Magnetic Kirchhoff–Schrödinger equations with critical exponential growth and indefinite potential in R 2

Abstract

In this paper, we are concerned with the following magnetic nonlinear equation of Kirchhoff type with critical exponential growth and an indefinite potential in $R^{2}$ $m (\int_{R^{2}} [{| \frac{1}{i} \nabla u - A (x) u |}^{2} + V (x) | u |^{2}] d x) [{(\frac{1}{i} \nabla - A (x))}^{2} u + V (x) u] = B (x) f (| u |^{2}) u,$ where $u \in H^{1} (R^{2}, C)$ , m is a Kirchhoff type function, $V : R^{2} \to R$ and $A : R^{2} \to R^{2}$ represent locally bounded potentials, while B denotes locally bounded and f exhibits critical exponential growth. By employing variational methods and utilizing the modified Trudinger–Moser inequality, we get ground state solutions or nontrivial solutions for the above equation. Furthermore, in the special case where m is a constant equal to 1, the equation is reduced to the following magnetic nonlinear Schrödinger equation, ${(\frac{1}{i} \nabla - A (x))}^{2} u + V (x) u = B (x) f ({| u |}^{2}) u in R^{2} .$ Applying analogous methods, we can also establish the existence of ground state solutions or nontrivial solutions to this equation.

Keywords

Kirchhoff–Schrödinger equation magnetic field indefinite potential minimization method Nehari manifold Trudinger–Moser inequality

1. Introduction and main results

In this paper, we study the following nonlinear Kirchhoff–Schrödinger equation in $R^{2}$

\begin{aligned} m (\int_{R^{2}} [{| \frac{1}{i} \nabla u - A (x) u |^{2} + V (x) | u |}^{2}] d x) [{(\frac{1}{i} \nabla - A (x))}^{2} u + V (x) u] \\ = B (x) f ({| u |}^{2}) u, \end{aligned}

(1.1)

where

m : [0, \infty) \to (0, \infty)

and the nonlinearity

f : R \to [0, \infty)

are the continuous functions,

u \in H^{1} (R^{2}, C)

V, B \in L_{loc}^{\infty} (R^{2}, R)

, and the magnetic potential

A \in L_{loc}^{\infty} (R^{2}, R^{2})

. The potential V may be non-positive on some subset with positive measure and f has critical exponential growth.

Generally speaking, the presence of the term

m (\int_{R^{2}} [{| \frac{1}{i} \nabla u - A (x) u |}^{2} + V (x) | u |^{2}] d x),

which depends on the gradient of u with respect to the magnetic field, renders the equation nonlocal and distinguishes it from pointwise identities. This nonlocal character makes the study of equation (1.1) particularly difficult and interesting. Firstly, Kirchhoff [25] conducted pioneering research on the lateral vibrations of elastic strings, culminating in the formulation of a hyperbolic equation that describes this dynamic behavior. The equation takes the form:

\frac{\partial^{2} u}{\partial t^{2}} - (k_{1} + k_{2} \int_{0}^{L} {| \frac{\partial u}{\partial x} |}^{2} d x) \frac{\partial^{2} u}{\partial x^{2}} = 0, k_{1}, k_{2} > 0; L > 0.

This equation accounts for the nonuniform distribution of vibrational energy along the string, introducing the integral term to capture the string’s dynamic response. For further insight into the early developments of Kirchhoff equations, readers are encouraged to explore the works cited in [11,22,24,31]. Additionally, there are many papers on the nonlocal Kirchhoff problems, see [3,14] and the references therein.

In particular, for the case that $A \equiv 0$ , if $R^{2}$ is replaced by $R^{N}$ , Li and Yang [26] studied the following N-Kirchhoff problem

\begin{aligned} {\begin{cases} M (\int_{R^{N}} (| \nabla u |^{N} + V (x) | u |^{N}) d x) (- Δ_{N} u + V (x) | u |^{N - 2} u) = λ A (x) | u |^{p - 2} u + f (u), & x \in R^{N}, \\ u \in W^{1, N} (R^{N}), \end{cases} \end{aligned}

(1.2)

where

M (s) = s^{k}

for

s ⩾ 0

k > 0

Δ_{N} u = div (| \nabla u |^{N - 2} \nabla u)

is the N-Laplacian operator of u, λ is a real positive parameter,

1 < p < N

0 < A (x) \in L^{σ} (R^{N})

with

σ = \frac{N}{N - p}

and

f \in C (R^{N}, R)

with critical exponential growth. By establishing suitable assumptions on the functions V and f, they demonstrated the existence of a positive constant Λ such that problem (1.2) admits at least two positive solutions for any

λ \in (0, Λ)

Generally, for the case of $A ≢ 0$ , Xiang, Pucci, Squassina and Zhang [36] dealt with the existence and multiplicity of solutions to the following Kirchhoff–Schrödinger equation involving an external magnetic potential

(a + b {[u]}_{s, A}^{2 ϑ - 2}) {(- Δ)}_{A}^{s} u + V (x) u = f (x, | u |) u in R^{N},

where

0 < s < 1

N > 2 s

a, b \in R_{0}^{+}

ϑ \in [1, \frac{N}{N - 2 s})

A : R^{N} \to R^{N}

is a magnetic potential,

{(- Δ)}_{A}^{s}

is the fractional magnetic operator and

V : R^{N} \to R^{+}

denotes an electric potential.

Within our research framework in $R^{2}$ , a considerable body of literature has explored the existence and multiplicity of bound state solutions for equations similar to (1.1). When $A \equiv 0$ , Furtado and Zanata [23] investigated the following analogous nonlinear Kirchhoff equation

\begin{aligned} m (\int_{R^{2}} (| \nabla u |^{2} + b (x) u^{2}) d x) (- Δ u + b (x) u) = B (x) f (u) in R^{2}, \end{aligned}

(1.3)

where

u \in H^{1} (R^{2}, R)

b, B \in L_{loc}^{\infty} (R^{2}, R)

, the Kirchhoff function

m : [0, \infty) \to (0, \infty)

is a continuous function and the nonlinearity

f : R \to [0, \infty)

is a continuous function with critical exponential growth. By establishing certain assumptions on the functions m, b, B and f, they demonstrated that the equation (1.3) possesses at least one nonnegative ground state solution or a nonnegative nontrivial solution. In the special case where of

m = 1

, equation (1.3) reduces to the following nonlinear Schrödinger equation

- Δ u + b (x) u = B (x) f (u) in R^{2},

for which they also obtained similar results.

In addition, for the case of $A ≢ 0$ , when $m \equiv 1$ , equation (1.1) simplifies to the following magnetic Schrödinger equation

\begin{aligned} {(\frac{1}{i} \nabla - A (x))}^{2} u + V (x) u = B (x) f (| u |^{2}) u in R^{2}, \end{aligned}

(1.4)

which means that our problem has become a local situation. Equation (1.4) appears when seeking the standing wave solutions

ψ (x, t) := e^{- i V t / ℏ} u (x)

, with

V \in R

, of the planar Schrödinger equation

i ℏ \frac{\partial ψ}{\partial t} = {(\frac{ℏ}{i} \nabla - A (x))}^{2} ψ + U (x) ψ - f ({| ψ |}^{2}) ψ in R^{2} \times R .

And the operator

ψ \mapsto (i ℏ \frac{\partial}{\partial t} - U (x)) ψ - {(\frac{ℏ}{i} \nabla - A (x))}^{2} ψ

appears when the interaction between the material field ψ and the external electromagnetic field with potential U and magnetic potential A is studied through the minimum coupling rule. Additionally, d’Avenia and Ji [17,18] investigated the following magnetic nonlinear Schrödinger equation similar to (1.4)

\begin{aligned} {(\frac{ε}{i} \nabla - A (x))}^{2} u + V (x) u = f ({| u |}^{2}) u in R^{2}, \end{aligned}

(1.5)

where

ε > 0

is a parameter,

V : R^{2} \to R

and

A : R^{2} \to R^{2}

are continuous potentials and the nonlinearity

f : R \to R

exhibits critical growth. Under certain assumptions on the potential V, they employed variational methods and Ljusternick–Schnirelmann theory to prove the existence of a multiplicity of solutions and their concentration for small

ε

in (1.5).

In summary, the magnetic nonlinear Schrödinger equations have been the subject of extensive research by numerous authors, who have employed a variety of appropriate methods to study them. See [4,7–9,15,16,21,27,35] and the references therein for a comprehensive overview. In particular, Ambrosio and d’Avenia [6] focused on the following magnetic nonlinear fractional Schrödinger equation

\begin{aligned} ε^{2 s} {(- Δ)}_{A / ε}^{s} u + V (x) u = f (| u |^{2}) u in R^{N}, \end{aligned}

(1.6)

where

ε

is a positive parameter,

0 < s < 1

{(- Δ)}_{A}^{s}

denotes the fractional magnetic Laplacian,

N ⩾ 3

A : R^{N} \to R^{N}

and

V : R^{N} \to R

are continuous potential and

f : R^{N} \to R

is a subcritical nonlinearity. Using variational methods and Ljusternick–Schnirelmann theory, they proved that the existence and multiplicity of solutions to (1.6) for

ε

small. Ambrosio [5] explored a fractional magnetic Schrödinger equation with exponential critical growth in

R

and established a multiplicity result. Additionally, an existing result of a magnetic Schrödinger equation with periodic magnetic potential satisfying a local integrability condition was studied by Bégout [10]. However, due to the presence of the magnetic potential A and the nonlocal term, our equation (1.1) becomes a complex-valued problem, and the weak limit of the energy functional’s (

P S

) sequence may not correspond to a solution of the original problem, which means that more dedicated estimates are needed. Furthermore, relatively few papers have investigated magnetic nonlinear nonlocal equations with critical exponential growth, a topic that greatly interests us and motivates our current research.

Drawing upon the work in [18], we offer some insightful comments and discuss the functional space that will be employed in our study. For $u : R^{2} \to C$ , we define

\nabla_{A} u := (\frac{\nabla}{i} - A) u

and

H_{A, V}^{1} (R^{2}, C) := {u \in H_{A}^{1} (R^{2}, C) : \int_{R^{2}} V (x) | u |^{2} d x < + \infty},

where

H_{A}^{1} (R^{2}, C) = {u \in L^{2} (R^{2}, C) : | \nabla_{A} u | \in L^{2} (R^{2}, R)} .

Clearly,

H_{A, V}^{1} (R^{2}, C)

is a subspace of

H_{A}^{1} (R^{2}, C)

and due to [28], we know that

H_{A}^{1} (R^{2}, C)

is a Hilbert space with the inner product and norm

⟨ u, v ⟩_{A} = Re \int_{R^{2}} (\nabla_{A} u \bar{\nabla_{A} v} + u \bar{v}) d x, ‖ u ‖_{A} = {⟨ u, u ⟩}_{A}^{\frac{1}{2}} .

Here, Re denotes the real part of a complex number, and the bar represents complex conjugation. Owing to the fact that

A \in L_{loc}^{2} (R^{2}, R^{2})

, we can invoke [28, Theorem 7.21] to assert the diamagnetic inequality

\begin{aligned} | \nabla_{A} u (x) | ⩾ | \nabla | u (x) | |, \end{aligned}

(1.7)

for each

u \in H_{A}^{1} (R^{2}, C)

. This inequality is a crucial tool in our subsequent analysis.

Focusing on potential V, we have the following notations

λ_{1} := inf {\int_{R^{2}} (| \nabla_{A} u |^{2} + V (x) | u |^{2}) d x : u \in H_{A, V}^{1} (R^{2}, C) ~and~ \int_{R^{2}} | u |^{2} d x = 1}

and for any open domain

Ω \subset R^{2}

ν_{r} (Ω) := {\begin{cases} inf {\int_{R^{2}} (\nabla_{A} u |^{2} + V (x) | u |^{2}) d x : u \in H_{A, V}^{0, 1} (Ω, C) a n d | u |_{L^{r} (Ω)} = 1}, & if Ω \neq \emptyset, \\ + \infty, & if Ω = \emptyset, \end{cases}

where

r ⩾ 2

and

H_{A, V}^{0, 1} (Ω, C)

is the closure of

C_{c}^{\infty} (Ω, C) := {φ \in C^{\infty} (Ω, C) : supp φ is a compactsubset~of Ω}

H_{A, V}^{1} (R^{2}, C)

Next, we give some assumptions about equation (1.1). The assumptions on the potential $V \in L_{loc}^{\infty} (R^{2}, R)$ are: $(V_{1})$

$λ_{1} > 0$ ;

(V_{2})

$lim_{s \to \infty} ν_{2} (R^{2} ∖ \bar{{x \in R^{2} : | x | < s}}) = + \infty$ ;

(V_{3})

there exists $V_{0} > 0$ such that

V (x) ⩾ - V_{0}, for all x \in R^{2} .

The assumptions on the potential

B \in L_{loc}^{\infty} (R^{2}, R)

are:

(B_{1})

$B (x) ⩾ 1$ for any $x \in R^{2}$ ;

(B_{2})

there exist $β > 1$ , $C_{0} > 0$ and $R_{0} > 0$ such that

B (x) ⩽ C_{0} [1 + (V^{+} (x))^{1 / β}], for all x \in R^{2} ∖ B_{R_{0}} (0),

where

V^{+} (x) := max {0, V (x)}

Assumptions (

V_{1}

)–(

V_{3}

) and (

B_{1}

)–(

B_{2}

) were originally introduced by Sirakov [33] in the context of subcritical Schrödinger equations in

R^{N}

with

N ⩾ 3

. We will utilize these assumptions to ensure that

H_{A, V}^{1} (R^{2}, C)

is a Hilbert space and to derive some useful Sobolev embedding for our subsequent analysis below.

Now, let’s consider the nonlocal term $m : [0, \infty) \to (0, \infty)$ and its associated assumptions: $(m_{1})$

$m_{0} := inf_{t ⩾ 0} m (t) > 0$ ;

(m_{2})

for every $t_{1}, t_{2} ⩾ 0$ , we have

M (t_{1} + t_{2}) ⩾ M (t_{1}) + M (t_{2}),

where

M (t) = \int_{0}^{t} m (s) d s

t ⩾ 0

;

(m_{3})

$\frac{m (t)}{t}$ is decreasing in $(0, \infty)$ .

Next, we give two notations (with more details provided in Lemma 2.5 and the definition of ζ in Lemma 2.1) and the assumptions that the nonlinearity f satisfies

\begin{aligned} S_{p} : & = inf_{u \in H_{A, V}^{1} (R^{2}, C) ∖ {0}} \frac{\int_{R^{2}} (| \nabla_{A} u |^{2} + V (x) | u |^{2}) d x}{| u |_{p}^{2}}, p ⩾ 2; \\ C_{p} & := inf {C > 0 : p M (t^{2} S_{p}) - 2 C t^{p} ⩽ p M (\frac{4 π ζ}{a_{0}}), for any t > 0}, p > 4; \end{aligned}

(f_{1})

$f (t) = 0$ if $t ⩽ 0$ ;

(f_{2})

there exists $a_{0} > 0$ such that

lim_{s \to \infty} \frac{s f (s^{2})}{e^{a s^{2}}} = {\begin{cases} 0. & if a > a_{0}, \\ + \infty, & if a < a_{0}; \end{cases}

(f_{3})

there exists $θ > 2$ such that

0 < θ F (s) ⩽ s f (s), for all s > 0,

where

F (s) = \int_{0}^{s} f (t) d t

s \in R

;

(f_{4})

$\frac{f (s)}{s}$ is non-decreasing in $(0, \infty)$ ;

(f_{5})

there exist $S_{0}, K_{0} > 0$ such that

F (s) ⩽ K_{0} f (s), for all s ⩾ S_{0};

(f_{6})

there exists $p_{0} > 4$ such that

f (s) ⩾ C_{p_{0}} s^{\frac{p_{0} - 2}{2}}, for all s > 0;

(f_{7})

there exists $λ > 0$ such that

\underset{s \to \infty}{lim_{_}} \frac{s f (s)}{e^{a_{0} s}} ⩾ λ > \frac{4}{a_{0}} m (\frac{4 π}{a_{0}}) inf_{R > 0} {R^{- 2} e^{\frac{R^{2} M_{R}}{2}}},

where

M_{R} := M_{A, R} + M_{V, R}, M_{A, R} := | A |_{L^{\infty} (B_{R} (0))}^{2} and M_{V, R} := | V |_{L^{\infty} (B_{R} (0))} .

Conditions (

m_{3}

) and (

f_{4}

) guarantee that the solutions obtained from Theorems 1.1 and 1.2 are ground states. Nevertheless, as we will demonstrate in the proofs if we replace (

m_{3}

) and (

f_{4}

) with the weaker conditions (

m_{3}^{*}

) and the monotonicity result from Lemma 2.10 in the subsequent chapter, we can still obtain nontrivial solutions to problem (1.1) that are not necessarily ground states. The condition (

m_{3}^{*}

) is as follows:

(m_{3}^{*})

for any $t > 1$ , we can obtain that there exists $a_{1} > 0$ such that

\begin{aligned} m (t) ⩽ a_{1} t . \end{aligned}

(1.8)

In virtue of (

f_{3}

), we know that

\frac{F (s)}{s^{θ}}

is non-decreasing and positive for

s > 0

. Consequently, for any

S > 0

, there exists

C_{S} > 0

such that

\begin{aligned} F (s) ⩾ C_{S} s^{θ}, for all s ⩾ S \end{aligned}

(1.9)

and

\begin{aligned} f (s) ⩾ θ C_{S} s^{θ - 1}, for all s ⩾ S . \end{aligned}

(1.10)

The main results of this paper are as follows.

Theorem 1.1.

Under assumptions ( $m_{1}$ )–( $m_{3}$ ), ( $V_{1}$ )–( $V_{3}$ ), ( $B_{1}$ )–( $B_{2}$ ), ( $f_{1}$ )–( $f_{6}$ ), problem ( 1.1 ) possesses a ground state solution.

Theorem 1.2.

Under assumptions ( $m_{1}$ )–( $m_{3}$ ), ( $V_{1}$ )–( $V_{2}$ ), ( $V_{4}$ ), ( $B_{1}$ )–( $B_{2}$ ), ( $f_{1}$ )–( $f_{5}$ ) and ( $f_{7}$ ), problem (1.1) has a ground state solution, where condition ( $V_{4}$ ) will be introduced in Section 2.

Before starting the main results for the local case, we introduce a notation

{\tilde{C}}_{q} := inf {C > 0 : q t^{2} S_{q} - 2 C t^{q} ⩽ \frac{4 π q ζ}{a_{0}}, for any t > 0}, q > 2.

It is evident that

{\tilde{C}}_{q}

is finite and

{\tilde{C}}_{q} = S_{q}^{\frac{q}{2}} {(\frac{a_{0} (q - 2)}{4 π q ζ})}^{\frac{q - 2}{2}}

. In the local case, we can obtain similar results by adapting the approach used for nonlocal problems. Instead of conditions (

f_{3}

), (

f_{4}

), (

f_{6}

) and (

f_{7}

), we consider the following assumptions:

({\tilde{f}}_{3})

there exists $θ > 2$ such that

0 < \frac{θ}{2} F (s) ⩽ s f (s), for all s > 0,

where

F (s) = \int_{0}^{s} f (t) d t

s \in R

;

({\tilde{f}}_{4})

$f (s)$ is non-decreasing in $(0, \infty)$ ;

({\tilde{f}}_{6})

there exists $q > 2$ such that

f (s) ⩾ {\tilde{C}}_{q} s^{\frac{q - 2}{2}}, for all s > 0;

({\tilde{f}}_{7})

there exists $\tilde{λ} > 0$ such that

\underset{s \to \infty}{lim_{_}} \frac{s f (s)}{e^{a_{0} s}} ⩾ \tilde{λ} > \frac{4}{a_{0}} inf_{R > 0} {R^{- 2} e^{\frac{R^{2} M_{R}}{2}}} .

Clearly, condition (

{\tilde{f}}_{3}

) is weaker than (

f_{3}

) and condition (

{\tilde{f}}_{4}

) is weaker than (

f_{4}

). Our main results about (1.4) are the following.

Theorem 1.3.

Under assumptions ( $V_{1}$ )–( $V_{3}$ ), ( $B_{1}$ )–( $B_{2}$ ) and ( $f_{1}$ )–( ${\tilde{f}}_{3}$ ), ( $f_{5}$ ) and ( ${\tilde{f}}_{6}$ ), problem ( 1.4 ) has a nontrivial solution. In addition, if f satisfies ( ${\tilde{f}}_{4}$ ), the solution is a ground state.

Theorem 1.4.

Under assumptions ( $V_{1}$ )–( $V_{2}$ ), ( $V_{4}$ ), ( $B_{1}$ )–( $B_{2}$ ), ( $f_{1}$ )–( ${\tilde{f}}_{3}$ ), ( $f_{5}$ ) and ( ${\tilde{f}}_{7}$ ), problem ( 1.4 ) has a nontrivial solution. In addition, if f satisfies ( ${\tilde{f}}_{4}$ ), the solution is a ground state.

This paper is organized as follows. We present several notations and technical lemmas in Section 2. In Section 3, we show the variational framework of problem (1.1). Minimax estimates are given in Section 4. In Section 5, we prove Theorems 1.1–1.2. Finally, in the last section, we prove Theorems 1.3–1.4.

2. Preliminaries

Hereafter, we use $‖ \cdot ‖$ , $| \cdot |_{p}$ , $‖ \cdot ‖_{A}$ and $‖ \cdot ‖_{A, V}$ to denote the norm of the space $H^{1} (R^{2}, C)$ , $L^{p} (R^{2}, C)$ , $H_{A}^{1} (R^{2}, C)$ and $H_{A, V}^{1} (R^{2}, C)$ , respectively. $B_{R} (x)$ denotes the open ball centered at $x \in R^{2}$ with radius $R > 0$ and $B_{R}^{c} (x)$ denotes the complement of $B_{R} (x)$ in $R^{2}$ . The support of a function φ is denoted by $supp φ$ . The constants C, $C_{1}, \dots$ are positive and their exact values are inessential, varying from line to line. Assumptions ( $V_{1}$ )–( $V_{3}$ ) and ( $B_{1}$ )–( $B_{2}$ ) will be assumed to hold throughout the remainder of the paper. Moreover, we denote by → (resp. ⇀) the strong (resp. weak) convergence and introduce the following weighted Lebesgue space

L_{B}^{p} (R^{2}, C) := {u : R^{2} \to C : \int_{R^{2}} B (x) | u |^{p} d x < + \infty}, p ⩾ 1,

which is a Banach space with the norm

| u |_{L_{B}^{p}} := {(\int_{R^{2}} B (x) | u |^{p} d x)}^{1 / p}

. By condition (

B_{1}

), we know that the embedding

L_{B}^{p} (R^{2}, C) ↪ L^{p} (R^{2}, C)

is continuous for

p ⩾ 1

About the nonlinearity, fixed $q ⩾ 1$ , for any $ξ > 0$ and $a > a_{0}$ , according to ( $f_{1}$ ) and ( $f_{2}$ ), there exists a constant $C = C (a, ξ, q) > 0$ such that

\begin{aligned} f (t) ⩽ ξ + C t^{(q - 2) / 2} (e^{a t} - 1) for any t ⩾ 0 \end{aligned}

(2.1)

and from condition (

{\tilde{f}}_{3}

), we obtain

\begin{aligned} F (t) ⩽ ξ t + C t^{q / 2} (e^{a t} - 1) for any t ⩾ 0. \end{aligned}

(2.2)

Moreover, using (2.1) and (2.2), we get

\begin{aligned} f (t^{2}) t^{2} ⩽ ξ t^{2} + C | t |^{q} (e^{a t^{2}} - 1) for any t \in R \end{aligned}

(2.3)

and

\begin{aligned} F (t^{2}) ⩽ ξ t^{2} + C | t |^{q} (e^{a t^{2}} - 1) for any t \in R . \end{aligned}

(2.4)

Similar to [33, Lemma 2.1], we have the following lemma to illustrate the space $H_{A, V}^{1} (R^{2}, C)$ .

Lemma 2.1.

There exists a $ζ > 0$ such that for any $u \in H_{A, V}^{1} (R^{2}, C)$ ,

\begin{aligned} \int_{R^{2}} (| \nabla_{A} u |^{2} + V (x) | u |^{2}) d x ⩾ ζ \int_{R^{2}} | \nabla_{A} u |^{2} d x . \end{aligned}

(2.5)

Proof.

By contradiction, assume that there exists a sequence ${u_{n}} \subset H_{A, V}^{1} (R^{2}, C)$ satisfies

\int_{R^{2}} {| \nabla_{A} u_{n} |}^{2} d x = 1 and \int_{R^{2}} ({| \nabla_{A} u_{n} |}^{2} + V (x) {| u_{n} |}^{2}) d x ⩽ \frac{1}{n} .

It follows from the

λ_{1} > 0

that

0 ⩽ λ_{1} | u_{n} |_{2}^{2} ⩽ \int_{R^{2}} (| \nabla_{A} u_{n} |^{2} + V (x) | u_{n} |^{2}) d x ⩽ \frac{1}{n},

which implies that

u_{n} \to 0

L^{2} (R^{2}, C)

. Due to the condition (

V_{3}

), it is a contradiction with the fact that

o (1) = - V_{0} \int_{R^{2}} {| u_{n} |}^{2} d x ⩽ \int_{R^{2}} V (x) {| u_{n} |}^{2} d x ⩽ \frac{1}{n} - 1.

From this Lemma, we obtain that the space $H_{A, V}^{1} (R^{2}, C)$ is a Hilbert space equipped with the inner product and norm

⟨ u, v ⟩_{A, V} = Re \int_{R^{2}} (\nabla_{A} u \bar{\nabla_{A} v} + V (x) u \bar{v}) d x, ‖ u ‖_{A, V} = {⟨ u, u ⟩}_{A, V}^{\frac{1}{2}} .

From now on, we know that

λ_{1}

is always nonnegative and equation (1.1) can be equivalently expressed in the form:

m (‖ u ‖_{A, V}^{2}) (\nabla_{A}^{2} u + V (x) u) = B (x) f (| u |^{2}) u .

Moreover, according to the diamagnetic inequality (1.7), (2.5) and

λ_{1} > 0

, for

u \in H_{A, V}^{1} (R^{2}, C)

, we have

‖ u ‖_{A, V}^{2} ⩾ ζ \int_{R^{2}} | \nabla_{A} u |^{2} d x ⩾ ζ \int_{R^{2}} | \nabla | u | |^{2} d x ~and~ ‖ u ‖_{A, V}^{2} ⩾ λ_{1} | u |_{2}^{2}

. Combining these inequalities, we obtain

‖ u ‖_{A, V}^{2} ⩾ \frac{ζ | \nabla | u | |_{2}^{2} + λ_{1} | u |_{2}^{2}}{2} ⩾ \frac{min {λ_{1}, ζ}}{2} ‖ | u | ‖^{2}, for each u \in H_{A, V}^{1} (R^{2}, C),

which means that if

u \in H_{A, V}^{1} (R^{2}, C)

, then

| u | \in H^{1} (R^{2}, R)

. Consequently, it follows from the embedding

H^{1} (R^{2}, R) ↪ L^{r} (R^{2}, R)

being continuous for

r ⩾ 2

that the space

H_{A, V}^{1} (R^{2}, C)

is continuously embedded in

L^{r} (R^{2}, C)

for

r ⩾ 2

, too. Similarly, for any domain

Ω \subset R^{2}

, the embedding

H_{A, V}^{0, 1} (Ω, C) ↪ L^{r} (Ω, C)

is also continuous for

r ⩾ 2

Since V may be negative on some sets with positive measure, we can not obtain $ζ > 1$ . However, we can take $ζ = 1$ if we replace ( $V_{3}$ ) by stronger condition, namely:

(

V_{4}

)

$V (x) ⩾ 0$ for any $x \in R^{2}$ .

Similar to [33, Lemma 2.2], we have the following lemma to further illustrate the condition ( $V_{2}$ ).

Lemma 2.2.

Let Ω be an open subset of $R^{2}$ and $q > 2$ . There exists $C = C (q, λ_{1})$ such that

ν_{q} (Ω) ⩾ C (ν_{2} (Ω))^{a},

where

a \in (0, 1)

is a fixed number.

Proof.

Due to the continuously embedding $H_{A, V}^{0, 1} (Ω, C) ↪ L^{r} (Ω, C)$ for $r ⩾ 2$ and Gagliardo–Nirenbery inequality, there exist $a \in (0, 1)$ and $C > 0$ such that

{(\int_{Ω} | u |^{q} d x)}^{\frac{1}{q}} ⩽ {(\int_{Ω} | u |^{2} d x)}^{\frac{a}{2}} {(\int_{Ω} | u |^{q + 1} d x)}^{\frac{1 - a}{q + 1}} ⩽ C {(\int_{Ω} | u |^{2} d x)}^{\frac{a}{2}} ‖ u ‖_{A, V}^{1 - a}

for each

u \in H_{A, V}^{0, 1} (Ω, C)

. Consequently, we obtain

\begin{aligned} ν_{q} (Ω) & = inf_{u \in H_{A, V}^{0, 1} (Ω, C) ∖ {0}} \frac{\int_{R^{2}} (| \nabla_{A} u |^{2} + V (x) | u |^{2}) d x}{{(\int_{Ω} | u |^{q} d x)}^{\frac{2}{q}}} ⩾ inf_{u \in H_{A, V}^{0, 1} (Ω, C) ∖ {0}} \frac{\int_{R^{2}} (| \nabla_{A} u |^{2} + V (x) | u |^{2}) d x}{C {(\int_{Ω} | u |^{2} d x)}^{a} ‖ u ‖_{A, V}^{2 (1 - a)}} \\ = \frac{1}{C} inf_{u \in H_{A, V}^{0, 1} (Ω, C) ∖ {0}} {(\frac{\int_{R^{2}} (| \nabla_{A} u |^{2} + V (x) | u |^{2}) d x}{\int_{Ω} | u |^{2} d x})}^{a} = \frac{1}{C} (ν_{2} (Ω))^{a}, \end{aligned}

which means if

lim_{s \to \infty} ν_{2} (R^{2} ∖ \bar{{x \in R^{2} : | x | < s}}) = + \infty

, then

lim_{s \to \infty} ν_{r} (R^{2} ∖ \bar{{x \in R^{2} : | x | < s}}) = + \infty

for any

r ⩾ 2

For the space $H_{A, V}^{1} (R^{2}, C)$ , similar to [18, Lemma 2.1], we have the following compact property.

Lemma 2.3.

The space $H_{A, V}^{1} (R^{2}, C)$ is compactly embedded in $L_{loc}^{r} (R^{2}, C)$ for $r ⩾ 1$ . Moreover, $H_{A, V}^{1} (R^{2}, C)$ is compactly embedded in $L^{r} (R^{2}, C)$ for $r ⩾ 2$ .

Proof.

(i) To begin with, we prove $H_{A, V}^{1} (R^{2}, C)$ is continuously embedded in $H^{1} (Ω, C)$ , where $Ω \subset R^{2}$ is an open subset with compact closure. Indeed, it follows from (2.5) and $λ_{1} > 0$ that

\begin{aligned} ‖ u ‖_{H^{1} (Ω, C)}^{2} & = \int_{Ω} (| \nabla u |^{2} + | u |^{2}) d x = \int_{Ω} (| \frac{\nabla}{i} u |^{2} + | u |^{2}) d x \\ ⩽ C \int_{Ω} (| \nabla_{A} u |^{2} + | A u |^{2} + | u |^{2}) d x ⩽ C (| \nabla_{A} u |_{2}^{2} + | u |_{2}^{2}) \\ ⩽ C (\frac{1}{ζ} + \frac{1}{λ_{1}}) ‖ u ‖_{A, V}^{2} ⩽ C ‖ u ‖_{A, V}^{2} . \end{aligned}

Using the classical method in [12], the rest proof of (ii) is similar to the proof of [18, Lemma 2.1] and we can get the embedding

H_{A, V}^{1} (R^{2}, C) ↪ L_{loc}^{r} (R^{2}, C)

is compact for

r ⩾ 1

(ii) Without loss of generality, we suppose that $u_{n} ⇀ 0$ in $H_{A, V}^{1} (R^{2}, C)$ . According to (i), up to a subsequence, we know that $u_{n} \to 0$ in $L_{loc}^{s} (R^{2}, C)$ for all $s ⩾ 1$ . Then for any $R > 0$ and all $r ⩾ 2$ , we take a function $ψ \in C^{\infty} (R^{2}, R)$ with $0 ⩽ ψ ⩽ 1$ , $ψ \equiv 0$ on $B_{R} (0)$ and $ψ \equiv 1$ on $B_{R + 1}^{c} (0)$ . For each $u \in H_{A, V}^{1} (R^{2}, C)$ , due to

\nabla_{A} (ψ u) = (\frac{\nabla}{i} - A) (ψ u) = \frac{ψ \nabla u + u \nabla ψ}{i} - A (ψ u) = ψ \nabla_{A} u + \frac{u \nabla ψ}{i} = ψ \nabla_{A} u - i u \nabla ψ

and Cauchy’s inequality, we have

\begin{aligned} ‖ ψ u ‖_{A, V}^{2} & = \int_{R^{2}} (| ψ \nabla_{A} u - i u \nabla ψ |^{2} + V (x) | ψ u |^{2}) d x ⩽ C (| \nabla_{A} u |_{2}^{2} + | u |_{2}^{2} + \int_{R^{2}} V (x) | u |^{2} d x) \\ ⩽ C (\frac{1}{ζ} + \frac{1}{λ_{1}} + 1) ‖ u ‖_{A, V}^{2} ⩽ C ‖ u ‖_{A, V}^{2}, \end{aligned}

where we have used (2.5) and

λ_{1} > 0

. Similar to the consequence of [1], we know that

ψ u \in H_{A, V}^{0, 1} (R^{2} ∖ \bar{B_{R} (0)}, C)

. Indeed, for any

u \in H_{A, V}^{1} (R^{2}, C)

, it is obvious that

ψ u \in H_{A, V}^{1} (R^{2}, C)

. Moreover, due to

C_{c}^{\infty} (R^{2}, C) \subset H_{A, V}^{1} (R^{2}, C) \subset H_{A}^{1} (R^{2}, C)

and the consequence of [28, Theorem 2.2], that is,

C_{c}^{\infty} (R^{2}, C)

is dense in

H_{A}^{1} (R^{2}, C)

, we know

C_{c}^{\infty} (R^{2}, C)

is dense in

H_{A, V}^{1} (R^{2}, C)

, which means that there exists a sequence

{φ_{n}} \subset C_{c}^{\infty} (R^{2}, C)

such that

‖ φ_{n} - u ‖_{A, V} \to 0

. Moreover, we have

‖ ψ φ_{n} - ψ u ‖_{A, V} = ‖ ψ (φ_{n} - u) ‖_{A, V} ⩽ C ‖ φ_{n} - u ‖_{A, V} \to 0

, and from

ψ φ_{n} \in C_{c}^{\infty} (R^{2} ∖ \bar{B_{R} (0)}, C)

, we obtain

ψ u \in H_{A, V}^{0, 1} (R^{2} ∖ \bar{B_{R} (0)}, C)

. Consequently, integrating the above inequalities, we obtain that

\begin{aligned} | u_{n} |_{r} & ⩽ | (1 - ψ) u_{n} |_{r} + | ψ u_{n} |_{r} \\ ⩽ | u_{n} |_{L^{r} (B_{R + 1} (0))} + \frac{\int_{R^{2} ∖ \bar{B_{R} (0)}} (| \nabla_{A} (ψ u_{n}) |^{2} + V (x) | ψ u_{n} |^{2}) d x}{ν_{r} (R^{2} ∖ \bar{B_{R} (0)})} \\ ⩽ | u_{n} |_{L^{r} (B_{R + 1} (0))} + \frac{C ‖ u_{n} ‖_{A, V}^{2}}{ν_{r} (R^{2} ∖ \bar{B_{R} (0)})} = a_{n} + β_{R} . \end{aligned}

Due to the consequence of (i), that is, up to a subsequence,

u_{n} \to 0

L_{loc}^{r} (R^{2}, C)

, the boundedness of

{u_{n}}

H_{A, V}^{1} (R^{2}, C)

, the assumption (

V_{2}

) and Lemma 2.2, we obtain

a_{n} \to 0

n \to \infty

and

β_{R} \to 0

R \to \infty

, which means that

u_{n} \to 0

L^{r} (R^{2}, C)

and the embedding

H_{A, V}^{1} (R^{2}, C) ↪ L^{r} (R^{2}, C)

is compact.

Similar to [33, Proposition 3.1], we have the following more complicated compact property.

Lemma 2.4.

The space $H_{A, V}^{1} (R^{2}, C)$ is continuously embedded in $L_{B}^{r} (R^{2}, C)$ for $r ⩾ 2$ . Moreover, this embedding is also compact for $r ⩾ 2$ .

Proof.

(i) We first prove that the embedding is continuous. For any $u \in H_{A, V}^{1} (R^{2}, C)$ and $r ⩾ 2$ , under conditions ( $B_{2}$ ), ( $V_{1}$ ) and ( $V_{3}$ ), using Hölder’s inequality, we get

\begin{aligned} \int_{R^{2}} B (x) | u |^{r} d x & ⩽ (C_{0} + max_{| x | ⩽ R_{0}} B (x)) | u |_{r}^{r} + C_{0} \int_{R^{2}} (V^{+} (x))^{\frac{1}{β}} | u |^{r} d x \\ ⩽ C (‖ u ‖_{A, V}^{r} + {(\int_{R^{2}} V^{+} (x) | u |^{2} d x)}^{\frac{1}{β}} {(\int_{R^{2}} | u |^{\frac{r β - 2}{β - 1}} d x)}^{\frac{β - 1}{β}}) \\ ⩽ C (‖ u ‖_{A, V}^{r} + {(‖ u ‖_{A, V}^{2} - \int_{{x : V (x) ⩽ 0}} V (x) | u |^{2} d x)}^{\frac{1}{β}} | u |_{\frac{r β - 2}{β - 1}}^{\frac{r β - 2}{β}}) \\ ⩽ C (‖ u ‖_{A, V}^{r} + {(‖ u ‖_{A, V}^{2} + V_{0} | u |_{2}^{2})}^{\frac{1}{β}} ‖ u ‖_{A, V}^{\frac{r β - 2}{β}}) \\ ⩽ C (1 + {(1 + \frac{V_{0}}{λ_{1}})}^{\frac{1}{β}}) ‖ u ‖_{A, V}^{r}, \end{aligned}

where we have used Lemma 2.3, together with the fact that

\frac{r β - 2}{β - 1} ⩾ 2

(ii) Next, we prove this continuous embedding is also compact. Without loss of generality, we suppose that $u_{n} ⇀ 0$ in $H_{A, V}^{1} (R^{2}, C)$ . By Lemma 2.3, up to a subsequence, we know that $u_{n} \to 0$ in $L^{r} (R^{2}, C)$ for all $r ⩾ 2$ . Then, similar to the proof of (i), we get

\int_{R^{2}} B (x) | u_{n} |^{r} d x ⩽ C (| u_{n} |_{r}^{r} + {(1 + \frac{V_{0}}{λ_{1}})}^{\frac{1}{β}} ‖ u_{n} ‖_{A, V}^{\frac{2}{β}} | u_{n} |_{\frac{r β - 2}{β - 1}}^{\frac{r β - 2}{β}}) .

Furthermore, because

\frac{r β - 2}{β - 1} ⩾ 2

and

{u_{n}}

is bounded in

H_{A, V}^{1} (R^{2}, C)

, we can obtain that

u_{n} \to 0

L_{B}^{r} (R^{2}, C)

and the embedding

H_{A, V}^{1} (R^{2}, C) ↪ L_{B}^{r} (R^{2}, C)

is also compact for

r ⩾ 2

Lemma 2.5.

Under assumptions ( $m_{1}$ ) and ( $m_{3}^{*}$ ), we can obtain that both $S_{p}$ and $C_{p}$ are finite for $p ⩾ 2$ and $p > 4$ , respectively. Moreover, $S_{p} > 0$ and can be achieved.

Proof.

(i) $S_{p} < + \infty$ for $p ⩾ 2$ is a natural consequence of Lemma 2.3, so we only prove that $C_{p} < + \infty$ for $p > 4$ .

Fixed $C_{p, 1} > 0$ , due to the continuity of M and $lim_{t \to 0} M (t) = 0$ , there exists $0 < δ < 1$ small enough such that

p M (t^{2} S_{p}) - 2 C_{p, 1} t^{p} ⩽ p M (\frac{4 π ζ}{a_{0}}), for any t \in (0, \sqrt{δ / S_{p}]} .

In the case of

δ < t^{2} S_{p} ⩽ 1

, since m is always positive, we have

M (t^{2} S_{p}) ⩽ M (1)

and there exists

C_{p, 2} > 0

such that

p M (t^{2} S_{p}) - p M (\frac{4 π ζ}{a_{0}}) ⩽ p M (1) - p M (\frac{4 π ζ}{a_{0}}) ⩽ 2 C_{p, 2} t^{p}, for any t \in (\sqrt{δ / S_{p}}, \sqrt{1 / S_{p}]} .

In addition, in the case of

t^{2} S_{p} ⩾ 1

, combining (1.8) and

p > 4

, there exists

C_{p, 3} > 0

such that

\begin{aligned} p M (t^{2} S_{p}) - p M (\frac{4 π ζ}{a_{0}}) & = p M (1) + p \int_{1}^{t^{2} S_{p}} m (s) d s - p M (\frac{4 π ζ}{a_{0}}) \\ ⩽ p M (1) + p \int_{1}^{t^{2} S_{p}} a_{1} s d s - p M (\frac{4 π ζ}{a_{0}}) \\ ⩽ 2 C_{p, 3} t^{p}, for any t \in [\sqrt{1 / S_{p}}, \infty) . \end{aligned}

In conclusion, using the above inequalities, we can conclude

C_{p} ⩽ max {C_{p, 1}, C_{p, 2}, C_{p, 3}} < + \infty

for

p > 4

(ii) It is clear that $S_{p} ⩾ 0$ for $p ⩾ 2$ . If $S_{p} = 0$ , there exists a minimizing sequence ${u_{n}} \subset H_{A, V}^{1} (R^{2}, C)$ such that $| u_{n} |_{p} = 1$ and $‖ u_{n} ‖_{A, V} \to 0$ . Due to Lemma 2.3, up to a subsequence, we obtain that $u_{n} \to 0$ in $L^{p} (R^{2}, C)$ for $p ⩾ 2$ . It is a contradiction with the fact that $lim_{n \to \infty} | u_{n} |_{p} = 1$ .

Moreover, for $p ⩾ 2$ , there exists a minimizing sequence ${v_{n}} \subset H_{A, V}^{1} (R^{2}, C)$ such that $| v_{n} |_{p} = 1$ and $‖ v_{n} ‖_{A, V}^{2} \to S_{p}$ . Up to a subsequence, there exists $v \in H_{A, V}^{1} (R^{2}, C)$ such that $v_{n} ⇀ v$ in $H_{A, V}^{1} (R^{2}, C)$ Furthermore, by the compact embedding $H_{A, V}^{1} (R^{2}, C) ↪ L^{p} (R^{2}, C)$ , there exists a subsequence of ${v_{n}}$ relabeled as ${v_{n}}$ such that

v_{n} \to v in L^{p} (R^{2}, C) ~as~ n \to \infty and 1 = lim_{n \to \infty} | v_{n} |_{p} = | v |_{p} .

In addition, from weakly lower semi-continuity of the norm

‖ \cdot ‖_{A, V}

and the definition of

S_{p}

, we can obtain that

S_{p} ⩽ ‖ v ‖_{A, V}^{2} ⩽ \underset{n \to \infty}{lim_{_}} ‖ v_{n} ‖_{A, V}^{2} = S_{p}

Next, we recall the following Trudinger–Moser inequality, whose proof was given in [2] (see also [13] or [20]).

Lemma 2.6.

If $a > 0$ and $u \in H^{1} (R^{2}, R)$ , we have $\int_{R^{2}} (e^{a u^{2}} - 1) d x < + \infty$ . Additionally, if $| \nabla u |_{2} ⩽ 1$ , $| u |_{2} ⩽ M < + \infty$ and $a < 4 π$ , there exists a positive constant $C = C (a, M)$ such that

\int_{R^{2}} (e^{a u^{2}} - 1) d x ⩽ C .

Similar to [23, Lemmas 2.2–2.3], we have the following two lemmas for space $H_{A, V}^{1} (R^{2}, C)$ , which are useful later.

Lemma 2.7.

If $a > 0$ , $u \in H_{A, V}^{1} (R^{2}, C)$ and $r \in [1, β)$ , we have that

\int_{R^{2}} B {(x)}^{r} (e^{a | u |^{2}} - 1)^{r} d x < + \infty

with β given in assumption (

B_{2}

Proof.

For $u \in H_{A, V}^{1} (R^{2}, C)$ and $r \in [1, β)$ , it follows from $B \in L_{loc}^{\infty} (R^{2}, R)$ that there exists $C_{1} > 0$ such that

\begin{aligned} \int_{R^{2}} B {(x)}^{r} (e^{a | u |^{2}} - 1)^{r} d x ⩽ \int_{R^{2} ∖ B_{R_{0}} (0)} B {(x)}^{r} (e^{r a | u |^{2}} - 1) d x + C_{1} \int_{B_{R_{0}} (0)} (e^{r a | u |^{2}} - 1) d x, \end{aligned}

(2.6)

where

R_{0} > 0

has been given in assumption (

B_{2}

) and we have used the inequality

\begin{aligned} (e^{t} - 1)^{s} ⩽ e^{t s} - 1, for~ t ⩾ 0 ~and~ s > 1. \end{aligned}

(2.7)

It follows from

| u | \in H^{1} (R^{2}, R)

and Lemma 2.6 that

\begin{aligned} \int_{B_{R_{0}} (0)} (e^{r a | u |^{2}} - 1) d x ⩽ \int_{R^{2}} (e^{r a | u |^{2}} - 1) d x < + \infty, \end{aligned}

(2.8)

which implies we only need to estimate the first integral on the right-hand of (2.6). Noticing

\begin{aligned} \int_{R^{2} ∖ B_{R_{0}} (0)} B {(x)}^{r} (e^{r a | u |^{2}} - 1) d x & = \int_{R^{2} ∖ B_{R_{0}} (0)} B {(x)}^{r} \overset{\infty}{\sum_{m = 1}} \frac{(r a | u |^{2})^{m}}{m!} d x \\ = \overset{\infty}{\sum_{m = 1}} \frac{(r a)^{m}}{m!} \int_{R^{2} ∖ B_{R_{0}} (0)} B {(x)}^{r} | u |^{2 m} d x, \end{aligned}

(2.9)

similar to the proof of Lemma 2.4 (i), by (

B_{2}

) and Hölder’s inequality, we have

\begin{aligned} \int_{R^{2} ∖ B_{R_{0}} (0)} B {(x)}^{r} | u |^{2 m} d x & ⩽ C_{2} | u |_{2 m}^{2 m} + C_{3} \int_{R^{2} ∖ B_{R_{0}} (0)} (V^{+} (x))^{\frac{r}{β}} | u |^{2 m} d x \\ ⩽ C_{2} | u |_{2 m}^{2 m} + C_{3} {(\int_{R^{2}} V^{+} (x) | u |^{2} d x)}^{\frac{r}{β}} {(\int_{R^{2}} | u |^{\frac{2 (m β - r)}{β - r}} d x)}^{\frac{β - r}{β}} \\ ⩽ C_{4} (1 + {(1 + \frac{V_{0}}{λ_{1}})}^{\frac{r}{β}}) ‖ u ‖_{A, V}^{2 m} ⩽ C_{5} ‖ u ‖_{A, V}^{2 m} . \end{aligned}

(2.10)

Consequently, due to (2.6), (2.8)–(2.10), we obtain that

\begin{aligned} \int_{R^{2}} B {(x)}^{r} (e^{a | u |^{2}} - 1)^{r} d x & ⩽ C_{5} \overset{\infty}{\sum_{m = 1}} \frac{(r a ‖ u ‖_{A, V}^{2})^{m}}{m!} + C_{1} \int_{B_{R_{0}} (0)} (e^{r a | u |^{2}} - 1) d x \\ = C_{5} (e^{r a ‖ u ‖_{A, V}^{2}} - 1) + C_{1} \int_{B_{R_{0}} (0)} (e^{r a | u |^{2}} - 1) d x < + \infty, \end{aligned}

(2.11)

which concludes the conclusion.

Lemma 2.8.

For $a > 0$ , $q > 0$ and any $w, v \in H_{A, V}^{1} (R^{2}, C)$ , we have

\int_{R^{2}} B (x) | w |^{q} (e^{a | v |^{2}} - 1) d x < + \infty .

Moreover, for ζ be taken in Lemma 2.1, if

a < 4 π ζ

and

‖ v ‖_{A, V} ⩽ 1

, there exists a positive constant

C = C (a, q)

such that

\int_{R^{2}} B (x) | w |^{q} (e^{a | v |^{2}} - 1) d x ⩽ C ‖ w ‖_{A, V}^{q} .

Proof.

We choose

r \in {\begin{cases} (1, β), & if q ⩾ 2, \\ (1, min {β, \frac{2}{2 - q}}), & if q < 2, \end{cases}

such that

r \in (1, β)

and

q r^{'} ⩾ 2

, where

r^{'} = \frac{r}{r - 1}

. It follows from Hölder’s inequality, the continuous embedding

H_{A, V}^{1} (R^{2}, C) ↪ L^{q r^{'}} (R^{2}, C)

and Lemma 2.7 that

\begin{aligned} \int_{R^{2}} B (x) | w |^{q} (e^{a | v |^{2}} - 1) d x & ⩽ | w |_{q r^{'}}^{q} {(\int_{R^{2}} B {(x)}^{r} (e^{a | v |^{2}} - 1)^{r} d x)}^{\frac{1}{r}} \\ ⩽ C_{1} ‖ w ‖_{A, V}^{q} {(\int_{R^{2}} B {(x)}^{r} (e^{a | v |^{2}} - 1)^{r} d x)}^{\frac{1}{r}} < + \infty \end{aligned}

(2.12)

and we have the first conclusion.

If $a < 4 π ζ$ and $‖ v ‖_{A, V} ⩽ 1$ , we can choose

r \in {\begin{cases} (1, min {β, \frac{4 π ζ}{a}}), & if q ⩾ 2, \\ (1, min {β, \frac{4 π ζ}{a}, \frac{2}{2 - q}}), & if q < 2, \end{cases}

such that

r \in (1, β)

q r^{'} ⩾ 2

and

r a < 4 π ζ

. Combining with (2.11)–(2.12) and noting

{| v |}^{2} = ζ^{- 1} (ζ^{\frac{1}{2}} | v |)^{2}

, we obtain that

\begin{aligned} \int_{R^{2}} B (x) | w |^{q} (e^{a | v |^{2}} - 1) d x & ⩽ C_{1} ‖ w ‖_{A, V}^{q} {(\int_{R^{2}} B {(x)}^{r} (e^{a | v |^{2}} - 1)^{r} d x)}^{\frac{1}{r}} \\ ⩽ C_{2} ‖ w ‖_{A, V}^{q} {(e^{r a ‖ v ‖_{A, V}^{2}} - 1 + \int_{B_{R_{0}} (0)} (e^{r a ζ^{- 1} (ζ^{\frac{1}{2}} | v |)^{2}} - 1) d x)}^{1 / r} . \end{aligned}

Due to (1.7) and (2.5), we obtain that

| \nabla (ζ^{\frac{1}{2}} | v |) |_{2}^{2} ⩽ ζ | \nabla_{A} v |_{2}^{2} ⩽ ‖ v ‖_{A, V}^{2} ⩽ 1

. Moreover, we have

| ζ^{\frac{1}{2}} | v | |_{2}^{2} ⩽ ζ ‖ | v | ‖^{2} ⩽ C_{3} ζ ‖ v ‖_{A}^{2} ⩽ M

for some positive constant M independent of v. Consequently, from

r a ζ^{- 1} < 4 π

and the above inequalities, it follows from

ζ^{\frac{1}{2}} | v | \in H^{1} (R^{2}, R)

and Lemma 2.6 that

\int_{R^{2}} B (x) | w |^{q} (e^{a | v |^{2}} - 1) d x ⩽ C ‖ w ‖_{A, V}^{q},

which means the proof is completed.

Now, we give a similar result to Lions [29, Subsec. I.7] for our space $H_{A, V}^{1} (R^{2}, C)$ .

Lemma 2.9.

Let $q > 0$ and let ${w_{n}}, {v_{n}} \subset H_{A, V}^{1} (R^{2}, C)$ be such that ${w_{n}}$ is bounded in $H_{A, V}^{1} (R^{2}, C)$ , $v_{n} ⇀ v$ in $H_{A, V}^{1} (R^{2}, C)$ and $‖ v_{n} ‖_{A, V} = 1$ for any $n \in N$ . Then, if $‖ v ‖_{A, V} < 1$ , it holds

sup_{n \in N} \int_{R^{2}} B (x) {| w_{n} |}^{q} (e^{p | v_{n} |^{2}} - 1) d x < + \infty, for all 0 < p < \frac{4 π ζ}{1 - ‖ v ‖_{A, V}^{2}} .

The same holds if

‖ v ‖_{A, V} = 1

and

0 < p < \infty

Proof.

For given $a, b \in C$ and $ε > 0$ , using Young’s inequality, we obtain

\begin{aligned} | a |^{2} & = (| a | - | b |)^{2} + | b |^{2} + 2 ε (| a | - | b |) | b | ε^{- 1} \\ ⩽ (| a | - | b |)^{2} + | b |^{2} + 2 (\frac{ε^{2} (| a | - | b |)^{2}}{2} + \frac{| b |^{2} ε^{- 2}}{2}) \\ = (1 + ε^{2}) (| a | - | b |)^{2} + (1 + ε^{- 2}) | b |^{2} ⩽ (1 + ε^{2}) | a - b |^{2} + (1 + ε^{- 2}) | b |^{2} . \end{aligned}

Thus, choosing

r_{1}, r_{2} > 1

such that

\frac{1}{r_{1}} + \frac{1}{r_{2}} = 1

, and using Young’s inequality again, we get

\begin{aligned} B (x) | w_{n} |^{q} e^{p | v_{n} |^{2}} & ⩽ (B (x) | w_{n} |^{q})^{1 / r_{1}} e^{p (1 + ε^{2}) | v_{n} - v |^{2}} (B (x) | w_{n} |^{q})^{1 / r_{2}} e^{p (1 + ε^{- 2}) | v |^{2}} \\ ⩽ \frac{1}{r_{1}} B (x) | w_{n} |^{q} e^{p r_{1} (1 + ε^{2}) | v_{n} - v |^{2}} + \frac{1}{r_{2}} B (x) | w_{n} |^{q} e^{p r_{2} (1 + ε^{- 2}) | v |^{2}} . \end{aligned}

Furthermore, similar to (2.12), we obtain

\begin{aligned} \int_{R^{2}} B (x) | w_{n} |^{q} (e^{p | v_{n} |^{2}} - 1) d x & ⩽ \frac{1}{r_{1}} \int_{R^{2}} B (x) | w_{n} |^{q} (e^{p r_{1} (1 + ε^{2}) | v_{n} - v |^{2}} - 1) d x \\ + \frac{1}{r_{2}} \int_{R^{2}} B (x) | w_{n} |^{q} (e^{p r_{2} (1 + ε^{- 2}) | v |^{2}} - 1) d x \\ ⩽ \frac{1}{r_{1}} \int_{R^{2}} B (x) | w_{n} |^{q} (e^{p r_{1} (1 + ε^{2}) ‖ v_{n} - v ‖_{A}^{2} | \frac{v_{n} - v}{‖ v_{n} - v ‖_{A, V}} |^{2}} - 1) d x \\ + \frac{C}{r_{2}} ‖ w_{n} ‖_{A, V}^{q} {(\int_{R^{2}} B {(x)}^{r} (e^{p r_{2} (1 + ε^{- 2}) | v |^{2}} - 1)^{r} d x)}^{\frac{1}{r}} . \end{aligned}

(2.13)

Because

{w_{n}}

is bounded in

H_{A, V}^{1} (R^{2}, C)

and Lemma 2.7, the second part on the right-hand of (2.13) is bounded independently of n, which means that we only need to estimate the first part. Due to

v_{n} ⇀ v

H_{A, V}^{1} (R^{2}, C)

n \to \infty

and

‖ v_{n} ‖_{A, V} = 1

for any

n \in N

, we get

lim_{n \to \infty} p ‖ v_{n} - v ‖_{A, V}^{2} = lim_{n \to \infty} p (‖ v_{n} ‖_{A, V}^{2} - 2 ⟨ v_{n}, v ⟩_{A, V} + ‖ v ‖_{A, V}^{2}) = p (1 - ‖ v ‖_{A, V}^{2}) < 4 π ζ .

Then by taking

r_{1} > 1

sufficiently close to 1 and

ε > 0

small enough, there exists

n_{0} \in N

such that

p r_{1} (1 + ε^{2}) ‖ v_{n} - v ‖_{A, V}^{2} < 4 π ζ, for all n > n_{0} .

Consequently, the result follows from (2.13) Lemma 2.8 and the above inequality.

Next, we give the very important monotonic conditions.

Lemma 2.10.

Under assumptions ( $m_{3}$ ) and ( $f_{4}$ ), we have the following two statements. (i)

The function $L (t) := \frac{1}{2} M (t) - \frac{1}{4} m (t) t$ is increasing in $[0, \infty)$ . Particularly, $L (t) > L (0) = 0$ for each $t > 0$ .

(ii)

The function $G (s) := s f (s) - 2 F (s)$ is non-decreasing in $[0, \infty)$ . Particularly, $G (s) ⩾ G (0) = 0$ for each $s > 0$ .

Proof.

The proof of (i) has been given in [23], so we only prove (ii) here. Let $s_{1}, s_{2} \in R$ with $0 < s_{1} < s_{2}$ . From ( $f_{4}$ ), we obtain

\begin{aligned} s_{1} f (s_{1}) - 2 F (s_{1}) & = - 2 F (s_{2}) + 2 \int_{s_{1}}^{s_{2}} \frac{f (t)}{t} t d t + \frac{f (s_{1})}{s_{1}} s_{1}^{2} \\ ⩽ - 2 F (s_{2}) + \frac{f (s_{2})}{s_{2}} (s_{2}^{2} - s_{1}^{2}) + \frac{f (s_{2})}{s_{2}} s_{1}^{2} = s_{2} f (s_{2}) - 2 F (s_{2}), \end{aligned}

which means that the function

G (s)

is non-decreasing in

(0, \infty)

. It follows from the continuity at

t = 0

that

G (s)

is non-decreasing in

[0, \infty)

Finally, we give a similar continuous result in [19]. Lemma 2.11.

Let Ω be an open subset of $R^{2}$ . If $f \in C (Ω \times R, R^{+})$ and ${u_{n}} \subset L^{1} (Ω, C)$ satisfies

u_{n} \to u ~in~ L^{1} (Ω, C) ~as~ n \to \infty, f (\cdot, | u_{n} |^{2}), f (\cdot, | u |^{2}) \in L^{1} (Ω, R), \int_{Ω} f (x, | u_{n} |^{2}) | u_{n} |^{2} d x ⩽ C_{1},

where

C_{1} > 0

is a constant, then

f (\cdot, | u_{n} |^{2}) \to f (\cdot, | u |^{2}) ~in~ L^{1} (Ω, R) ~as~ n \to \infty

Proof.

It suffices to prove that $\int_{Ω} f (x, | u_{n} |^{2}) d x \to \int_{Ω} f (x, | u |^{2}) d x$ as $n \to \infty$ . For given $ε > 0$ , since $f (\cdot, | u |^{2}) \in L^{1} (Ω, R)$ , there exists a $δ > 0$ such that

\begin{aligned} \int_{A} f (x, | u |^{2}) d x ⩽ ε \end{aligned}

(2.14)

for all measurable subset

A \subset Ω

with

| A | ⩽ δ

. Next, using the fact that

u \in L^{1} (Ω, C)

, there exists

M_{1} > 0

such that

| {x \in Ω : | u (x) |^{2} ⩾ M_{1}} | ⩽ δ

. Let

M = max {M_{1}, \frac{C_{1}}{ε}}

. we write

\begin{aligned} | \int_{Ω} f (x, | u_{n} |^{2}) d x - \int_{Ω} f (x, | u |^{2}) d x | & ⩽ \int_{Ω \cap {| u_{n} |^{2} ⩾ M}} f (x, | u_{n} |^{2}) d x + \int_{Ω \cap {| u |^{2} ⩾ M}} f (x, | u |^{2}) d x \\ + | \int_{Ω \cap {| u_{n} |^{2} < M}} f (x, | u_{n} |^{2}) d x - \int_{Ω \cap {| u |^{2} < M}} f (x, | u |^{2}) d x | = I_{1} + I_{2} + I_{3} \end{aligned}

and estimate each integral separately. First, due to

\int_{Ω} f (x, | u_{n} |^{2}) | u_{n} |^{2} d x ⩽ C_{1}

and

M ⩾ \frac{C_{1}}{ε}

, we have

I_{1} = \int_{Ω \cap {| u_{n} |^{2} ⩾ M}} \frac{f (x, | u_{n} |^{2}) | u_{n} |^{2}}{| u_{n} |^{2}} d x ⩽ \frac{C_{1}}{M} ⩽ ε

. In addition, by the choices made above and (2.14), we get

I_{2} = \int_{Ω \cap {| u |^{2} ⩾ M}} f (x, | u |^{2}) d x ⩽ ε

. Finally, we claim that

I_{3} = | \int_{Ω} f (x, | u_{n} |^{2}) χ_{(| u_{n} |^{2} < M)} - f (x, | u |^{2}) χ_{(| u |^{2} < M)} d x | \to 0 as n \to \infty .

Indeed, up to a subsequence,

g_{n} (x) := f (x, | u_{n} |^{2}) χ_{(| u_{n} |^{2} < M)} - f (x, | u |^{2}) χ_{(| u |^{2} < M)} \to 0

a.e. in Ω as

n \to \infty

. Moreover,

| g_{n} (x) | ⩽ {\begin{cases} f (x, | u |^{2}), & if | u_{n} |^{2} ⩾ M, \\ C_{M} + f (x, | u |^{2}), & if | u_{n} |^{2} < M, \end{cases}

where

C_{M} = sup {f (t) : | t | < M}

. Consequently, the claim follows from the Lebesgue’s dominated convergence theorem.

3. The variational framework

We say that $u \in H_{A, V}^{1} (R^{2}, C)$ is a weak solution to (1.1), if for any $v \in H_{A, V}^{1} (R^{2}, C)$ ,

m (‖ u ‖_{A, V}^{2}) Re \int_{R^{2}} (\nabla_{A} u \bar{\nabla_{A} v} + V (x) u \bar{v}) d x = Re \int_{R^{2}} B (x) f ({| u |}^{2}) u \bar{v} d x .

Under assumptions (

f_{1}

)–(

f_{3}

), it follows from (2.4), Lemma 2.8 and the continuous embedding

H_{A, V}^{1} (R^{2}, C) ↪ L_{B}^{2} (R^{2}, C)

that the following energy I to (1.1) is well defined and belongs to

C^{1} (H_{A, V}^{1} (R^{2}, C), R)

I (u) := \frac{1}{2} M (‖ u ‖_{A, V}^{2}) - \frac{1}{2} \int_{R^{2}} B (x) F ({| u |}^{2}) d x, u \in H_{A, V}^{1} (R^{2}, C)

(see the proof of Lemma 3.1 for more details). In addition, for

u, v \in H_{A, V}^{1} (R^{2}, C)

, there holds

⟨ I^{'} (u), v ⟩ = m (‖ u ‖_{A, V}^{2}) Re \int_{R^{2}} (\nabla_{A} u \bar{\nabla_{A} v} + V (x) u \bar{v}) d x - Re \int_{R^{2}} B (x) f (| u |^{2}) u \bar{v} d x .

The functional I satisfies the Mountain Pass geometry, namely we have the following two lemmas.

Lemma 3.1.
Under assumptions ( $m_{1}$ ), ( $f_{1}$ )–( $f_{3}$ ), there exist $ρ, σ > 0$ such that $I (u) ⩾ σ$ if $‖ u ‖_{A, V} = ρ$ .
Proof.
Fix $ξ > 0$ and $a > a_{0}$ . If $0 < ρ_{1} < {(\frac{4 π ζ}{a})}^{1 / 2}$ , for $u \in H_{A, V}^{1} (R^{2}, C)$ with $‖ u ‖_{A, V} ⩽ ρ_{1}$ , by (2.4) with $q > 2$ , Lemma 2.8 and the continuous embedding $H_{A, V}^{1} (R^{2}, C) ↪ L_{B}^{2} (R^{2}, C)$ , we get
$\begin{aligned} \int_{R^{2}} B (x) F (| u |^{2}) d x & ⩽ ξ \int_{R^{2}} B (x) | u |^{2} d x + C \int_{R^{2}} B (x) | u |^{q} (e^{a | u |^{2}} - 1) d x \\ ⩽ C_{1} ξ ‖ u ‖_{A, V}^{2} + C \int_{R^{2}} B (x) | u |^{q} (e^{a ‖ u ‖_{A, V}^{2} | \frac{u}{‖ u ‖_{A, V}} |^{2}} - 1) d x \\ ⩽ C_{1} ξ ‖ u ‖_{A, V}^{2} + C_{2} ‖ u ‖_{A, V}^{q} . \end{aligned}$
Combining with ( $m_{1}$ ), we obtain that
$I (u) = \frac{1}{2} M (‖ u ‖_{A, V}^{2}) - \frac{1}{2} \int_{R^{2}} B (x) F ({| u |}^{2}) d x ⩾ ‖ u ‖_{A, V}^{2} (\frac{m_{0}}{2} - ξ C_{1} - C_{2} ‖ u ‖_{A, V}^{q - 2}) .$
Because of $q > 2$ , we can choose $ξ > 0$ and $0 < ρ ⩽ ρ_{1}$ small enough such that $\frac{m_{0}}{2} - ξ C_{1} - C_{2} ρ^{q - 2} > 0$ . Therefore, for any $u \in H_{A, V}^{1} (R^{2}, C)$ with $‖ u ‖_{A, V} = ρ$ , we have
$I (u) ⩾ ρ^{2} (\frac{m_{0}}{2} - ξ C_{1} - C_{2} ρ^{q - 2}) := σ > 0.$

Lemma 3.2.
Under assumptions ( $m_{3}^{}$ ), (* $f_{1}$ )–( $f_{3}$ ), there exists $e \in H_{A, V}^{1} (R^{2}, C)$ such that $I (e) < 0$ and $‖ e ‖_{A, V} > ρ$ with ρ taken in Lemma 3.1.
Proof.
In virtue of the continuity of m and (1.8), there exists $C_{1} > 0$ such that
$\begin{aligned} M (t) ⩽ C_{1} t + a_{1} \frac{t^{2}}{2}, for any t ⩾ 0. \end{aligned}$
(3.1)
On the other hand, in virtue of the continuity of F and (1.9), there exist $C_{2}, C_{3} > 0$ such that for all $s ⩾ 0$ , $F (s) ⩾ C_{2} s^{θ} - C_{3}$ . Now choosing $φ \in C_{c}^{\infty} (R^{2}, C) ∖ {0}$ , there exists $Ω \subset R^{2}$ such that Ω contains $supp φ$ . Integrating ( $B_{1}$ ) and the above inequalities, we obtain that
$\begin{aligned} I (t φ) & = \frac{1}{2} M (t^{2} ‖ φ ‖_{A, V}^{2}) - \frac{1}{2} \int_{R^{2}} B (x) F (t^{2} | φ |^{2}) d x \\ ⩽ C_{1} t^{2} \frac{‖ φ ‖_{A, V}^{2}}{2} + a_{1} t^{4} \frac{‖ φ ‖_{A, V}^{4}}{4} - \frac{C_{2}}{2} t^{2 θ} \int_{R^{2}} | φ |^{2 θ} d x + \frac{C_{3}}{2} | Ω |, for any t ⩾ 0. \end{aligned}$
Since $\int_{R^{2}} | φ |^{2 θ} d x > 0$ and $2 θ > 4$ , we can conclude that $I (t φ) \to - \infty$ as $t \to + \infty$ . Consequently we can take $e = t_{0} φ$ with $t_{0}$ large enough such that $I (e) < 0$ .
Remark 3.3.
For the sake of clarity and completeness, we offer an alternative proof for the preceding lemma. By condition ( $f_{3}$ ), we are aware that $F (s) > 0$ for any $s > 0$ . Considering ( $B_{1}$ ), for any $w \in H_{A, V}^{1} (R^{2}, C) ∖ {0}$ , we have $\int_{R^{2}} B (x) F (| w |^{2}) d x > 0$ . Additionally, for every $s \in R$ , we define
$Φ_{s} (t) := t^{- 2 θ} F (t^{2} s^{2}) - F (s^{2}), t > 0,$
where $θ > 2$ defined in ( $f_{3}$ ). From ( $f_{3}$ ), we know that for $t > 0$ ,
$Φ_{s}^{^{'}} (t) = 2 t^{- 2 θ} t s^{2} f (t^{2} s^{2}) - 2 θ t^{- 2 θ - 1} F (t^{2} s^{2}) = 2 t^{- 2 θ - 1} [t^{2} s^{2} f (t^{2} s^{2}) - θ F (t^{2} s^{2})] ⩾ 0,$
which means that for any $t ⩾ 1$ , $Φ_{s} (t) ⩾ Φ_{s} (1) = 0$ and $F (t^{2} s^{2}) ⩾ t^{2 θ} F (s^{2})$ . Thus, due to (3.1), we obtain
$I (t w) ⩽ C_{1} t^{2} \frac{‖ w ‖_{A, V}^{2}}{2} + a_{1} t^{4} \frac{‖ w ‖_{A, V}^{4}}{4} - \frac{t^{2 θ}}{2} \int_{R^{2}} B (x) F ({| w |}^{2}) d x, t > 1$
and $I (t w) \to - \infty$ as $t \to + \infty$ .

Under assumptions ( $m_{1}$ ), ( $m_{3}^{}$ ), ( $f_{1}$ )–( $f_{3}$ ), in virtue of Lemmas 3.1–3.2, using the Mountain Pass Theorem, there exists a sequence ${u_{n}} \subset H_{A, V}^{1} (R^{2}, C)$ such that
$I (u_{n}) \to c_{V} := inf_{γ \in Γ} max_{t \in [0, 1]} I (γ (t)) and I^{'} (u_{n}) \to 0 as n \to \infty,$
where
$Γ := {γ \in C ([0, 1], H_{A, V}^{1} (R^{2}, C)) : γ (0) = 0 ~and~ I (γ (1)) < 0} .$
From Lemma 3.1, we know that $c_{V} ⩾ δ > 0$ .

For finding the ground state state solution to equation (1.1), we consider the Nehari manifold
$N := {u \in H_{A, V}^{1} (R^{2}, C) ∖ {0} : ⟨ I^{'} (u), u ⟩ = 0} and d_{V} = inf_{u \in N} I (u) .$
In addition, there is the following useful property.
Lemma 3.4.
Under assumptions ( $m_{1}$ ), ( $m_{3}^{}$ ), ( $f_{1}$ )–( $f_{4}$ ), for fixed $u \in H_{A, V}^{1} (R^{2}, C) ∖ {0}$ , there exists a unique $t (u)$ such that $I (t (u) u) = max_{t ⩾ 0} I (t u)$ and $t (u) u \in N$ .
Proof.
Defining $h (t) := I (t u)$ for $t > 0$ , we obtain that
$h^{'} (t) = 0 \Leftrightarrow ⟨ I^{'} (t u), u ⟩ = 0 \Leftrightarrow t u \in N \Leftrightarrow m (t^{2} ‖ u ‖_{A, V}^{2}) t ‖ u ‖_{A, V}^{2} = \int_{R^{2}} B (x) f (t^{2} | u |^{2}) t | u |^{2} d x .$
For any $ξ > 0$ and $a > a_{0}$ , by (2.1) with $q = 3$ and the continuous embedding $H_{A, V}^{1} (R^{2}, C) ↪ L_{B}^{2} (R^{2}, C)$ , it follows from Lemma 2.8 and $a t^{2} ‖ u ‖_{A, V}^{2} ⩽ 4 π ζ$ for t small enough that
$\begin{aligned} \int_{R^{2}} B (x) f (t^{2} | u |^{2}) t | u |^{2} d x & ⩽ ξ t \int_{R^{2}} B (x) | u |^{2} d x + C t^{2} \int_{R^{2}} B (x) | u |^{3} (e^{a t^{2} | u |^{2}} - 1) d x \\ ⩽ C_{1} t ξ ‖ u ‖_{A, V}^{2} + C t^{2} \int_{R^{2}} B (x) | u |^{3} (e^{a t^{2} ‖ u ‖_{A, V}^{2} | \frac{u}{‖ u ‖_{A, V}} |^{2}} - 1) d x \\ ⩽ C_{1} t ξ ‖ u ‖_{A, V}^{2} + C_{2} t^{2} ‖ u ‖_{A, V}^{3} \end{aligned}$
for t small enough. Combining with assumptions ( $m_{1}$ ), for t small enough, we obtain
$h^{'} (t) = m (t^{2} ‖ u ‖_{A, V}^{2}) t ‖ u ‖_{A, V}^{2} - \int_{R^{2}} B (x) f (t^{2} | u |^{2}) t | u |^{2} d x ⩾ t ‖ u ‖_{A, V}^{2} (m_{0} - ξ C_{1} - t C_{2} ‖ u ‖_{A, V}) .$
Then, we can choose $ξ > 0$ small enough such that $h^{'} (t) > 0$ for t small enough.

In virtue of the continuity of m and (1.8), there exists $C_{1} > 0$ such that
$m (t) ⩽ C_{1} + a_{1} t, for any t ⩾ 0.$
Furthermore, according to (1.10) and ( $B_{1}$ ),
$\begin{aligned} h^{'} (t) & = m (t^{2} ‖ u ‖_{A, V}^{2}) t ‖ u ‖_{A, V}^{2} - \int_{R^{2}} B (x) f (t^{2} | u |^{2}) t | u |^{2} d x \\ ⩽ C_{1} t ‖ u ‖_{A, V}^{2} + a_{1} t^{3} ‖ u ‖_{A, V}^{4} - θ C_{S} t^{2 θ - 1} \int_{R^{2}} | u |^{2 θ} d x \end{aligned}$
as t big enough, which means that $h^{'} (t) \to - \infty$ as $t \to + \infty$ . Consequently, there exists at least one critical point for $h (t)$ . Due to ( $m_{3}$ ) and ( $f_{4}$ ), there is a unique critical point $t (u)$ such that $h^{'} (t (u)) = 0$ . Indeed, we suppose that $t_{0}$ is the smallest critical point of $h (t)$ and so
$m (t_{0}^{2} ‖ u ‖_{A, V}^{2}) ‖ u ‖_{A, V}^{2} = \int_{R^{2}} B (x) f (t_{0}^{2} | u |^{2}) | u |^{2} d x .$
Because of ( $m_{3}$ ) and ( $f_{4}$ ), for any $t > t_{0}$ , we have that
$\int_{R^{2}} B (x) f (t_{0}^{2} | u |^{2}) | u |^{2} d x ⩽ \frac{t_{0}^{2}}{t^{2}} \int_{R^{2}} B (x) f (t^{2} | u |^{2}) | u |^{2} d x$
and
$\begin{aligned} m (t_{0}^{2} ‖ u ‖_{A, V}^{2}) ‖ u ‖_{A, V}^{2} & = \frac{m (t_{0}^{2} ‖ u ‖_{A, V}^{2})}{t_{0}^{2} ‖ u ‖_{A, V}^{2}} t_{0}^{2} ‖ u ‖_{A, V}^{4} > \frac{m (t^{2} ‖ u ‖_{A, V}^{2})}{t^{2} ‖ u ‖_{A, V}^{2}} t_{0}^{2} ‖ u ‖_{A, V}^{4} \\ = \frac{t_{0}^{2}}{t^{2}} m (t^{2} ‖ u ‖_{A, V}^{2}) ‖ u ‖_{A, V}^{2}, \end{aligned}$
which means that
$\begin{aligned} h^{'} (t) & = m (t^{2} ‖ u ‖_{A, V}^{2}) t ‖ u ‖_{A, V}^{2} - \int_{R^{2}} B (x) f (t^{2} | u |^{2}) t | u |^{2} d x \\ < \frac{t^{3}}{t_{0}^{2}} (m (t_{0}^{2} ‖ u ‖_{A, V}^{2}) ‖ u ‖_{A, V}^{2} - \int_{R^{2}} B (x) f (t_{0}^{2} | u |^{2}) | u |^{2} d x) = 0, for any t > t_{0} . \end{aligned}$
Consequently, $t (u) = t_{0}$ is the unique point such that $h (t (u)) = I (t (u) u) = max_{t ⩾ 0} I (t u)$ and $t (u) u \in N$ .

Similar to [18], we have the following lemma.
Lemma 3.5.
Under assumptions ( $m_{1}$ ), ( $m_{3}$ ) and ( $f_{1}$ )–( $f_{3}$ ), there exist $C_{1}, C_{2} > 0$ such that
$‖ u ‖_{A, V} ⩾ C_{1} a n d I (u) ⩾ C_{2}, for any u \in N .$

Proof.
By contradiction, if $inf_{u \in N} ‖ u ‖_{A, V} = 0$ , there exists a minimizing sequence ${u_{n}} \subset N$ such that $‖ u_{n} ‖_{A, V} \to 0$ as $n \to \infty$ . Moreover, for any $ξ > 0$ and $a > a_{0}$ , by (2.3) with $q > 2$ , and the continuous embedding $H_{A, V}^{1} (R^{2}, C) ↪ L_{B}^{2} (R^{2}, C)$ , it follows from Lemma 2.8 that
$\begin{aligned} \int_{R^{2}} B (x) f (| u_{n} |^{2}) | u_{n} |^{2} d x & ⩽ ξ \int_{R^{2}} B (x) | u_{n} |^{2} d x + C \int_{R^{2}} B (x) | u_{n} |^{q} (e^{a | u_{n} |^{2}} - 1) d x \\ ⩽ C_{1} ξ ‖ u_{n} ‖_{A, V}^{2} + C \int_{R^{2}} B (x) | u_{n} |^{q} (e^{a ‖ u_{n} ‖_{A, V}^{2} | \frac{u_{n}}{‖ u_{n} ‖_{A, V}} |^{2}} - 1) d x \\ ⩽ C_{1} ξ ‖ u_{n} ‖_{A, V}^{2} + C_{2} ‖ u_{n} ‖_{A, V}^{q} \end{aligned}$
for n big enough. Combining with ( $m_{1}$ ) and ${u_{n}} \subset N$ , for n big enough, we obtain that
$m_{0} ‖ u_{n} ‖_{A, V}^{2} ⩽ m (‖ u_{n} ‖_{A, V}^{2}) ‖ u_{n} ‖_{A, V}^{2} = \int_{R^{2}} B (x) f (| u_{n} |^{2}) | u_{n} |^{2} d x ⩽ C_{1} ξ ‖ u_{n} ‖_{A, V}^{2} + C_{2} ‖ u_{n} ‖_{A, V}^{q} .$
Choosing ξ small enough such that $m_{0} - C_{1} ξ > 0$ , we obtain that
$(m_{0} - C_{1} ξ) ‖ u_{n} ‖_{A, V}^{2} ⩽ C_{2} ‖ u_{n} ‖_{A, V}^{q} as n \to \infty,$
which is a contradiction with $‖ u_{n} ‖_{A, V} \to 0$ as $n \to \infty$ and $q > 2$ .

For any $u \in N$ , according to Lemma 2.10 (i), assumptions ( $f_{3}$ ) and $(m_{1})$ , we have
$\begin{aligned} I (u) & = I (u) - \frac{1}{2 θ} ⟨ I^{'} (u), u ⟩ \\ = \frac{1}{2} M (‖ u ‖_{A, V}^{2}) - \frac{1}{2} \int_{R^{2}} B (x) F (| u |^{2}) d x - \frac{1}{2 θ} m (‖ u ‖_{A, V}^{2}) ‖ u ‖_{A, V}^{2} \\ + \frac{1}{2 θ} \int_{R^{2}} B (x) f (| u |^{2}) | u |^{2} d x \\ ⩾ \frac{1}{2} M (‖ u ‖_{A, V}^{2}) - \frac{1}{4} m (‖ u ‖_{A, V}^{2}) ‖ u ‖_{A, V}^{2} + \frac{θ - 2}{4 θ} m (‖ u ‖_{A, V}^{2}) ‖ u ‖_{A, V}^{2} ⩾ \frac{θ - 2}{4 θ} m_{0} ‖ u ‖_{A, V}^{2} . \end{aligned}$
The second result is a consequence of the first item.

Then, arguing as [34, Theorem 4.2] (or [32]), we have the following Lemma. Lemma 3.6.
Under assumptions ( $m_{1}$ ), ( $m_{3}$ ) and ( $f_{1}$ )–( $f_{4}$ ), we obtain that
$c_{V} = inf_{u \in H_{A}^{1} (R^{2}, C) ∖ {0}} max_{t ⩾ 0} I (t u) = d_{V} > 0.$

Proof.
Similar to Remark 3.3, for any $u \in H_{A}^{1} (R^{2}, C) ∖ {0}$ , we obtain that $I (t u) \to - \infty$ as $t \to + \infty$ . It follows the definition of $c_{V}$ that $inf_{u \in H_{A}^{1} (R^{2}, C) ∖ {0}} max_{t ⩾ 0} I (t u) ⩾ c_{V}$ . Due to Lemma 3.4, there exists a unique $t (u)$ such that $I (t (u) u) = max_{t ⩾ 0} I (t u)$ and $t (u) u \in N$ . Then, taking infimum for $u \in H_{A}^{1} (R^{2}, C) ∖ {0}$ and due to
$I (v) ⩾ inf_{u \in H_{A}^{1} (R^{2}, C) ∖ {0}} I (t (u) u) for all v \in N; I (t (u) u) ⩾ d_{V} for each u \in H_{A}^{1} (R^{2}, C) ∖ {0},$
we can conclude that $d_{V} = inf_{u \in H_{A}^{1} (R^{2}, C) ∖ {0}} I (t (u) u) = inf_{u \in H_{A}^{1} (R^{2}, C) ∖ {0}} max_{t ⩾ 0} I (t u) ⩾ c_{V}$ .

On the other hand, the manifold N separates $H_{A, V}^{1} (R^{2}, C)$ into two components. According to the proof of Lemma 3.4, the component containing 0 also contains a small ball around 0. Moreover, from Lemma 3.1, Lemma 3.5 and $⟨ I^{'} (t u), u ⟩ ⩾ 0$ for all $t \in (0, t (u))$ , we know that $I (u) > 0$ for all u in this component. Consequently, any $γ \in Γ$ has to cross the N and $d_{V} ⩽ c_{V}$ .

4. Minimax estimates

In this section, we will give an upper estimate for the ground state energy of equation (1.1).

Proposition 4.1.
Under assumptions ( $m_{3}^{}$ ), (* $f_{1}$ )–( $f_{3}$ ) and ( $f_{6}$ ), $c_{V} < \frac{1}{2} M (\frac{4 π ζ}{a_{0}})$ .
Proof.
Let $p_{0} > 4$ be given in ( $f_{6}$ ). From Lemma 2.5, there exists a $v_{p_{0}} \in H_{A, V}^{1} (R^{2}, C) ∖ {0}$ such that $‖ v_{p_{0}} ‖_{A}^{2} = S_{p_{0}}$ and $| v_{p_{0}} |_{p_{0}} = 1$ . Due to Remark 3.3, we get that $I (t v_{p_{0}}) \to - \infty ~as~ t \to + \infty$ . In addition, it follows from the definition of $c_{V}$ that $c_{V} ⩽ max_{t > 0} I (t v_{p_{0}})$ . In virtue of assumption ( $f_{6}$ ), we obtain $F (s) ⩾ \frac{2}{p_{0}} C_{p_{0}} s^{\frac{p_{0}}{2}}$ for $s > 0$ . Combining with ( $B_{1}$ ), we get
$I (t v_{p_{0}}) < \frac{1}{2} M (t^{2} ‖ v_{p_{0}} ‖_{A, V}^{2}) - \frac{C_{p_{0}}}{p_{0}} t^{p_{0}} \int_{R^{2}} | v_{p_{0}} |^{p_{0}} d x = \frac{1}{2} M (t^{2} S_{p_{0}}) - \frac{C_{p_{0}}}{p_{0}} t^{p_{0}}, for any t > 0.$
Consequently, according to the definition of $C_{p_{0}}$ , we obtain that
$c_{V} ⩽ \underset{t > 0 max}{I} (t v_{p_{0}}) < max_{t > 0} {\frac{1}{2} M (t^{2} S_{p_{0}}) - \frac{C_{p_{0}}}{p_{0}} t^{p_{0}}} ⩽ \frac{1}{2} M (\frac{4 π ζ}{a_{0}}),$
which completes the proof.

If we replace the condition ( $V_{3}$ ) with a stronger condition ( $V_{4}$ ), we can take $ζ = 1$ . Moreover, similar to [23], for $n ⩾ 2$ and $R > 0$ , we can define the following scaled and truncated Green’s functions sequence (see Moser [30]):
${\tilde{G}}_{n} (x) = \frac{1}{\sqrt{2 π}} {\begin{cases} {(\log n)}^{\frac{1}{2}}, & if | x | ⩽ R / n, \\ \frac{\log (R / | x |)}{{(\log n)}^{1 / 2}}, & if R / n ⩽ | x | ⩽ R, \\ 0, & if | x | ⩾ R . \end{cases}$
Next, we give some properties about the truncated Green’s functions.
Lemma 4.2.
${\tilde{G}}_{n} (x) \in H_{A, V}^{1} (R^{2}, C)$ and $1 ⩽ ‖ {\tilde{G}}_{n} ‖_{A}^{2} ⩽ 1 + \frac{R^{2} M_{R}}{4 \log n}$ , where $M_{R} = M_{A, R} + M_{V, R}$ , $M_{A, R} = | A |_{L^{\infty} (B_{R} (0))}^{2}$ and $M_{V, R} = | V |_{L^{\infty} (B_{R} (0))}$ .
Proof.
Similar to [23], we have
$\begin{aligned} \int_{R^{2}} | \nabla {\tilde{G}}_{n} (x) |^{2} d x & = \frac{1}{2 π \log n} \int_{R / n ⩽ | x | ⩽ R} | x |^{- 2} d x = \frac{1}{\log n} \int_{R / n}^{R} r^{- 1} d r = 1, \\ \int_{R^{2}} | {\tilde{G}}_{n} (x) |^{2} d x & = \frac{\log n}{2 π} \int_{B_{R / n} (0)} d x + \frac{1}{2 π \log n} \int_{R / n ⩽ | x | ⩽ R} \log^{2} (\frac{R}{| x |}) d x ⩽ \frac{R^{2}}{4 \log n} \end{aligned}$
and therefore
$\int_{R^{2}} V (x) | {\tilde{G}}_{n} (x) |^{2} d x = \int_{B_{R} (0)} V (x) | {\tilde{G}}_{n} (x) |^{2} d x ⩽ \frac{R^{2} M_{V, R}}{4 \log n} .$
On the other hand,
$\begin{aligned} \int_{R^{2}} | \nabla_{A} {\tilde{G}}_{n} |^{2} d x & = \int_{R^{2}} {| \frac{\nabla {\tilde{G}}_{n}}{i} - A {\tilde{G}}_{n} |}^{2} d x = \int_{R^{2}} {| \frac{1}{i} (\nabla {\tilde{G}}_{n} - i A {\tilde{G}}_{n}) |}^{2} d x \\ = \int_{R^{2}} (| \nabla {\tilde{G}}_{n} |^{2} + 2 Re (i \nabla {\tilde{G}}_{n} \cdot A {\tilde{G}}_{n}) + | A {\tilde{G}}_{n} |^{2}) d x \\ = \int_{R^{2}} (| \nabla {\tilde{G}}_{n} |^{2} + | A {\tilde{G}}_{n} |^{2}) d x ⩽ 1 + M_{A, R} \int_{R^{2}} | {\tilde{G}}_{n} |^{2} d x ⩽ 1 + \frac{R^{2} M_{A, R}}{4 \log n} . \end{aligned}$
From the above inequalities, we obtain that $1 ⩽ ‖ {\tilde{G}}_{n} ‖_{A}^{2} ⩽ 1 + \frac{R^{2} M_{A, R}}{4 \log n} + \frac{R^{2} M_{V, R}}{4 \log n} ⩽ 1 + \frac{R^{2} M_{R}}{4 \log n}$ .

Due to Proposition 4.1, we obtain that $‖ {\tilde{G}}_{n} ‖_{A}^{2} \to 1$ . In addition, similar to [23, Lemma 4.2], we consider the sequence of functions $G_{n} := \frac{{\tilde{G}}_{n}}{‖ {\tilde{G}}_{n} ‖_{A, V}}$ and have the following technical result.
Lemma 4.3.
The functions sequence ${G_{n}}$ satisfies that
$\underset{n \to \infty}{lim_{_}} \int_{B_{R} (0)} e^{4 π G_{n}^{2}} d x ⩾ π R^{2} e^{\frac{- R^{2} M_{R}}{2}} + π R^{2} .$

Now, for the case that $V (x) ⩾ 0$ for any $x \in R^{2}$ , we can use the previous two lemmas to obtain a similar estimate for Proposition 4.1 with the condition ( $f_{7}$ ) rather than ( $f_{6}$ ). Proposition 4.4.
Under assumptions ( $m_{3}^{}$ ), (* $V_{4}$ ), ( $f_{1}$ )–( $f_{3}$ ) and ( $f_{7}$ ), $c_{V} < \frac{1}{2} M (\frac{4 π}{a_{0}})$ .
Proof.
Due to Remark 3.3, we obtain $I (t G_{n}) \to - \infty$ as $t \to + \infty$ . By the definition of $c_{V}$ , we know $c_{V} ⩽ max_{t > 0} I (t G_{n})$ . Because $G_{n} \in H_{A, V}^{1} (R^{2}, C) ∖ {0}$ and I has the Mountain Pass geometry, there exists $t_{n} > 0$ such that
$\begin{aligned} I (t_{n} G_{n}) = max_{t ⩾ 0} I (t G_{n}) and t_{n} G_{n} \in N, for all n \in N . \end{aligned}$
(4.1)
Next, we claim that there exists at least a $2 ⩽ n \in N$ such that $I (t_{n} G_{n}) < \frac{1}{2} M (\frac{4 π}{a_{0}})$ . By contradiction, if the above inequality does not hold, noting $‖ G_{n} ‖_{A, V} = 1$ , we obtain
$I (t_{n} G_{n}) = \frac{1}{2} M (t_{n}^{2}) - \frac{1}{2} \int_{R^{2}} B (x) F (t_{n}^{2} G_{n}^{2}) d x ⩾ \frac{1}{2} M (\frac{4 π}{a_{0}}) .$
Because both B and F are nonnegative, we have $M (t_{n}^{2}) ⩾ M (\frac{4 π}{a_{0}})$ . Moreover, since m is positive, we get M is an non-decreasing function, which implies that
$\begin{aligned} t_{n}^{2} ⩾ \frac{4 π}{a_{0}} . \end{aligned}$
(4.2)

On the other hand, from $t_{n} G_{n} \in N$ , ( $B_{1}$ ), $f ⩾ 0$ and the definition of $G_{n}$ , we have
$\begin{aligned} m (t_{n}^{2}) t_{n}^{2} = \int_{B_{R} (0)} B (x) f (t_{n}^{2} G_{n}^{2}) t_{n}^{2} G_{n}^{2} d x ⩾ \int_{B_{R / n} (0)} f (\frac{t_{n}^{2} \log n}{2 π ‖ {\tilde{G}}_{n} ‖_{A, V}^{2}}) \frac{t_{n}^{2} \log n}{2 π ‖ {\tilde{G}}_{n} ‖_{A, V}^{2}} d x . \end{aligned}$
(4.3)
It follows from $‖ {\tilde{G}}_{n} ‖_{A, V} \to 1$ as $n \to \infty$ and $t_{n}^{2} ⩾ \frac{4 π}{a_{0}} > 0$ that
$\begin{aligned} \frac{t_{n}^{2} \log n}{2 π ‖ {\tilde{G}}_{n} ‖_{A, V}^{2}} \to + \infty as n \to \infty . \end{aligned}$
(4.4)
In addition, fixed $0 < δ < λ$ , from ( $f_{7}$ ), there exists $S_{δ} > 0$ such that
$\begin{aligned} f (s) s ⩾ (λ - δ) e^{a_{0} s}, for all s ⩾ S_{δ} . \end{aligned}$
(4.5)
If ${t_{n}}$ is unbounded, there exists a subsequence of ${t_{n}}$ relabeled as ${t_{n}}$ such that $t_{n} \to + \infty$ as $n \to \infty$ . Then, combining (1.8) and (4.3)–(4.5), for n large enough, we conclude that
$a_{1} t_{n}^{4} ⩾ m (t_{n}^{2}) t_{n}^{2} ⩾ \int_{B_{R / n} (0)} (λ - δ) e^{\frac{a_{0} t_{n}^{2} \log n}{2 π ‖ {\tilde{G}}_{n} ‖_{A, V}^{2}}} d x = π R^{2} (λ - δ) e^{(\frac{a_{0} t_{n}^{2}}{2 π ‖ {\tilde{G}}_{n} ‖_{A, V}^{2}} - 2) \log n},$
which is a contradiction. Consequently, the sequence ${t_{n}}$ is bounded in $R$ and there exists a subsequence of ${t_{n}}$ relabeled as ${t_{n}}$ and $t_{0} ⩾ \frac{4 π}{a_{0}}$ such that $t_{n} \to t_{0}$ . Integrating the above inequality, there exists $C > 0$ such that $C ⩾ m (t_{n}^{2}) t_{n}^{2} ⩾ π R^{2} (λ - δ) e^{(\frac{a_{0} t_{n}^{2}}{2 π ‖ {\tilde{G}}_{n} ‖_{A, V}^{2}} - 2) \log n}$ for n large enough, which implies that $lim_{n \to \infty} \frac{a_{0} t_{n}^{2}}{2 π ‖ {\tilde{G}}_{n} ‖_{A, V}^{2}} - 2 = 2 (\frac{a_{0}}{4 π} t_{0}^{2} - 1) ⩽ 0$ . Combining with (4.2), we obtain that
$\begin{aligned} lim_{n \to \infty} t_{n}^{2} = t_{0}^{2} = \frac{4 π}{a_{0}} . \end{aligned}$
(4.6)

Next, for each $n ⩾ 2$ , we define the sets
$D_{n, δ} := {x \in B_{R} (0) : t_{n}^{2} G_{n}^{2} ⩾ S_{δ}} and E_{n, δ} := B_{R} (0) ∖ D_{n, δ} .$
By assumption ( $B_{1}$ ) and (4.3)–(4.5), we have
$\begin{aligned} m (t_{n}^{2}) t_{n}^{2} & = \int_{B_{R} (0)} B (x) f (t_{n}^{2} G_{n}^{2}) t_{n}^{2} G_{n}^{2} d x ⩾ \int_{D_{n, δ}} f (t_{n}^{2} G_{n}^{2}) t_{n}^{2} G_{n}^{2} d x + \int_{E_{n, δ}} f (t_{n}^{2} G_{n}^{2}) t_{n}^{2} G_{n}^{2} d x \\ ⩾ (λ - δ) (\int_{B_{R} (0)} e^{a_{0} t_{n}^{2} G_{n}^{2}} d x - \int_{E_{n, δ}} e^{a_{0} t_{n}^{2} G_{n}^{2}} d x) + \int_{E_{n, δ}} f (t_{n}^{2} G_{n}^{2}) t_{n}^{2} G_{n}^{2} d x . \end{aligned}$
(4.7)
Since $G_{n} (x) \to 0$ a.e. in $B_{R} (0)$ as $n \to \infty$ , we obtain $χ_{E_{n, δ}} (x) \to 1$ a.e. in $B_{R} (0)$ as $n \to \infty$ , where $χ_{E_{n, δ}}$ denotes the characteristic function of $E_{n, δ}$ . Noting $t_{n}^{2} G_{n}^{2} < S_{δ}$ in $E_{n, δ}$ , we know
$e^{a_{0} t_{n}^{2} G_{n}^{2}} ⩽ e^{a_{0} S_{δ}} and f (t_{n}^{2} G_{n}^{2}) t_{n}^{2} G_{n}^{2} ⩽ S_{δ} sup_{| t | ⩽ S_{δ}} f (t) in E_{n, δ} .$
Then, it follows from the Lebesgue’s dominated convergence theorem that
$\int_{E_{n, δ}} e^{a_{0} t_{n}^{2} G_{n}^{2}} d x \to π R^{2} and \int_{E_{n, δ}} f (t_{n}^{2} G_{n}^{2}) t_{n}^{2} G_{n}^{2} d x \to 0 as n \to \infty .$
Hence, combining (4.2), (4.6)–(4.7), Lemma 4.3 and the continuity of $m (t)$ , we have
$\begin{aligned} m (\frac{4 π}{a_{0}}) \frac{4 π}{a_{0}} & = \underset{n \to \infty}{lim_{_}} m (t_{n}^{2}) t_{n}^{2} ⩾ (λ - δ) (\underset{n \to \infty}{lim_{_}} \int_{B_{R} (0)} e^{a_{0} t_{n}^{2} G_{n}^{2}} d x - π R^{2}) \\ ⩾ (λ - δ) (\underset{n \to \infty}{lim_{_}} \int_{B_{R} (0)} e^{4 π G_{n}^{2}} d x - π R^{2}) ⩾ (λ - δ) π R^{2} e^{\frac{- R^{2} M_{R}}{2}} . \end{aligned}$
(4.8)
Because $0 < δ < λ$ is arbitrary, we can let $δ \to 0^{+}$ in (4.8) and get $λ ⩽ \frac{4}{a_{0}} m (\frac{4 π}{a_{0}}) R^{- 2} e^{\frac{R^{2} M_{R}}{2}}$ . At the end, if we take the infimum for $R > 0$ , we will get a contradiction with ( $f_{7}$ ), which concludes the proof.

5. Proof of Theorems 1.1-1.2

In this section, we start to prove our main results. Firstly, we give the following compact conclusion.

Proposition 5.1.
Under assumptions ( $m_{1}$ ), ( $m_{3}$ ), ( $f_{1}$ )–( $f_{3}$ ), ( $f_{5}$ ). If ${u_{n}} \subset H_{A, V}^{1} (R^{2}, C)$ is a $(P S)_{c}$ sequence for the energy functional I with $c \in R$ , then ${u_{n}}$ is bounded in $H_{A, V}^{1} (R^{2}, C)$ . Moreover, up to a subsequence, there exists $u \in H_{A, V}^{1} (R^{2}, C)$ such that (i)
$\int_{Ω} B (x) f (| u_{n} |^{2}) d x \to \int_{Ω} B (x) f (| u |^{2}) d x$ as $n \to \infty$ for any bounded domain $Ω \subset R^{2}$ ;
(ii)
$\int_{R^{2}} B (x) F (| u_{n} |^{2}) d x \to \int_{R^{2}} B (x) F (| u |^{2}) d x$ as $n \to \infty$ .

Proof.
Notice ${u_{n}} \subset H_{A, V}^{1} (R^{2}, C)$ is a $(P S)_{c}$ sequence for the energy functional I, that is
$I (u_{n}) \to c and I^{'} (u_{n}) \to 0 as n \to \infty .$
Combining with ( $m_{1}$ ), ( $f_{3}$ ) and Lemma 2.10 (i), we have
$\begin{aligned} c + o (1) + ‖ u ‖_{A, V} & ⩾ I (u_{n}) - \frac{1}{2 θ} ⟨ I^{'} (u_{n}), u_{n} ⟩ \\ = \frac{1}{2} M (‖ u_{n} ‖_{A, V}^{2}) - \frac{1}{2} \int_{R^{2}} B (x) F (| u_{n} |^{2}) d x \\ - \frac{1}{2 θ} m (‖ u_{n} ‖_{A, V}^{2}) ‖ u_{n} ‖_{A, V}^{2} + \frac{1}{2 θ} \int_{R^{2}} B (x) f (| u_{n} |^{2}) | u_{n} |^{2} d x \\ ⩾ \frac{1}{2} M (‖ u_{n} ‖_{A, V}^{2}) - \frac{1}{4} m (‖ u_{n} ‖_{A, V}^{2}) ‖ u_{n} ‖_{A, V}^{2} + \frac{θ - 2}{4 θ} m (‖ u_{n} ‖_{A, V}^{2}) ‖ u_{n} ‖_{A, V}^{2} \\ ⩾ \frac{θ - 2}{4 θ} m_{0} ‖ u_{n} ‖_{A, V}^{2} as n \to \infty \end{aligned}$
with $θ > 2$ given in assumption $(f_{3})$ , which means the sequence ${u_{n}}$ is bounded in $H_{A, V}^{1} (R^{2}, C)$ and there exists a subsequence of ${u_{n}}$ relabeled as ${u_{n}}$ and $u \in H_{A, V}^{1} (R^{2}, C)$ such that $u_{n} ⇀ u$ in $H_{A, V}^{1} (R^{2}, C)$ as $n \to \infty$ .

Let $Ω \subset R^{2}$ be any bounded domain. Up to a subsequence, it follows from Lemmas 2.3–2.4 that
$u_{n} \to u in L^{1} (Ω, C) and u_{n} \to u in L_{B}^{2} (R^{2}, C) ~as~ n \to \infty .$
In addition, since ${u_{n}}$ is bounded in $H_{A, V}^{1} (R^{2}, C)$ , we have $⟨ I^{'} (u_{n}), u_{n} ⟩ \to 0$ as $n \to \infty$ and so
$\begin{aligned} \int_{Ω} f (| u_{n} |^{2}) | u_{n} |^{2} d x & ⩽ \int_{R^{2}} B (x) f (| u_{n} |^{2}) | u_{n} |^{2} d x \\ = m (‖ u_{n} ‖_{A, V}^{2}) ‖ u_{n} ‖_{A, V}^{2} - ⟨ I^{'} (u_{n}), u_{n} ⟩ ⩽ C . \end{aligned}$
(5.1)
From (2.1) with $q = 2$ , for any $ξ > 0$ and $a > a_{0}$ , there exists a constant $C = C (a, ξ) > 0$ such that
$f (| u_{n}^{2} |) ⩽ ξ + C (e^{a | u_{n} |^{2}} - 1) and f (| u^{2} |) ⩽ ξ + C (e^{a | u^{2} |} - 1) .$
Moreover, it follows from Lemma 2.6 that
$\int_{Ω} f (| u_{n} |^{2}) d x ⩽ ξ | Ω | + C \int_{Ω} (e^{a | u_{n}^{2} |} - 1) d x < + \infty, for all v \in N; \int_{Ω} f (| u |^{2}) d x < + \infty,$
which imply that $f (| u_{n} |^{2}), f (| u |^{2}) \in L^{1} (Ω, R)$ . By Lemma 2.11 and (5.1), we conclude that $f (| u_{n} |^{2}) \to f (| u |^{2})$ in $L^{1} (Ω, R)$ as $n \to \infty$ . And so
$| \int_{Ω} B (x) (f (| u_{n} |^{2}) d x - f (| u |^{2})) d x | ⩽ | B |_{L^{\infty} (Ω)} \int_{Ω} | f (| u_{n} |^{2}) d x - f (| u |^{2}) | d x \to 0$
as $n \to \infty$ , which proves (i).

For the item (ii), taking any fixed $r > 0$ , due to (i) and [34, Lemma A.1], there exists $g_{r} \in L^{1} (B_{r} (0), R)$ such that $B (x) f (| u_{n} |^{2}) ⩽ g_{r} (x)$ a.e. in $B_{r} (0)$ . Then, using ( $f_{5}$ ), we get
$\begin{aligned} B (x) F (| u_{n} |^{2}) & ⩽ | B |_{L^{\infty} (B_{r} (0))} max_{0 ⩽ s^{2} ⩽ S_{0}} F (s^{2}) + K_{0} B (x) f (| u_{n} |^{2}) \\ ⩽ | B |_{L^{\infty} (B_{r} (0))} F (S_{0}) + K_{0} g_{r} (x) \end{aligned}$
for a.e. $x \in B_{r} (0)$ . Up to a subsequence, we have $u_{n} (x) \to u (x)$ a.e. in $B_{r} (0)$ as $n \to \infty$ , by the continuity of F and the Lebesgue’s dominated convergence theorem we obatin
$\int_{B_{r} (0)} B (x) F (| u_{n} |^{2}) d x \to \int_{B_{r} (0)} B (x) F (| u |^{2}) d x as n \to \infty .$
Thus, for proving the item (ii), it is enough to verify that, for fixed $δ > 0$ , there exists $r > 0$ such that
$\begin{aligned} \int_{R^{2} ∖ B_{r} (0)} B (x) F (| u_{n} |^{2}) d x < δ, for all n \in N; \int_{R^{2} ∖ B_{r} (0)} B (x) F (| u |^{2}) d x < δ . \end{aligned}$
(5.2)
It follows from $B (x) F (| u |^{2}) \in L^{1} (R^{2}, R)$ that the second inequality holds for $r > 0$ large enough. For the first inequality, we can use ( $f_{1}$ ) and ( $f_{5}$ ) to obtain there exist $C_{1}, C_{2} > 0$ such that
$F (s) ⩽ C_{1} s + C_{2} f (s), for all s ⩾ 0.$
Furthermore, fixed $K > 0$ , according to the continuous embedding $H_{A, V}^{1} (R^{2}, C) ↪ L_{B}^{3} (R^{2}, C)$ , the boundedness of ${u_{n}}$ in $H_{A, V}^{1} (R^{2}, C)$ and (5.1), we can take K large enough such that
$\begin{aligned} \int_{{| u_{n} | > K} \cap (R^{2} ∖ B_{r} (0))} B (x) F (| u_{n} |^{2}) d x & ⩽ C_{1} \int_{{| u_{n} | > K} \cap (R^{2} ∖ B_{r} (0))} B (x) | u_{n} |^{2} d x + C_{2} \int_{{| u_{n} | > K} \cap (R^{2} ∖ B_{r} (0))} B (x) f (| u_{n} |^{2}) d x \\ ⩽ \frac{C_{1}}{K} \int_{R^{2}} B (x) | u_{n} |^{3} d x + \frac{C_{2}}{K^{2}} \int_{R^{2}} B (x) f (| u_{n} |^{2}) | u_{n} |^{2} d x \\ ⩽ \frac{C_{3}}{K} + \frac{C_{4}}{K^{2}} ⩽ \frac{δ}{2}, for all n \in N . \end{aligned}$
(5.3)
On the other hand, from the inequality (2.4) with $q = 2$ , for any $a > a_{0}$ and $s \in R$ with $| s | ⩽ K$ , we get $F (s^{2}) ⩽ C_{5} (s^{2} + s^{2} (e^{a s^{2}} - 1)) ⩽ C_{5} e^{a K^{2}} s^{2} ⩽ C_{6} s^{2}$ , where $C_{6} = C_{6} (a, K) > 0$ is constant. Furthermore, because $u_{n} \to u$ in $L_{B}^{2} (R^{2}, C)$ , there exists $g \in L^{1} (R^{2}, R)$ such that $B (x) | u_{n} |^{2} ⩽ g (x)$ a.e. in $R^{2}$ . So, by taking $r > 0$ large enough, we obtain
$\begin{aligned} \int_{{| u_{n} | ⩽ K} \cap (R^{2} ∖ B_{r} (0))} B (x) F (| u_{n} |^{2}) d x & ⩽ C_{6} \int_{{| u_{n} | ⩽ K} \cap (R^{2} ∖ B_{r} (0))} B (x) | u_{n} |^{2} d x \\ ⩽ C_{6} \int_{R^{2} ∖ B_{r} (0)} g (x) d x ⩽ \frac{δ}{2}, for all n \in N . \end{aligned}$
Combining (5.3) and the above estimates, we conclude (5.2), which completes the proof of the item (ii).

Now, we are in position to prove Theorems 1.1–1.2. Proof Proof of Theorem 1.1.

In virtue of Lemmas 3.1–3.2, using the Mountain Pass Theorem, there exists a sequence ${u_{n}} \subset H_{A, V}^{1} (R^{2}, C)$ such that

\begin{aligned} I (u_{n}) \to c_{V} and I^{'} (u_{n}) \to 0 as n \to \infty . \end{aligned}

(5.4)

Due to Proposition 5.1, the sequence

{u_{n}}

is bounded in

H_{A, V}^{1} (R^{2}, C)

, and from Lemma 2.4, there exists a subsequence of

{u_{n}}

relabeled as

{u_{n}}

and

u_{0} \in H_{A, V}^{1} (R^{2}, C)

such that

\begin{aligned} u_{n} ⇀ u_{0} in H_{A, V}^{1} (R^{2}, C), u_{n} \to u_{0} in L_{B}^{2} (R^{2}, C) ~as~ n \to \infty . \end{aligned}

(5.5)

Next, we will use the method of contradiction to prove that

\begin{aligned} I (u_{0}) ⩾ 0. \end{aligned}

(5.6)

Otherwise, if

I (u_{0}) < 0

, then

u_{0} \neq 0

. Similar to Lemma 3.4, we define

h_{0} (t) := I (t u_{0}), t \in R_{0}^{+} = [0, + \infty) .

It is clear that

h_{0} (0) = 0

and

h_{0} (1) < 0

. Similar to the proof of Lemmas 3.1–3.2, we know that

h_{0} (t) > 0

for t small enough and

h_{0} (t) \to - \infty

t \to + \infty

. Consequently, there exists

t_{0} \in (0, 1)

such that

h^{'} (t_{0}) = ⟨ I^{'} (t_{0} u_{0}), u_{0} ⟩ = 0 and t_{0} u_{0} \in N .

Indeed, according to Lemma 3.4, we know that

t_{0}

is the unique critical point and

h_{0} (t_{0}) = I (t_{0} u_{0}) = max_{t ⩾ 0} I (t u_{0})

. Thus, by the definition of

c_{V}

B (x) ⩾ 1

and Lemma 2.10 (i)–(ii), it follows from the weakly lower semi-continuity of

‖ \cdot ‖_{A, V}

and Fatou’s lemma that

\begin{aligned} c_{V} & ⩽ h_{0} (t_{0}) = h_{0} (t_{0}) - \frac{1}{4} h^{'} (t_{0}) t_{0} \\ = \frac{1}{2} M (‖ t_{0} u_{0} ‖_{A, V}^{2}) - \frac{1}{4} m (‖ t_{0} u_{0} ‖_{A, V}^{2}) ‖ t_{0} u_{0} ‖_{A, V}^{2} \\ + \frac{1}{4} \int_{R^{2}} B (x) (f (| t_{0} u_{0} |^{2}) | t_{0} u_{0} |^{2} - 2 F (| t_{0} u_{0} |^{2})) d x \\ < \frac{1}{2} M (‖ u_{0} ‖_{A, V}^{2}) - \frac{1}{4} m (‖ u_{0} ‖_{A, V}^{2}) ‖ u_{0} ‖_{A, V}^{2} + \frac{1}{4} \int_{R^{2}} B (x) (f (| u_{0} |^{2}) | u_{0} |^{2} - 2 F (| u_{0} |^{2})) d x \\ ⩽ \frac{1}{2} M (\underset{n \to \infty}{lim_{_}} ‖ u_{n} ‖_{A, V}^{2}) - \frac{1}{4} m (\underset{n \to \infty}{lim_{_}} ‖ u_{n} ‖_{A, V}^{2}) \underset{n \to \infty}{lim_{_}} ‖ u_{n} ‖_{A, V}^{2} \\ + \underset{n \to \infty}{lim_{_}} \frac{1}{4} \int_{R^{2}} B (x) (f (| u_{n} |^{2}) | u_{n} |^{2} - 2 F (| u_{n} |^{2})) d x \\ ⩽ \underset{n \to \infty}{lim_{_}} (I (u_{n}) - \frac{1}{4} I^{'} (u_{n}) u_{n}) = c_{V}, \end{aligned}

which is a contradiction.

Finally, we will claim that $I (u_{0}) = c_{V}$ and $I^{'} (u_{0}) = 0$ . Due to the boundedness of ${u_{n}}$ in $H_{A, V}^{1} (R^{2}, C)$ , there exists $ρ_{0} ⩾ 0$ such that $‖ u_{n} ‖_{A, V} \to ρ_{0} ~as~ n \to \infty$ . From the weakly lower semi-continuity of $‖ \cdot ‖_{A, V}$ and (5.5), we obatin $‖ u_{0} ‖_{A, V} ⩽ \underset{n \to \infty}{lim_{_}} ‖ u_{n} ‖_{A, V} = ρ_{0}$ . If $‖ u_{0} ‖_{A, V} = ρ_{0}$ , then the proof is completed. Otherwise, if $‖ u_{0} ‖_{A, V} < ρ_{0}$ , we define

v_{n} := \frac{u_{n}}{‖ u_{n} ‖_{A, V}} and v_{0} := \frac{u_{0}}{ρ_{0}} .

Up to a subsequence, we have that

v_{n} ⇀ v_{0}

H_{A, V}^{1} (R^{2}, C)

n \to \infty

and

‖ v_{0} ‖_{A, V} < 1

. Consequently, from Lemma 2.9, we obtain

\begin{aligned} sup_{n \in N} \int_{R^{2}} B (x) | u_{n} - u_{0} |^{q} (e^{p | v_{n} |^{2}} - 1) d x < + \infty, for all q > 0 ~and~ 0 < p < \frac{4 π ζ}{1 - ‖ v_{0} ‖_{A, V}^{2}} . \end{aligned}

(5.7)

Moreover, from (5.4), (5.6), Proposition 4.1, Proposition 5.1 (ii) and assumption (

m_{2}

), we obtain

\begin{aligned} M (ρ_{0}^{2}) & = lim_{n \to \infty} M (‖ u_{n} ‖_{A, V}^{2}) = lim_{n \to \infty} (2 I (u_{n}) + \int_{R^{2}} B (x) F (| u_{n} |^{2}) d x) \\ = 2 c_{V} + \int_{R^{2}} B (x) F (| u_{0} |^{2}) d x = 2 c_{V} + M (‖ u_{0} ‖_{A, V}^{2}) - 2 I (u_{0}) \\ < M (\frac{4 π ζ}{a_{0}}) + M (‖ u_{0} ‖_{A, V}^{2}) ⩽ M (\frac{4 π ζ}{a_{0}} + ‖ u_{0} ‖_{A, V}^{2}) . \end{aligned}

It follows from M is monotonously increasing that

ρ_{0}^{2} < \frac{4 π ζ}{a_{0}} + ‖ u_{0} ‖_{A, V}^{2}

. Hence, due to

ρ_{0}^{2} = \frac{ρ_{0}^{2} - ‖ u_{0} ‖_{A, V}^{2}}{1 - ‖ v_{0} ‖_{A, V}^{2}}

, we obtain that

lim_{n \to \infty} a_{0} ‖ u_{n} ‖_{A, V}^{2} = a_{0} ρ_{0}^{2} = \frac{a_{0} (ρ_{0}^{2} - ‖ u_{0} ‖_{A, V}^{2})}{1 - ‖ v_{0} ‖_{A, V}^{2}} < \frac{4 π ζ}{1 - ‖ v_{0} ‖_{A, V}^{2}}

. Then, there is

η > 0

such that

a_{0} ‖ u_{n} ‖_{A, V}^{2} < η < \frac{4 π ζ}{1 - ‖ v_{0} ‖_{A, V}^{2}}

for n large enough. Furthermore, for n large enough, we can take

a > a_{0}

close to

a_{0}

and

r > 1

close to 1 such that

r a ‖ u_{n} ‖_{A, V}^{2} < η < \frac{4 π ζ}{1 - ‖ v_{0} ‖_{A, V}^{2}}

and by (5.7),

\begin{aligned} \int_{R^{2}} B (x) | u_{n} - u_{0} |^{2 - r} (e^{r a | u_{n} |^{2}} - 1) d x & = \int_{R^{2}} B (x) | u_{n} - u_{0} |^{2 - r} (e^{r a ‖ u_{n} ‖_{A, V}^{2} | v_{n} |^{2}} - 1) d x \\ ⩽ \int_{R^{2}} B (x) | u_{n} - u_{0} |^{2 - r} (e^{η | v_{n} |^{2}} - 1) d x \\ ⩽ sup_{n \in N} \int_{R^{2}} B (x) | u_{n} - u_{0} |^{2 - r} (e^{η | v_{n} |^{2}} - 1) d x < + \infty . \end{aligned}

(5.8)

Consequently, from (2.3) with

q = 1

, (2.7), (5.5) and (5.8), according to Hölder’s inequality and the continuous embedding

H_{A, V}^{1} (R^{2}, C) ↪ L_{B}^{2} (R^{2}, C)

, we conclude that

\begin{aligned} | Re \int_{R^{2}} B (x) f (| u_{n} |^{2}) u_{n} \bar{(u_{n} - u_{0})} d x | \\ ⩽ \int_{R^{2}} B (x) f (| u_{n} |^{2}) | u_{n} |^{2} \frac{| u_{n} - u_{0} |}{| u_{n} |} d x \\ ⩽ C_{2} \int_{R^{2}} B (x) | u_{n} | | u_{n} - u_{0} | d x + C_{3} \int_{R^{2}} B (x) | u_{n} - u_{0} | (e^{a | u_{n} |^{2}} - 1) d x \\ = C_{2} \int_{R^{2}} \sqrt{B (x)} | u_{n} | \sqrt{B (x)} | u_{n} - u_{0} | d x \\ + C_{3} \int_{R^{2}} (B (x) | u_{n} - u_{0} |^{2})^{\frac{r - 1}{r}} (B (x) | u_{n} - u_{0} |^{2 - r})^{\frac{1}{r}} (e^{a | u_{n} |^{2}} - 1) d x \\ ⩽ C_{4} ‖ u_{n} ‖_{A, V} | u_{n} - u_{0} |_{L_{B}^{2}} + C_{3} | u_{n} - u_{0} |_{L_{B}^{2}}^{2 (r - 1) / r} {(\int_{R^{2}} B (x) | u_{n} - u_{0} |^{2 - r} (e^{r a | u_{n} |^{2}} - 1) d x)}^{\frac{1}{r}} \\ ⩽ C_{5} | u_{n} - u_{0} |_{L_{B}^{2}} + C_{6} | u_{n} - u_{0} |_{L_{B}^{2}}^{2 (r - 1) / r} \to 0 as n \to \infty . \end{aligned}

Combining with

⟨ I^{'} (u_{n}), u_{n} - u_{0} ⟩ \to 0

‖ u_{n} ‖_{A, V} \to ρ_{0}

and

u_{n} ⇀ u_{0}

H_{A, V}^{1} (R^{2}, C)

, we obtain

\begin{aligned} 0 & = lim_{n \to \infty} (⟨ I^{'} (u_{n}), u_{n} - u_{0} ⟩ + Re \int_{R^{2}} B (x) f (| u_{n} |^{2}) u_{n} \bar{(u_{n} - u_{0})} d x) \\ = lim_{n \to \infty} m (‖ u_{n} ‖_{A, V}^{2}) ⟨ u_{n}, u_{n} - u_{0} ⟩_{A, V} = m (ρ_{0}^{2}) (ρ_{0}^{2} - ‖ u_{0} ‖_{A, V}^{2}) > 0, \end{aligned}

which is a contradiction. Thus, we get that

lim_{n \to \infty} ‖ u_{n} ‖_{A, V} = ‖ u_{0} ‖_{A, V} = ρ_{0}

and so

u_{n} \to u_{0} ~in~ H_{A, V}^{1} (R^{2}, C)

. Combining

I \in C^{1} (H_{A, V}^{1} (R^{2}, C), R)

and (5.4), we have

I (u_{0}) = c_{V} > 0

and

I^{'} (u_{0}) = 0

. Recalling that Lemma 3.6,

u_{0}

is a ground state solution.

Proof Proof of Theorem 1.2.

In this case, $ζ = 1$ . Using Proposition 4.4 instead of Proposition 4.1, similar to the proof of the Theorem 1.1, we can draw the conclusion.

6. The proof of Theorems 1.3-1.4

In the final section, we consider the local case. From now on, we always assume that $m \equiv 1$ , which means equation (1.1) becomes the following Schrödinger equation (1.4)

{(\frac{1}{i} \nabla - A (x))}^{2} u + V (x) u = B (x) f (| u |^{2}) u in R^{2} .

We say that

u \in H_{A, V}^{1} (R^{2}, C)

is a weak solution to (1.4), if for any

v \in H_{A, V}^{1} (R^{2}, C)

Re \int_{R^{2}} (\nabla_{A} u \bar{\nabla_{A} v} + V (x) u \bar{v}) d x = Re \int_{R^{2}} B (x) f (| u |^{2}) u \bar{v} d x .

Under assumptions (

f_{1}

)–(

{\tilde{f}}_{3}

), it follows from (2.4), Lemma 2.8 and the continuous embedding

H_{A, V}^{1} (R^{2}, C) ↪ L_{B}^{2} (R^{2}, C)

that the following energy J is well defined and belongs to

C^{1} (H_{A, V}^{1} (R^{2}, C), R)

J (u) := \frac{1}{2} ‖ u ‖_{A, V}^{2} - \frac{1}{2} \int_{R^{2}} B (x) F (| u |^{2}) d x, u \in H_{A, V}^{1} (R^{2}, C) .

Moreover, for

u, v \in H_{A, V}^{1} (R^{2}, C)

, there holds

⟨ J^{'} (u), v ⟩ = Re \int_{R^{2}} (\nabla_{A} u \bar{\nabla_{A} v} + V (x) u \bar{v}) d x - Re \int_{R^{2}} B (x) f (| u |^{2}) u \bar{v} d x .

Similar to Lemmas 3.1–3.2, we can prove that J satisfies the Mountain Pass geometry and there exists a sequence

{u_{n}} \subset H_{A, V}^{1} (R^{2}, C)

such that

J (u_{n}) \to c_{V}^{*} := inf_{γ \in Λ} max_{t \in [0, 1]} J (γ (t)) > 0 and J^{'} (u_{n}) \to 0 as n \to \infty,

where

Λ := {γ \in C ([0, 1], H_{A, V}^{1} (R^{2}, C)) : γ (0) = 0 ~and~ J (γ (1)) < 0} .

For finding the ground state solution to equation (1.4), we consider the Nehari manifold

N_{J} = {u \in H_{A, V}^{1} (R^{2}, C) ∖ {0} : ⟨ J^{'} (u), u ⟩ = 0} and d_{V}^{*} = inf_{u \in N_{J}} J (u) .

Using (

f_{1}

)–(

{\tilde{f}}_{3}

) and (

{\tilde{f}}_{4}

), for each fixed

u \in H_{A, V}^{1} (R^{2}, C) ∖ {0}

, we define

h_{J} (t) := J (t u)

and have

h_{J}^{^{'}} (t) = 0 \Leftrightarrow ⟨ J^{'} (t u), u ⟩ = 0 \Leftrightarrow t u \in N_{J} \Leftrightarrow ‖ u ‖_{A, V}^{2} = \int_{R^{2}} B (x) f (t^{2} | u |^{2}) | u |^{2} d x .

Similar to the proof of Lemma 3.4 and Lemma 3.6, due to

f (s)

is non-decreasing in

(0, \infty)

, we obtain that there exists

t_{J} (u)

such that

J (t_{J} (u) u) max_{t ⩾ 0} J (t u) and t_{J} (u) u \in N_{J}; c_{V}^{*} = inf_{u \in H_{A}^{1} (R^{2}, C) ∖ {0}} max_{t ⩾ 0} J (t u) = d_{V}^{*} > 0.

Similar to Proposition 4.1 and Proposition 4.4, we have the following estimates for

c_{V}^{*}

Proposition 6.1.
The following statements are true. (i)
Under assumptions ( $f_{1}$ )–( ${\tilde{f}}_{3}$ ) and ( ${\tilde{f}}_{6}$ ), $c_{V}^{} < \frac{2 π ζ}{a_{0}}$ .
(ii)
Under assumptions (* $V_{4}$ ), ( $f_{1}$ )–( ${\tilde{f}}_{3}$ ) and ( ${\tilde{f}}_{7}$ ), $c_{V}^{} < \frac{2 π}{a_{0}}$ .

And similar to Proposition 5.1, we have
Proposition 6.2.
Under assumptions (* $f_{1}$ )–( ${\tilde{f}}_{3}$ ) and ( $f_{5}$ ). If ${u_{n}} \subset H_{A, V}^{1} (R^{2}, C)$ is a $(P S)_{c}$ sequence for the energy functional J with $c \in R$ , then ${u_{n}}$ is bounded in $H_{A, V}^{1} (R^{2}, C)$ . Moreover, up to a subsequence, there exists $u \in H_{A, V}^{1} (R^{2}, C)$ such that (i)
$\int_{Ω} B (x) f (| u_{n} |^{2}) d x \to \int_{Ω} B (x) f (| u |^{2}) d x$ as $n \to \infty$ for any bounded domain $Ω \subset R^{2}$ ;
(ii)
$\int_{R^{2}} B (x) F (| u_{n} |^{2}) d x \to \int_{R^{2}} B (x) F (| u |^{2}) d x$ as $n \to \infty$ .

Proof.
Notice ${u_{n}} \subset H_{A, V}^{1} (R^{2}, C)$ is a $(P S)_{c}$ sequence for the energy functional J, that is
$J (u_{n}) \to c and J^{'} (u_{n}) \to 0 as n \to \infty .$
Combining with ( ${\tilde{f}}_{3}$ ), we have
$\begin{aligned} c + o (1) + ‖ u_{n} ‖_{A, V} & ⩾ J (u_{n}) - \frac{1}{θ} ⟨ J^{'} (u_{n}), u_{n} ⟩ \\ = \frac{θ - 2}{2 θ} ‖ u_{n} ‖_{A, V}^{2} + \frac{1}{θ} \int_{R^{2}} B (x) (f (| u_{n} |^{2}) | u_{n} |^{2} - \frac{θ}{2} F (| u_{n} |^{2})) d x \\ ⩾ \frac{θ - 2}{2 θ} ‖ u_{n} ‖_{A, V}^{2} as n \to \infty \end{aligned}$
with $θ > 2$ given in assumption ( ${\tilde{f}}_{3}$ ), which means that the sequence ${u_{n}}$ is bounded in $H_{A, V}^{1} (R^{2}, C)$ . The rest proof is similar to the proof of Proposition 5.1, so we omit it here.

Now, we are in position to prove Theorems 1.3–1.4.
Proof Proof of Theorem 1.3.

Since J satisfies the Mountain Pass geometry, we can obtain there exists a sequence ${u_{n}} \subset H_{A, V}^{1} (R^{2}, C)$ such that

\begin{aligned} J (u_{n}) \to c_{V}^{*} and J^{'} (u_{n}) \to 0 as n \to \infty . \end{aligned}

(6.1)

By Proposition 6.2, we obtain the sequence

{u_{n}}

is bounded in

H_{A, V}^{1} (R^{2}, C)

, and from Lemma 2.4 there exists a subsequence of

{u_{n}}

relabeled as

{u_{n}}

and

u_{0} \in H_{A, V}^{1} (R^{2}, C)

such that

\begin{aligned} u_{n} ⇀ u_{0} in H_{A, V}^{1} (R^{2}, C), u_{n} \to u_{0} in L_{B}^{2} (R^{2}, C) ~as~ n \to \infty . \end{aligned}

(6.2)

Next, up to a subsequence, we claim that for any $φ \in C_{c}^{\infty} (R^{2}, C)$ ,

\begin{aligned} Re \int_{R^{2}} B (x) f (| u_{n} |^{2}) u_{n} \bar{φ} d x \to Re \int_{R^{2}} B (x) f (| u_{0} |^{2}) u_{0} \bar{φ} d x as n \to \infty . \end{aligned}

(6.3)

Up to a subsequence, from (6.2), we have that

u_{n} \to u_{0}

a.e. in

R^{2}

n \to \infty

and so there exists

C_{1} > 0

such that

| u_{n} | ⩽ C_{1}

a.e. in

R^{2}

. Due to Proposition 6.2 (i), we know

\int_{supp φ} B (x) f (| u_{n} |^{2}) d x \to \int_{supp φ} B (x) f (| u_{0} |^{2}) d x as n \to \infty,

and from [34, Lemma A.1] there exists

k \in L^{1} (supp φ, R)

such that

B (x) f (| u_{n} |^{2}) ⩽ k (x)

a.e. in

supp φ

. Thus, for any

φ \in C_{c}^{\infty} (R^{2}, C)

, we have

| Re (B (x) f (| u_{n} |^{2}) u_{n} \bar{φ}) | ⩽ C_{1} k (x) | φ |

for a.e.

x \in R^{2}

with

Re (B (x) f (| u_{n} |^{2}) u_{n} \bar{φ}) \to Re (B (x) f (| u_{0} |^{2}) u_{0} \bar{φ}) a . e . in R^{2} ~as~ n \to \infty .

Using the Lebesgue’s convergence dominated theorem, we can get (6.3). Furthermore, according to (6.1)–(6.3), we obtain that

\begin{aligned} 0 & = lim_{n \to \infty} ⟨ J^{'} (u_{n}), φ ⟩ = lim_{n \to \infty} (⟨ u_{n}, φ ⟩_{A, V} - Re \int_{R^{2}} B (x) f (| u_{n} |^{2}) u_{n} \bar{φ} d x) \\ = ⟨ u_{0}, φ ⟩_{A, V} - Re \int_{R^{2}} B (x) f (| u_{0} |^{2}) u_{0} \bar{φ} d x = ⟨ J^{'} (u_{0}), φ ⟩, for all φ \in C_{c}^{\infty} (R^{2}, C) . \end{aligned}

Due to the argument of Lemma 2.3, we know that

C_{c}^{\infty} (R^{2}, C)

is dense in

H_{A, V}^{1} (R^{2}, C)

, from which we obtain

⟨ J^{'} (u_{0}), u_{0} ⟩ = 0

. In addition, due to (

{\tilde{f}}_{3}

), we have that

\begin{aligned} J (u_{0}) & = \frac{1}{2} ‖ u_{0} ‖_{A, V}^{2} - \frac{1}{2} \int_{R^{2}} B (x) F (| u_{0} |^{2}) d x \\ ⩾ \frac{1}{θ} (‖ u_{0} ‖_{A, V}^{2} - \int_{R^{2}} B (x) f (| u_{0} |^{2}) | u_{0} |^{2} d x) = \frac{1}{θ} ⟨ J^{'} (u_{0}), u_{0} ⟩ = 0 \end{aligned}

with

θ > 2

taken in assumption (

{\tilde{f}}_{3}

). Consequently, we can use the estimate that

c_{V}^{*} < \frac{2 π ζ}{a_{0}}

and proceed as in the proof of Theorem 1.1.

Proof Proof of Theorem 1.4.

In this case, $ζ = 1$ . Using the estimate $c_{V}^{*} < \frac{2 π}{a_{0}}$ rather than $c_{V}^{*} < \frac{2 π ζ}{a_{0}}$ , similar to the proof of theorem of Theorem 1.3, we can get the ideal result.

Footnotes

Acknowledgements

This research was partially supported by the NSFC (12071192, 12371242) and Natural Science Foundation of Gansu Province of China (22JR5RA473).

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