Existence of nontrivial solutions for a quasilinear Schrödinger–Poisson system in $\mathbb{R}^3$ with periodic potentials

Wei, Chongqing, Shanxi University, Taiyuan, China
Li, Anran, Shanxi University, Taiyuan, China
Zhao, Leiga, School of Mathematics and Statistics, Beijing Technology and Business University, Beijing, P. R. China

Electron. J. Qual. Theory Differ. Equ. 2023, No. 48, 1-15. DOI: https://doi.org/10.14232/ejqtde.2023.1.48

Communicated by  Mugnai, Dimitri

Received on 2023-04-12 Appeared on 2023-12-08

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Abstract: In this paper, we study the following quasilinear Schrödinger–Poisson system in $\mathbb{R}^3$
$$\begin{equation*} \begin{cases} -\Delta u+V(x)u+\lambda \phi u=f(x,u),&x\in{\mathbb{R}^3},\\ -\Delta \phi-\varepsilon^4\Delta_4\phi=\lambda u^2,&x\in{\mathbb{R}^3}, \end{cases} \end{equation*}$$
where $\lambda$ and $\varepsilon$ are positive parameters, $\Delta_4u=\hbox{div}(|\nabla u|^2\nabla u)$, $V$ is a continuous and periodic potential function with positive infimum, $f(x,t)\in C( \mathbb{R}^3\times \mathbb{R},\mathbb{R})$ is periodic with respect to $x$ and only needs to satisfy some superquadratic growth conditions with respect to $t$. One nontrivial solution is obtained for $\lambda$ small enough and $\varepsilon$ fixed by a combination of variational methods and truncation technique.

Keywords: quasilinear Schrödinger–Poisson system, periodic potential, variational methods, truncation technique, nontrivial solution

Mathematics Subject Classification: 35B38, 35D30, 35J50

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