## Nontrivial Solution for Schrödinger-Poisson type Systems with Double Critical Terms

Feng Xiaojing,

 基金资助: 国家自然科学基金.  11571209国家自然科学基金.  11671239山西省自然科学基金.  201801D211001山西省自然科学基金.  201801D121002山西省自然科学基金.  201801D221012山西省留学基金.  020-005

 Fund supported: the NSFC.  11571209the NSFC.  11671239NSF of Shanxi Province.  201801D211001NSF of Shanxi Province.  201801D121002NSF of Shanxi Province.  201801D221012the Shanxi Scholarship Council.  020-005

Abstract

The existence of a nontrivial solution to the Schrödinger-Poisson type system with both nonlinear critical growth and nonlocal critical growth is obtained by applying the concentration-compactness principle and mountain pass theorem. The double critical growth in the system presents an obstacle when showing the convergence of the bounded (PS) sequences. At the same time, it is difficult to estimate the critical level of the mountain pass. The key ingredient of the paper is to show that the critical level of the mountain pass is below the non-compactness level of the associated energy functional.

Keywords： Schrödinger-Poisson type system ; Critical exponent ; Variational method

Feng Xiaojing. Nontrivial Solution for Schrödinger-Poisson type Systems with Double Critical Terms. Acta Mathematica Scientia[J], 2020, 40(6): 1590-1598 doi:

## 1 引言及主要结果

$\left\{\begin{array}{ll} -\Delta u+u-\phi |u|^3u = |u|^4u+f(u), & x\in {{\Bbb R}} ^3, \\-\Delta \phi = |u|^5, & x\in {{\Bbb R}} ^3 \end{array}\right.$

$\left\{\begin{array}{ll} -\Delta u+V(x)u+K(x)\phi u = f(x, u), & x\in {{\Bbb R}} ^3, \\-\Delta \phi = K(x)u^2, & x\in {{\Bbb R}} ^3, \end{array}\right.$

($f_2$) $\lim\limits_{s\to 0}f(s)/s = 0$;

($f_3$) $0<4 F(s) = 4\int_0^sf(t){\rm d}t\leq sf(s), \ s\in{{\Bbb R}} \setminus\{0\};$

($f_4$) $\lim\limits_{s\to\infty}F(s)/s^{4} = \infty$.

## 2 预备知识

\begin{align} S = \inf\limits_{u\in D^{1, 2}\backslash\{0\}}\frac{\int_{{{\Bbb R}} ^3}|\nabla u|^2} {\left(\int_{{{\Bbb R}} ^3}|u|^{6}\right)^\frac{1}{3}}. \end{align}

\begin{align} \phi_u(x) = \frac{1}{4\pi}\int_{{{\Bbb R}} ^3}\frac{|u(y)|^5}{|x-y|}{\rm d}y, \ x\in{{\Bbb R}} ^3, \end{align}

(ⅱ) 对任意的t>0 u\in H^1({{\Bbb R}} ^3) , 有 \phi_{tu} = t^{5}\phi_u ; (ⅲ) 对任意的 u\in H^1({{\Bbb R}} ^3) , 有 \|\phi_u\|_{D^{1, 2}}\leq S^{-1/2}\|u\|^{5}_{6} 以及 \begin{align} \int_{{{\Bbb R}} ^3}\phi_u |u|^{5}\leq S^{-1}\|u\|^{10}_{6}, \end{align} 其中 S 为最佳Sobolev嵌入常数; (ⅳ) 若 u_n H^1({{\Bbb R}} ^3)中弱收敛于$u$, 则存在子列使得$\phi_{u_n} $$D^{1, 2} 中弱收敛于 u . 此外, 系统 (1.1) 的弱解为定义在 H^1({{\Bbb R}} ^3) 上泛函 的临界点.由条件( f_1 )和( f_2 ), 可知 J$$ H^1({{\Bbb R}} ^3)$上有定义, 且$J\in C^1(H^1({{\Bbb R}} ^3), {{\Bbb R}} )$,

\begin{align} |F(s)|\leq\frac{\varepsilon}{2}|s|^2+\frac{C_\varepsilon}{q }|s|^{q}, \ s\in {{\Bbb R}} . \end{align}

\begin{align} \left|\int_{{{\Bbb R}} ^3}F(u)\right|\leq\frac{\varepsilon}{2}\int_{{{\Bbb R}} ^3}u^2 +\frac{C_\varepsilon}{q}\int_{{{\Bbb R}} ^3}|u|^q \leq \frac{\varepsilon}{2}\|u\|^2+\frac{CC_\varepsilon}{q}\|u\|^q, \end{align}

## 3 定理1.1的证明

\begin{align} c = \inf\limits_{\gamma\in \Gamma}\max\limits_{t\in[0, 1]}J(\gamma(t))>0, \end{align}

\begin{align} J(u_n)\to c, \ J'(u_n)\to 0, \ n\to \infty. \end{align}

$\{u_n\} $$H^1({{\Bbb R}} ^3) 中有界, 则存在 \{y_n\}\subset{{\Bbb R}} ^3 以及 R, \delta>0 , 使得 引理的详细证明可参见文献[9].假设结论不成立, 首先由文献[25, lemma 1.21], u_n$$ L^p({{\Bbb R}} ^3)$中强收敛到$0\ (2<p<6)$.由条件(f_1) (f_2) , 对任意的 \varepsilon >0 , 存在 C_\varepsilon>0 , 使得 并且 因而, 我们得到 \begin{align} \lim\limits_{n\to\infty}\int_{{{\Bbb R}} ^3}f(u_n)u_n = \lim\limits_{n\to\infty}\int_{{{\Bbb R}} ^3}F(u_n) = 0. \end{align} 进一步结合(3.3)式以及当 n\to\infty \langle J'(u_n), u_n\rangle = o(1), 推得当$n\to\infty$时, 有

\begin{align} \|u_n\|^2-\int_{{{\Bbb R}} ^3}\phi_{u_n}|u_n|^{5}-\int_{{{\Bbb R}} ^3}|u_n|^{6} = o(1). \end{align}

当$n$充分大时, 由条件$(f_3)$, 可得

\begin{align} \limsup\limits_{n\to \infty}\int_{B_R(0)}|v_n(x)|^2\geq \delta. \end{align}

\begin{align} \int_{{{\Bbb R}} ^3}(\phi_ {v_n}- \phi_v)|v|^3v\varphi\to 0, \ n\to \infty. \end{align}

v_n(x) {{\Bbb R}} ^3 中几乎处处收敛于 v(x) 以及 推得 即当 n\to\infty 时, 有 成立.进一步结合(3.10)式, 有 \begin{align} \int_{{{\Bbb R}} ^3}\phi_{v_n}|v_n|^3v_n\varphi \to \int_{{{\Bbb R}} ^3}\phi_{v}|v|^3v\varphi , \ n\to\infty. \end{align} 另一方面, 显然有 \begin{align} \int_{{{\Bbb R}} ^3}|v_n|^4v_n\varphi \to \int_{{{\Bbb R}} ^3}|v|^4v\varphi , \ n\to\infty. \end{align} 根据条件 (f_1) (f_2), 存在$C>0$, 使得

$\int_{{{\Bbb R}} ^3}|v_n||\varphi|\to \int_{{{\Bbb R}} ^3}|v||\varphi|$以及$\int_{{{\Bbb R}} ^3}|v_n|^{q-1}|\varphi|\to \int_{{{\Bbb R}} ^3}|v|^{q-1}|\varphi|$, $n\to\infty$, 我们有

\begin{align} \int_{{{\Bbb R}} ^3}f(v_n)\varphi \to \int_{{{\Bbb R}} ^3}f(v)\varphi, \ {\rm}\ n\to\infty. \end{align}

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