## Ground-State Solutions for Schrödinger-Maxwell Equations in the Critical Growth

Fang Liwan,1, Huang Wennian,2, Wang Minqing,2

 基金资助: 广西师范大学科学研究基金.  2014ZD001广西自然科学基金.  2015GXNSFBA1390182017广西研究生教育创新计划项目.  XYCZ2017074

 Fund supported: the Science Research Fund of Guangxi Normal University.  2014ZD001the Natural Science Foundation of Guangxi.  2015GXNSFBA139018the Postgraduate Education Innovation Plan Project of Guangxi in 2017.  XYCZ2017074

Abstract

In this paper, we study the existence of the ground state solutions for the following Schrödinger-Maxwell equations

where β is a positive constant. Under some assumptions on V, K and b(x), by using the variational method and critical point theorem, we prove that such a class of equations has at least a ground state solution for α < 0 and p ∈ (3, 4).

Keywords： Schrödinger-Maxwell equations ; Critical point theorem ; Critical growth ; Ground state solution ; Nehari manifold

Fang Liwan, Huang Wennian, Wang Minqing. Ground-State Solutions for Schrödinger-Maxwell Equations in the Critical Growth. Acta Mathematica Scientia[J], 2019, 39(3): 475-483 doi:

## 1 引言

$$$\left\{\begin{array}{ll} -\triangle u+V(x)u-(K(x)+\alpha)\phi u=\beta|u|^{4}u+b(x)|u|^{p-1}u, &(x, u)\in(\mathbb{R}^{3}, \mathbb{R}), \\ \triangle\phi=(K(x)+\alpha)u^{2}, &(x, u)\in(\mathbb{R}^{3}, \mathbb{R}), \end{array} \right.$$$

$(V_{1})$$V\in C(\mathbb{R}^{3}, \mathbb{R}),$$\inf\limits_{x\in\mathbb{R}^{3}}V(x)>0$,对任一$M>0, meas\{x\in\mathbb{R}^{3}|V(x)\leq M\}<\infty.$

$(V_{2})$$V(\infty)=\lim\limits_{|x|\rightarrow\infty}\inf V(x)\geq V(x),$$V(x)\not\equiv V(\infty)$.

(K) $K(x)\in L^{\infty}(\mathbb{R}^{3}, \mathbb{R}), $$K(x)\leq0, x\in\mathbb{R}^{3}. (B) b(x):\mathbb{R}^{3}\rightarrow\mathbb{R}^{+},且b(x)\in L^{p+2}(\mathbb{R}^{3}). K(x)\equiv0, x\in\mathbb{R}^{3}, Zhang在文献[1]中研究如下系统 $$\left\{\begin{array}{ll} -\triangle u+V(x)u+\mu\phi u=f(u), & (x, u)\in(\mathbb{R}^{3}, \mathbb{R}), \\ -\triangle\phi=\mu u^{2}, & (x, u)\in(\mathbb{R}^{3}, \mathbb{R}), \end{array} \right.$$ 其中V, \mu>0, f(u)临界增长且为奇函数,运用变分形式山路定理,得到了系统(1.2)的正径向解和基态解的存在性.而Liu和Guo在文献[2]中运用变分法研究下列临界点情形的Schrödinger-Maxwell系统 $$\left\{\begin{array}{ll} -\triangle u+V(x)u+\lambda\phi u=\mu|u|^{q-1}u+|u|^{4}u, & (x, u)\in(\mathbb{R}^{3}, \mathbb{R}), \\ -\triangle\phi=u^{2}, &(x, u)\in(\mathbb{R}^{3}, \mathbb{R}), \end{array} \right.$$ 其中\mu是一个正参数.在V的适当假设下,运用变分形式的山路定理,当q\in(3, 5)$$q\in(2, 3], $$\mu足够大的时候,对任一\lambda>0,系统(1.3)存在基态解(u, \phi_{u})\in H^{1}(\mathbb{R}^{3})\times D^{1, 2}(\mathbb{R}^{3}). \alpha=0时, Yang和Han在文献[3]中研究如下系统 $$\left\{\begin{array}{ll} -\triangle u+V(x)u+K(x)\phi u=f(x, u), & (x, u)\in(\mathbb{R}^{3}, \mathbb{R}), \\ -\triangle\phi=K(x)u^{2}, &(x, u)\in(\mathbb{R}^{3}, \mathbb{R}), \end{array} \right.$$ 其中V>0, K(x)\geq0.存在a_{2}>0, p\in(4, 2^{*}),使得|f(x, u)|\leq a_{2}(1+|u|^{p-1}),以及\lim\limits_{|u|\rightarrow0}\frac{f(x, u)}{u}=0, \lim\limits_{|u|\rightarrow\infty}\frac{f(x, u)}{|u|^{4}}=+\infty,当u>0时, \frac{f(x, u)}{u^{3}}递增, u<0时, \frac{f(x, u)}{u^{3}}递减,且存在\gamma>0,使得F(x, u)\geq-\gamma u^{4}(F(x, u)=\int^{u}_{0}f(x, s){\rm d}s.运用山路定理和喷泉定理可得系统(1.4)在f(x, u)关于u为奇函数的条件下平凡解和高能解的存在性和多重性. 自从系统(1.1)在文献[5]中提出来以后,就有很多学者研究其解的存在性和多重性,而更多的结论是关于非线性项只有一项,而且是次临界情形的,可详见文献[5-12].本文的主要工作是运用山路定理研究临界情形下,具有两项非线性项的系统(1.1)基态解的存在性. 下面是本文的主要结论: 定理1.1 若(V_{1}), (V_{2}), (K)以及(B)成立, p\in(3, 4),且\beta>0,则对每个\alpha<0, 系统(1.1)至少存在一个基态解. 注1.1 定理1.1是扩展[1-3]的结论.相对于系统(1.2), (1.3)和(1.4),系统(1.1)所满足的条件较复杂,这使得证明基态解的存在性时较困难.本文研究的系统与系统(1.2), (1.3)和(1.4)比较,创新点是同时满足以下两点: a) Scrödinger-Maxwell方程有两个参量K(x)$$\alpha$,且都小于0.

b)非线性项是非自治的,含有两项,且其中一项临界增长.

$E$上赋予內积及由该內积导出的范数为

$(V_{1})$知,下面的嵌入是紧嵌入:

$$$\Delta\phi_{u}=(K(x)+\alpha)u^{2}.$$$

$(1.5)$式,有

$$$\Phi(u_{n})\rightarrow c, \ \ \Phi'(u_{n})\rightarrow 0.$$$

任取$\{u_{n}\}$满足(2.1)式,则$-\langle \Phi'(u_{n}), u_{n}\rangle\leq o(1)\|u_{n}\|,$且存在$M>0,$使得$|\Phi(u_{n})|\leq M$.由(K)和(B),有

根据引理2.6可知,满足(2.1)式的$\{u_{n}\}$有界,即$\|u_{n}\|<\infty$.根据$E$的自反性,设$\{u_{n}\}$的子列$\{u^{*}_{n}\}$满足

$$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u^{*}_{n}\rightharpoonup u^{*},\ \ \ u^{*} \in E.\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u^{*}_{n}\rightarrow u^{*},\ \ \ u^{*} \in L^{t}(\mathbb{R}^{3}),\ \ t\in(2,6).\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u^{*}_{n}\rightarrow u^{*},\ \ \ {\it a.e.}\ \ u^{*} \in \mathbb{R}^{3}.$$$

$v^{*}_{n}=u^{*}_{n}-u^{*}$.根据上面的讨论和Brezi-Lieb引理[16],有

$$$o(1)=\langle\Phi'(u^{*}_{n}),u^{*}_{n}\rangle-\langle\Phi'(u^{*}),u^{*}\rangle=\|v^{*}_{n}\|^{2}-\beta\int_{\mathbb{R}^{3}}(v^{*}_{n})^{6}{\rm d}x.$$$

$S=\inf\limits_{u_{n}\in D^{1, 2}\setminus\{0\}}\frac{\|\nabla u_{n}\|^{2}}{\|u_{n}\|^{2}_{6}}.$类似文献[1]的讨论,对任一$\epsilon, r>0$,定义

$l>0$,由Sobolev嵌入定理,有$S\leq\frac{\|v^{*}_{n}\|^{2}}{(\int_{\mathbb{R}^{3}}|v^{*}_{n}|^{6}{\rm d}x)^{\frac{1}{3}}}.$从而

$\{v_{n}\}$$\Phi的非平凡临界点序列,满足 因为\Phi(v_{n})有界,故由引理2.6的讨论知, \{v_{n}\}$$E$上有界.因此$\{v_{n}\}$$\Phi$$(PS)_{c}$序列.由系统(2.1)和引理2.7,有

$$$\Phi(v_{n})\rightarrow\Phi(v_{0})+\sum\limits_{i = 1}^{i = k} {{\Phi ^\infty }} ({w^i})=c+o(1),$$$

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