We study the existence of solutions to the following Chern-Simons-Schrödinger system (CSS system) in $H^{1}({\mathbb R^{2}})$

$ \begin{equation} \label{eq-V} \left\{ \begin{array}{ll} -\Delta u+ V(x)u+ A_{0}u+\sum\limits_{j=1}^{2}A_{j}^{2}u=f(u), \\ \partial_{1}A_0=A_{2}|u|^{2}, \quad \partial_{2}A_0=-A_{1}|u|^{2}, \\ \partial_1A_2-\partial_2A_1=-\frac{1}{2}u^2, \quad \partial_1A_{1}+\partial_{2}A_{2}=0, \end{array} \right. \end{equation} $

(1.1)

where $V(x)$ and $f(u)$ satisfy

(V1) $V(x)\in C({\mathbb{R}^{2}}, {\mathbb{R}})$ and $V(x)\geq V_0>0$ for all $x\in{\mathbb{R}^{2}}$;

(V2) the function $[V(x)]^{-1}$ belongs to $L^1(\mathbb{R}^2)$;

(F1) $f\in C({\mathbb{R}}, {\mathbb{R}})$ and $f(0)=0$;

(F2) $\lim\limits_{s\rightarrow +\infty}\frac{f(s)}{e^{\alpha s^2}}=0$ for all $\alpha>0$;

(F3) there exist $\theta>6$ and $s_1>0$ such that for all $|s|\geq s_1$,

$ 0 <\theta F(s)\doteq \theta \displaystyle\int^s_0 f(t)\, dt\leq sf(s); $

(F4) $\lim\limits_{s\rightarrow 0}2F(s)s^{-2} <\lambda_1$, where

$ \lambda_1=\inf\limits_{u\in E\setminus\{0\}}\frac{\displaystyle\int_{\mathbb{R}^{2}}|\nabla u|^2 +V(x)u^2\, dx}{\displaystyle\int_{\mathbb{R}^{2}}u^2\, dx}\geq V_0>0, $

$E=\{u\in H^1 (\mathbb{R}^{2}):\displaystyle\int_{\mathbb{R}^{2}}V(x)u^2\, dx <\infty\}$ is a subspace of $H^1 (\mathbb{R}^{2})$ and also a Hilbert space endowed with the following inner product

$ \langle u, \, v\rangle=\displaystyle\int_{\mathbb{R}^{2}}(\nabla u\cdot\nabla v +V(x)uv)\, dx, \, \, \, \forall u, \, v\in E. $

$f(u)=\lambda (2s+s^2)e^s$ for $0 <\lambda <\frac{\lambda_1}{2}$ is an example that $f(u)$ satisfies assumptions (F1)-(F4), which was given in [1].

The CSS system received much attention recently, which describes the dynamics of large number of particles in a electromagnetic field. About the detail of its physical background, we refer to the references we mentioned below and references therein.

The CSS system arises from the Euler-Lagrange equations which are given by

$ \begin{equation} \label{eq1} \left\{ \begin{array}{ll} iD_0\phi+(D_1D_1+D_2D_2)\phi=f(\phi), \\ \partial_0A_1-\partial_1A_0=-\text{Im}(\bar\phi D_2\phi), \\ \partial_0A_2-\partial_2A_0=\text{Im}(\bar\phi D_1\phi), \\ \partial_1A_2-\partial_2A_1=-\frac{1}{2}|\phi|^2. \end{array} \right. \end{equation} $

(1.2)

This system was proposed in [2-4]. Berge, De Bouard, Saut [5] and Huh [6] studied the blowing up time-dependent solutions of problem (1.2) as well as Liu, Smith, Tataru [7] considered the local wellposedness.

We assume that the Coulomb gauge condition $\partial_{0}A_{0}+\partial_1A_{1}+\partial_{2}A_{2}=0$ holds, then the standing wave $\psi(x, t)=e^{i\omega t}\ u$ of problem (1.2) satisfies

$ \begin{equation} \label{eqn-oemga1} \left\{ \begin{array}{ll} -\Delta u+\omega u+A_{0}u+A_{1}^{2}u+A_{2}^{2}u=f(u), \\ \partial_{1}A_0=A_{2} u^{2}, \quad \partial_{2}A_0=-A_{1} u^{2}, \\ \partial_1A_2-\partial_2A_1=-\frac{1}{2}|u|^2, \quad \partial_1A_{1}+\partial_{2}A_{2}=0. \end{array} \right. \end{equation} $

(1.3)

Under some radial assumptions, on the one hand, the existence, non-existence, and multiplicity of standing waves to the nonlinear CSS systems were investigated by [8-12] etc.

On the other hand, the existence of solitary waves was considered by [13-16] etc. For problem (1.1) with $f(u)=|u|^{p-2}u, \, p>4$, without the radial assumptions we mentioned above, by the concentration compactness principle with $V(x)$ is a constant and the argument of global compactness with $V\in C(\mathbb{R}^{2})$ and $0 <V_{0} <V(x) <V_{\infty}$, the existence of nontrivial solutions to Chern-Simons-Schrödinger systems (1.1) was obtained in [17].

Inspired by [1] and [17], the purpose of the present paper is to study the existence of solutions for systems (1.1) with exponential nonlinearities. The main difficult of systems (1.1) is that the non-local term $A_j, \, j=0, 1, 2$ depend on $u$ and there is a lack of compactness in ${\mathbb {R}^{2}}$. Using the mountain pass theorem, we have the following main result.

Theorem 1.1 Suppose (V1), (V2), (F1), (F2), (F3) and (F4) hold, then problem (1.1) has a solution.

This paper is organized as follows. In Section 2 we introduce the workframe and some technical lemmas. In Section 3 we prove the mountain pass construction and (PS) condition, which yields Theorem 1.1.

In this section, we outline the variational workframe for the future study.

Let $H^{1}(\mathbb{R}^{2})$ denote the usual Sobolev space with

$ \|u\|=\big(\displaystyle\int_{\mathbb{R}^{2}}|\nabla u|^{2}+V(x)|u|^{2}\, dx\big)^{1/2}. $

We consider the following subspace of $H^1(\mathbb{R}^2)$,

$ E=\{u\in H^1(\mathbb{R}^2):\, \displaystyle\int\limits_{\mathbb{R}^2} V(x)u^2\, dx <\infty\}. $

Condition (V1) implies that the embedding $E\hookrightarrow H^{1}(\mathbb{R}^{2})$ is continuous. Assumption (V2) and Hölder inequality yield that

$ \begin{equation}\label{L1} \|u\|_{L^1(\mathbb{R}^2)}\leq \Big(\displaystyle\int\limits_{\mathbb{R}^2}V(x)^{-1}\, dx\Big)^{\frac{1}{2}}\|u\|. \end{equation} $

(2.1)

Consequently,

$ \begin{equation}\label{embedding} E\hookrightarrow L^q (\mathbb{R}^2), \, \, \, 1\leq q <\infty \end{equation} $

(2.2)

are continuous. Furthermore, by condition (V2), the above embeddings are compact (see [18, 19]).

Define the functional

$ \begin{align*} J(u) &=\frac{1}{2}\displaystyle\int_{\mathbb R^2} \Big(|\nabla u |^{2} +V(x)|u|^{2}+A^2_{1}|u|^{2}+A^2_{2}|u|^{2}\Big)\, dx -\displaystyle\int_{\mathbb R^2}F(u)\, dx, \end{align*} $

where $F(u)=\displaystyle\int^u_0 f(t)\, dt$. Note that

$ \begin{align}\nonumber &\displaystyle\int_{\mathbb R^2} A_0 |u|^2\, dx =-2\displaystyle\int_{\mathbb R^2} A_0(\partial_1 A_2-\partial_2 A_1)\, dx\\ =&2\displaystyle\int_{\mathbb R^2} (A_2 \partial_1 A_0-A_1 \partial_2 A_0)\, dx =2\displaystyle\int_{\mathbb R^2} (A^2_1 +A^2_2 )|u|^2\, dx.\label{eqn-relA0A12} \end{align} $

(2.3)

We have the derivative of $J$ in $H^{1}(\mathbb{R}^{2})$ as follows

$ \begin{align*} &\langle J^{\prime}(u), \, \eta\rangle\\ =&\displaystyle\int_{\mathbb R^2} \Big(\nabla u \nabla \eta + V(x)u\eta+(A^2_1(u) +A^2_2(u) )u\eta+A_{0}u\eta-f(u)\eta\Big)\, dx \end{align*} $

for all $\eta\in C^\infty_0({\mathbb R^2})$. Especially, from (2.3), we obtain that

$ \begin{align*} \langle J^{\prime}(u), \, u\rangle =\displaystyle\int_{\mathbb R^2} \Big(|\nabla u|^2 + V(x)|u|^2 +3\big(A^2_1(u)|u|^{2} +A^2_2(u)\big)|u|^{2} -f(u)u\Big)\, dx. \end{align*} $

By (1.1), we have that $A_{j}$ satisfy

$ \begin{align*} \Delta A_{1}=\partial_{2}(\frac{|u|^{2}}{2}), \quad \Delta A_{2}=-\partial_{1}(\frac{|u|^{2}}{2}), \quad \Delta A_{0}=\partial_{1}(A_{2}|u|^{2})-\partial_{2}(A_{1}|u|^{2}), \end{align*} $

which provide

$ A_{1}=A_{1}(u)= K_{2}*(\frac{|u|^{2}}{2}) =-\frac{1}{2\pi}\displaystyle\int_{\mathbb R^2} \frac{x_2-y_2}{|x-y|^{2}}\frac{|u|^2(y)}{2} \, dy, $

(2.4)

$ A_{2}=A_{2}(u)= - K_{1}*(\frac{|u|^{2}}{2}) =\frac{1}{2\pi}\displaystyle\int_{\mathbb R^2} \frac{x_1-y_1}{|x-y|^{2}}\frac{|u|^2(y)}{2}\, dy, $

(2.5)

$ A_{0}=A_{0}(u)= K_{1}*(A_{1}|u|^{2})-K_{2}*(A_{2}|u|^{2}), $

(2.6)

where $K_{j}=\frac{-x_{j}}{2\pi|x|^{2}}$ for $j=1, 2$ and $*$ denotes the convolution.

We know that $J$ is well defined in $H^{1}(\mathbb{R}^{2})$, $J\in C^{1}(H^{1}(\mathbb{R}^{2}))$, and the weak solution of (1.1) is the critical point of the functional $J$ from the following properties.

Proposition 2.1 (see [17]) Let $1 <s <2$ and $\frac{1}{s}-\frac{1}{q}=\frac{1}{2}$.

(ⅰ) Then there is a constant $C$ depending only on $s$ and $q$ such that

$ \big(\displaystyle\int_{\mathbb{R}^{2}}\big|Tu(x)\big|^{q}\, dx\big)^{\frac{1}{q}}\le C\big(\displaystyle\int_{\mathbb{R}^{2}}|u(x)|^{s}\, dx\big)^{\frac{1}{s}}, $

where the integral operator $T$ is given by

$ Tu(x):= \displaystyle\int_{\mathbb{R}^{2}}\frac{u(y)}{|x-y|}\, dy. $

(ⅱ) If $u\in H^{1}(\mathbb{R}^{2})$, then we have that for $j=1, 2$, $ \|A_{j}^{2}(u)\|_{L^{q}(\mathbb{R}^{2})}\le C\|u\|_{L^{2s}(\mathbb{R}^{2})}^{2}$ and

$ \|A_{0}(u)\|_{L^{q}(\mathbb{R}^{2})}\le C\|u\|_{L^{2s}(\mathbb{R}^{2})}^{2} \|u\|_{L^{4}(\mathbb{R}^{2})}^{2}. $

(ⅲ) For $q'=\frac{q}{q-1}$, $j=1, 2$,

$ \|A_{j}(u)u\|_{L^{2}(\mathbb{R}^{2})}\le \||A_{j}(u)|^{2}\|_{L^{q}(\mathbb{R}^{2})} \|u\|_{L^{2q'}(\mathbb{R}^{2})}^{2}. $

We will use the following mountain pass theorem to obtain our main result.

Theorem 2.2 (see [20]) Let ${\mathbb E}$ be a real Banach space and suppose that $I\in C^1({\mathbb E}, {\mathbb R})$ satisfies the following conditions

(ⅰ) $I(0)=0$ and there exist $\rho>0$, $\alpha>0$ such that $I_{\partial B_{\rho}(0)}\geq\alpha$;

(ⅱ) there exists $e\in E\backslash\overline{B_{\rho}(0)}$ such that $I(e)\leq 0$.

Let $c=\inf\limits_{\gamma\in\Lambda}\max\limits_{0\leq \tau\leq 1} I(\gamma(\tau)), $ where $\Lambda =\{\gamma\in C([0, 1], {\mathbb E}): \gamma(0)=0, \gamma(1)=e\}$ is the set of continuous paths joining 0 and $e$. Then $c\geq\alpha$ and there exists a sequence $\{u_{k}\}\subset {\mathbb E}$ such that

$ I(u_{k}) \stackrel{k}\rightarrow c \; \text{and} \; \|I'(u_{k})\|_{{\mathbb E}^*} \stackrel{k}\rightarrow 0. $

Moreover, if $I$ satisfies (PS) condition, then $c$ is a critical value of $I$ in $E$.

The following result on $A_j, \, j=0, 1, 2$ is important to prove the compactness.

Proposition 2.3 (see [17]) Suppose that $u_{n}$ converges to $u$ a.e. in $\mathbb{R}^{2}$ and $u_{n}$ converges weakly to $u$ in $H^{1}(\mathbb{R}^{2})$. Let $A_{j, n}:=A_{j}(u_{n}(x))$, $j=0, 1, 2$. Then

(ⅰ) $A_{j, n}$ converges to $A_{j}(u(x))$ a.e. in $\mathbb{R}^{2}$.

(ⅱ) $\displaystyle\int_{\mathbb{R}^{2}} A_{i, n}^{2}u_{n}u\, dx$, $\displaystyle\int_{\mathbb{R}^{2}} A_{i, n}^{2}|u|^{2}\, dx$ and $\displaystyle\int_{\mathbb{R}^{2}} A_{i, n}^{2}|u_{n}|^{2}\, dx$ converge to$\displaystyle\int_{\mathbb{R}^{2}} A_{i}^{2}|u|^{2}\, dx$ for $i=1, 2$; $\displaystyle\int_{\mathbb{R}^{2}} A_{0, n}u_{n}u\, dx$ and $\displaystyle\int_{\mathbb{R}^{2}} A_{0, n}|u_{n}|^{2}\, dx$ converge to$\displaystyle\int_{\mathbb{R}^{2}} A_{0}|u|^{2}\, dx$.

(ⅲ) $ \displaystyle\int_{\mathbb{R}^{2}} |A_{i}(u_{n}-u)|^{2} |u_{n}-u|^{2}\, dx= \displaystyle\int_{\mathbb{R}^{2}} |A_{i}(u_{n})|^{2}|u_{n}|^{2}\, dx -\displaystyle\int_{\mathbb{R}^{2}} |A_{i}(u)|^{2}|u|^{2} \, dx+o_{n}(1)$ for $i=1, 2$.

In order to prove the mountain pass construction, we need the following results in [1, 21, 22].

Proposition 2.4 (ⅰ) If $\alpha>0$ and $u\in H^1(\mathbb{R}^2)$ then $\displaystyle\int\limits_{\mathbb{R}^2} \big (e^{\alpha u^2}-1\big )\, dx <\infty.$ Moreover, if $\|\nabla u\|^2_2\leq 1$, $\|u\|_2\leq M <\infty$ and $\alpha <4\pi$, then there exists a constant $C=C(M, \, \alpha)$ such that $\displaystyle\int\limits_{\mathbb{R}^2} \big (e^{\alpha u^2}-1\big )\, dx <C(M, \, \alpha).$

(ⅱ) Let $\beta>0$ and $r>1$. Then for each $\alpha>r$ there exists a positive constant $C=C(\alpha)$ such that for all $s\in \mathbb{R}$, $\big(e^{\beta s^2}-1\big )^r\leq C\big(e^{\alpha\beta s^2}-1\big).$ In particular, if $u\in H^1(\mathbb{R}^2)$, then $\big(e^{\beta u^2}-1\big )^r$ belongs to $L^1(\mathbb{R}^2)$.

(ⅲ) If $v\in E$, $\beta >0$, $q>0$ and $\|v\|\leq M$ with $\beta M^2 < 4\pi$, then there exists $C=C(\beta, \, M, \, q)>0$ such that

$ \begin{equation} \displaystyle\int\limits_{\mathbb{R}^2} \big (e^{\beta v^2}-1\big ) |v|^q\, dx\leq C\|v\|^q. \end{equation} $

(2.7)

First of all, we prove the mountain pass structure.

Lemma 3.1 Assume (F2), (F3), and (F4) hold. Then there exist $\rho >0, \, \alpha>0$ such that $J(u)>\alpha$ for all $\|u\|=\rho$.

Proof From (F4), there exist $\varepsilon, \, \delta>0$, such that

$ \begin{align}\label{m1} |F(s)|\leq \frac{\lambda_1-\varepsilon}{2}|s|^2, \, \, \, \forall |s|\leq \delta. \end{align} $

(3.1)

By (F2) and (F3), we have $\forall q>2$, there exists $C=C(q, \delta)$ such that

$ \begin{align}\label{m2} |F(s)|\leq C|s|^q(e^{\alpha s^2-1}), \, \, \, \forall |s|\geq \delta. \end{align} $

(3.2)

By (3.1) and (3.2), we have

$ \begin{equation}\label{mountain3} |F(s)|\leq \frac{\lambda_1-\varepsilon}{2}|s|^2+ C|s|^q(e^{\alpha s^2-1}), \, \, \, \forall s\in \mathbb{R}, \, q>2. \end{equation} $

(3.3)

From (ⅲ) of Proposition 2.4, the definition of $\lambda_1$, and the continuous embeddings (2.2), we obtain

$ \begin{align*} J(u)\geq \frac{1}{2}\|u\|^2-\frac{\lambda_1-\varepsilon}{2}\|u\|^2-C\|u\|^{q}\geq \frac{1}{2}\big(1-\frac{\lambda_1-\varepsilon}{\lambda_1}\big)\|u\|^2-C\|u\|^{q}. \end{align*} $

Hence, we have

$ J(u)\geq \|u\|[\frac{1}{2}\big(1-\frac{\lambda_1-\varepsilon}{\lambda_1}\big)\|u\|-c\|u\|^{q-1}]. $

By $\varepsilon>0$ and $q>2$, we can choose $\rho>0$ and $\alpha>0$ such that $J_{\partial B_\rho(0)}\geq\alpha>0.$

Lemma 3.2 Assume that $f$ satisfies (F3). Then there exists $e\in E$ with $\|e\|>\rho$ such that $J(e) < 0$.

Proof Let $u\in H^1(\mathbb{R}^2)$ such that $u\equiv s_1$ in $B_1$, $u\equiv 0$ in $B^c_2$ and $u\geq 0$. Define $k={\rm supp}(u)$. From (F3), there exist positive constants $C_1, \, C_2$ such that for all $s\in\mathbb{R}$,

$ \begin{equation} F(s)\geq C_1|s|^\theta-C_2. \end{equation} $

(3.4)

Then we have for $t>1$,

$ J(tu)\leq \frac{t^2}{2}\|u\|^2+Ct^6\|u\|^6-Ct^\theta\displaystyle\int\limits_{\{x:\, t|u(x)|\geq s_1\}}u^\theta\, dx+C_1|k|. $

Since $\theta >6$, we have $J(tu)\rightarrow -\infty$ as $t\rightarrow +\infty$. Let $e=tu$ with $t$ sufficiently large, the proof is completed.

By Theorem 2.2, the functional $J$ has a (PS)$_c$ sequence. Next, we show this (PS)$_c$ sequence is bounded.

Lemma 3.3 Assume (F2) and (F3) hold. Let $(u_n)$ is a (PS)$_c$ sequence of $J$ in $E$, that is, $J(u_n)\rightarrow c$ and $J^\prime (u_n)\rightarrow 0$. Then $\|u_n\|\leq C$ for some positive constant $C$.

Proof We know

$ \begin{align*} \frac{1}{2}\|u_n\|^2+\frac{1}{2}\displaystyle\int\limits_{\mathbb{R}^2}\big(A^2_{1, n}|u_n|^2+A^2_{2, n}|u_n|^2\big)\, dx-\displaystyle\int\limits_{\mathbb{R}^2}F(u_n)\, dx=c+o_n(1) \end{align*} $

and for any $\varphi\in E$, we have

$ \begin{align*} \displaystyle\int\limits_{\mathbb{R}^2}\big(\nabla u_n \nabla\varphi+V(x)u_n\varphi\big)\, dx+\displaystyle\int\limits_{\mathbb{R}^2}\big(A^2_{1, n}+A^2_{2, n}+A_{0, n}\big)u_n\varphi\, dx -\displaystyle\int\limits_{\mathbb{R}^2}f(u_n)\varphi\, dx=o_n(\|\varphi\|). \end{align*} $

By (F3) and $\theta>6$, we get

$ \begin{align*} \theta c+\varepsilon_n\|u_n\|\geq &(\frac{\theta}{2}-1)\|u_n\|^2+(\frac{\theta}{2}-3)\displaystyle\int\limits_{\mathbb{R}^2}\big(A^2_{1, n}|u_n|^2+A^2_{2, n}|u_n|^2\big)\, dx\\ &-\displaystyle\int\limits_{\mathbb{R}^2}\big(\theta F(u_n)-f(u_n)u_n\big)\, dx\\ \geq &(\frac{\theta}{2}-1)\|u_n\|^2-\displaystyle\int\limits_{\{x:|u_n(x)| <s_1\}}\big(\theta F(u_n)-f(u_n)u_n\big)\, dx, \end{align*} $

where $\varepsilon_n\rightarrow 0$ as $n\rightarrow\infty$. From $|f(s)s-F(s)|\leq c_1|s|$ for all $|s|\leq s_1$ and inequality (\ref{L1}), we obtain $\theta c+\varepsilon_n\|u_n\|\geq(\frac{\theta}{2}-1)\|u_n\|^2-c_1\|u_n\|, $ which implies that $\|u_n\|\leq C$.

Now we are going to prove (PS) condition.

Lemma 3.4 The functional $J$ satisfies (PS) condition.

Proof Let $\{u_n\}$ be a (PS)$_c$ sequence of $J$, that is, $J(u_{n})\rightarrow c$ and $J'(u_{n})=0$. By Lemma 3.3, $\{u_n\}$ is bounded, up to a subsequence, we may assume that $u_n\rightharpoonup u_0$ in $E$, $u_n\rightarrow u_0$ in $L^q(\mathbb{R}^2)$ for all $q\geq 1$ and $u_n\rightarrow u_0$ almost everywhere in $\mathbb{R}^2$, as $n\rightarrow\infty$. If $f(s)$ satisfies (F2) and (F4), we have for each $\alpha>0$, there exist $b_1, \, b_2>0$ such that for all $s\in \mathbb{R}$,

$ \begin{align*} |f(s)|\leq b_1 |s|+b_2 \big( e^{\alpha s^2}-1\big). \end{align*} $

Then we have

$ \begin{align*} |f(u_n)-f(u_0)||u_n-u_0| \leq C[|u_n|+|u_0|+(e^{\alpha u^2_n}-1)+(e^{\alpha u^2_0}-1)]|u_n-u_0|. \end{align*} $

By (ⅰ) and (ⅱ) of Proposition 2.4 and Hölder inequality, we obtain

$ \begin{align}\label{psps1} \lim\limits_{n\rightarrow\infty}\displaystyle\int_{\mathbb{R}^{2}}(f(u_n)-f(u_0))(u_n-u_0)\, dx =0. \end{align} $

(3.5)

By Proposition 2.3, we have

$ \begin{align}\label{ps24} \lim\limits_{n\rightarrow\infty}\displaystyle\int_{\mathbb{R}^{2}}[(A_1^2(u_n)+A_2^2(u_n))u_n-(A_1^2(u_0)+A_2^2(u_0))u_0](u_n-u_0)\, dx =0, \end{align} $

(3.6)

and

$ \begin{align}\label{psps2} \lim\limits_{n\rightarrow\infty}\displaystyle\int_{\mathbb{R}^{2}}[(A_0(u_n)u_n+A_0(u_0)u_0)](u_n-u_0)\, dx =0. \end{align} $

(3.7)

From (3.5), (3.6), and (3.7), we have

$ \begin{align*} \|u_n-u_0\|_{E}^2 =&\langle J^\prime(u_n)-J^\prime(u_0), u_n-u_0\rangle\\ &+\displaystyle\int_{\mathbb{R}^{2}}[(A_1^2(u_0)+A_2^2(u_0))u_0-(A_1^2(u_n)+A_2^2(u_n))u_n](u_n-u_0)\, dx\\ &+\displaystyle\int_{\mathbb{R}^{2}}[(A_0(u_0)u_0-A_0(u_n)u_n)](u_n-u_0)\, dx\\ &+\displaystyle\int_{\mathbb{R}^{2}}(f(u_n)-f(u_0))(u_n-u_0)\, dx. \end{align*} $

We obtain that $u_n\rightarrow u_0$ as $n\rightarrow \infty$ in $E$.

Proof of Theorem 1.1 By Theorem 2.2, Lemma 3.1, Lemma 3.2 and Lemma 3.4, we obtain that functional $J$ has a critical point $u_0$ at the minimax level

$ c=\inf\limits_{\gamma\in\Lambda}\max\limits_{0\leq \tau\leq 1} I(\gamma(\tau)), $

where $\Lambda =\{\gamma\in C([0, 1], {\mathbb E}): \gamma(0)=0, \gamma(1)=e\}$.