On solutions for a class of Kirchhoff systems involving critical growth in R 2

Abstract

In this work we study the existence of solutions for the following class of elliptic systems involving Kirchhoff equations in the plane: $\begin{matrix} \{\begin{matrix} m (‖ u ‖^{2}) [- Δ u + u] = λ f (u, v), & x \in R^{2}, \\ ℓ (‖ v ‖^{2}) [- Δ v + v] = λ g (u, v), & x \in R^{2}, \end{matrix} \end{matrix}$ where $λ > 0$ is a parameter, $m, ℓ : [0, + \infty) \to [0, + \infty)$ are Kirchhoff-type functions, $‖ \cdot ‖$ denotes the usual norm of the Sobolev space $H^{1} (R^{2})$ and the nonlinear terms f and g have exponential critical growth of Trudinger–Moser type. Moreover, when f and g are odd functions, we prove that the number of solutions increases when the parameter λ becomes large.

Keywords

Kirchhoff systems exponential critical growth Trudinger–Moser inequality

1. Introduction

This article is devoted to study the existence of ground state and multiplicity of solutions for the following class of elliptic systems involving Kirchhoff equations: $\begin{matrix} (S_{λ}) & \{\begin{matrix} m (‖ u ‖^{2}) [- Δ u + u] = λ f (u, v), & x \in R^{2}, \\ ℓ (‖ v ‖^{2}) [- Δ v + v] = λ g (u, v), & x \in R^{2}, \end{matrix} \end{matrix}$ where $λ > 0$ is a parameter, $m, ℓ : [0, + \infty) \to [0, + \infty)$ are continuous functions, $‖ \cdot ‖$ denotes the usual norm of the Sobolev space $H^{1} (R^{2})$ and there exists a function $F \in C^{1} (R \times R, R)$ such that $\nabla F = (f, g)$ . Here, our intention is to deal with System ( S λ ) when the nonlinear terms f and g have exponential critical growth of Trudinger–Moser type. In order to motivate our results, we begin by giving a brief survey on this subject.

1.1. Motivation and related results

Recently, many authors have studied the problem $\begin{matrix} (1.1) & \{\begin{matrix} - (a + b \int_{Ω} | \nabla u |^{2} d x) Δ u = f (x, u), & x \in Ω, \\ u (x) = 0, & x \in \partial Ω, \end{matrix} \end{matrix}$ where $Ω \subset R^{N}$ is a bounded domain, $a, b > 0$ and $f : Ω \times R \to R$ is a continuous function. This class of problems is related to the stationary problem of a model introduced by Kirchhoff (see [15]) in the study on transverse vibrations of elastic strings proposed by the hyperbolic equation $\begin{matrix} (1.2) & ρ \frac{\partial^{2} u}{\partial t^{2}} - (\frac{P_{0}}{h} + \frac{E}{2 L} \int_{0}^{L} {| \frac{\partial u}{\partial x} |}^{2} d x) \frac{\partial^{2} u}{\partial x^{2}} = 0, (x, t) \in [0, L] \times [0, \infty), \end{matrix}$ where the parameters in the equation have the following meanings: L is the length of the string, h is the area of cross-section, E is the Young modulus of the material, ρ is the mass density and $P_{0}$ is the initial tension. Equation (1.2) is a generalization of the classical d’Alembert’s wave equation, by considering the effects of the changes in the length of the strings during the vibrations. We also refer the classical works of S. Bernstein [6], S. Pohozaev [25] and J.-L. Lions [18]. In [18], it was introduced a functional analysis approach to study a class of problems similar to (1.2). Motivated by the physical interest and impulsed by the abstract framework proposed by J.-L. Lions, various classes of Kirchhoff problems have been studied by many authors, see for instance [1,5,10,12,16,17,19,20] and references therein.

The study of the Kirchhoff problem (1.1) can be naturally extended to the following class of problems: $\begin{matrix} (1.3) & \{\begin{matrix} - m (‖ \nabla u ‖_{2}^{2}) Δ u = f (x, u) & in Ω, \\ u = 0 & on \partial Ω, \end{matrix} \end{matrix}$ where $Ω \subset R^{N}$ ( $N ⩾ 2$ ) is a domain, $m : [0, \infty) \to [0, \infty)$ is a continuous function satisfying $m (t) ⩾ m_{0} > 0$ for all $t ⩾ 0$ and $f : Ω \times R \to R$ is continuous. It is well known that if $N ⩾ 3$ , then the maximal growth on the nonlinear term $f (x, u)$ which allows us to treat elliptic problems variationally in the usual Sobolev space $H^{1} (Ω)$ is given by $| u |^{2^{*} - 1}$ as $| u | \to \infty$ , where $2^{*} : = 2 N / (N - 2)$ is the critical Sobolev exponent. If $N = 2$ then $2^{*}$ becomes infinite and $H^{1} (Ω)$ is continuously embedded into $L^{q} (Ω)$ for all $q \in [1, \infty)$ , but it is not continuously embedded into $L^{\infty} (Ω)$ . If Ω is a smooth bounded domain in $R^{2}$ , then the maximum growth that allows to study elliptic problems variationally is motivated by the Trudinger–Moser inequality (see [22]):

Theorem A.
There exists a constant $C > 0$ such that $\begin{matrix} sup_{{u \in W_{0}^{1, 2} (Ω) : ‖ \nabla u ‖_{2} ⩽ 1}} \int_{Ω} e^{α u^{2}} d x ⩽ C | Ω |, \forall α ⩽ 4 π . \end{matrix}$ Moreover, $4 π$ is the best constant, that is, the supreme just above is $+ \infty$ if $α > 4 π$ .

Though there have been some works on the existence of ground states for Kirchhoff problems involving nonlinear terms of polynomial growth, not much has been done when the nonlinear terms have exponential critical growth, for instance, we can cite [3,4,12,14,21,23]. In [12], the authors have studied the existence of positive ground state solution for (1.3) when Ω is a smooth bounded domain in $R^{2}$ and the nonlinear term $f (x, u)$ has exponential critical growth motivated by means of the Trudinger–Moser inequality (Theorem A). Precisely, the authors have assumed that there exists $α_{0} > 0$ such that $\begin{matrix} lim_{u \to + \infty} \frac{f (x, u)}{e^{α u^{2}}} = \{\begin{matrix} 0, & if α > α_{0}, \\ + \infty, & if α < α_{0}, \end{matrix} \end{matrix}$ uniformly in $x \in Ω$ . For this purpose, they have used minimax techniques combined with Theorem A. However, the boundedness of the domain plays a very important role in the work, since it implies that the embedding $H^{1} (Ω) ↪ L^{p} (Ω)$ , for $p \in [1, \infty)$ , is compact. In the same direction, in [23], the authors study problem (1.3) with $m (t) = 1 + α t$ ( $α > 0$ ) and the nonlinear term $f (x, u)$ has exponential critical growth. They prove the existence of solutions and a multiplicity result induced by the nonlocal dependence.

In the work [4], S. Aouaoui studies the Kirchhoff equation $\begin{matrix} (a + b ‖ u ‖^{2}) [- Δ u + u] = g (x) f (u), x \in R^{2}, \end{matrix}$ where $a, b > 0$ , the weight function $g (x)$ is positive and belongs to $L^{2} (R^{2}) \cap L^{\infty} (R^{2})$ and $f : R \to R$ is a nondecreasing continuous function with exponential critical growth and satisfies others appropriate conditions. The author proves the existence of at least three nontrivial solutions and the existence of infinitely many sign-changing solutions when f is odd.

In [14], the authors proved the existence of ground state solution for the nonlocal problem $\begin{matrix} m (\int_{R^{2}} (| \nabla u |^{2} + b (x) u^{2}) d x) [- Δ u + b (x) u] = A (x) f (u), x \in R^{2}, \end{matrix}$ where $m : [0, \infty) \to [0, \infty)$ is a continuous function satisfying $m (t) ⩾ m_{0} > 0$ for all $t ⩾ 0$ , $f : R \to R$ is continuous and has exponential critical growth, $A \in L_{loc}^{\infty} (R^{2})$ with $A (x) ⩾ 1$ for all $x \in R^{2}$ and the potential $b (x)$ is bounded from below.

In the work [13], the authors considered the Kirchhoff system $\begin{matrix} (1.4) & \{\begin{matrix} - m (\int_{Ω} | \nabla u |^{2}) Δ u = F_{u} (x, u, v) + μ_{1} | u |^{4} u, & x \in Ω, \\ - l (\int_{Ω} | \nabla v |^{2}) Δ v = F_{v} (x, u, v) + μ_{2} | v |^{4} v, & x \in Ω, \end{matrix} \end{matrix}$ where $Ω \subset R^{3}$ is a smooth bounded domain, m, l are positive Kirchhoff functions type and the nonlinearity F is subcritical and locally superlinear at infinity. By using the Symmetric Mountain Pass Theorem, they obtained multiple solutions for small values of $μ_{1}$ and $μ_{2}$ .

Thus, inspired by the works previously mentioned, our first purpose here is to consider the class of Kirchhoff systems ( S λ ), defined in the whole space $R^{2}$ and involving nonlinearities with exponential critical growth. As we know, the notion of criticality in all $R^{2}$ is expressed by the following Trudinger–Moser inequality which was considered by J.M. do Ó in [11] and D.M. Cao in [7]:
Theorem B.
If $α > 0$ and $u \in H^{1} (R^{2})$ , then $\begin{matrix} \int_{R^{2}} (e^{α u^{2}} - 1) d x < \infty . \end{matrix}$ Moreover, if $0 < α < 4 π$ , $‖ \nabla u ‖_{2} ⩽ 1$ and $‖ u ‖_{2} ⩽ \tilde{C}$ , then there exists a constant $C = C (α, \tilde{C}) > 0$ , depending only on α and $\tilde{C}$ , such that $\begin{matrix} (1.5) & \int_{R^{2}} (e^{α u^{2}} - 1) d x ⩽ C . \end{matrix}$

Motivated by Theorem B, we assume that the nonlinearities $f, g : R^{2} \to R$ in System ( S λ ) have exponential critical growth, that is, we suppose that there exists $α_{0} > 0$ such that $\begin{matrix} (CG) & lim_{\begin{array}{c} u \to + \infty \\ v \to + \infty \end{array}} \frac{| f (u, v) |}{e^{α (u^{2} + v^{2})}} = \{\begin{matrix} 0, & if α > α_{0}, \\ + \infty, & if α < α_{0}, \end{matrix} lim_{\begin{array}{c} u \to + \infty \\ v \to + \infty \end{array}} \frac{| g (u, v) |}{e^{α (u^{2} + v^{2})}} = \{\begin{matrix} 0, & if α > α_{0}, \\ + \infty, & if α < α_{0} . \end{matrix} \end{matrix}$

Due to the critical growth of the nonlinear terms f, g and the unboundedness of the domain, we deal with “lack of compactness”. In order to overcome this difficulty, Theorem B plays a crucial role. Moreover, we mention that the equations in System ( S λ ) are called nonlocal due to the presence of the Kirchhoff terms $m (‖ u ‖^{2})$ and $ℓ (‖ v ‖^{2})$ . This type of equations take care of the behavior of the solution in the whole space, which implies that the equations are no longer a point-wise identity. Nonlocal problems appear in several situations, for instance, they can describe the density of a population (for example, a bacterial population) subject to spreading (for more information on nonlocal problems, we can cite [8,9,27]). These characteristics give rise to additional difficulties, which make the problem more interesting from a mathematical point of view. For these reasons, we may consider suitable assumptions to treat System ( S λ ) variationally. As we will see below, we consider a broad class of Kirchhoff functions m and ℓ, with very general assumptions about them. To the best of our knowledge, there is no work in the literature concerned with the existence of ground state and multiplicity of solutions for Kirchhoff systems in the plane involving nonlinearities with exponential critical growth.
1.2. Assumptions and main results

Let us define $M (t) : = \int_{0}^{t} m (s) d s$ and $L (t) : = \int_{0}^{t} ℓ (s) d s$ . We suppose that $m, ℓ : [0, + \infty) \to [0, + \infty)$ are continuous and satisfy the following assumptions: $(H_{1})$

there exist constants $m_{0}, ℓ_{0} > 0$ such that $m (t) ⩾ m_{0}$ and $ℓ (t) ⩾ ℓ_{0}$ , for all $t ⩾ 0$ ;

(H_{2})

M, L are convex functions and there hold $\begin{matrix} M (t) ⩾ \frac{1}{2} m (t) t and L (t) ⩾ \frac{1}{2} ℓ (t) t, \forall t ⩾ 0 . \end{matrix}$

We observe that if m, ℓ are nondecreasing then M, L are convex functions.

With respect to the nonlinearities f and g, besides of the conditions (CG), we require the following hypotheses: $(F_{1})$

there hold $\begin{matrix} lim_{(u, v) \to (0, 0)} \frac{f (u, v)}{| (u, v) |} = 0 and lim_{(u, v) \to (0, 0)} \frac{g (u, v)}{| (u, v) |} = 0; \end{matrix}$

(F_{2})

there exists $θ > 4$ such that $\begin{matrix} θ F (u, v) ⩽ \nabla F (u, v) . (u, v), \forall (u, v) \in R^{2}; \end{matrix}$

(F_{3})

$\underset{t \to 0^{+}}{lim inf} \frac{F (t, t)}{t^{θ}} = : ξ > 0$ .

Remark 1.1.
By deriving the quotient $F (t, t) / t^{θ}$ for $t > 0$ and by using condition ( $F_{2}$ ), it is easily seen that $F (t, t) / t^{θ}$ is nondecreasing for $t > 0$ . Thus, by virtue of condition ( $F_{3}$ ), we obtain $\begin{matrix} (1.6) & F (t, t) ⩾ ξ t^{θ}, \forall t ⩾ 0 . \end{matrix}$ This condition will be used to estimate the minimax level of the energy functional associated to System ( S λ ). We point out that condition $(F_{3})$ is just local.
Hereafter, let us consider the space $E : = H_{rad}^{1} (R^{2}) \times H_{rad}^{1} (R^{2})$ endowed with the norm $‖ (u, v) ‖^{2} = ‖ u ‖^{2} + ‖ v ‖^{2}$ . A pair $(u, v) \in E$ is said to be a solution (in the weak sense) of System ( S λ ) if there holds $\begin{matrix} (1.7) & \begin{matrix} m (‖ u ‖^{2}) \int_{R^{2}} (\nabla u \nabla φ + u φ) d x + ℓ (‖ v ‖^{2}) \int_{R^{2}} (\nabla v \nabla ψ + v ψ) d x \\ = λ \int_{R^{2}} f (u, v) φ d x + λ \int_{R^{2}} g (u, v) ψ d x, \forall φ, ψ \in C_{0}^{\infty} (R^{2}) . \end{matrix} \end{matrix}$

We observe that, by $(F_{1})$ , $f (0, 0) = g (0, 0) = 0$ . Hence, $(u, v) = (0, 0)$ is the trivial solution for System ( S λ ). Now, we are in position to formulate our first main result.
Theorem 1.2.
Suppose that $(H_{1})$ , $(H_{2})$ , $(F_{1})$ – $(F_{3})$ hold and f, g have exponential critical growth. Then, for each $λ > 0$ , System ( S λ ) admits at least one nontrivial solution, provided that $ξ > 0$ , given in $(F_{3})$ , is large enough.

We are also concerned with the existence of ground state (or least energy) solution, i.e., a solution whose energy is minimal among the energy of all nontrivial solutions. For this purpose, we replace hypothesis $(H_{2})$ by the assumption $(\hat{H_{2}})$
M, L are convex functions and $\frac{m (t)}{t}$ , $\frac{ℓ (t)}{t}$ are non-increasing for $t > 0$ ;
It is not difficult to check that ( $\hat{H_{2}}$ ) implies condition ( $H_{2}$ ). Furthermore, we assume the following additional hypothesis on the function $F (u, v)$ : $(F_{4})$
For each $(u, v) \neq (0, 0)$ , the map $t \mapsto \frac{\nabla F (t u, t v) (u, v)}{t^{3}}$ is nondecreasing for $t > 0$ .
Thus, we are ready to state our second main result:
Theorem 1.3.
Suppose that $(H_{1})$ , $(\hat{H_{2}})$ , $(F_{1})$ – $(F_{4})$ hold and f, g have critical exponential growth. Then, for each $λ > 0$ , System ( S λ ) admits at least one ground state solution, provided that $ξ > 0$ , given in $(F_{3})$ , is large enough.

Our third result ensures that if f and g are odd functions, then the number of solutions increases when the parameter λ becomes large. More precisely, we have:
Theorem 1.4.
Suppose that $(H_{1})$ , $(H_{2})$ , $(F_{1})$ – $(F_{3})$ hold and f, g are odd functions with critical exponential growth. Then, for any $m \in N$ , there exists $Λ_{m} > 0$ such that System ( S λ ) admits at least m pairs of nontrivial solutions, provided that $λ > Λ_{m}$ .
Remark 1.5.
Typical examples of functions $m, ℓ : [0, + \infty) \to [0, + \infty)$ that verify our assumptions are given by $m (t) = a_{1} + b_{1} t^{σ_{1}}$ and $ℓ (t) = a_{2} + b_{2} t^{σ_{2}}$ , where $a_{1}, a_{2} ⩾ 0$ , $b_{1}, b_{2} > 0$ and $σ_{1}, σ_{2} \in [0, 1]$ (as $σ_{1} = σ_{2} = 1$ , we have the standard Kirchhoff functions which were considered in [15] for the scalar case). An example of function $F (u, v)$ satisfying our assumptions is given by $\begin{matrix} F (u, v) = ξ u^{2 η} + (e^{u^{2}} - 1) u^{2 η} + ξ v^{2 η} + (e^{v^{2}} - 1) v^{2 η}, \end{matrix}$ for some $η > 2$ and $ξ > 0$ sufficiently large.
Remark 1.6.
By supposing $\nabla F (u, v) . (u, v) ⩾ 0$ for all $(u, v) \in R^{2}$ , we observe that $λ > 0$ is a necessary condition for the existence of nontrivial solution for System ( S λ ). Indeed, if $λ ⩽ 0$ and $(u_{0}, v_{0}) \in E$ is a weak solution for ( S λ ), then taking $(φ, ψ) = (u_{0}, v_{0})$ in (1.7) and using assumption $(m_{1})$ we obtain $\begin{matrix} m_{0} ‖ u_{0} ‖^{2} + ℓ_{0} ‖ v_{0} ‖^{2} ⩽ λ \int_{R^{2}} \nabla F (u_{0}, v_{0}) . (u_{0}, v_{0}) d x ⩽ 0, \end{matrix}$ which implies that $‖ u_{0} ‖ = ‖ v_{0} ‖ = 0$ , that is, $(u_{0}, v_{0}) = (0, 0)$ .
Remark 1.7.
It is worthwhile to mention that if $f (0, v) > 0$ for all $v \neq 0$ and $g (u, 0) > 0$ for all $u \neq 0$ , then a nontrivial solution $(u_{0}, v_{0}) \in E$ for System ( S λ ) is not semitrivial, i.e., $u_{0} \neq 0$ and $v_{0} \neq 0$ . Moreover, under suitable additional assumptions, one can adapt some arguments from [26] to obtain a positive ground state solution.
Remark 1.8.
In order to overcome the difficulty imposed by the lack of compactness, we prove that the value of the minimax level associated to System ( S λ ) is related with the parameter ξ introduced in $(F_{3})$ . In Theorems 1.2 and 1.3, we prove the existence of solutions provided that ξ is sufficiently large. Precisely, we consider $\begin{matrix} ξ > ξ_{0} : = max {ξ_{1}, {(\frac{2 ξ_{1}}{θ})}^{\frac{θ}{2}} {(\frac{α_{0} λ (θ - 2)}{ν_{0} (θ - 4)})}^{\frac{θ - 2}{2}}}, \end{matrix}$ where $ξ_{1} : = [M (8 π) + L (8 π)] / (2 λ π)$ and $ν_{0} : = min {m_{0}, ℓ_{0}}$ . If $ξ > ξ_{0}$ then we are able to apply the Trudinger–Moser inequality (Theorem B) to prove that the energy functional associated to ( S λ ) satisfies the Palais–Smale condition in a convenient interval. Regarding to the multiplicity of solutions in Theorem 1.4, the parameter $ξ > 0$ can be arbitrary, because the value of the minimax level and the number of solutions is related to the parameter λ, which we may consider it large enough.

The remainder of this paper is organized as follows. In the forthcoming section, we introduce the variational framework. In Section 3, we prove that the energy functional associated to the system satisfies the mountain pass geometry. In Section 4, we show that under a suitable hypothesis, the energy functional satisfies the Palais–Smale condition in an appropriate interval. Section 5 is concerned with minimax estimates and the proof of Theorem 1.2. In Section 6, we prove Theorem 1.3 and finally Section 7 presents the proof of Theorem 1.4.

Throughout this paper, the symbols C, $\tilde{C}$ , $C_{0}$ , $C_{1}$ , $C_{2}$ , …represent several (possibly different) positive constants. Moreover, $o_{n} (1)$ denotes a sequence that converges to 0 as $n \to \infty$ . The norm in $L^{p} (R^{2})$ ( $1 ⩽ p < \infty$ ) and $L^{\infty} (R^{2})$ will be denoted respectively by $‖ \cdot ‖_{p}$ and $‖ \cdot ‖_{\infty}$ . The Euclidean norm in $R^{2}$ will be denoted by $| \cdot |$ .
2. Variational framework

In this section, we introduce the variational framework to study System ( S λ ). Associated to System ( S λ ), we have the energy functional $I_{λ} : E \to R$ defined by $\begin{matrix} I_{λ} (u, v) : = \frac{1}{2} M (‖ u ‖^{2}) + \frac{1}{2} L (‖ v ‖^{2}) - λ \int_{R^{2}} F (u, v) d x . \end{matrix}$ In view of (CG) and $(F_{1})$ , for $ε > 0$ , $r ⩾ 1$ and $α > α_{0}$ , there exists $C = C (ε, r, α) > 0$ such that $\begin{matrix} (2.1) & | f (u, v) | + | g (u, v) | ⩽ ε | (u, v) | + C (e^{α (u^{2} + v^{2})} - 1) {| (u, v) |}^{r - 1}, \end{matrix}$ which implies $\begin{matrix} (2.2) & F (u, v) ⩽ \frac{ε}{2} {| (u, v) |}^{2} + C (e^{α (u^{2} + v^{2})} - 1) {| (u, v) |}^{r}, \end{matrix}$ for all $(u, v) \in R^{2}$ . Notice that $\begin{matrix} (2.3) & a b - 1 ⩽ \frac{1}{2} (a^{2} - 1) + \frac{1}{2} (b^{2} - 1), \forall a, b ⩾ 0 . \end{matrix}$ In view of Theorem B and (2.3), for all $(u, v) \in E$ one has $\begin{matrix} \int_{R^{2}} (e^{α (u^{2} + v^{2})} - 1) d x ⩽ \frac{1}{2} \int_{R^{2}} (e^{2 α u^{2}} - 1) d x + \frac{1}{2} \int_{R^{2}} (e^{2 α v^{2}} - 1) d x < \infty . \end{matrix}$ Thus, the energy functional $I_{λ}$ is well defined on E. Moreover, by standard arguments we conclude that $I_{λ} \in C^{1} (E, R)$ and its derivative is given by $\begin{matrix} I_{λ}^{'} (u, v) (φ, ψ) & = m (‖ u ‖^{2}) \int_{R^{2}} (\nabla u \nabla φ + u φ) d x + ℓ (‖ v ‖^{2}) \int_{R^{2}} (\nabla v \nabla ψ + v ψ) d x \\ - λ \int_{R^{2}} f (u, v) φ d x - λ \int_{R^{2}} g (u, v) ψ d x, \end{matrix}$ for $(u, v), (φ, ψ) \in E$ . Therefore, if $(u_{0}, v_{0}) \in E$ is a critical point of $I_{λ}$ , then by applying the Palais Symmetric Criticality Principle (see [24]), we can conclude that the pair $(u_{0}, v_{0})$ is a weak solution of System ( S λ ) (see more details in Section 5).

3. Mountain pass geometry

In this section, we prove that the functional $I_{λ}$ has the mountain mass geometry. For this purpose, we obtain the following auxiliary lemma:

Lemma 3.1.
Suppose that ( CG ) and $(F_{1})$ hold. If $r ⩾ 2$ , $α > α_{0}$ and $‖ (u, v) ‖ ⩽ ρ < \sqrt{π / α}$ , then $\begin{matrix} (3.1) & \int_{R^{2}} (e^{α (u^{2} + v^{2})} - 1) {| (u, v) |}^{r} d x ⩽ C {‖ (u, v) ‖}^{r}, \end{matrix}$ for some $C = C (r, α) > 0$ .
Proof.
In view of (2.3) and Hölder inequality, we deduce that $\begin{matrix} \int_{R^{2}} (e^{α (u^{2} + v^{2})} - 1) {| (u, v) |}^{r} d x ⩽ C {[\frac{1}{2} \int_{R^{2}} (e^{4 α u^{2}} - 1) d x + \frac{1}{2} \int_{R^{2}} (e^{4 α v^{2}} - 1) d x]}^{1 / 2} {‖ (u, v) ‖}^{r} . \end{matrix}$ Since $‖ (u, v) ‖ ⩽ ρ < \sqrt{π / α}$ , it follows that $4 α ‖ u ‖^{2} < 4 π$ and $4 α ‖ v ‖^{2} < 4 π$ . In light of Theorem B, one has $\begin{matrix} \int_{R^{2}} (e^{4 α u^{2}} - 1) d x ⩽ \int_{R^{2}} (e^{4 α ‖ u ‖^{2} {(\frac{u}{‖ u ‖})}^{2}} - 1) d x ⩽ C_{1} \end{matrix}$ and $\begin{matrix} \int_{R^{2}} (e^{4 α v^{2}} - 1) d x ⩽ \int_{R^{2}} (e^{4 α ‖ v ‖^{2} {(\frac{v}{‖ v ‖})}^{2}} - 1) d x ⩽ C_{2}, \end{matrix}$ which imply (3.1) and the proof is done. □

Now, let us prove the mountain pass geometry. Lemma 3.2.
Suppose that $(H_{1})$ , $(H_{2})$ , $(F_{1})$ – $(F_{3})$ and ( CG ) hold. Then, we have: $(I_{1})$
there exist $τ > 0$ and $ϱ > 0$ such that $I_{λ} (u, v) ⩾ τ$ whenever $‖ (u, v) ‖ = ϱ$ ;
$(I_{2})$
There exists $e \in H_{rad}^{1} (R^{2})$ with $‖ (e, e) ‖ > ϱ$ such that $I_{λ} (e, e) < 0$ .

Proof.
For $0 < ε < min {m_{0}, ℓ_{0}}$ , $r > 2$ and $α > α_{0}$ , it follows from $(H_{1})$ and (2.2) that $\begin{matrix} I_{λ} (u, v) ⩾ \frac{1}{2} (min {m_{0}, ℓ_{0}} - λ ε) {‖ (u, v) ‖}^{2} - C \int_{R^{2}} (e^{α (u^{2} + v^{2})} - 1) {| (u, v) |}^{r} d x, \end{matrix}$ for all $(u, v) \in E$ . By considering $‖ (u, v) ‖ = ϱ < \sqrt{π / α}$ , from Lemma 3.1 we get $\begin{matrix} I_{λ} (u, v) ⩾ \frac{1}{2} (min {m_{0}, ℓ_{0}} - λ ε) ϱ^{2} - C ϱ^{r} . \end{matrix}$ By taking $ϱ < \sqrt{π / α}$ sufficiently small so that $(min {m_{0}, ℓ_{0}} - λ ε) / 2 - C ϱ^{r - 2} > 0$ , we conclude that $\begin{matrix} I_{λ} (u, v) ⩾ τ : = ϱ^{2} [\frac{1}{2} (min {m_{0}, ℓ_{0}} - λ ε) - C ϱ^{r - 2}], for ‖ (u, v) ‖ = ϱ, \end{matrix}$ which proves $(I_{1})$ . In order to show $(I_{2})$ , let $φ \in C_{0, rad}^{\infty} (R^{2}) ∖ {0}$ be such that $φ ⩾ 0$ in $R^{2}$ and we denote $K = supp (φ)$ . According to estimate (1.6), we have $\begin{matrix} (3.2) & I_{λ} (t φ, t φ) ⩽ \frac{1}{2} M (t^{2} ‖ u ‖^{2}) + \frac{1}{2} L (t^{2} ‖ v ‖^{2}) - λ ξ \int_{K} t^{θ} φ^{θ} d x . \end{matrix}$ In view of $(H_{2})$ , we have $M (t) / t^{2}$ is non-increasing for $t > 0$ . Thus, $M (t) ⩽ a_{1} t^{2}$ for all $t ⩾ 1$ , where $a_{1} = M (1) > 0$ . On the other hand, $M (t) ⩽ a_{0} t$ for all $0 ⩽ t ⩽ 1$ , where $a_{0} = {max}_{s \in [0, 1]} m (s) > 0$ . Therefore, $\begin{matrix} (3.3) & M (t) ⩽ a_{0} t + a_{1} t^{2}, \forall t ⩾ 0 . \end{matrix}$ Analogously, there exist $b_{0}, b_{1} > 0$ such that $\begin{matrix} (3.4) & L (t) ⩽ b_{0} t + b_{1} t^{2}, \forall t ⩾ 0 . \end{matrix}$ Combining (3.2), (3.3) and (3.4), we obtain $\begin{matrix} I_{λ} (t φ, t φ) ⩽ \frac{t^{2}}{2} ‖ φ ‖^{2} (a_{0} + b_{0}) + \frac{t^{4}}{2} ‖ φ ‖^{4} (a_{1} + b_{1}) - λ ξ t^{θ} \int_{K} φ^{θ} d x . \end{matrix}$ Since $θ > 4$ we conclude that $I_{λ} (t φ, t φ) \to - \infty$ , as $t \to + \infty$ . Therefore, $(I_{2})$ follows by taking $e = t φ$ for $t > 0$ sufficiently large. □

4. Palais–Smale sequences

Let X be a Banach space and $J : X \to R$ a functional of $C^{1}$ -class. A sequence ${u_{n}} \subset X$ is said to be a Palais–Smale sequence at the level $c \in R$ for the functional J (in short ${(PS)}_{c}$ sequence), whenever $J (u_{n}) \to c$ and $‖ J^{'} (u_{n}) ‖_{X^{'}} \to 0$ as $n \to \infty$ . The functional J satisfies the Palais–Smale condition at level $c \in R$ , whenever any ${(PS)}_{c}$ sequence for the functional J admits a convergent subsequence.

In this section, we obtain some properties of ${(PS)}_{c}$ sequences associated to the functional $I_{λ}$ .

Lemma 4.1.
If ${(u_{n}, v_{n})} \subset E$ is a ${(PS)}_{c}$ sequence for the functional $I_{λ}$ , then ${(u_{n}, v_{n})}$ is bounded in E.
Proof.
Let ${(u_{n}, v_{n})}$ be a ${(PS)}_{c}$ sequence for $I_{λ}$ . According to $(H_{2})$ , we have $\begin{array}{l} c + o_{n} (1) ‖ (u_{n}, v_{n}) ‖ & ⩾ I_{λ} (u_{n}, v_{n}) - \frac{1}{θ} I_{λ}^{'} (u_{n}, v_{n}) (u_{n}, v_{n}) \\ ⩾ (\frac{1}{4} - \frac{1}{θ}) m (‖ u_{n} ‖^{2}) ‖ u_{n} ‖^{2} + (\frac{1}{4} - \frac{1}{θ}) ℓ (‖ v_{n} ‖^{2}) ‖ v_{n} ‖^{2} \\ + \frac{λ}{θ} \int [\nabla F (u_{n}, v_{n}) (u_{n}, v_{n}) - θ F (u_{n}, v_{n})] d x, \end{array}$ which jointly with $(H_{1})$ and $(F_{2})$ implies $\begin{matrix} c + o_{n} (1) ‖ (u_{n}, v_{n}) ‖ ⩾ (\frac{θ - 4}{4 θ}) ν_{0} {‖ (u_{n}, v_{n}) ‖}^{2}, \end{matrix}$ where $ν_{0} : = min {m_{0}, ℓ_{0}}$ . Therefore, from this inequality ${(u_{n}, v_{n})}$ is bounded in E. □

As an immediate consequence of the previous lemma, we have: Corollary 4.2.
If ${(u_{n}, v_{n})} \subset E$ is a ${(PS)}_{c}$ sequence for the functional $I_{λ}$ , then $\begin{matrix} (4.1) & \underset{n \to + \infty}{lim sup} {‖ (u_{n}, v_{n}) ‖}^{2} ⩽ \frac{4 θ}{(θ - 4) ν_{0}} c . \end{matrix}$
Lemma 4.3.
Let ${(u_{n}, v_{n})}$ be a ${(PS)}_{c}$ sequence for $I_{λ}$ and suppose that $\begin{matrix} (4.2) & c < \frac{ν_{0} (θ - 4) π}{2 α_{0} θ}, \end{matrix}$ where $ν_{0} : = min {m_{0}, ℓ_{0}}$ . If $(u_{n}, v_{n}) ⇀ (u_{0}, v_{0})$ weakly in E, then $\begin{matrix} (4.3) & \int_{R^{2}} f (u_{n}, v_{n}) (u_{n} - u_{0}) d x \to 0 and \int_{R^{2}} g (u_{n}, v_{n}) (v_{n} - v_{0}) d x \to 0, \end{matrix}$ as $n \to \infty$ .
Proof.
Let us prove the first convergence in (4.3), the other one is similar. Let $ε > 0$ , $α > α_{0}$ , $1 < σ < 2$ to be choose later and $σ^{'} = σ / (σ - 1)$ . In view of (2.1) and Hölder’s inequality, we deduce $\begin{matrix} | \int_{R^{2}} f (u_{n}, v_{n}) (u_{n} - u_{0}) d x | ⩽ ε {‖ (u_{n}, v_{n}) ‖}_{2} ‖ u_{n} - u_{0} ‖_{2} + C {[\int_{R^{2}} (e^{α σ (u_{n}^{2} + v_{n}^{2})} - 1) d x]}^{1 / σ} ‖ u_{n} - u_{0} ‖_{σ^{'}} . \end{matrix}$ Taking into account (2.3), Lemma 4.1 and the compact embedding $H_{rad}^{1} (R^{2}) ↪ L^{σ^{'}} (R^{2})$ , we obtain $\begin{matrix} (4.4) & | \int_{R^{2}} f (u_{n}, v_{n}) (u_{n} - u_{0}) d x | ⩽ ε C + C {[\frac{1}{2} \int_{R^{2}} (e^{2 α σ u_{n}^{2}} - 1) d x + \frac{1}{2} \int_{R^{2}} (e^{2 α σ v_{n}^{2}} - 1) d x]}^{\frac{1}{σ}} o_{n} (1) . \end{matrix}$ In order to conclude the proof, it is sufficient to show that $\begin{matrix} (4.5) & \int_{R^{2}} (e^{2 α σ u_{n}^{2}} - 1) d x ⩽ C and \int_{R^{2}} (e^{2 α σ v_{n}^{2}} - 1) d x ⩽ C . \end{matrix}$ Notice that $\begin{matrix} (4.6) & \int_{R^{2}} (e^{2 α σ u_{n}^{2}} - 1) d x ⩽ \int_{R^{2}} (e^{2 α σ ‖ (u_{n}, v_{n}) ‖^{2} {(\frac{u_{n}}{‖ u_{n} ‖})}^{2}} - 1) d x . \end{matrix}$ Using (4.1) and assumption (4.2), for some $δ > 0$ one has $2 α_{0} ‖ (u_{n}, v_{n}) ‖^{2} ⩽ 4 π - δ$ , for $n \in N$ sufficiently large. Thus, for $α > α_{0}$ close to $α_{0}$ , $σ > 1$ close to 1, we have $2 α σ ‖ (u_{n}, v_{n}) ‖^{2} < 4 π$ , for $n \in N$ large enough. Hence, the uniform boundedness in (4.6) follows from Theorem B. Analogously, we obtain the second inequality in (4.5). Therefore, (4.4) and (4.5) imply the first convergence in (4.3) and this finishes the proof. □
Proposition 4.4.
If the estimate ( 4.2 ) holds, then the functional $I_{λ}$ satisfies the Palais–Smale condition at level c.
Proof.
For the sake of simplicity, let us define $\begin{matrix} Φ (u, v) : = \frac{1}{2} M (‖ u ‖^{2}) + \frac{1}{2} L (‖ v ‖^{2}) . \end{matrix}$ Since $(H_{2})$ holds, $Φ (u, v)$ is convex. Let ${(u_{n}, v_{n})} \subset E$ be a ${(PS)}_{c}$ sequence for $I_{λ}$ , such that $c \in R$ satisfies (4.2). From Lemma 4.1, we may suppose, up to a subsequence, that $(u_{n}, v_{n}) ⇀ (u_{0}, v_{0})$ weakly in E. By the lower weak semicontinuity and the convexity of M and L, we have $\begin{matrix} (4.7) & \frac{1}{2} M (‖ u_{0} ‖^{2}) + \frac{1}{2} L (‖ v_{0} ‖^{2}) ⩽ \frac{1}{2} \underset{n \to + \infty}{lim inf} M (‖ u_{n} ‖^{2}) + \frac{1}{2} \underset{n \to + \infty}{lim inf} L (‖ v_{n} ‖^{2}) . \end{matrix}$ On the other hand, it follows from Lemma 4.3 that $\begin{matrix} \int_{R^{2}} f (u_{n}, v_{n}) (u_{n} - u_{0}) d x \to 0 and \int_{R^{2}} g (u_{n}, v_{n}) (v_{n} - v_{0}) d x \to 0, \end{matrix}$ as $n \to + \infty$ . From the convexity of $Φ (u, v)$ , we deduce that $\begin{array}{rcl} Φ (u_{0}, v_{0}) - Φ (u_{n}, v_{n}) & ⩾ & Φ^{'} (u_{n}, v_{n}) (u_{0} - u_{n}, v_{0} - v_{n}) \\ = & I_{λ}^{'} (u_{n}, v_{n}) (u_{0} - u_{n}, v_{0} - v_{n}) + λ \int_{R^{2}} f (u_{n}, v_{n}) (u_{0} - u_{n}) d x \\ + λ \int_{R^{2}} g (u_{n}, v_{n}) (v_{0} - v_{n}) d x, \end{array}$ which implies $Φ (u_{0}, v_{0}) + o_{n} (1) ⩾ Φ (u_{n}, v_{n})$ . Thus, one has $\begin{matrix} (4.8) & \begin{matrix} \frac{1}{2} M (‖ u_{0} ‖^{2}) + \frac{1}{2} L (‖ v_{0} ‖^{2}) & ⩾ \underset{n \to + \infty}{lim sup} Φ (u_{n}, v_{n}) \\ ⩾ \frac{1}{2} \underset{n \to + \infty}{lim inf} M (‖ u_{n} ‖^{2}) + \frac{1}{2} \underset{n \to + \infty}{lim inf} L (‖ v_{n} ‖^{2}) . \end{matrix} \end{matrix}$ Combining (4.7) and (4.8), up to a subsequence, we have $\begin{matrix} \frac{1}{2} M (‖ u_{n} ‖^{2}) \to \frac{1}{2} M (‖ u_{0} ‖^{2}) and \frac{1}{2} L (‖ v_{n} ‖^{2}) \to \frac{1}{2} L (‖ v_{0} ‖^{2}), as n \to + \infty . \end{matrix}$ Since $M (t)$ and $L (t)$ are strictly increasing on $t > 0$ , we conclude that $‖ u_{n} ‖^{2} \to ‖ u_{0} ‖^{2}$ and $‖ v_{n} ‖^{2} \to ‖ v_{0} ‖^{2}$ , as $n \to + \infty$ . Therefore, $(u_{n}, v_{n}) \to (u_{0}, v_{0})$ strongly in E and this finishes the proof. □

5. Proof of Theorem 1.2

Let us consider the minimax level $\begin{matrix} c^{*} : = inf_{γ \in Γ} max_{t \in [0, 1]} I_{λ} (γ (t)), \end{matrix}$ where $\begin{matrix} Γ : = {γ \in C ([0, 1], E) : γ (0) = (0, 0) and I_{λ} (γ (1)) < 0} . \end{matrix}$ In this section, we prove that if ξ is sufficiently large, then $c^{*}$ satisfies the minimax estimate (4.2). Precisely, we obtain the following lemma:

Lemma 5.1.
Suppose that $\begin{matrix} (5.1) & ξ > max {ξ_{1}, {(\frac{2 ξ_{1}}{θ})}^{\frac{θ}{2}} {(\frac{α_{0} λ (θ - 2)}{ν_{0} (θ - 4)})}^{\frac{θ - 2}{2}}}, \end{matrix}$ where $ξ_{1} : = [M (8 π) + L (8 π)] / (2 λ π)$ and $ν_{0} : = min {m_{0}, ℓ_{0}}$ . Then, there holds $\begin{matrix} (5.2) & c^{} < \frac{ν_{0} (θ - 4) π}{2 α_{0} θ} . \end{matrix}$
Proof.
Let $φ \in C_{0, rad}^{\infty} (R^{2}) ∖ {0}$ be such that $0 ⩽ φ ⩽ 1$ , $supp (φ) \subset B_{2} (0)$ , $φ \equiv 1$ in $B_{1} (0)$ and $| \nabla φ | ⩽ 1$ . Thus, one has $\begin{matrix} ‖ φ ‖^{2} = \int_{B_{2} (0)} | \nabla φ |^{2} d x + \int_{B_{2} (0)} φ^{2} d x ⩽ 2 | B_{2} (0) | = 8 π . \end{matrix}$ Moreover, since $ξ > ξ_{1}$ , by $(F_{3})$ we obtain $\begin{matrix} (5.3) & I_{λ} (φ, φ) < \frac{M (8 π) + L (8 π)}{2} - λ ξ_{1} π = 0 . \end{matrix}$ Let $γ : [0, 1] \to E$ be the path defined by $γ (t) : = (t φ, t φ)$ . Hence, by (5.3) we have $γ \in Γ$ and we can conclude that $\begin{array}{rcl} c^{} & ⩽ & max_{t \in [0, 1]} [\frac{1}{2} M (t^{2} ‖ φ ‖^{2}) + \frac{1}{2} L (t^{2} ‖ φ ‖^{2}) - λ \int_{R^{2}} F (t φ, t φ) d x] \\ ⩽ & max_{t \in [0, 1]} [(\frac{M (8 π) + L (8 π)}{2}) t^{2} - λ ξ t^{θ} \int_{B_{1} (0)} φ^{θ} d x] \\ ⩽ & λ π max_{t \in [0, + \infty)} [ξ_{1} t^{2} - ξ t^{θ}], \end{array}$ where we have used that M and L are convex. By standard computations, it follows that $\begin{matrix} max_{t \in [0, + \infty)} [ξ_{1} t^{2} - ξ t^{θ}] = \frac{1}{ξ^{\frac{2}{θ - 2}}} (θ - 2) 2^{\frac{2}{θ - 2}} {(\frac{ξ_{1}}{θ})}^{\frac{θ}{θ - 2}}, \end{matrix}$ and therefore $\begin{matrix} c^{} ⩽ \frac{λ π}{ξ^{\frac{2}{θ - 2}}} (θ - 2) 2^{\frac{2}{θ - 2}} {(\frac{ξ_{1}}{θ})}^{\frac{θ}{θ - 2}} . \end{matrix}$ Thus, if ξ satisfies (5.1) then (5.2) holds and the proof is finished. □
Conclusion of the proof of Theorem 1.2.
In view of Lemma 3.2, $I_{λ}$ has the mountain pass geometry. Moreover, according to Lemma 4.4, the functional $I_{λ}$ satisfies the Palais–Smale condition at level c whenever (4.2) holds. Since the minimax level $c^{} \in (0, [ν_{0} (θ - 4) π] / 2 α_{0} θ)$ whenever $ξ > 0$ is large, it follows from the Mountain Pass Theorem that $I_{λ}$ admits a critical point $(u_{0}, v_{0}) \in E$ such that $I_{λ} (u_{0}, v_{0}) = c^{*}$ . Now, let $O (2)$ be the group of the orthogonal transformations of $R^{2}$ . Taking into account that the action of the topological group $G : = O (2) \times O (2)$ on the Hilbert space $H : = H^{1} (R^{2}) \times H^{1} (R^{2})$ , defined by $(g_{1}, g_{2}) . (u (x), v (x)) = (u (g_{1}^{- 1} x), v (g_{2}^{- 1} x))$ , is isometric and the functional $I_{λ}$ is invariant by this action, by invoking the Palais Symmetric Criticality Principle (see [24]), we can conclude that $(u_{0}, v_{0})$ is a weak solution of System ( S λ ), since $Fix (G) = E$ .1
¹
$Fix (G)$ denotes the space of invariant points of the action of $G$ .

□

6. Proof of Theorem 1.3

In this section, we deal with System ( S λ ) but replacing hypothesis $(H_{2})$ by $(\hat{H_{2}})$ and considering condition $(F_{4})$ . Let us define $\begin{matrix} b : = inf_{(u, v) \in S} I_{λ} (u, v), where S : = {(u, v) \in E : (u, v) \neq (0, 0) and I_{λ}^{'} (u, v) = 0} . \end{matrix}$ In order to prove Theorem 1.3, it suffices to conclude that $c^{*} ⩽ b$ , because $I_{λ} (u_{0}, v_{0}) = c^{*}$ and $(u_{0}, v_{0}) \in S$ . Let $(u, v) \in S$ and $η : (0, + \infty) \to R$ be defined by $η (t) : = I_{λ} (t u, t v)$ . It is clear that η is differentiable and its derivative is given by $\begin{matrix} η^{'} (t) & = I_{λ}^{'} (t u, t v) (u, v) \\ = m (t^{2} ‖ u ‖^{2}) t ‖ u ‖^{2} + ℓ (t^{2} ‖ v ‖^{2}) t ‖ v ‖^{2} - λ \int_{R^{2}} f (t u, t v) u^{2} d x - λ \int_{R^{2}} g (t u, t v) v^{2} d x . \end{matrix}$ Since $(u, v) \in S$ , we have $\begin{matrix} m (‖ u ‖^{2}) ‖ u ‖^{2} + ℓ (‖ v ‖^{2}) ‖ v ‖^{2} - λ \int_{R^{2}} \nabla F (u, v) (u, v) d x = 0 . \end{matrix}$ Thus, we obtain $\begin{array}{l} η^{'} (t) & = t^{3} ‖ u ‖^{4} [\frac{m (t^{2} ‖ u ‖^{2})}{t^{2} ‖ u ‖^{2}} - \frac{m (‖ u ‖^{2})}{‖ u ‖^{2}}] + t^{3} ‖ v ‖^{4} [\frac{ℓ (t^{2} ‖ v ‖^{2})}{t^{2} ‖ v ‖^{2}} - \frac{ℓ (‖ v ‖^{2})}{‖ v ‖^{2}}] \\ + λ t^{3} \int_{R^{2}} [\nabla F (u, v) (u, v) - \frac{\nabla F (t u, t v) (u, v)}{t^{3}}] d x . \end{array}$ Notice that $η^{'} (1) = 0$ . Moreover, taking into account $(\hat{H_{2}})$ and $(F_{4})$ , it follows that $η^{'} (t) ⩾ 0$ for all $t \in (0, 1)$ and $η^{'} (t) ⩽ 0$ for all $t > 1$ . Therefore, $\begin{matrix} I_{λ} (u, v) = max_{t \in [0, + \infty)} I_{λ} (t u, t v) . \end{matrix}$ Let $γ_{0} : [0, 1] \to R$ be defined by $γ_{0} (t) = t t_{0} (u, v)$ , where $t_{0}$ is a positive number that satisfies $I_{λ} (t_{0} u, t_{0} v) < 0$ . Thus, $γ_{0} \in Γ$ and $\begin{matrix} c^{*} ⩽ max_{t \in [0, 1]} I_{λ} (γ_{0} (t)) ⩽ max_{t \in [0, + \infty)} I_{λ} (t u, t v) = I_{λ} (u, v) . \end{matrix}$ Once $(u, v) \in S$ is arbitrary, we conclude that $c^{*} ⩽ b$ and this finishes the proof of Theorem 1.3.

7. Proof of Theorem 1.4

In this section, we are concerned with the multiplicity of solutions for System ( S λ ). In order to prove Theorem 1.4, we shall use the following version of the Symmetric Mountain Pass Theorem, see [2].

Theorem 7.1.

Let X be a real Banach space, and let $J \in C^{1} (X, R)$ be an even functional satisfying $J (0) = 0$ and $(J_{1})$

there are constants $ρ, α > 0$ such that $J |_{\partial B_{ρ} (0)} ⩾ α$ ;

(J_{2})

there is $A > 0$ and a finite-dimensional subspace V of X such that $\begin{matrix} max_{u \in V} J (u) ⩽ A . \end{matrix}$

If the functional J satisfies the

{(PS)}_{d}

condition for all

d \in (0, A)

, then it possesses at least

dim V

pairs of nontrivial critical points.

In order to prove that the functional $I_{λ}$ satisfies the assumptions of the preceding theorem, we use the following technical lemma:

Lemma 7.2.

For any $λ > 0$ , let $h_{λ} : [0, + \infty) \to R$ be defined by $\begin{matrix} h_{λ} (t) = a t^{2} + b t^{4} - λ c t^{q}, \end{matrix}$ where $a, b, c > 0$ and $q > 4$ . If $A_{λ} : = {max}_{t ⩾ 0} h_{λ} (t)$ then $A_{λ} \in (0, + \infty)$ for all $λ > 0$ and ${lim}_{λ \to + \infty} A_{λ} = 0$ .

Proof.

It is not difficult to see that $h_{λ} (t) > 0$ for $t > 0$ sufficiently small and ${lim}_{t \to + \infty} h_{λ} (t) = - \infty$ . Thus, $A_{λ} \in (0, + \infty)$ . Moreover, for any $λ > 0$ , there exists $t_{0} = t_{0} (λ) > 0$ such that $A_{λ} = h_{λ} (t_{0})$ . Therefore, we have $\begin{matrix} 0 < A_{λ} ⩽ \{\begin{matrix} (a + b) t_{0}^{2} - λ c t_{0}^{q}, & if t_{0} ⩽ 1, \\ (a + b) t_{0}^{4} - λ c t_{0}^{q}, & if t_{0} > 1 . \end{matrix} \end{matrix}$ This fact shows that $\begin{matrix} A_{λ} ⩽ min {max_{t ⩾ 0} h_{1, λ} (t), max_{t ⩾ 0} h_{2, λ} (t)}, \end{matrix}$ where $h_{1, λ} (t) : = (a + b) t^{2} - λ c t^{q}$ and $h_{2, λ} (t) : = (a + b) t^{4} - λ c t^{q}$ . By standard computations, we obtain $\begin{matrix} max_{t ⩾ 0} h_{1, λ} (t) = (\frac{q - 2}{2}) {[\frac{2 (a + b)}{q}]}^{\frac{q}{q - 2}} \frac{c^{\frac{2}{2 - q}}}{λ^{\frac{2}{q - 2}}}, \end{matrix}$ and $\begin{matrix} max_{t ⩾ 0} h_{2, λ} (t) = (\frac{q - 4}{4}) {[\frac{4 (a + b)}{q}]}^{\frac{q}{q - 4}} \frac{c^{\frac{4}{4 - q}}}{λ^{\frac{4}{q - 4}}} . \end{matrix}$ Since $q > 4$ , one has $\begin{matrix} lim_{λ \to + \infty} max_{t ⩾ 0} h_{1, λ} (t) = 0 and lim_{λ \to + \infty} max_{t ⩾ 0} h_{2, λ} (t) = 0, \end{matrix}$ which implies that ${lim}_{λ \to + \infty} A_{λ} = 0$ . □

Conclusion of the proof of Theorem 1.4.

For any $m \in N$ , let us introduce the finite-dimensional subspace V of E defined by $\begin{matrix} V : = span {(φ_{1}, φ_{1}), (φ_{2}, φ_{2}), \dots, (φ_{m}, φ_{m})}, \end{matrix}$ where ${φ_{i}}_{i = 1}^{m} \subset C_{0, rad}^{\infty} (R^{2})$ is a collection of smooth functions with disjoint supports. It is worthwhile to mention that the results obtained in Sections 3 and 4 hold for any $λ > 0$ . Thus, in view of Lemma 3.2, it follows that $I_{λ}$ satisfies condition $(J_{1})$ . Since in V the norms are equivalent, there exists $c_{m} > 0$ , depending on m, such that $\begin{matrix} I_{λ} (u, u) ⩽ \frac{1}{2} M (‖ u ‖^{2}) + \frac{1}{2} L (‖ u ‖^{2}) - λ c_{m} ξ ‖ u ‖^{θ}, \forall (u, u) \in V, \end{matrix}$ where we have used (1.6). In view of (3.3) and (3.4), we obtain $\begin{matrix} I_{λ} (u, u) ⩽ (a_{0} + b_{0}) ‖ u ‖^{2} + \frac{(a_{1} + b_{1})}{2} ‖ u ‖^{4} - λ c_{m} ξ ‖ u ‖^{θ}, \forall (u, u) \in V . \end{matrix}$ By considering $a = a_{0} + b_{0}$ , $b = (a_{1} + b_{1}) / 2$ , $c = c_{m} ξ$ and $t = ‖ u ‖$ in Lemma 7.2, we conclude that $\begin{matrix} max_{(u, u) \in V} I_{λ} (u, u) ⩽ A_{λ}, \end{matrix}$ which implies that $I_{λ}$ satisfies condition $(J_{2})$ . Moreover, ${lim}_{λ \to + \infty} A_{λ} = 0$ . Thus, there exists $Λ_{m} > 0$ in such way that $\begin{matrix} A_{λ} < \frac{ν_{0} (θ - 4) π}{2 α_{0} θ}, \forall λ > Λ_{m} . \end{matrix}$ Therefore, in view of Proposition 4.4, we are able to apply Theorem 7.1 to obtain m pairs of nontrivial critical points of $I_{λ}$ , which are nontrivial solutions of System ( S λ ) after we apply the Palais Symmetric Criticality Principle. This finalizes the proof. □

Footnotes

Acknowledgements

The authors thank the editor and the referee for their valuable comments and suggestions.

Research supported in part by CAPES Cod 001 and CNPq grants 310747/2019-8 and 305726/2017-0.

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