state energy

m_{Ω}

. Thus Theorem 1.2 is proved.

Footnotes

Authors’ contributions

The authors contribute the manuscript equally.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project is supported by Yunnan Fundamental Research Projects (grant No: 202301AT070465) and Xingdian Talent Support Program for Young Talents of Yunnan Province.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Appendix A.1.

Assume that $\frac{1}{γ} < β + \frac{1}{β} - 1$ . Then $\frac{1}{1 + max {\frac{1}{p} + \frac{q - 1}{q}, \frac{1}{q} + \frac{p - 1}{p}}} > \frac{1}{1 + β}$ . Proof.

When $p \geq q, β = p, γ = q$ , we have

max {\frac{1}{p} + \frac{q - 1}{q}, \frac{1}{q} + \frac{p - 1}{p}} = \frac{1}{q} + \frac{p - 1}{p},

then

\frac{1}{q} + \frac{p - 1}{p} - p = \frac{p + p q - q - p^{2} q}{p q} = \frac{p - q + p q (1 - p)}{p q} .

p - q + p q (1 - p) < 0

, then

1 - p < \frac{1}{p} - \frac{1}{q}

, we can obtain

\frac{1}{q} < p + \frac{1}{p} - 1

When $p \leq q, β = q, γ = p$ , we have

max {\frac{1}{p} + \frac{q - 1}{q}, \frac{1}{q} + \frac{p - 1}{p}} = \frac{1}{p} + \frac{q - 1}{q},

then

\frac{1}{p} + \frac{q - 1}{q} - q = \frac{q + p q - p - p q^{2}}{p q} = \frac{q - p + p q (1 - q)}{p q} .

q - p + p q (1 - q) < 0

, then

1 - q < \frac{1}{q} - \frac{1}{p}

, we can obtain

\frac{1}{p} < q + \frac{1}{q} - 1

Thus when $\frac{1}{γ} < β + \frac{1}{β} - 1$ , we have $max {\frac{1}{p} + \frac{q - 1}{q}, \frac{1}{q} + \frac{p - 1}{p}} < β$ , that is, $\frac{1}{1 + max {\frac{1}{p} + \frac{q - 1}{q}, \frac{1}{q} + \frac{p - 1}{p}}} > \frac{1}{1 + β}$ . □

Appendix A.2.

Assume that $(F_{1})$ and $(F_{3})$ hold. then

| F (x, t, s) | \leq \frac{2 C_{1} (x)}{p} | t |^{p} + (\frac{C_{1} (x) (p - 1)}{p} + \frac{C_{1} (x)}{q}) | s |^{q} + \frac{2 C_{2} (x)}{r_{1}} | t |^{r_{1}} + (\frac{C_{2} (x) (r_{1} - 1)}{r_{1}} + \frac{C_{2} (x)}{r_{2}}) | s |^{r_{2}}

for all

x \in V

and

\forall (t, s) \in R^{2}

. Proof.

From $(F_{3})$ and Young inequality, we can get

\begin{aligned} | F (x, t, s) | \\ \leq \int_{0}^{| t |} | F_{t} (x, t, s) | d t + \int_{0}^{| s |} | F_{s} (x, 0, s) | d s \\ \leq \int_{0}^{| t |} [C_{1} (x) (| t |^{p - 1} + | s |^{\frac{q (p - 1)}{p}}) + C_{2} (x) (| t |^{r_{1} - 1} + | s |^{\frac{r_{2} (r_{1} - 1)}{r_{1}}})] d t + \int_{0}^{| s |} (C_{1} (x) | s |^{q - 1} + C_{2} (x) | s |^{r_{2} - 1}) d s \\ = \frac{C_{1} (x)}{p} | t |^{p} + C_{1} (x) | s |^{\frac{q (p - 1)}{p}} | t | + \frac{C_{2} (x)}{r_{1}} | t |^{r_{1}} + C_{2} (x) | s |^{\frac{r_{2} (r_{1} - 1)}{r_{1}}} | t | + \frac{C_{1} (x)}{q} | s |^{q} + \frac{C_{2} (x)}{r_{2}} | s |^{r_{2}} \\ \leq \frac{C_{1} (x)}{p} | t |^{p} + \frac{C_{1} (x) (p - 1)}{p} | s |^{q} + \frac{C_{1} (x)}{p} | t |^{p} + \frac{C_{2} (x)}{r_{1}} | t |^{r_{1}} + \frac{C_{2} (x) (r_{1} - 1)}{r_{1}} | s |^{r_{2}} + \frac{C_{2} (x)}{r_{1}} | t |^{r_{1}} + \frac{C_{1} (x)}{q} | s |^{q} + \frac{C_{2} (x)}{r_{2}} | s |^{r_{2}} \\ = \frac{2 C_{1} (x)}{p} | t |^{p} + (\frac{C_{1} (x) (p - 1)}{p} + \frac{C_{1} (x)}{q}) | s |^{q} + \frac{2 C_{2} (x)}{r_{1}} | t |^{r_{1}} + (\frac{C_{2} (x) (r_{1} - 1)}{r_{1}} + \frac{C_{2} (x)}{r_{2}}) | s |^{r_{2}} . \end{aligned}

Thus, the proof is completed. □

Appendix A.3.

Assume that $(F_{3})$ holds. then

| F_{t} (x, t, s) t + F_{s} (x, t, s) s | \leq C_{3} (x) (| t |^{p} + | s |^{q}) + C_{4} (x) (| t |^{r_{1}} + | s |^{r_{2}})

for all

x \in V

and

\forall (t, s) \in R^{2}

, where

C_{3} (x) = C_{1} (x) \cdot max {1 + \frac{1}{p} + \frac{q - 1}{q}, 1 + \frac{1}{q} + \frac{p - 1}{p}}

and

C_{4} (x) = C_{2} (x) \cdot max {1 + \frac{1}{r_{1}} + \frac{r_{2} - 1}{r_{2}}, 1 + \frac{1}{r_{2}} + \frac{r_{1} - 1}{r_{1}}}

. Proof.

It follows from $(F_{3})$ that

\begin{aligned} | F_{t} (x, t, s) t + F_{s} (x, t, s) s | \\ \leq C_{1} (x) (| t |^{p} + | s |^{\frac{q (p - 1)}{p}} | t |) + C_{2} (x) (| t |^{r_{1}} + | s |^{\frac{r_{2} (r_{1} - 1)}{r_{1}}} | t |) + C_{1} (x) (| t |^{\frac{p (q - 1)}{q}} | s | + | s |^{q}) + C_{2} (x) (| t |^{\frac{r_{1} (r_{2} - 1)}{r_{2}}} | s | + | s |^{r_{2}}) \\ \leq C_{1} (x) | t |^{p} + \frac{C_{1} (x) (p - 1)}{p} | s |^{q} + \frac{C_{1} (x)}{p} | t |^{p} + C_{2} (x) | t |^{r_{1}} + \frac{C_{2} (x) (r_{1} - 1)}{r_{1}} | s |^{r_{2}} + \frac{C_{2} (x)}{r_{1}} | t |^{r_{1}} \\ + \frac{C_{1} (x) (q - 1)}{q} | t |^{p} + \frac{C_{1} (x)}{q} | s |^{q} + C_{1} (x) | s |^{q} + \frac{C_{2} (x) (r_{2} - 1)}{r_{2}} | t |^{r_{1}} + \frac{C_{2} (x)}{r_{2}} | s |^{r_{2}} + C_{2} (x) | s |^{r_{2}} \\ = (C_{1} (x) + \frac{C_{1} (x)}{p} + \frac{C_{1} (x) (q - 1)}{q}) | t |^{p} + (C_{1} (x) + \frac{C_{1} (x)}{q} + \frac{C_{1} (x) (p - 1)}{p}) | s |^{q} \\ + (C_{2} (x) + \frac{C_{2} (x)}{r_{1}} + \frac{C_{2} (x) (r_{2} - 1)}{r_{2}}) | t |^{r_{1}} + (C_{2} (x) + \frac{C_{2} (x)}{r_{2}} + \frac{C_{2} (x) (r_{1} - 1)}{r_{1}}) | s |^{r_{2}} \\ \leq C_{1} (x) \cdot max {1 + \frac{1}{p} + \frac{q - 1}{q}, 1 + \frac{1}{q} + \frac{p - 1}{p}} (| t |^{p} + | s |^{q}) \\ + C_{2} (x) \cdot max {1 + \frac{1}{r_{1}} + \frac{r_{2} - 1}{r_{2}}, 1 + \frac{1}{r_{2}} + \frac{r_{1} - 1}{r_{1}}} (| t |^{r_{1}} + | s |^{r_{2}}) \\ = C_{3} (x) (| t |^{p} + | s |^{q}) + C_{4} (x) (| t |^{r_{1}} + | s |^{r_{2}}) . \end{aligned}

Thus, we finish the proof. □

Appendix A.4.

If $β = p, t_{λ} \in (0, 1)$ and $(F_{4})$ holds, then () holds. Proof.

Let $A (t_{λ}) = \frac{1}{p} F_{t_{λ} u_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) t_{λ} u_{λ} + \frac{1}{p} F_{t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) t_{λ} v_{λ} - F (x, t_{λ} u_{λ}, t_{λ} v_{λ})$ .

Then

\begin{aligned} A^{'} (t_{λ}) \\ = \frac{1}{p} F_{t_{λ} u_{λ}, t_{λ} u_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) t_{λ} u_{λ}^{2} + \frac{1}{p} F_{t_{λ} u_{λ}, t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) t_{λ} u_{λ} v_{λ} + \frac{1}{p} F_{t_{λ} u_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) u_{λ} \\ + \frac{1}{p} F_{t_{λ} v_{λ}, t_{λ} u_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) t_{λ} u_{λ} v_{λ} + \frac{1}{p} F_{t_{λ} v_{λ}, t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) t_{λ} v_{λ}^{2} + \frac{1}{p} F_{t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) v_{λ} \\ - F_{t_{λ} u_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) u_{λ} - F_{t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) v_{λ} \\ > \frac{p - 1}{p} F_{t_{λ} u_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) u_{λ} + \frac{1}{p} F_{t_{λ} u_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) u_{λ} + \frac{1}{p} F_{t_{λ} u_{λ}, t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) t_{λ} u_{λ} v_{λ} \\ + \frac{p - 1}{p} F_{t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) v_{λ} + \frac{1}{p} F_{t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) v_{λ} + \frac{1}{p} F_{t_{λ} v_{λ}, t_{λ} u_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) t_{λ} u_{λ} v_{λ} \\ - F_{t_{λ} u_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) u_{λ} - F_{t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) v_{λ} \\ = \frac{2}{p} F_{t_{λ} u_{λ}, t_{λ} v_{λ}} (x, t_{λ} u_{λ}, t_{λ} v_{λ}) t_{λ} u_{λ} v_{λ} \\ \geq 0. \end{aligned}

A (t_{λ})

is a strictly monotonically increasing function. Thus

A (1) > A (t_{λ})

. □

Appendix A.5.

$N_{Ω} \subseteq N_{λ}$ . Proof.

For any $(u_{0}, v_{0}) \in N_{Ω}$ , we have

\begin{aligned} \int_{{\bar{Ω}}_{a}} | \nabla u_{0} |^{p} d μ + \int_{Ω_{a}} | u_{0} |^{p} d μ + \int_{{\bar{Ω}}_{b}} | \nabla v_{0} |^{q} d μ + \int_{Ω_{b}} | v_{0} |^{q} d μ \\ - \int_{Ω_{a} \cup Ω_{b}} F_{u_{0}} (x, u_{0}, v_{0}) u_{0} d μ - \int_{Ω_{a} \cup Ω_{b}} F_{v_{0}} (x, u_{0}, v_{0}) v_{0} d μ = 0. \end{aligned}

Then by

(F_{1})

(A_{1})

and the fact that

W_{Ω_{a}} (W_{Ω_{b}})

is the completion of

C_{c} (Ω_{a}) (C_{c} (Ω_{b}))

, we have

\begin{aligned} ⟨ J_{λ}^{'} (u_{0}, v_{0}), (u_{0}, v_{0}) ⟩ \\ = \int_{V} (| \nabla u_{0} |^{p} + (λ a + 1) | u_{0} |^{p}) d μ + \int_{V} (| \nabla v_{0} |^{q} + (λ b + 1) | v_{0} |^{q}) d μ - \int_{V} (F_{u_{0}} (x, u_{0}, v_{0}) u_{0} + F_{v_{0}} (x, u_{0}, v_{0}) v_{0}) d μ \\ = \int_{V ∖ {\bar{Ω}}_{a}} | \nabla u_{0} |^{p} d μ + \int_{Ω_{a}} | \nabla u_{0} |^{p} d μ + \int_{\partial Ω_{a}} | \nabla u_{0} |^{p} d μ + \int_{V ∖ Ω_{a}} (λ a + 1) | u_{0} |^{p} d μ + \int_{Ω_{a}} (λ a + 1) | u_{0} |^{p} d μ \\ + \int_{V ∖ {\bar{Ω}}_{b}} | \nabla v_{0} |^{q} d μ + \int_{Ω_{b}} | \nabla v_{0} |^{q} d μ + \int_{\partial Ω_{b}} | \nabla v_{0} |^{q} d μ + \int_{V ∖ Ω_{b}} (λ b + 1) | v_{0} |^{q} d μ + \int_{Ω_{b}} (λ b + 1) | v_{0} |^{q} d μ \\ - \int_{V ∖ (Ω_{a} \cup Ω_{b})} F_{u_{0}} (x, u_{0}, v_{0}) u_{0} d μ - \int_{Ω_{a} \cup Ω_{b}} F_{u_{0}} (x, u_{0}, v_{0}) u_{0} d μ \\ - \int_{V ∖ (Ω_{a} \cup Ω_{b})} F_{v_{0}} (x, u_{0}, v_{0}) v_{0} d μ - \int_{Ω_{a} \cup Ω_{b}} F_{v_{0}} (x, u_{0}, v_{0}) v_{0} d μ \\ = \int_{{\bar{Ω}}_{a}} | \nabla u_{0} |^{p} d μ + \int_{Ω_{a}} | u_{0} |^{p} d μ + \int_{{\bar{Ω}}_{b}} | \nabla v_{0} |^{q} d μ + \int_{Ω_{b}} | v_{0} |^{q} d μ - \int_{Ω_{a} \cup Ω_{b}} (F_{u_{0}} (x, u_{0}, v_{0}) u_{0} d μ + F_{v_{0}} (x, u_{0}, v_{0}) v_{0}) d μ \\ = 0. \end{aligned}

Thus

(u_{0}, v_{0}) \in N_{λ}

. □

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