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

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