Existence of ground state solutions for the modified Chern-Simons-Schrödinger equations with general Choquard type nonlinearity

1.   Introduction

In this paper, we establish the existence of ground state solutions to the following modified Chern-Simons-Schrödinger equation

where $\kappa$, $q > 0$, $\alpha\in (0, 2)$ and $h(l) = \int_0^l{\varrho u^2(\varrho)}{\rm{d}}\varrho$ $(l\geq0)$, $u$ is a radially symmetric function, $I_\alpha$ is a Riesz potential defined by

and $\Gamma$ is the Gamma function, $F(t) = \int^t_0f(s){\rm{d}}s$, the potential $V$ is supposed to satisfies:

$(\mathcal{V}_1)$ $V\in \mathcal{C}({\mathbb R}^2, {\mathbb R})$;

$(\mathcal{V}_2)$ $V(|x|) = V(x)$ and there exists $\beta\geq\gamma > 0$, such that $\beta\geq V(|x|)\geq\gamma$ for all $x\in {\mathbb R}^2$.

Moreover, we assume that the nonlinearity $f: {\mathbb R}\rightarrow {\mathbb R}$ verifies:

$(f_1)$ $f\in \mathcal{C}({\mathbb R}^2, {\mathbb R})$, $f(t) = o(t)$;

$(f_2)$ There exist constant $p\in(2+\alpha, +\infty)$ and $C > 0$ such that

$(f_3)$ There exists a constant $\vartheta > 8$ such that

As we all know, Eq $\rm (1.1)$ originates from seeking the standing waves of the following nonlinear Chern-Simons-Schrödinger system

where $i$ denotes the imaginary unit, $\partial_0 = \frac{\partial}{\partial t}$, $\partial_1 = \frac{\partial}{\partial x_1}$, $\partial_2 = \frac{\partial}{\partial x_2}$ for $(t, x_1, x_2)\in {\mathbb R}^{1+2}$, $\phi: {\mathbb R}^{1+2}\rightarrow \mathbb{C}$ is the complex scalar field, $A_j: {\mathbb R}^{1+2}\rightarrow {\mathbb R}$ is the gauge field, $D_j = \partial_j+iA_j$ is the covariant derivative for $j = 0, 1, 2$. The system was first proposed by Jackiw and Pi, consisting of Schrödinger equation augmented by the gauge field. The two-dimensional Chern-Simons-Schrödinger system is a non-relativistic quantum model describing the dynamics of a large number of particles in the plane, in which these particles interact directly through the spontaneous magnetic field. Moreover, the important applications of the system are also reflected in the study of the high temperature superconductors and fractional quantum Hall effect and Aharovnov-Bohm scattering. For more physical backgrounds of $\rm (1.2)$, we refer readers to ^[16,17] and the references therein.

As far as we know, Byeon et al.'s ^[1] was the first article investigate the standing wave solutions of this system by the variational method. They considered the standing waves of system $\rm (1.2)$ with power type nonlinearity, that is, $f(u) = \lambda |u|^{p-1}u$, and obtained the existence and nonexistence results for $\rm (1.2)$ of type

where $w > 0$ is a given frequency, $\lambda > 0$ and $p > 1$, $u$, $k$, $h$ are real valued functions depending only on $|x|$. The ansatz $\rm (1.3)$ satisfies the Coulomb gauge condition $\partial_1A_1+\partial_2A_2 = 0$. After then, many researchers began to pay attention to this field. see e.g. ^{[2,3,5,7,11,12,13,15,20,21,29]} and the references therien. However, through the study of large number of literatures, it is found that there are few papers studying the modified Chern-Simons-Schrödinger equation, except for ^[8,23,24]. To best of our knowledge, there is no article to pay attention to general Choquard type nonlinearity for modified Chern-Simons-Schrödinger equation. Motivated by the previously mentioned paper [26], we shall study the existence of ground state solutions for Eq (1.1) using a change of variable and variational argument.

The problem $\rm (1.1)$ is the Euler-Lagrange equation of the energy functional

From the variational point of view, the main difficulty of this problem is the energy functional $\mathcal{I}$ can not be well defined for $u\in H_r^1({\mathbb R}^2)$. To solve this problem, we intend to adopt the Liu and Wang's ^[18] approach, considering the change of variable $g: {\mathbb R}\rightarrow {\mathbb R}$ given by

$g(0) = 0$ and $g(-t) = -g(t)$ on $(-\infty, 0]$. By the change of $u = g(v)$ of variable, Eq $\rm (1.1)$ is transformed into a semilinear problem

where

Furthermore, the functional $\mathcal{I}(u)$ can be reduced to

where

Obviously, the energy functional $\mathcal{J}(u)$ is well defined in $H_r^1({\mathbb R}^2)$. It is easy to see that $v$ is a critical point of $\mathcal{J}$,

for any $\psi\in H_r^1({\mathbb R}^2)$, then $v$ is a weak solution of $\rm (1.4)$, that is $u = g(v)$ solves $\rm (1.1)$. In particular, for $\tau = 2$ or $\tau = 4$, using the integrate by parts, one has

Note that by the Cauchy inequality, there exists a constant $C_0 > 0$ such that

Then for $v\in H_r^1({\mathbb R}^2)$, we have

Now, we give our result in the following.

Theorem 1.1. Under assumptions $(\mathcal{V}_1)$, $(\mathcal{V}_2)$ and $(f_1)$–$(f_3)$, problem $\rm (1.1)$ has a ground state solution.

Notations. To facilitate expression, hereafter, we recall the following basic notes:

$\bullet$ $H^1({\mathbb R}^2)$: = $\{u\in L^2({\mathbb R}^2), \nabla u\in L^2({\mathbb R}^2)\}$ with the norm $\|v\| = \left(\int_{ {\mathbb R}^2}(v^2+|\nabla v|^2)\right)^{1/2}$;

$\bullet$ $H^1_r({\mathbb R}^2)$: = $\{v\in H^1({\mathbb R}^2)$: $v(x) = v(|x|) \};$

$\bullet$ $L^s({\mathbb R}^2)$ denotes the Lebesgue space with the norm $\|v\|_{s} = \left(\int_{ {\mathbb R}^2}|v|^s\right)^{1/s}$, where $1\leq s < +\infty$;

$\bullet$ The embedding $H^1_r({\mathbb R}^2)\hookrightarrow L^s({\mathbb R}^2)$ is continuous for $2\leq s < +\infty$;

$\bullet$ The embedding $H^1_r({\mathbb R}^2)\hookrightarrow L^s({\mathbb R}^2)$ is compact for $2 < s < +\infty$;

$\bullet$ $H^1_r({\mathbb R}^2)\hookrightarrow L^\frac{4\upsilon}{2+\alpha}({\mathbb R}^2)$ if and only if $\frac{2+\alpha}{2}\leq\upsilon < +\infty$;

$\bullet$ $\int_{ {\mathbb R}^2}\spadesuit$ denotes $\int_{ {\mathbb R}^2}\spadesuit{\rm{d}}x$;

$\bullet$ The weak convergence in $H^1_r({\mathbb R}^2)$ is denoted by $\rightharpoonup$, and the strong convergence by $\rightarrow$;

$\bullet$ We use $C$, $C_0$ denote various positive constants.

The remainder of the paper is organized as follows. In section 2, we present some preliminary results. Section 3 devote to some required results and complete the proof details of Theorem 1.1.

2.   Preliminaries

In this section, we give some useful lemmas and proposition, which play an important role in the proof of our result. Next, let us recall some properties of the variable $g$, which are proved in ^[6,18,27].

Lemma 2.1. (^[6,18,27]) The function $g(t)$ and its derivative satisfy the following properties:

$(g_1)$ $|g(t)|\leq |t|$ for all $t\in {\mathbb R}$;

$(g_2)$ $|g(t)|\leq 2^{1/4}|t|^{1/2}$ for all $t\in {\mathbb R}$;

$(g_3)$ $g(t)/2\leq tg'(t)\leq g(t)$ for all $t\geq 0$;

$(g_4)$ $g^2(t)/2\leq tg(t)g'(t)\leq g^2(t)$ for all $t\in {\mathbb R}$;

$(g_5)$ $|g(t)g'(t)|\leq 1/\sqrt{2}$ for all $t\in {\mathbb R}$;

$(g_6)$ There exists a constant $C > 0$ such that

Next, the following inequality holds if and only if the functions in $H^1_r({\mathbb R}^2)$.

Proposition 2.2. (^[1]) For $v\in H^1_r({\mathbb R}^2)$, there holds

In order to achieve our main result, we would like to recall the well-known Hardy-Littlewood-Sobolev inequality in ^[19].

Lemma 2.3. (^[19]) Let $\mu$, $\nu > 1$ and $0 < \alpha < N(N = 1, 2...)$ be such that

Where $\zeta\in L^\mu({\mathbb R}^N)$ and $\eta\in L^\nu({\mathbb R}^N)$, there exists a constant $C$, independent of $\zeta$, $\eta$, such that

Finally, for functional $\mathcal{C}(v)$, $\mathcal{D}(v)$, we give the following compactness lemma:

Lemma 2.4. (^[8]) Suppose that a sequence $\{v_n\}$ converges weakly to a function $v$ in $H^1_r({\mathbb R}^2)$ as $n\rightarrow +\infty$. Then for each $\psi\in H^1_r({\mathbb R}^2)$, $\mathcal{C}(v_n)$, $\mathcal{C}'(v_n)\psi$ and $\mathcal{C}'(v_n)v_n$, $\mathcal{D}(v_n)$ and $\mathcal{D}'(v_n)\psi$, $\mathcal{D}'(v_n)v_n$ converges up to a subsequence to $\mathcal{C}(v)$, $\mathcal{C}'(v)\psi$ and $\mathcal{C}'(v)v$, $\mathcal{D}(v)$ and $\mathcal{D}'(v)\psi$, $\mathcal{D}'(v)v$, respectively, as $n\rightarrow +\infty$.

3.   Proof of Theorem 1.1

In this section, we would like to complete the proof of Theorem 1.1.

Theorem 3.1. (^[14]) Set $(E, \parallel\cdot\parallel)$ be a Banach space, $\mathbb{I}\subset {\mathbb R}^{+}$ be a real interval. Consider a family $\Psi_\eta$ of $C^1$-functional on $E$

where $B(v)$ is non-negative and when $\parallel v\parallel\rightarrow +\infty$, either $\mathcal{A}(v)\rightarrow +\infty$ or $\mathcal{B}(v)\rightarrow +\infty$. Assume that there exist two points $v_1$, $v_2$ holds

where

Then for a.e. $\lambda\in \mathbb{I}$, there exist a sequence $\{v_n\}\subset E$ such that

$(1)$ $\{v_n\}$ is bounded in $E$;

$(2)$ $\lim\limits_{n\rightarrow +\infty} \Psi_\lambda(v_n) = c_{\lambda}$;

$(3)$ $\lim\limits_{n\rightarrow +\infty} \Psi'_\lambda(v_n) = 0$ in the dual space $E^{-1}$ of $E$.

Furthermore, the map $\lambda\mapsto c_{\lambda}$ is non-increasing and left continuous.

Let $\mathbb{I} = [\frac{1}{2}, 1]$, we define the following energy functional

where $\lambda\in \mathbb{I}$. Moreover, let

and

Setting $\parallel v\parallel\rightarrow +\infty$, then $\mathcal{A}(v)\rightarrow +\infty$. Furthermore, $\mathcal{B}(v)\geq0$.

Next, we prove that the functional $\mathcal{J}$ exhibits the mountain pass geometry.

Lemma 3.2. Under assumptions $(\mathcal{V}_1)$ and $(\mathcal{V}_2)$, then there holds:

$(i)$ There exists $v\in H^1_r({\mathbb R}^2)\backslash\{0\}$ such that $\mathcal{J}_\lambda(v) < 0$ for all $\lambda\in \mathbb{I}$;

$(ii)$ $c_\lambda = \inf\limits_{\gamma\in\bar{\Gamma}_\lambda}\max\limits_{t\in[0, 1]}\mathcal{J}_\lambda(\gamma(t)) > \max\Big\{\mathcal{J}_\lambda(0), \mathcal{J}_\lambda(v) \Big\}$ for all $\lambda\in \mathbb{I}$, where

Proof. (i) Let $v\in H^1_r({\mathbb R}^2)\backslash\{0\}$ be fixed. For any $\lambda\in \mathbb{I} = [\frac{1}{2}, 1]$, we have

Arguing as in ^[4,9], we consider $\xi\in \mathcal{C}^\infty_0({\mathbb R}^2)$ which satisfies $0\leq\xi(x)\leq 1$, $\xi(x) = 0$ for $|x|\geq 2$ and $\xi(x) = 1$ for $|x|\leq 1$. By $(g_3)$, we can deduce that $g(t\xi(x))\geq g(t)\xi(x)$ for $t\geq0$. According to Yang et al. ^[25], from $(f_3)$ and $(g_4)$ that for $t > \frac{1}{\|\xi\|}$, we have

Thus from $(g_1)$, one has

for all $t > 0$. By $\vartheta > 8$, we deduce that $\mathcal{J}_\lambda(t\xi)\rightarrow -\infty$ as $t\rightarrow +\infty$. Thus, there exists a $t_0 > 0$ such that $\mathcal{J}_\lambda(t_0\xi) < 0$. Then taking a function $v = t_0\xi$, we have $\mathcal{J}_\lambda(v) < 0$ for all $\lambda\in \mathbb{I}$.

(ii) By Chen et al. ^[4] and Fang et al. ^[], there exists $\rho' > 0$ such that

for all $\|v\|\leq\rho'$. From $(g_2)$, Lemma 2.3 and Sobolev imbedding inequality, for $\varepsilon > 0$ sufficiently small, one has

Since $p > 2+\alpha$, we get $\mathcal{J}_\lambda(v) > 0$ if $\rho'$ is small enough. Hence, $\mathcal{J}_\lambda(0)$ is strict local minimum, $c_\lambda > 0$.

By Theorem 3.1, it is easy to know that for any almost everywhere $\lambda\in \mathbb{I}$, there exists a bounded sequence $\{w_n\}\subset H^1_r({\mathbb R}^2)$ such that $\mathcal{J}'_\lambda(w_{n})\rightarrow 0$ and $\mathcal{J}_\lambda(w_n)\rightarrow c_{\lambda}$, which is called $(PS)$ sequence.

Lemma 3.3. Assume that $\{w_n\} \subset H^1_r({\mathbb R}^2)$ is a sequence of obtain above. Then, for almost $\lambda\in\mathbb{I}$ there exists $w_\lambda\in H^1_r({\mathbb R}^2)\backslash\{0\}$, such that $\mathcal{J}'_\lambda(w_\lambda) = 0$ and $\mathcal{J}_\lambda(w_\lambda) = c_\lambda$.

Proof. By Theorem 3.1 and Lemma 3.2, we know that $\{w_n \}\subset H^1_r({\mathbb R}^2)$ is bounded, then up to a subsequence, there exists $w_\lambda\in H^1_r({\mathbb R}^2)\backslash\{0\}$ such that $w_n\rightharpoonup w_\lambda$ in $H^1_r({\mathbb R}^2)$, $w_n\rightarrow w_\lambda$ in $L^s({\mathbb R}^2)$ $(s > 2)$ and $w_{n}\rightarrow w_\lambda$ a.e. in ${\mathbb R}^2$. By the Lebesgue-dominated convergence theorem, it is easy to check that $\mathcal{J}'_\lambda(w_\lambda) = 0$. Similar to ^[9,10,22,28], we get

$(f_1)$ and $(f_2)$ imply that for each $\varepsilon > 0$, there exists a constant $C_\varepsilon > 0$ such that

Furthermore, using Lemma 2.3, the Hölder inequality and $(g_2)$, $(g_5)$, one obtains

In the same way, we can prove that

Thus, it follows from $\rm (1.7)$, $\rm (3.1)$–$\rm (3.4)$ and Lemma 2.4 that

then, we deduce that $w_{n}\rightarrow w_\lambda$ in $H_r^1({\mathbb R}^2)$. Thus, $w_\lambda$ is a nontrivial critical point of $\mathcal{J}_\lambda$ with $\mathcal{J}_\lambda(w_\lambda) = c_\lambda$. This completes the proof.

Proof of Theorem 1.1. At first, by Theorem 3.1, for a.e. $\lambda\in \mathbb{I}$, there exists $w_\lambda \in H_r^1({\mathbb R}^2)$ such that $w_n\rightharpoonup w_\lambda\neq0$ in $H_r^1({\mathbb R}^2)$, $J'_\lambda(w_n)\rightarrow 0$ and $\mathcal{J}_\lambda(w_n)\rightarrow c_\lambda$. By Lemma 3.3, one obtains $\mathcal{J}'_\lambda(w_\lambda) = 0$, $\mathcal{J}_\lambda(w_\lambda) = c_\lambda$. Then, take $\{\lambda_n\}\subset \mathbb{I}$ such that $\lim\limits_{n\rightarrow +\infty}\lambda_n = 1$, $w_{\lambda_n}\in H_r^1({\mathbb R}^2)$ and $\mathcal{J}'_{\lambda_n}(w_{\lambda_n}) = 0$, $\mathcal{J}_{\lambda_n}(w_{\lambda_n}) = c_{\lambda_n}$. Next, we claim that $\|w_{\lambda_n}\|\leq C$. From $(f_3)$, $\rm (1.6)$, $\rm (1.7)$ and Lemma 3.2 and $\mathcal{J}_{\lambda_n}(w_{\lambda_n})\leq c_{\frac{1}{2}}$, $J'_{\lambda_n}(w_{\lambda_n}) = 0$, it follows that

$\rm (3.5)$ infer that $\int_{ {\mathbb R}^2}|\nabla w_{\lambda_n}|^2\leq C$. From $(g_1)$ and $(g_6)$, it holds

Then by $\rm (1.8)$, Proposition 2.2 and $\rm (3.5)$, we deduce that $\int_{ {\mathbb R}^2}|w_{\lambda_n}|^2\leq C$. Hence, there is a constant $C > 0$ independent of $n$ such that $\|w_n\| = \int_{ {\mathbb R}^2}(|\nabla w_n|^2+w_n^2)\leq C$. Next, we can suppose that the limit of $\mathcal{J}_{\lambda_n}(w_{\lambda_n})$ exists. By Theorem 3.1, we have $c_{\lambda_n}\rightarrow c_1$ is continuous from the left. So, we get $0\leq\lim\limits_{n\rightarrow +\infty}\mathcal{J}_{\lambda_n}(w_{\lambda_n}) = c_{\lambda_n}\leq c_{\frac{1}{2}}$. Thus, using the fact that

and for any $\psi\in H_r^1({\mathbb R}^2) \backslash \{0\}$, there holds

Then, up to a subsequence, $\{w_{\lambda_n}\}$ is a bounded $(PS)_{c_1}$ sequence of $\mathcal{J}$. Preceding the same method as Lemma 3.3, we get that the existence of a nontrivial solution $v_0\in H_r^1({\mathbb R}^2)$ for $\mathcal{J}$ satisfying $\mathcal{J}(v_0) = c_1$, $\mathcal{J}'(v_0) = 0$.

To seek the ground state solutions, we need to define $m_0 : = \inf\{\mathcal{J}(v_0): v_0\neq 0, \mathcal{J}'(v_0) = 0\}$. From $\rm (3.5)$, we have $m_0\geq0$. Set $\{v'_n\}$ be a sequence satisfies $\mathcal{J}'(v'_n) = 0$, $\mathcal{J}(v'_n)\rightarrow m_0$. Similar to the discussed above, one obtains $\{v'_n\}$ is a bounded $(PS)_{m_0}$ sequence of $\mathcal{J}$. Arguing as in Lemma 3.3, one obtains there exists a $\bar{v}_0\in H^1_r({\mathbb R}^2)$ such that $\mathcal{J}'(\bar{v}_0) = 0$, $\mathcal{J}(\bar{v}_0) = m_0$, which implies that $\bar{u}_0 = g(\bar{v}_0)$ is a ground state solution of $\rm (1.1)$. This completes the proof.

4.   Conclusions

In this paper, we have considered the modified Chern-Simons-Schrödinger equation involving radially symmetric variable potential $V$ and general Choquard type nonlinearity. By using a change of variable and variational argument, we obtain the existence of ground state solutions. It is hoped that the results obtained in this paper may be a new starting point for further research in this field.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant Nos. 11361042, 11771198, 11901276, 11961045), Jiangxi Provincial Natural Science Foundation (Grant Nos. 20202BAB201001 and 20202BAB211004) and Science and technology research project of Jiangxi Provincial Department of Education (Grant Nos. GJJ218419).

Conflict of interest

The authors declare that they have no competing interests.