# Existence and multiplicity of solutions to a Kirchhoff type elliptic system
with Trudinger-Moser growth

Shengbing Deng  S. Deng-School of Mathematics and Statistics, Southwest
University, Chongqing 400715, People’s Republic of China<EMAIL_ADDRESS>and  Xingliang Tian  X. Tian-School of Mathematics and Statistics, Southwest
University, Chongqing 400715, People’s Republic of China.
<EMAIL_ADDRESS>
Abstract: This paper deals with the existence and multiplicity of solutions
for a class of Kirchhoff type elliptic systems involving nonlinearities with
Trudinger-Moser exponential growth. We first study the existence of solutions
for the following system:

$\displaystyle\left\\{\begin{array}[]{ll}-\big{(}a_{1}+b_{1}\|u\|^{2(\theta_{1}-1)}\big{)}\Delta
u=\lambda H_{u}(x,u,v)&\ \mbox{in}\ \ \ \Omega,\\\\[5.69054pt]
-\big{(}a_{2}+b_{2}\|v\|^{2(\theta_{2}-1)}\big{)}\Delta v=\lambda
H_{v}(x,u,v)&\ \mbox{in}\ \ \ \Omega,\\\\[5.69054pt] u=0,v=0&\ \mbox{on}\ \ \
\partial\Omega,\end{array}\right.$

where $\Omega$ is a bounded domain in $\mathbb{R}^{2}$ with smooth boundary,
$\|w\|=\big{(}\int_{\Omega}|\nabla w|^{2}dx\big{)}^{1/2}$, $H_{u}(x,u,v)$ and
$H_{v}(x,u,v)$ behave like $e^{\beta|(u,v)|^{2}}$ when
$|(u,v)|\rightarrow\infty$ for some $\beta>0$, $a_{1},a_{2}>0$,
$b_{1},b_{2}>0$, $\theta_{1},\theta_{2}>1$ and $\lambda$ is a positive
parameter. In the later part of the paper, we also discuss a new multiplicity
result for the above system with a positive parameter induced by the nonlocal
dependence. The Kirchhoff term and the lack of compactness of the associated
energy functional due to the Trudinger-Moser embedding have to be overcome via
some new techniques.

Keywords: Kirchhoff type elliptic systems; existence of solutions;
multiplicity of solutions; Trudinger-Moser inequality.

_2010 Mathematics Subject Classification:_ 35J50, 35J57.

##  1\. Introduction

In this paper, we consider a class of systems of the form

(1.4)
$\displaystyle\left\\{\begin{array}[]{ll}-\big{(}a_{1}+b_{1}\|u\|^{2(\theta_{1}-1)}\big{)}\Delta
u=\lambda H_{u}(x,u,v)&\ \mbox{in}\ \ \ \Omega,\\\\[5.69054pt]
-\big{(}a_{2}+b_{2}\|v\|^{2(\theta_{2}-1)}\big{)}\Delta v=\lambda
H_{v}(x,u,v)&\ \mbox{in}\ \ \ \Omega,\\\\[5.69054pt] u=0,v=0&\ \mbox{on}\ \ \
\partial\Omega,\end{array}\right.$

where $\Omega$ is a bounded domain in $\mathbb{R}^{2}$ with smooth boundary,
$\|w\|=\big{(}\int_{\Omega}|\nabla w|^{2}dx\big{)}^{1/2}$, $a_{1},a_{2}>0$,
$b_{1},b_{2}>0$, $\theta_{1},\theta_{2}>1$ and $\lambda$ is a positive
parameter. $H_{u}$ and $H_{v}$ have the maximal growth which allows treating
(1.4) variationally in the Sobolev space $H^{1}_{0}(\Omega,\mathbb{R}^{2})$.
Here $H^{1}_{0}(\Omega,\mathbb{R}^{2})$ denotes the Sobolev space modeled in
$L^{2}(\Omega,\mathbb{R}^{2})$ with the scalar product

$\ \langle W,\Phi\rangle=\int_{\Omega}\nabla u\cdot\nabla\varphi
dx+\int_{\Omega}\nabla v\cdot\nabla\psi dx,$

where $W=(u,v)$ and $\Phi=(\varphi,\psi)$, to which corresponds the norm
$\|W\|=\langle W,W\rangle^{1/2}$. We shall consider the variational situation
in which

$\ \big{(}H_{u}(x,W),H_{v}(x,W)\big{)}=\nabla H(x,W)$

for a function $H:\Omega\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ of class
$C^{1}$, where $\nabla H$ stands for the gradient of $H$ in the variables
$W=(u,v)\in\mathbb{R}^{2}$.

We mention that a great attention has been focused on the study of problems
involving exponential growth nonlinearities, which are related to the famous
Trudinger-Moser inequality. Let $\Omega$ be a bounded domain in
$\mathbb{R}^{2}$, and denote by $H_{0}^{1}(\Omega)$ the standard first order
Sobolev space given by

$H_{0}^{1}(\Omega)=cl\Big{\\{}u\in C^{\infty}_{0}(\Omega)\ :\
\int_{\Omega}|\nabla u|^{2}dx<\infty\Big{\\}},$

with the inner product $\langle u,v\rangle=\int_{\Omega}\nabla u\cdot\nabla
vdx$ and the norm $\|u\|=\langle u,u\rangle^{1/2}$. This space is a limiting
case for the Sobolev embedding theorem, which yields
$H_{0}^{1}(\Omega)\hookrightarrow L^{p}(\Omega)$ for all $1\leq p<\infty$, but
one knows by easy examples that $H_{0}^{1}(\Omega)\not\subseteq
L^{\infty}(\Omega)$. Hence, one is led to look for a function
$g:\mathbb{R}\to\mathbb{R}^{+}$ with maximal growth such that

$\sup\limits_{u\in H_{0}^{1}(\Omega),\|u\|\leq 1}\int_{\Omega}g(u)dx<\infty.$

It was shown that by Trudinger [35] and Moser [31] that the maximal growth is
of exponential type. More precisely, $\exp(\beta u^{2})\in L^{1}(\Omega),\
\forall\ u\in H_{0}^{1}(\Omega),\ \forall\ \beta>0$, and

(1.5) $\displaystyle\sup\limits_{u\in H_{0}^{1}(\Omega),\|u\|\leq
1}\int_{\Omega}\exp(\beta u^{2})dx<\infty,\quad\mbox{if}\ \beta\leq 4\pi,$

where $4\pi$ is the sharp constant in the sense that the supreme in the left
is $+\infty$ if $\beta>4\pi$. Therefore, from this result they have naturally
associated notions of criticality and subcriticality, namely, they say that a
function $g:\Omega\times\mathbb{R}\to\mathbb{R}$ has subcritical growth if

$\lim_{|u|\rightarrow\infty}\frac{|g(x,u)|}{\exp\left(\beta
u^{2}\right)}=0,\text{uniformly on }x\in\Omega,\ \forall\beta>0,$

and $g$ has critical growth on $\Omega$ if there exists $\beta_{0}>0$ such
that

$\lim_{|u|\rightarrow\infty}\frac{|g(x,u)|}{\exp\left(\beta
u^{2}\right)}=\begin{cases}0,&\text{uniformly on }x\in\Omega,\
\forall\beta>\beta_{0},\\\\[3.0pt] \infty,&\text{uniformly on }x\in\Omega,\
\forall\beta<\beta_{0}.\end{cases}$

We should stress that some results about the existence of solutions for
elliptic problems involving exponential nonlinearities has been obtained by
Adimurthi, de Figueiredo-do Ó-Ruf, de Figueiredo-Miyagaki-Ruf, do Ó, Lam-Lu,
etc [1, 11, 12, 14, 23, 24, 25, 26] and the references therein. Moreover, de
Souza [13] has studied the existence of solutions for a singular class of
elliptic systems involving critical exponential growth in $\mathbb{R}^{2}$.

On the other hand, system (1.4) is related to the stationary analogue of the
equation

(1.6) $\displaystyle u_{tt}-\left(a+b\int_{\Omega}|\nabla
u|^{2}dx\right)\triangle u=g(x,u)$

proposed by Kirchhoff [28] as an extension of the classical D’Alembert’s wave
equation for free vibrations of elastic strings. Kirchhoff’s model takes into
account the changes in length of the string produced by transverse vibrations.
Equation (1.6) received much attention only after Lions [30] proposed an
abstract framework to the problem. Equation (1.6) is called nonlocal because
of the term $a+b\int_{\Omega}|\nabla u|^{2}dx$ which implies that the equation
in (1.6) is no longer a pointwise identity. The presence of the term
$a+b\int_{\Omega}|\nabla u|^{2}dx$ provokes some mathematical difficulties
which makes the study of such a class of problems particularly interesting. We
refer to [3, 7, 8, 9, 19, 18, 21, 22, 27, 32, 33] for more existence results
of the Kirchhoff type equations. In particular, let
$\Omega\subset\mathbb{R}^{2}$ be a bounded domain, Figueiredo and Severo [18]
studied the existence of positive ground state solutions for a nonlocal
Kirchhoff problem of the type

$\displaystyle\left\\{\begin{array}[]{ll}-m\left(\int_{\Omega}|\nabla
u|^{2}dx\right)\Delta u=f(x,u),&\mbox{in}\ \ \Omega,\\\\[5.69054pt]
u=0,&\mbox{on}\ \ \partial\Omega,\end{array}\right.$

where $m$ is a Kichhoff type function. Naimen and Tarsi [32] studied the
multiple solutions for the following problem

$\displaystyle\left\\{\begin{array}[]{ll}-\big{(}1+b\int_{\Omega}|\nabla
u|^{2}dx\big{)}\Delta u=\lambda f(x,u),\ u\geq 0&\mbox{in}\ \
\Omega,\\\\[5.69054pt] u=0,&\mbox{on}\ \ \partial\Omega,\end{array}\right.$

where $b,\ \lambda$ are positive parameters, and the authors adjusted $b$ and
$\lambda$ to get the multiple solutions. Arora etc. [2] studied a system of
Kichhoff equations with exponential nonlinearity of Choquard type, and de
Albuquerque etc. [10] have studied a class of Kirchhoff systems involving
critical growth in the whole $\mathbb{R}^{2}$. Recently, mainly inspired by
[13, 18, 32], we have studied system (1.4) on Heisenberg group for some
special Kirchhoff functions [16], and see also [17, 34] for system (1.4) with
$\lambda=1$ and abstract Kichhoff type functions.

Motivated by the above results, in the present paper, we will first prove the
existence of solutions for a Kirchhoff type elliptic system (1.4), and we will
treat both the subcritical case and the critical case, which we define next.
Let $h(x,W)$ be a function in $\overline{\Omega}\times\mathbb{R}^{2}$. We say
that

$(TM)^{sc}$: $h$ has subcritical growth at $\infty$ if

$\lim_{|W|\to\infty}\frac{|h(x,W)|}{\exp(\beta|W|^{2})}=0,\ \ \text{uniformly
on }x\in\Omega,\ \forall\beta>0.$

$(TM)^{c}$: $h$ has critical growth at $\infty$ if there exists $\beta_{0}>0$
such that

$\lim_{|W|\to\infty}\frac{|h(x,W)|}{\exp(\beta|W|^{2})}=\begin{cases}0,&\text{uniformly
on }x\in\Omega,\ \forall\beta>\beta_{0},\\\\[3.0pt] \infty,&\text{uniformly on
}x\in\Omega,\ \forall\beta<\beta_{0}.\end{cases}$

Let us introduce the precise assumptions under which our problem is studied.
Set

$\displaystyle\theta_{max}=$ $\displaystyle\max\\{\theta_{1},\theta_{2}\\},\
\theta_{min}=\min\\{\theta_{1},\theta_{2}\\}>1,$ $\displaystyle a_{max}=$
$\displaystyle\max\\{a_{1},a_{2}\\},\ a_{min}=\min\\{a_{1},a_{2}\\}>0,$
$\displaystyle b_{max}=$ $\displaystyle\max\\{b_{1},b_{2}\\},\
b_{min}=\min\\{b_{1},b_{2}\\}>0.$

We give some assumptions on $H$ as follows:

  * ($H_{1}$):

$H(x,\cdot,\cdot)\in C^{1}(\mathbb{R}^{2})$, $H(x,W)\geq 0$, $uH_{u}(x,W)\geq
0$, $vH_{v}(x,W)\geq 0$ for all $W=(u,v)\in\mathbb{R}^{2}$, uniformly on
$x\in\Omega$.

  * ($H_{2}$):

there exist constants $S_{0},\ M_{0}>0$ such that

$0<H(x,W)\leq M_{0}|\nabla H(x,W)|$, for all $|W|\geq S_{0}$ uniformly on
$x\in\Omega$.

  * ($\overline{H_{2}}$):

there exist $\mu\geq 2\theta_{max}$ and $R>0$ such that $0<\mu H(x,W)\leq
W\cdot\nabla H(x,W)$, for all $(x,|W|)\in\Omega\times[R,+\infty)$.

  * ($SL$):

$\limsup_{|W|\rightarrow 0}\frac{H(x,W)}{|W|^{2}}=0$ uniformly on
$x\in\Omega$; superlinearity at 0.

  * ($AL$):

$\limsup_{|W|\rightarrow 0}\frac{H(x,W)}{|W|^{2}}=\frac{a_{min}}{2}$ uniformly
on $x\in\Omega$; asymptotically linearity at 0.

######  Remark 1.1.

We observe that condition $(H_{2})$ implies that for each $\xi>0$, there
exists $R_{\xi}>0$ such that

(1.9) $0<\xi H(x,W)\leq|W||\nabla H(x,W)|,\ \
\forall(x,|W|)\in\Omega\times[R_{\xi},+\infty),$

then for some $C_{1},C_{2}>0$, it holds that

(1.10) $H(x,W)\geq C_{1}|W|^{\xi}-C_{2},\ \
\forall(x,W)\in\Omega\times\mathbb{R}^{2}.$

Furthermore, condition $(SL)$ or $(AL)$ implies $H(x,0,0)=0$ uniformly on
$x\in\Omega$. It is worth to mention that condition $(\overline{H_{2}})$ is
slightly stronger than (1.9).

We say that $W=(u,v)\in H^{1}_{0}(\Omega,\mathbb{R}^{2})$ is a weak solution
of problem (1.4) if

$\ \big{(}a_{1}+b_{1}\|u\|^{2(\theta_{1}-1)}\big{)}\int_{\Omega}\nabla
u\cdot\nabla\varphi
dx+\big{(}a_{2}+b_{2}\|v\|^{2(\theta_{2}-1)}\big{)}\int_{\Omega}\nabla
v\cdot\nabla\psi dx=\lambda\int_{\Omega}\Phi\cdot\nabla H(x,W)dx,$

for all $\Phi=(\varphi,\psi)\in H^{1}_{0}(\Omega,\mathbb{R}^{2})$. We first
give the results about the existence of solutions for both subcritical case
and critical case:

######  Theorem 1.2.

Suppose that $H_{u},\ H_{v}$ have subcritical growth $(TM)^{sc}$, and
$(H_{1})$, $(H_{2})$, $(\overline{H_{2}})$, $(SL)$ are satisfied. Then,
problem (1.4) has at least one nontrivial solution for each $\lambda>0$.

######  Theorem 1.3.

Suppose that $H_{u},\ H_{v}$ have critical growth $(TM)^{c}$, and $(H_{1})$,
$(H_{2}),(SL)$ are satisfied. Moreover, if

  * ($H_{3}$):

$0<2\theta_{max}H(x,W)\leq W\cdot\nabla H(x,W)$, for all
$(x,W)\in\Omega\times\mathbb{R}^{2}\backslash\\{(0,0)\\}$,

then there exists a constant $\lambda_{*}>0$ such that problem (1.4) admits at
least one nontrivial solution with positive energy for all
$\lambda\geq\lambda_{*}$.

Next, we consider the multiplicity of solutions for problem (1.4). Recently,
Fiscella, Pucci and Zhang in [20] obtained the existence of solution for a
$p$-fractional Hardy-Schrödinger-Kirchhoff system in $\mathbb{R}^{N}$ with
critical nonlinearities as follows

$\displaystyle\left\\{\begin{array}[]{ll}M(\|(u,v)\|^{p})(\mathcal{L}^{s}_{p}u+V(x)|u|^{p-2}u)-\sigma\frac{|u|^{p-2}u}{|x|^{ps}}=\lambda
H_{u}(x,u,v)+\frac{\psi}{p^{*}_{s}}|v|^{\omega}|u|^{\psi-2}u,\\\\[5.69054pt]
M(\|(u,v)\|^{p})(\mathcal{L}^{s}_{p}v+V(x)|v|^{p-2}v)-\sigma\frac{|v|^{p-2}v}{|x|^{ps}}=\lambda
H_{v}(x,u,v)+\frac{\omega}{p^{*}_{s}}|u|^{\psi}|v|^{\omega-2}v,\end{array}\right.$

where $0<s<1<p<\infty,\ sp<N,\ \psi>1$ and $\omega>1$ with
$\psi+\omega=p^{*}_{s}$ and $p^{*}_{s}=\frac{Np}{N-sp}$, and the authors
adjusted $\lambda$ (for $\lambda$ large enough) to get the existence of
solution.

Motivated by the works mentioned above and also [32], the main purpose of the
second part of this paper is to investigate the multiplicity of solutions for
subcritical case and another type solution for critical case which is
different from the Mountain pass solution. As stated in Lemma 6.1, we will
define $\Lambda_{0}(b_{min})$ (see (6.3)) depending on $b_{min}$ such that
$\Lambda_{0}(b_{min})\rightarrow\infty$ as $b_{min}\rightarrow\infty$. Then
there exists a value $b_{0}>0$ (see (6.4)) such that for all $b_{min}>b_{0}$,
it holds that $\Lambda_{0}(b_{min})>\frac{a_{max}}{a_{min}}\lambda_{1}$, where

(1.12) $\lambda_{1}:=\inf_{W\in
H^{1}_{0}(\Omega,\mathbb{R}^{2})\setminus\\{(0,0)\\}}\frac{\|W\|^{2}}{\int_{\Omega}|W|^{2}dx}>0.$

######  Theorem 1.4.

Suppose that $H_{u},\ H_{v}$ have subcritical growth $(TM)^{sc}$, and
$(H_{1})$, $(H_{2})$, $(\overline{H_{2}})$, $(AL)$ are satisfied. Then when
$b_{min}>b_{0}$, problem (1.4) has at least two nontrivial solutions for each
$\frac{a_{max}}{a_{min}}\lambda_{1}<\lambda<\Lambda_{0}(b_{min})$.

######  Theorem 1.5.

Suppose that $H_{u},\ H_{v}$ have critical growth $(TM)^{c}$, and $(H_{1})$,
$(H_{2}),(AL)$ are satisfied. Then when $b_{min}>b_{0}$, problem (1.4) has at
least one nontrivial solution with negative energy for each
$\frac{a_{max}}{a_{min}}\lambda_{1}<\lambda<\Lambda_{0}(b_{min})$.

Next, we make some remarks about above theorems.

######  Remark 1.6.

Theorem 1.3, we can quantify the number $\lambda_{*}$ for some special
examples. Such as for $H(x,W)=|W|^{\mu}\exp(\beta_{0}|W|^{2})$ with $\mu\geq
2\theta_{max}$ and $\beta_{0}>0$. We fix a function $W_{0}=(u_{0},v_{0})\in
H^{1}_{0}(\Omega,\mathbf{R}^{2})$ satisfying $\|W_{0}\|=1$ such that

$S_{\mu}:=\inf_{W\in
H^{1}_{0}(\Omega,\mathbf{R}^{2})\setminus\\{(0,0)\\}}\frac{\|W\|}{(\int_{\Omega}|W|^{\mu}dx)^{1/\mu}}=\left(\int_{\Omega}|W_{0}|^{\mu}dx\right)^{-1/\mu}.$

Therefore the mountain pass level $c_{M,\lambda}$ satisfies

$c_{M,\lambda}\leq\max_{t\geq
0}\left[\frac{a_{max}}{2}t^{2}+\frac{b_{max}}{2\theta_{min}}\left(t^{2\theta_{1}}\|u_{0}\|^{2\theta_{1}}+t^{2\theta_{2}}\|v_{0}\|^{2\theta_{2}}\right)-\lambda
t^{\mu}S^{-\mu}_{\mu}\right].$

It is enough to take

$\lambda_{*}=\max\left\\{\frac{2a_{max}}{\mu
S^{-\mu}_{\mu}}\left(\frac{b_{max}}{a_{max}\theta_{min}}\right)^{\frac{\mu-2}{2\theta_{max-2}}},\frac{2a_{max}}{\mu
S^{-\mu}_{\mu}},\Gamma_{*}\right\\}>0$

where

$\Gamma_{*}=\left\\{\frac{\min\bigg{\\{}\frac{1}{2}\Big{(}a_{1}\frac{2\pi}{\beta_{0}}+\frac{b_{1}}{\theta_{1}}\big{(}\frac{2\pi}{\beta_{0}}\big{)}^{\theta_{1}}\Big{)},\frac{1}{2}\Big{(}a_{2}\frac{2\pi}{\beta_{0}}+\frac{b_{2}}{\theta_{2}}\big{(}\frac{2\pi}{\beta_{0}}\big{)}^{\theta_{2}}\Big{)}\bigg{\\}}}{a_{max}\left(1-\frac{2}{\mu}\right)\left[\frac{2a_{max}}{\mu
S^{-\mu}_{\mu}}\right]^{\frac{2}{\mu-2}}}\right\\}^{\frac{2}{\mu-2}},$

such that for each $\lambda>\lambda_{*}$, it holds that

$c_{M,\lambda}<\min\bigg{\\{}\frac{1}{2}\Big{(}a_{1}\frac{2\pi}{\beta_{0}}+\frac{b_{1}}{\theta_{1}}\big{(}\frac{2\pi}{\beta_{0}}\big{)}^{\theta_{1}}\Big{)},\frac{1}{2}\Big{(}a_{2}\frac{2\pi}{\beta_{0}}+\frac{b_{2}}{\theta_{2}}\big{(}\frac{2\pi}{\beta_{0}}\big{)}^{\theta_{2}}\Big{)}\bigg{\\}}.$

######  Remark 1.7.

In Theorem 1.5, the reason why we couldn’t get another solution with positive
energy as in Theorem 1.4, is the difficulty of keeping the mountain pass value
less than some proper level. In fact, by the standard assumption

  * ($H_{4}$):

there exists $\eta>0$ such that

$\liminf_{|W|\rightarrow\infty}\frac{W\cdot\nabla
H_{u}(x,W)}{\exp(\beta_{0}|W|^{2})}\geq\eta>\max\left\\{\frac{4(a_{1}+b_{1}\big{(}\frac{4\pi}{\beta_{0}}\big{)}^{\theta_{1}-1})}{\beta_{0}d^{2}e},\frac{4(a_{2}+b_{2}\big{(}\frac{4\pi}{\beta_{0}}\big{)}^{\theta_{2}-1})}{\beta_{0}d^{2}e}\right\\},$

then by using some proper functions (see [11, 17]), we can prove the mountain
pass level $c_{M}$ satisfying

$\displaystyle\begin{split}c_{M}<\min\bigg{\\{}\frac{1}{2}\Big{(}a_{1}\frac{4\pi}{\beta_{0}}+\frac{b_{1}}{\theta_{1}}\big{(}\frac{4\pi}{\beta_{0}}\big{)}^{\theta_{1}}\Big{)},\frac{1}{2}\Big{(}a_{2}\frac{4\pi}{\beta_{0}}+\frac{b_{2}}{\theta_{2}}\big{(}\frac{4\pi}{\beta_{0}}\big{)}^{\theta_{2}}\Big{)}\bigg{\\}}.\end{split}$

However, it is not enough to make sure that functional satisfies Palais-Smale
condition by Lions’ concentration-compactness lemma, see (5.6) in Lemma 5.3.

######  Remark 1.8.

There are many functions satisfying conditions in each theorem:

  * ($i$)

in Theorem 1.2: $H(x,W)=|W|^{\mu}[\exp(\alpha_{0}|W|)-1]$ with $\mu\geq 2$ and
$\alpha_{0}>0$ which satisfies $(\overline{H_{2}})$ but does not satisfy
$({H_{3}})$ for $2\leq\mu<2\theta_{max}$;

  * ($ii$)

in Theorem 1.3:
$H(x,W)=\frac{|W|^{\mu}\exp(\beta_{0}|W|^{2})}{2^{\alpha_{0}|W|^{2}}}$ with
$\mu\geq 2\theta_{max}$ and $0\leq\alpha_{0}<\frac{\beta_{0}}{\log 2}$;

  * ($iii$)

in Theorem 1.4: $H(x,W)=\frac{a_{min}}{2}|W|^{2}\exp(\alpha_{0}|W|)$ with
$\alpha_{0}>0$ which satisfies $(\overline{H_{2}})$ but does not satisfy
$({H_{3}})$;

  * ($iv$)

in Theorem 1.5: $H(x,W)=\frac{a_{min}}{2}|W|^{2}\exp(\beta_{0}|W|^{2})$ with
$\beta_{0}>0$ which does not satisfy $({H_{3}})$.

Moreover, it is worth noting that if we add conditions in $(H_{1})$ that
$H_{u}(x,u,v)\equiv 0$ and $H_{v}(x,u,v)\equiv 0$ for $u\leq 0$ or $v\leq 0$,
uniformly on $x\in\Omega$, then by Maximum principle, we can proof that the
solutions we obtain in above theorems are entire positive, i.e. $u>0$ and
$v>0$ in $\Omega$, and the example functions only need to be defined as
$\overline{H}(x,W)=H(x,W)\mbox{sign}u\ \mbox{sign}v$, where $\mbox{sign}t=0$
for $t\leq 0$ and $\mbox{sign}t=1$ for $t>0$.

This paper is organized as follows: Section 2 contains some technical results.
In Section 3, we present the variational setting in which our problem will be
treated. Section 4 is devoted to showing some properties of the Palais-Samle
sequences. Then, we split Section 5 into two subsections for the subcritical
case and critical case, and we complete the proofs of Theorems 1.2-1.3. And
then, we will study the multiple solutions of problem (1.4). In section 6, we
construct an important variational framework and complete the proofs of
Theorems 1.3-1.4. Hereafter, $C,C_{0},C_{1},C_{2}...$ will denote positive
(possibly different) constants.

##  2\. Some preliminary results

We first introduce some famous inequalities as follows, and inspired by those
inequalities, we conclude some similar forms of inequalities.

######  Lemma 2.1.

[31] Let $\Omega$ be a bounded domain in $\mathbb{R}^{2}$ and $u\in
H^{1}_{0}(\Omega)$. Then for every $\beta>0$,

$\displaystyle\ \int_{\Omega}\exp(\beta|u|^{2})dx<\infty.$

Moreover, there exists a constant $C(\Omega)>0$ such that

$\sup_{u\in H^{1}_{0}(\Omega):\|u\|\leq
1}\int_{\Omega}\exp(\beta|u|^{2})dx\leq C(\Omega),\ \ \forall\beta\leq 4\pi.$

Furthermore, $4\pi$ is the best constant, that is, the supreme in the left is
$\infty$ if $\beta>4\pi$.

######  Lemma 2.2.

[29] Let $\\{u_{n}\\}$ be a sequence of functions in $H^{1}_{0}(\Omega)$ with
$\|\nabla u_{n}\|_{2}=1$ such that $u_{n}\rightharpoonup u\neq 0$ weakly in
$H^{1}_{0}(\Omega)$. Then for any $0<p<4\pi/(1-\|\nabla u\|^{2}_{2})$ we have

$\sup_{n}\int_{\Omega}\exp(p|u_{n}|^{2})dx<\infty.$

Now, by the above results, we begin to establish the Trudinger-Moser type
inequalities in $H^{1}_{0}(\Omega,\mathbb{R}^{2})$:

######  Lemma 2.3.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{2}$ and $(u,v)\in
H^{1}_{0}(\Omega,\mathbb{R}^{2})$. Then for every $\beta>0$,

(2.1) $\displaystyle\int_{\Omega}\exp(\beta|(u,v)|^{2})dx<\infty.$

Moreover, it holds that

(2.2) $\sup_{(u,v)\in H^{1}_{0}(\Omega,\mathbb{R}^{2}),\|(u,v)\|\leq
1}\int_{\Omega}\exp\left(\beta|(u,v)|^{2}\right)dx\leq C(\Omega),\ \
\forall\beta\leq 4\pi,$

where $C(\Omega)$ is given as in Lemma 2.1.

###### Proof.

For each $W=(u,v)\in H^{1}_{0}(\Omega,\mathbb{R}^{2})$, we have that
$|W|^{2}=|u|^{2}+|v|^{2}$. Then by using Hölder’s inequality and Lemma 2.1,

$\displaystyle\begin{split}\int_{\Omega}\exp(\beta|W|^{2})dx=&\int_{\Omega}\exp(\beta|u|^{2})\exp(\beta|v|^{2})dx\\\
\leq&\left(\int_{\Omega}\exp(2\beta|u|^{2})dx\right)^{1/2}\left(\int_{\Omega}\exp(2\beta|v|^{2})dx\right)^{1/2}<\infty.\end{split}$

Thus (2.1) holds.

Then we begin to prove (2.2). It is sufficiently enough to prove that

(2.3) $\sup_{(u,v)\in
H^{1}_{0}(\Omega,\mathbb{R}^{2}),\|(u,v)\|=1}\int_{\Omega}\exp\left(4\pi|(u,v)|^{2}\right)dx\leq
C(\Omega).$

Indeed, for each $(u,v)\in H^{1}_{0}(\Omega,\mathbb{R}^{2})$ with
$\|(u,v)\|=1$, then $\|(u,v)\|^{2}=\|u\|^{2}+\|v\|^{2}=1$. If $\|u\|=0$ or
$\|v\|=0$, then from (2.1), we know (2.3) holds also. Otherwise, $\|u\|\neq 0$
and $\|v\|\neq 0$, then we take $\frac{1}{p}=\|u\|^{2}$,
$\frac{1}{q}=\|v\|^{2}$ and by the Young’s inequality, we have

$\begin{split}\int_{\Omega}\exp\left(4\pi|(u,v)|^{2}\right)dx=&\int_{\Omega}\exp\left(4\pi
u^{2}\right)\exp\left(4\pi v^{2}\right)dx\\\
\leq&\int_{\Omega}\frac{\exp\left(4\pi
pu^{2}\right)}{p}dx+\int_{\Omega}\frac{\exp\left(4\pi qv^{2}\right)}{q}dx\\\
=&\|u\|^{2}\int_{\Omega}\exp\left(4\pi\left(\frac{u}{\|u\|}\right)^{2}\right)dx+\|v\|^{2}\int_{\Omega}\exp\left(4\pi\left(\frac{v}{\|v\|}\right)^{2}\right)dx\\\
\leq&\|u\|^{2}C(\Omega)+\|v\|^{2}C(\Omega)=C(\Omega),\end{split}$

thus (2.3) holds and the proof is complete. ∎

######  Lemma 2.4.

Let $\\{W_{n}\\}$ be a sequence of functions in
$H^{1}_{0}(\Omega,\mathbb{R}^{2})$ with $\|W_{n}\|=1$ such that
$W_{n}\rightharpoonup W_{0}\neq(0,0)$ weakly in
$H^{1}_{0}(\Omega,\mathbb{R}^{2})$. Then for any $0<p<4\pi/(1-\|W_{0}\|^{2})$,
we have

$\sup_{n}\int_{\Omega}\exp(p|W_{n}|^{2})dx<\infty.$

###### Proof.

Since $W_{n}\rightharpoonup W_{0}$ and $\|W_{n}\|=1$, we conclude that

$\displaystyle\|W_{n}-W_{0}\|^{2}=1-2\langle
W_{n},W_{0}\rangle+\|W_{0}\|^{2}\rightarrow 1-\|W_{0}\|^{2}<\frac{4\pi}{p}$

Thus, for large $n$ we have

$\displaystyle p\|W_{n}-W_{0}\|^{2}<4\pi.$

Now choosing $q>1$ close to 1 and $\epsilon>0$ satisfying

$\displaystyle qp(1+\epsilon^{2})\|W_{n}-W_{0}\|^{2}\leq 4\pi.$

Then Lemma 2.3 indicates that

(2.4)
$\displaystyle\int_{\Omega}\exp\left(qp(1+\epsilon^{2})|W_{n}-W_{0}|^{2}\right)dx\leq
C(\Omega).$

Moreover, since

$\displaystyle p|W_{n}|^{2}\leq
p(1+\epsilon^{2})|W_{n}-W_{0}|^{2}+p(1+1/\epsilon^{2})|W_{0}|^{2}$

which can be proved by Young’s inequality, it follows that

$\displaystyle\exp(p|W_{n}|^{2})\leq\exp(p(1+\epsilon^{2})|W_{n}-W_{0}|^{2})\exp(p(1+1/\epsilon^{2})|W_{0}|^{2}).$

Consequently, by Hölder’s inequality and (2.4), it holds that

$\displaystyle\int_{\Omega}\exp(p|W_{n}|^{2})dx\leq$
$\displaystyle\left(\int_{\Omega}\exp(qp(1+\epsilon^{2})|W_{n}-W_{0}|^{2})dx\right)^{\frac{1}{q}}\left(\int_{\Omega}\exp(rp(1+1/\epsilon^{2})|W_{0}|^{2})dx\right)^{\frac{1}{r}}$
$\displaystyle\leq$ $\displaystyle
C\left(\int_{\Omega}\exp(rp(1+1/\epsilon^{2})|W_{0}|^{2})dx\right)^{\frac{1}{r}},$

for large $n$, where $r=\frac{q}{q-1}$. By Lemma 2.3, we know the second term
in the last inequality is bounded, thus the proof is complete. ∎

######  Lemma 2.5.

If $W=(u,v)\in H^{1}_{0}(\Omega,\mathbb{R}^{2}),\beta>0,q>0$ and $\|W\|\leq N$
with $\beta N^{2}<2\pi$, there exists $C=C(\beta,N,q)>0$ such that

(2.5) $\displaystyle\int_{\Omega}|W|^{q}\exp(\beta|W|^{2})dx\leq C\|W\|^{q}.$

###### Proof.

We consider $r>1$ close to 1 such that $r\beta N^{2}\leq 2\pi$ and $sq\geq 1$,
where $s=\frac{r}{r-1}$. Using Hölder’s inequality and Young’s inequality, we
have

$\displaystyle\begin{split}\int_{\Omega}|W|^{q}\exp(\beta|W|^{2})dx\leq&\left(\int_{\Omega}\exp(r\beta|W|^{2})dx\right)^{\frac{1}{r}}\|W\|^{q}_{qs}\\\
\leq&\left(\int_{\Omega}\exp(2r\beta
N^{2}(\frac{u}{\|W\|})^{2})dx+\int_{\Omega}\exp(2r\beta
N^{2}(\frac{v}{\|W\|})^{2})dx\right)^{\frac{1}{r}}\|W\|^{q}_{qs}\end{split}$

Finally, using the continuous embedding
$H^{1}_{0}(\Omega,\mathbb{R}^{2})\hookrightarrow
L^{sq}(\Omega,\mathbb{R}^{2})$ and Lemma 2.1, we conclude that

$\displaystyle\int_{\Omega}|W|^{q}\exp(\beta|W|^{2})dx\leq C\|W\|^{q}.$

∎

Finally, in order to deal with the dual Kirchhoff functions, in Section 6 we
need the following algebraic inequality:

######  Lemma 2.6.

If $x,y\geq 0,\ t\geq 1$, then

(2.6)
$\displaystyle(x+y)^{t}\leq(2^{t}-1)(x^{t}+y^{t})\leq(2^{t}-1)(x+y)^{t}.$

###### Proof.

It is obvious that $x^{t}+y^{t}\leq(x+y)^{t}$. If $0\leq y\leq x$, then we fix
$y$ and define $g(x)=(x^{t}+y^{t})-\frac{(x+y)^{t}}{2^{t}-1}$. It holds that

$\displaystyle\begin{split}g(y)=&2y^{t}-\frac{2^{t}y^{t}}{2^{t}-1}=\frac{2^{t}-2}{2^{t}-1}y^{t}\geq
0,\\\
g^{\prime}(x)=&tx^{t-1}-\frac{t(x+y)^{t-1}}{2^{t}-1}\geq\frac{2^{t-1}-1}{2^{t}-1}(tx^{t-1})\geq
0,\end{split}$

thus $g(x)\geq 0$ for all $x\geq y$, i.e.
$(x+y)^{t}\leq(2^{t}-1)(x^{t}+y^{t})$. On the other hand, if $x\leq y$, the
result can be obtained by the same way. The proof is complete. ∎

##  3\. The variational framework

We now consider the functional $J_{\lambda}$ given by

$J_{\lambda}(W)=\frac{1}{2}\Big{(}a_{1}\|u\|^{2}+\frac{b_{1}}{\theta_{1}}\|u\|^{2\theta_{1}}\Big{)}+\frac{1}{2}\Big{(}a_{2}\|v\|^{2}+\frac{b_{2}}{\theta_{2}}\|v\|^{2\theta_{2}}\Big{)}-\lambda\int_{\Omega}H(x,W)dx,$

where $W=(u,v)\in H^{1}_{0}(\Omega,\mathbb{R}^{2})$.

######  Lemma 3.1.

Under our assumptions, we have that $J_{\lambda}$ is well defined and $C^{1}$
on $H^{1}_{0}(\Omega,\mathbb{R}^{2})$ for all $\lambda>0$. Moreover,

$\begin{split}\langle
J_{\lambda}^{\prime}(W),\Phi\rangle=&\big{(}a_{1}+b_{1}\|u\|^{2(\theta_{1}-1)}\big{)}\int_{\Omega}\nabla
u\cdot\nabla\varphi
dx+\big{(}a_{2}+b_{2}\|v\|^{2(\theta_{2}-1)}\big{)}\int_{\Omega}\nabla
v\cdot\nabla\psi dx\\\ &-\lambda\int_{\Omega}\Phi\cdot\nabla
H(x,W)dx,\end{split}$

where $\Phi=(\varphi,\psi)\in H^{1}_{0}(\Omega,\mathbb{R}^{2})$.

###### Proof.

We have that $H_{u}$ and $H_{v}$ are continuous and have subcritical (or
critical) growth $(TM)^{sc}$ (or $(TM)^{c}$). Thus, giving $\beta>0$ (or
$\beta>\beta_{0}$), there exists $C>0$ such that $|H_{u}(x,W)|,\
|H_{v}(x,W)|\leq Ce^{\beta|W|^{2}}$ for all
$(x,W)\in\Omega\times\mathbb{R}^{2}$. Then,

(3.1) $|\nabla H(x,W)|\leq|H_{u}(x,W)|+|H_{v}(x,W)|\leq
C_{1}\exp(\beta|W|^{2}),\ \ \text{for all
}(x,W)\in\Omega\times\mathbb{R}^{2}.$

By $(SL)$ (or $(AL)$), given $\epsilon>0$, there exists $\delta>0$ such that

(3.2) $|H(x,W)|\leq\frac{a_{min}+\epsilon}{2}|W|^{2}\ \ \text{always that
}|W|<\delta.$

Thus, using (3.1), (3.2) and $(H_{3})$, we have

$\displaystyle\int_{\Omega}|H(x,W)|dx\leq\frac{a_{min}+\epsilon}{2}\int_{\Omega}|W|^{2}dx+C_{1}\int_{\Omega}|W|\exp(\beta|W|^{2})dx.$

Considering the continuous imbedding
$H^{1}_{0}(\Omega,\mathbb{R}^{2})\hookrightarrow L^{s}(\Omega,\mathbb{R}^{2})$
for $s\geq 1$ and using Lemma 2.5, it follows that $H(x,W)\in L^{1}(\Omega)$
which implies that $J_{\lambda}$ is well defined. Using standard arguments
together with Lemma 4.3 underlying (see [4, Theorem A.VI] and [6]), we can see
that $J_{\lambda}\in
C^{1}\big{(}H^{1}_{0}(\Omega,\mathbb{R}^{2}),\mathbb{R}\big{)}$ with

$\begin{split}\langle
J_{\lambda}^{\prime}(W),\Phi\rangle=&\big{(}a_{1}+b_{1}\|u\|^{2(\theta_{1}-1)}\big{)}\int_{\Omega}\nabla
u\cdot\nabla\varphi
dx+\big{(}a_{2}+b_{2}\|v\|^{2(\theta_{2}-1)}\big{)}\int_{\Omega}\nabla
v\cdot\nabla\psi dx\\\ &-\lambda\int_{\Omega}\Phi\cdot\nabla
H(x,W)dx,\end{split}$

where $\Phi=(\varphi,\psi)\in H^{1}_{0}(\Omega,\mathbb{R}^{2})$. ∎

From Lemma 3.1, we have that the critical points of the functional
$J_{\lambda}$ are precisely the weak solutions of problem (1.4). In the next
two lemmas we check that the functional $J_{\lambda}$ satisfies the geometric
conditions of the Mountain-pass theorem.

######  Lemma 3.2.

Suppose that $(H_{2}),\ (SL)$ hold and the functions $H_{u},\ H_{v}$ have
subcritical (or critical) growth $(TM)^{sc}$ (or $(TM)^{c}$). Then, for each
$\lambda>0$, there exist positive number $\rho_{0}$ and $\tau_{0}$ such that

$\displaystyle J_{\lambda}(W)\geq\tau_{0},\ \forall W\in
H^{1}_{0}(\Omega,\mathbb{R}^{2})\ \ \mbox{with}\ \ \|W\|=\rho_{0}.$

###### Proof.

By $(SL)$, given $0<\epsilon<\frac{\lambda_{1}a_{min}}{2\lambda}$ there exists
$\delta>0$ such that

$\displaystyle H(x,W)\leq\epsilon|W|^{2},$

always that $|W|<\delta$. On the other hand, for $\beta>0$ (subcritical case)
or $\beta>\beta_{0}$ (critical case), we have that there exists $C_{1}>0$ such
that $|H_{u}(x,W)|,\ |H_{v}(x,W)|\leq C_{1}|W|^{q-1}\exp(\beta|W|^{2})$ for
all $|W|\geq\delta$ with $q>2$. Thus, by $(H_{2})$, we have

(3.3) $\displaystyle
H(x,W)\leq\epsilon|W|^{2}+C_{2}|W|^{q}\exp(\beta|W|^{2}),\
\forall(x,W)\in\Omega\times\mathbb{R}^{2}.$

Thus,

$\displaystyle\begin{split}J_{\lambda}(W)=&\frac{1}{2}\Big{(}a_{1}\|u\|^{2}+\frac{b_{1}}{\theta_{1}}\|u\|^{2\theta_{1}}\Big{)}+\frac{1}{2}\Big{(}a_{2}\|v\|^{2}+\frac{b_{2}}{\theta_{2}}\|v\|^{2\theta_{2}}\Big{)}-\lambda\int_{\Omega}H(x,W)dx\\\
\geq&\frac{1}{2}a_{min}\|W\|^{2}-\lambda\epsilon\int_{\Omega}|W|^{2}dx-\lambda
C_{2}\int_{\Omega}|W|^{q}\exp(\beta|W|^{2})dx\\\
\geq&\Big{(}\frac{a_{min}}{2}-\frac{\lambda\epsilon}{\lambda_{1}}\Big{)}\|W\|^{2}-\lambda
C_{2}\int_{\Omega}|W|^{q}\exp(\beta|W|^{2})dx.\end{split}$

By Lemma 2.5, there exists $N>0$ such that $\beta N^{2}<2\pi$ and we take
$\|W\|\leq N$, there exists $C>0$ such that
$\int_{\Omega}|W|^{q}\exp(\beta|W|^{2})dx\leq C\|W\|^{q}$. Therefore,

$\displaystyle
J_{\lambda}(W)\geq\Big{(}\frac{a_{min}}{2}-\frac{\lambda\epsilon}{\lambda_{1}}\Big{)}\|W\|^{2}-\lambda
C_{3}\|W\|^{q},$

Since $q>2$, there exists $\kappa_{1}>0$ such that
$\Big{(}\frac{a_{min}}{2}-\frac{\lambda\epsilon}{\lambda_{1}}\Big{)}\kappa^{2}_{1}-\lambda
C_{3}\kappa^{q}_{1}>0$. Consequently, taking
$\rho_{0}=\min\\{N,\kappa_{1}\\}>0$, we can get that
$J_{\lambda}(W)\geq\tau_{0}$ whenever $\|W\|=\rho_{0}$, where
$\tau_{0}=\Big{(}\frac{a_{min}}{2}-\frac{\lambda\epsilon}{\lambda_{1}}\Big{)}\rho^{2}_{0}-\lambda
C_{3}\rho^{q}_{0}>0$. Thus, the proof is finished. ∎

######  Lemma 3.3.

Suppose that $(H_{1})$, $(H_{2})$ hold and $H_{u},\ H_{v}$ have subcritical
(or critical) growth $(TM)^{sc}$ (or $(TM)^{c}$). Then there exists $E_{0}\in
H^{1}_{0}(\Omega,\mathbb{R}^{2})$ with $\|E_{0}\|>\rho_{0}$ such that
$J_{\lambda}(E_{0})<0$.

###### Proof.

(1.10) indicates that $H(x,W)\geq C_{1}|W|^{\xi}-C_{2}$ for all
$(x,W)\in\Omega\times\mathbb{R}^{2}$, and here we choose $\xi>2\theta_{max}$.
Thus,

$\displaystyle\begin{split}J_{\lambda}(tW)=&\frac{1}{2}\Big{(}a_{1}\|u\|^{2}t^{2}+\frac{b_{1}}{\theta_{1}}\|u\|^{2\theta_{1}}t^{2\theta_{1}}\Big{)}+\frac{1}{2}\Big{(}a_{2}\|v\|^{2}t^{2}+\frac{b_{2}}{\theta_{2}}\|v\|^{2\theta_{2}}t^{2\theta_{2}}\Big{)}-\lambda\int_{\Omega}H(x,W)dx\\\
\leq&\frac{1}{2}\Big{(}a_{1}\|u\|^{2}+a_{2}\|v\|^{2}\Big{)}t^{2}+\frac{b_{1}}{2\theta_{1}}\|u\|^{2\theta_{1}}t^{2\theta_{1}}+\frac{b_{2}}{2\theta_{2}}\|v\|^{2\theta_{2}}t^{2\theta_{2}}\\\
&-C_{1}t^{\xi}\int_{\Omega}|W|^{\xi}dx+C_{2}|\Omega|,\\\ \end{split}$

for all $t>0$, $W\in H^{1}_{0}(\Omega,\mathbb{R}^{2})\backslash\\{(0,0)\\}$,
and $|\Omega|$ is the Lebesgue measure of $\Omega$. This yields that
$J_{\lambda}(tW)\rightarrow-\infty$ as $t\rightarrow+\infty$. Setting
$E_{0}=tW$ with $t$ large enough such that $J_{\lambda}(E_{0})<0$ with
$\|E_{0}\|>\rho_{0}$. ∎

##  4\. On Palais-Smale sequences

In order to prove that a Palais-Smale sequence converges to a weak solution of
problem (1.4), we need to establish the following lemma:

######  Lemma 4.1.

Suppose $(H_{1})$, $(\overline{H_{2}})$ hold and $H_{u},\ H_{v}$ have
subcritical (or critical) growth $(TM)^{sc}$ (or $(TM)^{c}$). For each
$\lambda>0$, and let $\\{W_{n}\\}\ in\ H^{1}_{0}(\Omega,\mathbb{R}^{2})$ be
the Palais-Smale sequence for functional $J_{\lambda}$ at finite level. Then
there exists $C>0$ such that

(4.1) $\displaystyle\|W_{n}\|\leq C,\ \ \ \
\int_{\Omega}u_{n}H_{u}(x,W_{n})dx\leq C,\ \ \ \
\int_{\Omega}v_{n}H_{v}(x,W_{n})dx\leq C.$

###### Proof.

Let $\\{W_{n}\\}\subset H^{1}_{0}(\Omega,\mathbb{R}^{2})$ be a sequence such
that $J_{\lambda}(W_{n})\rightarrow c$ and
$J_{\lambda}^{\prime}(W_{n})\rightarrow 0$ with $|c|<\infty$, we can take this
as follows:

(4.2)
$\begin{split}J_{\lambda}(W_{n})=&\frac{1}{2}\Big{(}a_{1}\|u_{n}\|^{2}+\frac{b_{1}}{\theta_{1}}\|u_{n}\|^{2\theta_{1}}\Big{)}+\frac{1}{2}\Big{(}a_{2}\|v_{n}\|^{2}+\frac{b_{2}}{\theta_{2}}\|v_{n}\|^{2\theta_{2}}\Big{)}-\lambda\int_{\Omega}H(x,W_{n})dx\\\
=&c+\delta_{n},\ \ \end{split}$

where $\delta_{n}\rightarrow 0$ as $n\rightarrow\infty$, and

(4.3) $\begin{split}\langle
J_{\lambda}^{\prime}(W_{n}),W_{n}\rangle=&\big{(}a_{1}+b_{1}\|u_{n}\|^{2(\theta_{1}-1)}\big{)}\|u_{n}\|^{2}+\big{(}a_{2}+b_{2}\|v_{n}\|^{2(\theta_{2}-1)}\big{)}\|v_{n}\|^{2}\\\
&-\lambda\int_{\Omega}W_{n}\cdot\nabla H(x,W_{n})dx=o(\|W_{n}\|).\end{split}$

Thus for $n$ large enough, by $(\overline{H_{2}})$ we have

$\displaystyle\begin{split}C_{1}+\|W_{n}\|\geq&J_{\lambda}(W_{n})-\frac{1}{\mu}\langle
J_{\lambda}^{\prime}(W_{n}),W_{n}\rangle\\\
=&\frac{1}{2}\Big{(}a_{1}\|u_{n}\|^{2}+\frac{b_{1}}{\theta_{1}}\|u_{n}\|^{2\theta_{1}}\Big{)}+\frac{1}{2}\Big{(}a_{2}\|v_{n}\|^{2}+\frac{b_{2}}{\theta_{2}}\|v_{n}\|^{2\theta_{2}}\Big{)}\\\
&-\frac{1}{\mu}\big{(}a_{1}+b_{1}\|u_{n}\|^{2(\theta_{1}-1)}\big{)}\|u_{n}\|^{2}-\frac{1}{\mu}\big{(}a_{2}+b_{2}\|v_{n}\|^{2(\theta_{2}-1)}\big{)}\|v_{n}\|^{2}\\\
&+\int_{\Omega}\left(\frac{1}{\mu}W_{n}\cdot\nabla
H(x,W_{n})-H(x,W_{n})\right)dx\\\
\geq&\Big{(}\frac{1}{2}-\frac{1}{\mu}\Big{)}\Big{(}a_{1}\|u_{n}\|^{2}+a_{2}\|v_{n}\|^{2}\Big{)}-C_{\mu}|\Omega|\\\
\geq&\frac{\mu-2}{2\mu}a_{min}\|W_{n}\|^{2}-C_{\mu}|\Omega|,\end{split}$

for some $C_{1}>0$, where $C_{\mu}=\sup\\{|H(x,W)-\frac{1}{\mu}W\cdot\nabla
H(x,W)|:(x,|W|)\in\overline{\Omega}\times[0,R]\\}$. Then $\|W_{n}\|$ is
bounded since $\mu\geq 2\theta_{max}>2$ and $a_{min}>0$. From (4.2) and (4.3),
it can be concluded directly that there exists a constant $C>0$ such that
(4.1) holds since $(H_{1})$ holds. ∎

######  Lemma 4.2.

Suppose $\Omega$ is a bounded domain in $\mathbb{R}^{2}$. Let $\\{u_{n}\\}$ be
in $L^{1}(\Omega)$ such that $u_{n}\rightarrow u$ in $L^{1}(\Omega)$ and
$f(x,s)$ be a continuous function. Then $f(x,u_{n})\rightarrow f(x,u)$ in
$L^{1}(\Omega)$ provided that $f(x,u_{n})\in L^{1}(\Omega)$ for all $n$ and
$\int_{\Omega}|f(x,u_{n})u_{n}|dx\leq C$.

###### Proof.

This proof can be found, for instant, in [12]. ∎

Then we will use the following result which is a converse of the Lebesgue
dominated convergence theorem in the space $H^{1}_{0}(\Omega)$.

######  Lemma 4.3.

Let $\\{u_{n}\\}$ be a sequence of functions in $H^{1}_{0}(\Omega)$ strongly
convergent. Then there exist a subsequence $\\{u_{n_{k}}\\}$ of $\\{u_{n}\\}$
and $g\in H^{1}_{0}(\Omega)$ such that $|u_{n_{k}}(x)|\leq g(x)$ almost
everywhere in $\Omega$.

###### Proof.

See the proof in [15]. ∎

######  Remark 4.4.

Under the assumptions in Lemma 4.1, by Lemma 4.2, we can get that

(4.4) $H_{u}(x,W_{n})\rightarrow H_{u}(x,W)\ \ \mbox{and}\ \
H_{v}(x,W_{n})\rightarrow H_{v}(x,W)\ \ \mbox{in}\ \ L^{1}(\Omega),$

where $W$ is the weak limit of Palais-Smale sequence $\\{W_{n}\\}$.

######  Lemma 4.5.

Suppose that $(H_{1})-(H_{2})$, $(\overline{H_{2}})$ hold and $H_{u},\ H_{v}$
have subcritical (or critical) growth $(TM)^{sc}$ (or $(TM)^{c}$). If
$\\{W_{n}\\}\subset H^{1}_{0}(\Omega,\mathbb{R}^{2})$ is a Palais-Smale
sequence for $J_{\lambda}$ and $W$ is its weak limit then, up to a
subsequence,

(4.5) $H(x,W_{n})\rightarrow H(x,W)\ \ in\ \ L^{1}(\Omega).$

###### Proof.

From Remark 4.4, we have

$\displaystyle H_{u}(x,W_{n})\rightarrow H_{u}(x,W)\ \ \mbox{and}\ \
H_{v}(x,W_{n})\rightarrow H_{v}(x,W)\ \ \mbox{in}\ \ L^{1}(\Omega).$

Thus, by Lemma 4.3, there exist $F_{1},\ F_{2}\in L^{1}(\Omega)$ such that
$|H_{u}(x,W_{n})|\leq F_{1}(x),\ |H_{v}(x,W_{n})|\leq F_{2}(x)$ almost
everywhere in $\Omega$. From $(H_{2})$ we can conclude that

$\displaystyle\begin{split}|H(x,W_{n})|&\leq\sup_{(x,|W_{n}|)\in\Omega\times[0,S_{0}]}|H(x,W_{n}(x))|+M_{0}|\nabla
H(x,W_{n})|\\\
&\leq\sup_{(x,|W_{n}|)\in\Omega\times[0,S_{0}]}|H(x,W_{n}(x))|+M_{0}(F_{1}+F_{2})\
\ \mbox{a.e.}\ \ \mbox{in}\ \ \Omega.\end{split}$

Thus, by the generalized Lebesgue dominated convergence theorem

$H(x,W_{n})\rightarrow H(x,W)\ \ \mbox{in}\ \ L^{1}(\Omega).$

∎

##  5\. Existence results

According to Lemmas 3.2 and 3.3, let

(5.1)
$c_{M,\lambda}=\inf_{\gamma\in\Upsilon}\max_{t\in[0,1]}J_{\lambda}(\gamma(t))>0,$

be the minimax level for $J_{\lambda}$, where $\Upsilon=\\{\gamma\in
C\big{(}[0,1],H^{1}_{0}(\Omega,\mathbb{R}^{2})\big{)}:\gamma(0)=0,\gamma(1)=E_{0}\\}$,
and $E_{0}$ is given in Lemma 3.3. Therefore, by using the Mountain-pass
theorem, there exists a sequence $\\{W_{n}\\}\subset
H^{1}_{0}(\Omega,\mathbb{R}^{2})$ satisfying

(5.2) $J_{\lambda}(W_{n})\rightarrow c_{M,\lambda}\ \ \mbox{and}\ \
J_{\lambda}^{\prime}(W_{n})\rightarrow 0.$

###  5.1. Subcritical case: Proof of Theorem 1.2

######  Lemma 5.1.

Suppose that $H_{u},\ H_{v}$ have subcritical growth $(TM)^{sc}$, then
functional $J_{\lambda}$ satisfies the Palais-Smale condition at any finite
level.

###### Proof.

Let $\\{W_{n}\\}\subset H^{1}_{0}(\Omega,\mathbb{R}^{2})$ be a sequence such
that $J_{\lambda}(W_{n})\rightarrow c$ and
$J_{\lambda}^{\prime}(W_{n})\rightarrow 0$ with $|c|<\infty$. From Lemma 4.1,
we have $\\{W_{n}\\}$ is bounded and there exists $C_{0}>0$ such that
$\|W_{n}\|\leq C_{0}$, so we can get a subsequence still denoted by
$\\{W_{n}\\}$, for some $W=(u,v)\in H^{1}_{0}(\Omega,\mathbb{R}^{2})$

$\displaystyle\begin{split}&W_{n}\rightharpoonup W\ \ \mbox{in}\ \
H^{1}_{0}(\Omega,\mathbb{R}^{2}),\\\ &W_{n}\rightarrow W\ \ \mbox{in}\ \
L^{s}(\Omega,\mathbb{R}^{2}),\ \ \mbox{for all}\ \ s\geq 1.\end{split}$

Since

(5.3) $\begin{split}\langle
J_{\lambda}^{\prime}(W_{n}),W_{n}-W\rangle=&\big{(}a_{1}+b_{1}\|u_{n}\|^{2(\theta_{1}-1)}\big{)}\langle
u_{n},u_{n}-u\rangle+\big{(}a_{2}+b_{2}\|v_{n}\|^{2(\theta_{2}-1)}\big{)}\langle
v_{n},v_{n}-v\rangle\\\ &-\lambda\int_{\Omega}(W_{n}-W)\cdot\nabla
H(x,W_{n})dx.\end{split}$

From $J_{\lambda}^{\prime}(W_{n})\rightarrow 0$ in
$\big{(}H^{1}_{0}(\Omega,\mathbb{R}^{2})\big{)}_{*}$, we have $\langle
J_{\lambda}^{\prime}(W_{n}),W_{n}-W\rangle\rightarrow 0$. Meanwhile, by
subscritical growth condition $(TM)^{sc}$ and the Hölder’s inequality, it
follows that

$\displaystyle\begin{split}\int_{\Omega}|W_{n}-W|\cdot|\nabla
H(x,W_{n})|dx&\leq C\int_{\Omega}|W_{n}-W|\exp(\beta|W_{n}|^{2})dx\\\ &\leq
C\|W_{n}-W\|_{2}\left(\int_{\Omega}\exp(2\beta|W_{n}|^{2})dx\right)^{\frac{1}{2}},\end{split}$

for all $\beta>0$, $r>1$, and $s=\frac{r}{r-1}$. Choosing
$\beta=\frac{2\pi}{C^{2}_{0}}$, we have $2\beta\|W_{n}\|^{2}\leq 4\pi$, then
by Lemma 2.3 and the embedding
$H^{1}_{0}(\Omega,\mathbb{R}^{2})\hookrightarrow L^{s}(\Omega,\mathbb{R}^{2})$
is compact for $s\geq 2$, the second term in the last inequality converges to
zero. Thus, in (5.3), it must be that

$\displaystyle\big{(}a_{1}+b_{1}\|u_{n}\|^{2(\theta_{1}-1)}\big{)}\langle
u_{n},u_{n}-u\rangle+\big{(}a_{2}+b_{2}\|v_{n}\|^{2(\theta_{2}-1)}\big{)}\langle
v_{n},v_{n}-v\rangle\rightarrow 0$

as $n\rightarrow\infty$. Since $a_{1},\ a_{2}>0$, $b_{1},\ b_{2}>0$, and
$W_{n}\rightharpoonup W$ in $H^{1}_{0}(\Omega,\mathbb{R}^{2})$, it must be
that

$\displaystyle\langle u_{n},u_{n}-u\rangle\rightarrow 0\ \ \mbox{and}\ \
\langle v_{n},v_{n}-v\rangle\rightarrow 0,$

by the semicontinuity of norm, which means $\|u_{n}\|^{2}\rightarrow\|u\|^{2}$
and $\|v_{n}\|^{2}\rightarrow\|v\|^{2}$. Then by Brézis-Lieb Lemma [5], we
have $u_{n}\rightarrow u$ and $v_{n}\rightarrow v$ in $H^{1}_{0}(\Omega)$,
i.e. $W_{n}\rightarrow W$ in $H^{1}_{0}(\Omega,\mathbb{R}^{2})$. This proof is
complete. ∎

Proof of Theorem 1.2: By (5.2) and Lemma 5.1, using Minimax principle, there
exists a critical point $W_{0}$ for $J_{\lambda}$ at level $c_{M,\lambda}$. We
affirm that $W_{0}\neq(0,0)$. In fact, suppose by contradiction that
$W_{0}\equiv(0,0)$. Because $J_{\lambda}$ is continuous,
$J_{\lambda}(W_{n})\rightarrow c_{M,\lambda}$ and $W_{n}\rightarrow W_{0}$ in
$H^{1}_{0}(\Omega,\mathbb{R}^{2})$, we can know that
$0<c_{M,\lambda}=\lim_{n\rightarrow\infty}J_{\lambda}(W_{n})=J_{\lambda}(W_{0})=J_{\lambda}((0,0))=0$,
what is absurd. Thus, the proof of Theorem 1.2 is complete. ∎

###  5.2. Critical case: Proof of Theorem 1.3

######  Lemma 5.2.

Suppose that $H_{u},\ H_{v}$ have critical growth $(TM)^{c}$, and $(H_{1})$,
$(H_{2})$ are satisfied. If $\\{W_{n}\\}\subset
H^{1}_{0}(\Omega,\mathbb{R}^{2})$ is a bounded Palais-Smale sequence for
$J_{\lambda}$ at any finite level with

(5.4)
$\displaystyle\limsup_{n\rightarrow\infty}\|W_{n}\|^{2}<\frac{4\pi}{\beta_{0}},$

then $\\{W_{n}\\}$ possesses a strongly subsequence, i.e. the functional
$J_{\lambda}$ satisfies the Palais-Smale condition.

###### Proof.

Let $\\{W_{n}\\}\subset H^{1}_{0}(\Omega,\mathbb{R}^{2})$ is a bounded Palais-
Smale sequence for $J_{\lambda}$ satisfying (5.4), then up to a subsequence,
there is $W=(u,v)\in H^{1}_{0}(\Omega,\mathbb{R}^{2})$ such that

$\displaystyle\begin{split}&W_{n}\rightharpoonup W\ \ \mbox{in}\ \
H^{1}_{0}(\Omega,\mathbb{R}^{2}).\end{split}$

By $J^{\prime}_{\lambda}(W_{n})\rightarrow 0$ in
$\big{(}H^{1}_{0}(\Omega,\mathbb{R}^{2})\big{)}^{*}$, we can get

$\displaystyle\begin{split}o_{n}(1)=&\big{(}a_{1}+b_{1}\|u_{n}\|^{2(\theta_{1}-1)}\big{)}\langle
u_{n},u_{n}-u\rangle+\big{(}a_{2}+b_{2}\|v_{n}\|^{2(\theta_{2}-1)}\big{)}\langle
v_{n},v_{n}-v\rangle\\\ &-\lambda\int_{\Omega}(W_{n}-W)\cdot\nabla
H(x,W_{n})dx,\end{split}$

that is,

(5.5)
$\displaystyle\begin{split}&\big{(}a_{1}+b_{1}\|u_{n}\|^{2(\theta_{1}-1)}\big{)}(\|u_{n}\|^{2}-\|u\|^{2})+\big{(}a_{2}+b_{2}\|v_{n}\|^{2(\theta_{2}-1)}\big{)}(\|v_{n}\|^{2}-\|v\|^{2})\\\
=&\lambda\int_{\Omega}(W_{n}-W)\cdot\nabla H(x,W_{n})dx+o_{n}(1).\end{split}$

From (5.4), there exists $\zeta>0$ such that
$\beta_{0}\|W_{n}\|^{2}<\zeta<4\pi$ for $n$ sufficiently large and also, there
exist $\beta>\beta_{0}$ close to $\beta_{0}$ and $q>1$ close to $1$ such that
$q\beta\|W_{n}\|^{2}\leq\zeta<4\pi$ for $n$ sufficiently large. Then by
conditions $(TM)^{c}$ and $(H_{2})$, we have

$\displaystyle\begin{split}\left|\int_{\Omega}(W_{n}-W)\cdot\nabla
H(x,W_{n})dx\right|\leq
C_{1}\int_{\Omega}|W_{n}-W|\exp(\beta|W_{n}|^{2})dx,\end{split}$

and by the Hölder’s inequality, Lemma 2.3 and Sobolev embedding theorem, we
can get that

$\displaystyle\begin{split}\left|\int_{\Omega}(W_{n}-W)\cdot\nabla
H(x,W_{n})dx\right|\leq&C_{1}\|W_{n}-W\|_{\frac{q}{q-1}}\left(\int_{\Omega}|W_{n}-W|\exp(\beta|W_{n}|^{2})dx\right)^{\frac{1}{q}}\\\
\leq&C_{2}\|W_{n}-W\|_{\frac{q}{q-1}}\to 0,\end{split}$

as $n\to\infty$. Then combining with (5.5), we have $\|u_{n}\|\to\|u\|$ and
$\|v_{n}\|\to\|v\|$. Then by Brézis-Lieb Lemma [5], we have $u_{n}\rightarrow
u,\ v_{n}\rightarrow v$ in $H^{1}_{0}(\Omega)$, i.e. $W_{n}\rightarrow W$ in
$H^{1}_{0}(\Omega,\mathbb{R}^{2})$ and the result follows. ∎

######  Lemma 5.3.

Suppose that $H_{u},\ H_{v}$ have critical growth $(TM)^{c}$, and $(H_{1})$,
$(H_{2})$, $(H_{3})$, $(SL)$ are satisfied. Moreover, if

(5.6)
$\displaystyle\begin{split}c_{M,\lambda}<\min\bigg{\\{}\frac{1}{2}\Big{(}a_{1}\frac{2\pi}{\beta_{0}}+\frac{b_{1}}{\theta_{1}}\big{(}\frac{2\pi}{\beta_{0}}\big{)}^{\theta_{1}}\Big{)},\frac{1}{2}\Big{(}a_{2}\frac{2\pi}{\beta_{0}}+\frac{b_{2}}{\theta_{2}}\big{(}\frac{2\pi}{\beta_{0}}\big{)}^{\theta_{2}}\Big{)}\bigg{\\}}.\end{split}$

for some $\lambda>0$, then the functional $J_{\lambda}$ satisfies the Palais-
Smale condition at level $c_{M,\lambda}$.

###### Proof.

From (5.2) and Lemma 4.1, there exists a bounded Palais-Smale consequence
$\\{W_{n}\\}$ for $J_{\lambda}$ at level $c_{M,\lambda}$. Then up to a
subsequence, still denoted by $\\{W_{n}\\}$, for some $W_{0}\in
H^{1}_{0}(\Omega,\mathbb{R}^{2})$, one has

(5.7) $\displaystyle\begin{split}W_{n}&\rightharpoonup W_{0}\ \ \ \ \ \
\mbox{in}\ \ H^{1}_{0}(\Omega,\mathbb{R}^{2}),\\\ W_{n}&\rightarrow W_{0}\ \ \
\ \ \ \mbox{in}\ \ L^{s}(\Omega,\mathbb{R}^{2}),\ \ \mbox{for all}\ \ s\geq
1\\\ W_{n}(x)&\rightarrow W_{0}(x)\ \ \mbox{in}\ \ \mathbb{R}^{2},\ \
\mbox{for a.e.}\ \ x\in\Omega.\end{split}$

Next, we will make some claims as follows:

Claim 1. $W_{0}\neq(0,0)$.

###### Proof.

Supposing by contradiction that $W_{0}\equiv(0,0)$. Hence, by Lemma 4.5 and
$(SL)$, $\int_{\Omega}H(x,W_{n})\rightarrow 0$ and so

$\displaystyle\begin{split}&\frac{1}{2}\Big{(}a_{1}\|u_{n}\|^{2}+\frac{b_{1}}{\theta_{1}}\|u_{n}\|^{2\theta_{1}}\Big{)}+\frac{1}{2}\Big{(}a_{2}\|v_{n}\|^{2}+\frac{b_{2}}{\theta_{2}}\|v_{n}\|^{2\theta_{2}}\Big{)}\rightarrow
c_{M,\lambda}\\\
&<\min\bigg{\\{}\frac{1}{2}\Big{(}a_{1}\frac{2\pi}{\beta_{0}}+\frac{b_{1}}{\theta_{1}}\big{(}\frac{2\pi}{\beta_{0}}\big{)}^{\theta_{1}}\Big{)},\frac{1}{2}\Big{(}a_{2}\frac{2\pi}{\beta_{0}}+\frac{b_{2}}{\theta_{2}}\big{(}\frac{2\pi}{\beta_{0}}\big{)}^{\theta_{2}}\Big{)}\bigg{\\}}.\end{split}$

We can get that for $n$ large enough $\|u_{n}\|^{2},\
\|v_{n}\|^{2}<\frac{2\pi}{\beta_{0}}$, so
$\|W_{n}\|^{2}=\|u_{n}\|^{2}+\|v_{n}\|^{2}<\frac{4\pi}{\beta_{0}}$. Then by
Lemma 5.2, we have $\|W_{n}\|^{2}\rightarrow 0$. Therefore
$J_{\lambda}(W_{n})\rightarrow 0$, what is absurd for $c_{M,\lambda}$ and we
must have $W_{0}\neq(0,0)$. ∎

Claim 2. Let $\lim\inf_{n\rightarrow\infty}\|u_{n}\|^{2}=\sigma^{2},\
\lim\inf_{n\rightarrow\infty}\|v_{n}\|^{2}=\tau^{2}$. Then
$W_{0}=(u_{0},v_{0})$ is a weak solution of

(5.11)
$\displaystyle\left\\{\begin{array}[]{ll}-\big{(}a_{1}+b_{1}\sigma^{2(\theta_{1}-1)}\big{)}\Delta
u=\lambda H_{u}(x,u,v)&\ \mbox{in}\ \ \ \Omega,\\\\[5.69054pt]
-\big{(}a_{2}+b_{2}\tau^{2(\theta_{2}-1)}\big{)}\Delta v=\lambda
H_{v}(x,u,v)&\ \mbox{in}\ \ \ \Omega,\\\\[5.69054pt] u=0,v=0&\ \mbox{on}\ \ \
\partial\Omega.\end{array}\right.$

###### Proof.

By $J_{\lambda}^{\prime}(W_{n})\rightarrow 0$ and Remark 4.4, we get that

$\displaystyle\begin{split}\big{(}a_{1}+b_{1}\sigma^{2(\theta_{1}-1)}\big{)}\int_{\Omega}\nabla
u_{0}\nabla\varphi+\big{(}a_{2}+b_{2}\tau^{2(\theta_{2}-1)}\big{)}\int_{\Omega}\nabla
v_{0}\nabla\psi-\lambda\int_{\Omega}\Phi\cdot\nabla H(x,W_{0})=0,\end{split}$

$\mbox{for all}\ \Phi=(\varphi,\psi)\in
C^{\infty}_{0}(\Omega,\mathbb{R}^{2})$. Since
$C^{\infty}_{0}(\Omega,\mathbb{R}^{2})$ is dense in
$H^{1}_{0}(\Omega,\mathbb{R}^{2})$, we conclude this claim. ∎

Claim 3.
$\lim_{n\rightarrow\infty}\|W_{n}\|^{2}-\|W_{0}\|^{2}<\frac{4\pi}{\beta_{0}}$.

###### Proof.

Suppose by contradiction that
$\lim_{n\rightarrow\infty}\|W_{n}\|^{2}-\|W_{0}\|^{2}=\sigma^{2}-\|u_{0}\|^{2}+\tau^{2}-\|v_{0}\|^{2}\geq\frac{4\pi}{\beta_{0}}$
where $\sigma$ and $\tau$ are given in Claim 2. Thus, at least one between
$\sigma^{2}-\|u_{0}\|^{2}$ and $\tau^{2}-\|v_{0}\|^{2}$ is not less then
$\frac{2\pi}{\beta_{0}}$, let
$\sigma^{2}-\|u_{0}\|^{2}\geq\frac{2\pi}{\beta_{0}}$. Therefore, from (5.2),
we have

$\displaystyle\begin{split}c_{M,\lambda}=&\lim_{n\rightarrow\infty}\big{[}J_{\lambda}(W_{n})-\frac{1}{2\theta_{max}}\langle
J_{\lambda}^{\prime}(W_{n}),W_{n}\rangle\big{]}\\\
=&\lim_{n\rightarrow\infty}\Bigg{[}\frac{1}{2}\Big{(}a_{1}\|u_{n}\|^{2}+\frac{b_{1}}{\theta_{1}}\|u_{n}\|^{2\theta_{1}}\Big{)}+\frac{1}{2}\Big{(}a_{2}\|v_{n}\|^{2}+\frac{b_{2}}{\theta_{2}}\|v_{n}\|^{2\theta_{2}}\Big{)}\\\
&-\frac{1}{2\theta_{max}}\big{(}a_{1}+b_{1}\|u_{n}\|^{2(\theta_{1}-1)}\big{)}\|u_{n}\|^{2}-\frac{1}{2\theta_{max}}\big{(}a_{2}+b_{2}\|v_{n}\|^{2(\theta_{2}-1)}\big{)}\|v_{n}\|^{2}\\\
&+\lambda\int_{\Omega}\left(\frac{1}{2\theta_{max}}W_{n}\cdot\nabla
H(x,W_{n})-H(x,W_{n})\right)dx\Bigg{]}\\\
\geq&\lim\inf_{n\rightarrow\infty}\Big{[}\big{(}\frac{1}{2}-\frac{1}{2\theta_{max}}\big{)}a_{1}\|u_{n}\|^{2}+\big{(}\frac{1}{2\theta_{1}}-\frac{1}{2\theta_{max}}\big{)}b_{1}\|u_{n}\|^{2\theta_{1}}\Big{]}\\\
&+\lim\inf_{n\rightarrow\infty}\Big{[}\big{(}\frac{1}{2}-\frac{1}{2\theta_{max}}\big{)}a_{2}\|v_{n}\|^{2}+\big{(}\frac{1}{2\theta_{1}}-\frac{1}{2\theta_{max}}\big{)}b_{2}\|v_{n}\|^{2\theta_{2}}\Big{]}\\\
&+\lambda\lim\inf_{n\rightarrow\infty}\int_{\Omega}\left(\frac{1}{2\theta_{max}}W_{n}\cdot\nabla
H(x,W_{n})-H(x,W_{n})\right)dx\\\
\geq&\lim\inf_{n\rightarrow\infty}\Big{[}\big{(}\frac{1}{2}-\frac{1}{2\theta_{max}}\big{)}a_{1}\|u_{n}\|^{2}+\big{(}\frac{1}{2\theta_{1}}-\frac{1}{2\theta_{max}}\big{)}b_{1}\|u_{n}\|^{2\theta_{1}}\Big{]}\\\
&+\frac{\lambda}{2\theta_{max}}\lim\inf_{n\rightarrow\infty}\int_{\Omega}\big{(}W_{n}\cdot\nabla
H(x,W_{n})-2\theta_{max}H(x,W_{n})\big{)}dx.\\\ \end{split}$

Then by $(H_{3})$, Remark 4.4 and Lemma 4.5, we obtain

$\displaystyle\begin{split}c_{M,\lambda}\geq&\Big{[}\big{(}\frac{1}{2}-\frac{1}{2\theta_{max}}\big{)}a_{1}\frac{2\pi}{\beta_{0}}+\big{(}\frac{1}{2\theta_{1}}-\frac{1}{2\theta_{max}}\big{)}b_{1}(\frac{2\pi}{\beta_{0}})^{\theta_{1}}\Big{]}\\\
&+\frac{\lambda}{2\theta_{max}}\lim\inf_{n\rightarrow\infty}\int_{\Omega}\big{(}W_{n}\cdot\nabla
H(x,W_{n})-W_{0}\cdot\nabla H(x,W_{0})\big{)}dx.\\\
=&\frac{1}{2}\Big{[}a_{1}\frac{2\pi}{\beta_{0}}+\frac{b_{1}}{\theta_{1}}(\frac{2\pi}{\beta_{0}})^{\theta_{1}}\Big{]}\\\
&-\frac{1}{2\theta_{max}}\Big{[}a_{1}\frac{2\pi}{\beta_{0}}+b_{1}(\frac{2\pi}{\beta_{0}})^{\theta_{1}}-\lambda\lim\inf_{n\rightarrow\infty}\int_{\Omega}\big{(}W_{n}\cdot\nabla
H(x,W_{n})-W_{0}\cdot\nabla H(x,W_{0})\big{)}dx\Big{]}.\end{split}$

Here, we assert

$\displaystyle 0\geq
a_{1}\frac{2\pi}{\beta_{0}}+b_{1}(\frac{2\pi}{\beta_{0}})^{\theta_{1}}-\lambda\lim\inf_{n\rightarrow\infty}\int_{\Omega}\big{(}W_{n}\cdot\nabla
H(x,W_{n})-W_{0}\cdot\nabla H(x,W_{0})\big{)}dx.$

Indeed, $\langle J_{\lambda}^{\prime}(W_{n}),W_{n}\rangle\rightarrow 0$ and
Claim 2 indicate that

$\displaystyle\begin{split}0&=\Big{(}a_{1}+b_{1}\sigma^{2(\theta_{1}-1)}\Big{)}\sigma^{2}+\Big{(}a_{2}+b_{2}\tau^{2(\theta_{2}-1)}\Big{)}\tau^{2}-\lambda\lim\inf_{n\rightarrow\infty}\int_{\Omega}W_{n}\cdot\nabla
H(x,W_{n})dx.\\\
0&=\Big{(}a_{1}+b_{1}\sigma^{2(\theta_{1}-1)}\Big{)}\|u_{0}\|^{2}+\Big{(}a_{2}+b_{2}\tau^{2(\theta_{1}-1)}\Big{)}\|v_{0}\|^{2}-\lambda\int_{\Omega}W_{0}\cdot\nabla
H(x,W_{0})dx.\end{split}$

Subtracting the second equality from the first one, we get

$\displaystyle\begin{split}0=&\Big{(}a_{1}+b_{1}\sigma^{2(\theta_{1}-1)}\Big{)}(\sigma^{2}-\|u_{0}\|^{2})+\Big{(}a_{2}+b_{2}\tau^{2(\theta_{2}-1)}\Big{)}(\tau^{2}-\|v_{0}\|^{2})\\\
&-\lambda\lim\inf_{n\rightarrow\infty}\int_{\Omega}\big{(}W_{n}\cdot\nabla
H(x,W_{n})-W_{0}\cdot\nabla H(x,W_{0})\big{)}dx\\\
\geq&\Big{(}a_{1}+b_{1}\sigma^{2(\theta_{1}-1)}\Big{)}\frac{2\pi}{\beta_{0}}-\lambda\lim\inf_{n\rightarrow\infty}\int_{\Omega}\big{(}W_{n}\cdot\nabla
H(x,W_{n})-W_{0}\cdot\nabla H(x,W_{0})\big{)}dx\\\
\geq&a_{1}\frac{2\pi}{\beta_{0}}+b_{1}(\frac{2\pi}{\beta_{0}})^{\theta_{1}}-\lambda\lim\inf_{n\rightarrow\infty}\int_{\Omega}\big{(}W_{n}\cdot\nabla
H(x,W_{n})-W_{0}\cdot\nabla H(x,W_{0})\big{)}dx.\end{split}$

This shows the assertion. Then we can conclude that

$\displaystyle
c_{M,\lambda}\geq\frac{1}{2}\Big{[}a_{1}\frac{2\pi}{\beta_{0}}+\frac{b_{1}}{\theta_{1}}(\frac{2\pi}{\beta_{0}})^{\theta_{1}}\Big{]},$

which contradicts the assumption. Thus this claim is proved. ∎

Claim 4. $J_{\lambda}(W_{0})=c_{M,\lambda}$.

###### Proof.

By using Lemma 4.5, and semicontinuity of norm, we have that
$J_{\lambda}(W_{0})\leq c_{M,\lambda}$. We will show that the case
$J_{\lambda}(W_{0})<c_{M,\lambda}$ can not occur. We suppose
$J_{\lambda}(W_{0})<c_{M,\lambda}$, and define $\sigma$ and $\tau$ as in Claim
2, then we get $\sigma>\|u_{0}\|$ or $\tau>\|v_{0}\|$. Next, defining
$Z_{n}=\frac{W_{n}}{\|W_{n}\|}$ and
$Z_{0}=\frac{W_{0}}{\sqrt{\sigma^{2}+\tau^{2}}}=\frac{W_{0}}{\lim_{n\rightarrow\infty}\|W_{n}\|}$,
we have $Z_{n}\rightharpoonup Z_{0}$ in $H^{1}_{0}(\Omega,\mathbb{R}^{2})$ and
$\|Z_{0}\|<1$, then Lemma 2.4 which shows that

(5.12) $\displaystyle\sup_{n}\int_{\Omega}\exp(p|Z_{n}|^{2})dx<\infty,\ \
\forall p<\frac{4\pi}{1-\|Z_{0}\|^{2}}.$

Since
$\sigma^{2}+\tau^{2}=\frac{\sigma^{2}+\tau^{2}-\|W_{0}\|^{2}}{1-\|Z_{0}\|^{2}}$,
by Claim 3, it follows that

$\displaystyle\lim_{n\rightarrow\infty}\|W_{n}\|^{2}<\frac{4\pi/\beta_{0}}{1-\|Z_{0}\|^{2}}.$

Thus, there exists $\alpha>0$ such that
$\beta_{0}\|W_{n}\|^{2}<\alpha<\frac{4\pi}{1-\|Z_{0}\|^{2}}$ for $n$
sufficiently large. For $q>1$ close to 1 and $\beta>\beta_{0}$ close to
$\beta_{0}$ we still have
$q\beta\|W_{n}\|^{2}\leq\alpha<\frac{4\pi}{1-\|Z_{0}\|^{2}}$, and provoking
(5.12), for some $C>0$ and $n$ large enough, we conclude that

$\displaystyle\int_{\Omega}\exp(q\beta|W_{n}|^{2})dx\leq\int_{\Omega}\exp(\alpha|Z_{n}|^{2})dx\leq
C.$

Then, by using $(H_{2})$ and critical growth condition $(TM)^{c}$, (5.7),
Hölder’s inequality and Sobolev imbedding, we can obtain

$\displaystyle\begin{split}\left|\int_{\Omega}(W_{n}-W_{0})\cdot\nabla
H(x,W_{n})dx\right|\leq&C_{1}\int_{\Omega}|W_{n}|\cdot|W_{n}-W_{0}|dx\\\
&+C_{2}\int_{\Omega}|W_{n}-W_{0}|\exp(\beta|W_{n}|^{2})dx\\\
\leq&C_{1}\|W_{n}-W_{0}\|_{2}\|W_{n}\|_{2}+C_{3}\|W_{n}-W_{0}\|_{\frac{q}{q-1}}\rightarrow
0.\end{split}$

Since $\langle J_{\lambda}^{\prime}(W_{n}),W_{n}-W_{0}\rangle\rightarrow 0$,
it follows that

$\displaystyle\big{(}a_{1}+b_{1}\|u_{n}\|^{2(\theta_{1}-1)}\big{)}\langle
u_{n},u_{n}-u_{0}\rangle+\big{(}a_{2}+b_{2}\|v_{n}\|^{2(\theta_{2}-1)}\big{)}\langle
v_{n},v_{n}-v_{0}\rangle\rightarrow 0.$

On the other hand,

$\displaystyle\begin{split}\big{(}a_{1}+b_{1}\|u_{n}\|^{2(\theta_{1}-1)}\big{)}\langle
u_{n},u_{n}-u_{0}\rangle&=\big{(}a_{1}+b_{1}\|u_{n}\|^{2(\theta_{1}-1)}\big{)}\|u_{n}\|^{2}-\big{(}a_{1}+b_{1}\|u_{n}\|^{2(\theta_{1}-1)}\big{)}\langle
u_{n},u_{0}\rangle\\\
&\rightarrow\big{(}a_{1}+b_{1}\sigma^{2(\theta_{1}-1)}\big{)}\sigma^{2}-\big{(}a_{1}+b_{1}\sigma^{2(\theta_{1}-1)}\big{)}\|u_{0}\|^{2},\\\
\big{(}a_{2}+b_{2}\|v_{n}\|^{2(\theta_{2}-1)}\big{)}\langle
v_{n},v_{n}-v_{0}\rangle&=\big{(}a_{2}+b_{2}\|v_{n}\|^{2(\theta_{2}-1)}\big{)}\|v_{n}\|^{2}-\big{(}a_{2}+b_{2}\|v_{n}\|^{2(\theta_{2}-1)}\big{)}\langle
v_{n},v_{0}\rangle\\\
&\rightarrow\big{(}a_{2}+b_{2}\tau^{2(\theta_{2}-1)}\big{)}\tau^{2}-\big{(}a_{2}+b_{2}\tau^{2(\theta_{2}-1)}\big{)}\|v_{0}\|^{2},\end{split}$

which implies that $\sigma=\|u_{0}\|$ and $\tau=\|v_{0}\|$, i.e.
$\|W_{n}\|\rightarrow\|W_{0}\|$. Then, we must have
$J_{\lambda}(W_{0})=c_{M,\lambda}$, what is an absurd. Thus, this claim is
proved. ∎

Finalizing the proof of Lemma 5.3: By Lemma 4.5 and Claim 4, we can know that

$\displaystyle\begin{split}\Big{(}a_{1}\sigma^{2}+\frac{b_{1}}{\theta_{1}}\sigma^{2\theta_{1}}\Big{)}+&\Big{(}a_{2}\tau^{2}+\frac{b_{2}}{\theta_{2}}\tau^{2\theta_{2}}\Big{)}=\Big{(}a_{1}\|u_{0}\|^{2}+\frac{b_{1}}{\theta_{1}}\|u_{0}\|^{2\theta_{1}}\Big{)}+\Big{(}a_{2}\|v_{0}\|^{2}+\frac{b_{2}}{\theta_{2}}\|v_{0}\|^{2\theta_{2}}\Big{)}.\end{split}$

By the semicontinuity of norm, we have $\sigma=\|u_{0}\|,\ \tau=\|v_{0}\|$.
Then by Brézis-Lieb Lemma [5], we have $u_{n}\rightarrow u_{0},\
v_{n}\rightarrow v_{0}$ in $H^{1}_{0}(\Omega)$, i.e. $W_{n}\rightarrow W_{0}$
in $H^{1}_{0}(\Omega,\mathbb{R}^{2})$. ∎

######  Lemma 5.4.

Suppose that $H_{u},\ H_{v}$ have the critical growth $(TM)^{c}$ and
$(H_{1})$, $(H_{2})$, $(H_{3})$, $(SL)$ are satisfied. Then, we have
$0<c_{M,\lambda}\rightarrow 0$ as $\lambda\rightarrow+\infty$.

###### Proof.

Previous lemmas and the definition of $c_{M,\lambda}$ (see (5.1)) imply
$c_{M,\lambda}>0$ for each $\lambda>0$. Fixing any nonnegative functions
$\Phi=(\phi,\varphi)\in
C^{\infty}_{0}(\Omega,\mathbb{R}^{2})\backslash\\{(0,0)\\}$, then there exists
$t_{\lambda}>0$ such that
$J_{\lambda}(t_{\lambda}\Phi)=\max_{t>0}J_{\lambda}(t\Phi)$. Since
$\frac{dJ_{\lambda}(t\Phi)}{dt}\big{|}_{t=t_{\lambda}}=0$, then by $(H_{3})$
and (1.10) we have

(5.13)
$\displaystyle\begin{split}a_{1}t_{\lambda}^{2}\|\phi\|^{2}+a_{2}t_{\lambda}^{2}\|\varphi\|^{2}+b_{1}t_{\lambda}^{2\theta_{1}}\|\phi\|^{2\theta_{1}}+b_{2}t_{\lambda}^{2\theta_{2}}\|\varphi\|^{2\theta_{2}}=&\lambda\int_{\Omega}t_{\lambda}\Phi\cdot\nabla
H(x,t_{\lambda}\Phi)dx\\\
\geq&\lambda\left(C_{1}t_{\lambda}^{\xi}\int_{\Omega}|\Phi|^{\xi}dx-
C_{2}\right)\end{split}$

for some $C_{1},\ C_{2}>0$ with $\xi>2\theta_{max}$. This implies that
$t_{\lambda}$ is bounded from above uniformly in $\lambda$. In addition, we
claim $t_{\lambda}\rightarrow 0$ as $\lambda\rightarrow+\infty$. If not, we
can choose a sequence $\\{\lambda_{n}\\}\subset\mathbb{R}^{+}$ and a constant
$t_{0}>0$ such that $\lambda_{n}\rightarrow+\infty$ and
$t_{\lambda_{n}}\rightarrow t_{0}>0$. Since $\Phi\in
C^{\infty}_{0}(\Omega,\mathbb{R}^{2})$, we have $\nabla
H(x,t_{\lambda_{n}}\Phi)\rightarrow\nabla H(x,t_{0}\Phi)$ in
$L^{1}(\Omega,\mathbb{R}^{2})$ by [12, Lemma 2.1]. Then we get

$\displaystyle\begin{split}\int_{\Omega}\Phi\cdot\nabla
H(x,t_{\lambda_{n}}\Phi)dx\rightarrow\int_{\Omega}\Phi\cdot\nabla
H(x,t_{0}\Phi)dx.\end{split}$

Noting that the right side is strictly positive by $(H_{3})$. Then it follows
from (5.13) that

$\displaystyle\begin{split}a_{1}t_{\lambda_{n}}^{2}\|\phi\|^{2}+a_{2}t_{\lambda_{n}}^{2}\|\varphi\|^{2}+b_{1}t_{\lambda_{n}}^{2\theta_{1}}\|\phi\|^{2\theta_{1}}+b_{2}t_{\lambda_{n}}^{2\theta_{2}}\|\varphi\|^{2\theta_{2}}\rightarrow+\infty\end{split}$

as $n\rightarrow\infty$, this is impossible. Therefore by $(H_{1})$, we can
conclude that

$\displaystyle\begin{split}c_{M,\lambda}\leq J_{\lambda}(t_{\lambda}\Phi)\leq
t^{2}_{\lambda}\frac{a_{1}\|\phi\|^{2}}{2}+t^{2}_{\lambda}\frac{a_{2}\|\varphi\|^{2}}{2}+t_{\lambda}^{2\theta_{1}}\frac{b_{1}\|\phi\|^{2\theta_{1}}}{2\theta_{1}}+t_{\lambda}^{2\theta_{2}}\frac{b_{2}\|\varphi\|^{2\theta_{2}}}{2\theta_{2}}\rightarrow
0\end{split}$

as $\lambda\rightarrow+\infty$. Thus, this proof is complete. ∎

Proof of Theorem 1.3: By Lemma 5.4: $0<c_{M,\lambda}\rightarrow 0$ as
$\lambda\rightarrow+\infty$, thus, there exists a constant $\lambda_{*}>0$
large enough such that for all $\lambda\geq\lambda_{*}$, it holds that

$\displaystyle
c_{M,\lambda}<\min\bigg{\\{}\frac{1}{2}\Big{(}a_{1}\frac{2\pi}{\beta_{0}}+\frac{b_{1}}{\theta_{1}}\big{(}\frac{2\pi}{\beta_{0}}\big{)}^{\theta_{1}}\Big{)},\frac{1}{2}\Big{(}a_{2}\frac{2\pi}{\beta_{0}}+\frac{b_{2}}{\theta_{2}}\big{(}\frac{2\pi}{\beta_{0}}\big{)}^{\theta_{2}}\Big{)}\bigg{\\}}.$

Lemma 5.3 indicts that the functional $J_{\lambda}$ satisfies the Palais-Smale
condition at level $c_{M,\lambda}$ for each $\lambda\geq\lambda_{*}$.

Then by using Minimax principle, there exists a critical point $W_{*}$ for
$J_{\lambda}$ at level $c_{M,\lambda}$ for each $\lambda\geq\lambda_{*}$. We
affirm that $W_{*}\neq(0,0)$. In fact, suppose by contradiction that
$W_{*}\equiv(0,0)$. Because $J_{\lambda}$ is continuous,
$J_{\lambda}(W_{n})\rightarrow c_{M,\lambda}$ and $W_{n}\rightarrow W_{*}$ in
$H^{1}_{0}(\Omega,\mathbb{R}^{2})$, we can know that
$0<c_{M,\lambda}=\lim_{n\rightarrow\infty}J_{\lambda}(W_{n})=J_{\lambda}(W_{*})=J_{\lambda}((0,0))=0$,
what is absurd. Thus, the proof of Theorem 1.3 is complete. ∎

##  6\. Multiplicity results

The next lemma motivates us to investigate the multiplicity of solutions in
the asymptotically case.

######  Lemma 6.1.

Suppose that $(H_{1})$, $(H_{2})$, $(AL)$ hold and $H_{u},\ H_{v}$ have
subcritical growth $(TM)^{sc}$ (or critical growth $(TM)^{c}$). Then, there
exists a constant $b_{0}>0$ such that for every $b_{min}>b_{0}$, we have a
constant $\Lambda_{0}(b_{min})>\frac{a_{max}}{a_{min}}\lambda_{1}$ such that
there exist constants $\tau_{1}>0$ and $0<\rho_{1}<\rho_{2}<1$ such that

$\displaystyle J_{\lambda}(W)\geq\tau_{1},\ \ \forall\ W\in
H^{1}_{0}(\Omega,\mathbb{R}^{2})\ \mbox{with}\ \rho_{1}\leq\|W\|\leq\rho_{2},$

and

$\displaystyle-\infty<\inf_{\|W\|\leq\rho_{1}}J_{\lambda}(W)<0,$

for all $\frac{a_{max}}{a_{min}}\lambda_{1}<\lambda<\Lambda_{0}(b_{min})$,
where $b_{min}>b_{0}$. Furthermore, we can take the constant
$\Lambda_{0}(b_{min})$ depending only on $b_{min}$ such that
$\Lambda_{0}(b_{min})\rightarrow\infty$ as $b_{min}\rightarrow\infty$.

###### Proof.

By assumptions $(AL)$, $(TM)^{sc}$ (or $(TM)^{c}$) and $(H_{2})$, we can
choose $\beta=\max\\{1,2\beta_{0}\\}$ (that is, for subcritical case we take
$\beta=1$ and for critical case we take $\beta=2\beta_{0}$) such that

$\displaystyle
H(x,W)\leq\frac{(a_{min}+1)|W|^{2}}{2}+C|W|^{2}e^{\beta|W|^{2}},\
\forall(x,W)\in\Omega\times\mathbb{R}^{2}.$

Thus by Lemmas 2.4, 2.6 and Hölder’s inequality, we obtain

(6.1)
$\displaystyle\begin{split}J_{\lambda}(W)=&\frac{1}{2}\Big{(}a_{1}\|u\|^{2}+\frac{b_{1}}{\theta_{1}}\|u\|^{2\theta_{1}}\Big{)}+\frac{1}{2}\Big{(}a_{2}\|v\|^{2}+\frac{b_{2}}{\theta_{2}}\|v\|^{2\theta_{2}}\Big{)}-\lambda\int_{\Omega}H(x,W)dx\\\
\geq&\frac{b_{min}}{2\theta_{max}}\big{(}\|u\|^{2\theta_{max}}+\|v\|^{2\theta_{max}}\big{)}+\frac{a_{min}}{2}\big{(}\|u\|^{2}+\|v\|^{2}\big{)}\\\
&-\frac{\lambda(a_{min}+1)}{2}\int_{\Omega}|W|^{2}dx-\lambda
C\int_{\Omega}|W|^{2}\exp(\beta|W|^{2})dx\\\
\geq&\frac{b_{min}}{2\theta_{max}(2^{\theta_{max}}-1)}\|W\|^{2\theta_{max}}-\left[\frac{\lambda(a_{min}+1)}{2\lambda_{1}}-\frac{a_{min}}{2}\right]\|W\|^{2}\\\
&-\lambda
C\Big{(}\int_{\Omega}|W|^{4}dx\Big{)}^{1/2}\left(\int_{\Omega}\exp\left(2\beta\|W\|^{2}(W/\|W\|)^{2}\right)dx\right)^{1/2},\end{split}$

for all $W\in H^{1}_{0}(\Omega,\mathbb{R}^{2})$ with $\|W\|\leq 1$. If
$\|W\|^{2}\leq\min\\{1,\frac{\pi}{\beta_{0}}\\}$, then from Lemma 2.3, it
holds that

(6.2)
$\displaystyle\begin{split}J_{\lambda}(W)\geq&\frac{b_{min}}{2\theta_{max}(2^{\theta_{max}}-1)}\|W\|^{2\theta_{max}}-\Big{[}\frac{\lambda(a_{min}+1)}{2\lambda_{1}}-\frac{a_{min}}{2}\Big{]}\|W\|^{2}-\lambda
C^{\prime}\|W\|^{2}\\\
\geq&\frac{b_{min}}{2\theta_{max}(2^{\theta_{max}}-1)}\|W\|^{2\theta_{max}}-\lambda
C_{*}\|W\|^{2}\\\
=&\frac{b_{min}}{2\theta_{max}(2^{\theta_{max}}-1)}\|W\|^{2}\left[\|W\|^{2\theta_{max}-2}-\frac{2\theta_{max}(2^{\theta_{max}}-1)\lambda
C_{*}}{b_{min}}\right],\end{split}$

for some constant $C_{*}>0$. Then we choose

(6.3)
$\displaystyle\Lambda_{0}(b_{min}):=\frac{b_{min}}{2\theta_{max}(2^{\theta_{max}}-1)C_{*}}\left(\min\\{1,\frac{\pi}{\beta_{0}}\\}\right)^{\theta_{max}-1}.$

It follows that there exists a constant

(6.4)
$b_{0}:=2\theta_{max}(2^{\theta_{max}}-1)C_{*}\left(\min\\{1,\frac{\pi}{\beta_{0}}\\}\right)^{1-\theta_{max}}\frac{a_{max}}{a_{min}}\lambda_{1}>0$

such that

$\Lambda_{0}(b_{min})>\frac{a_{max}}{a_{min}}\lambda_{1}\quad\mbox{for
all}\quad b_{min}>b_{0}.$

Taking $b_{min}>b_{0}$ and
$\frac{a_{max}}{a_{min}}\lambda_{1}\leq\lambda<\Lambda_{0}(b_{min})$ we have

$\frac{2\theta_{max}(2^{\theta_{max}}-1)C_{*}\lambda}{b_{min}}<\left(\min\\{1,\frac{\pi}{\beta_{0}}\\}\right)^{\theta_{max}-1}.$

Then from (6.2), we can choose $0<\rho_{1}<\rho_{2}<1$ satisfying

$\left[\frac{2\theta_{max}(2^{\theta_{max}}-1)C_{*}\lambda}{b_{min}}\right]^{\frac{1}{2(\theta_{max}-1)}}<\rho_{1}<\rho_{2}<\left(\min\\{1,\frac{\pi}{\beta_{0}}\\}\right)^{\frac{1}{2}}$

such that

(6.5) $\displaystyle J_{\lambda}(W)\geq\tau_{1},\ \ \mbox{for all}\ \ W\in
H^{1}_{0}(\Omega,\mathbb{R}^{2})\ \mbox{with}\ \rho_{1}\leq\|W\|\leq\rho_{2},$

for some constant $\tau_{1}>0$. On the other hand, we claim

(6.6) $\displaystyle J_{\lambda}(t\Phi_{1})<0,$

if $\frac{a_{max}}{a_{min}}\lambda_{1}<\lambda$ and $t>0$ is sufficiently
small, where $\Phi_{1}=(\phi_{1},\varphi_{1})$ is the eigenfunction of
$\lambda_{1}$, i.e.
$\lambda_{1}=\frac{\|\Phi_{1}\|^{2}}{\int_{\Omega}|\Phi_{1}|^{2}dx}$.
Actually, fix $\epsilon>0$ such that
$\lambda(a_{min}-\epsilon)>\lambda_{1}a_{max}$. $(AL)$ implies that there
exists a constant $S_{\epsilon}$ such that

$\displaystyle H(x,W)\geq\frac{a_{min}-\epsilon}{2}|W|^{2},\ \mbox{if}\
0\leq|W|\leq S_{\epsilon}\ \mbox{ uniformly for}\ x\in\Omega.$

Set $\kappa_{\epsilon}>0$ such that
$\kappa_{\epsilon}\sup_{x\in\Omega}|\Phi_{1}|\leq S_{\epsilon}$. Consequently,
by $(H_{1})$ we can get that

$\displaystyle\begin{split}J_{\lambda}(t\Phi_{1})\leq&\frac{t^{2}a_{max}}{2}\|\Phi_{1}\|^{2}-\frac{t^{2}\lambda(a_{min}-\epsilon)}{2}\int_{\Omega}|\Phi_{1}|^{2}dx+t^{2\theta_{1}}\frac{b_{1}\|\phi_{1}\|^{2\theta_{1}}}{2\theta_{1}}+t^{2\theta_{2}}\frac{b_{2}\|\varphi_{1}\|^{2\theta_{2}}}{2\theta_{2}}\\\
=&-\frac{t^{2}}{2}\left[\frac{\lambda(a_{min}-\epsilon)}{\lambda_{1}a_{max}}-1\right]a_{max}\|\Phi_{1}\|^{2}+t^{2\theta_{1}}\frac{b_{1}\|\phi_{1}\|^{2\theta_{1}}}{2\theta_{1}}+t^{2\theta_{2}}\frac{b_{2}\|\varphi_{1}\|^{2\theta_{2}}}{2\theta_{2}},\end{split}$

for all $0<t<\kappa_{\epsilon}$. Since $2\theta_{1},2\theta_{2}>2$, this
concludes the claim. Thus, (6.5) and (6.6) end the proof. ∎

In the rest of this section, we always suppose that
$\frac{a_{max}}{a_{min}}\lambda_{1}<\lambda<\Lambda_{0}(b_{min})$, where
$\Lambda_{0}(b_{min})$ is obtained in Lemma 6.1 and $b_{min}>b_{0}$.

In order to obtain a weak solution with negative energy, by Lemma 6.1, we
define that

$-\infty<c_{0,\lambda}:=\inf_{\|W\|\leq\rho_{1}}J_{\lambda}(W)<0,$

where $\rho_{1}$ is given as in Lemma 6.1. Since $\overline{B}_{\rho_{1}}$ is
a complete metric space with the metric given by the norm of
$H^{1}_{0}(\Omega,\mathbb{R}^{2})$, convex and the functional $J_{\lambda}$ is
of class $C^{1}$ and bounded below on $\overline{B}_{\rho_{1}}$, by the
Ekeland variational principle, there exists a sequence $\\{W^{0}_{n}\\}$ in
$\overline{B}_{\rho_{1}}$ such that

(6.7) $J_{\lambda}(W^{0}_{n})\rightarrow
c_{0,\lambda}=\inf_{\|W\|\leq\rho_{1}}J_{\lambda}(W)<0\ \ \mbox{and}\ \
J_{\lambda}^{\prime}(W^{0}_{n})\rightarrow 0.$

And in order to obtain another weak solution with positive energy, according
to Lemmas 3.3 and 6.1, let

$c_{*,\lambda}=\inf_{\gamma\in\Gamma_{\lambda}}\max_{t\in[0,1]}J_{\lambda}(\gamma(t))>0$

be the minimax level of $J_{\lambda}$, where $\Gamma_{\lambda}=\\{\gamma\in
C\big{(}[0,1],H^{1}_{0}(\Omega,\mathbb{R}^{2})\big{)}:\gamma(0)=0,\gamma(1)=E_{0}\\}$,
where $E_{0}$ is given in Lemma 3.3 and here we choose
$\|E_{0}\|>\max\\{\rho_{0},\rho_{2}\\}$. Therefore, by using the Mountain-pass
theorem, there exists a sequence $\\{W^{*}_{n}\\}\subset
H^{1}_{0}(\Omega,\mathbb{R}^{2})$ satisfying

(6.8) $J_{\lambda}(W^{*}_{n})\rightarrow c_{*,\lambda}\ \ \mbox{and}\ \
J_{\lambda}^{\prime}(W^{*}_{n})\rightarrow 0.$

Proof of Theorem 1.4: Similarly to the proof of Lemma 5.1, by using the
subcritical growth condition $(TM)^{sc}$, we can get that the functional
$J_{\lambda}$ satisfies the Palais-Smale condition. Then, similarly to the
proof of Theorem 1.2, by (6.7), (6.8), using Minimax principle, there exist
two nontrivial critical points $W_{0,\lambda},\ W_{*,\lambda}$ for
$J_{\lambda}$ at the levels of $c_{0,\lambda}$ and $c_{*,\lambda}$,
respectively. By compactness condition and the definitions of $c_{0,\lambda}$
and $c_{*,\lambda}$, it is easy to verify that $W_{0,\lambda},\ W_{*,\lambda}$
are different. Thus, the proof of Theorem 1.4 is complete. ∎

Proof of Theorem 1.5: From (6.7) and Lemma 6.1, there exists a sequence
$\\{W^{0}_{n}\\}$ in $\overline{B}_{\rho_{1}}$ such that

$\displaystyle J_{\lambda}(W^{0}_{n})\rightarrow
c_{0,\lambda}=\inf_{\|W\|\leq\rho_{1}}J_{\lambda}(W)<0\ \ \mbox{and}\ \
J_{\lambda}^{\prime}(W^{0}_{n})\rightarrow 0,$

where $\rho^{2}_{1}\leq\min\\{1,\frac{\pi}{\beta_{0}}\\}$ is given in Lemma
6.1. Thus
$\|W^{0}_{n}\|^{2}\leq\min\\{1,\frac{\pi}{\beta_{0}}\\}<\frac{4\pi}{\beta_{0}}$
for all $n\in\mathbb{N}$. Then by Lemma 5.2, the functional $J_{\lambda}$
satisfies the Palais-Smale condition at level $c_{0,\lambda}$. Thus, we can
get one nontrivial critical point $W_{0,\lambda}$ for $J_{\lambda}$ at level
$c_{0,\lambda}$ and $J_{\lambda}(W_{0,\lambda})=c_{0,\lambda}<0$. Therefore,
the proof of Theorem 1.5 is complete. ∎

Acknowledgements:

The authors have been supported by National Nature Science Foundation of China
11971392, Natural Science Foundation of Chongqing, China cstc2019jcyjjqX0022
and Fundamental Research Funds for the Central Universities XDJK2019TY001.

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