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 R 3 { − Δ u + V ( x ) u + λ ϕ u = f ( x , u ) , x ∈ R 3 , − Δ ϕ − ε 4 Δ 4 ϕ = λ u 2 , x ∈ R 3 , where λ and ε are positive parameters, Δ 4 u = div ( | ∇ u | 2 ∇ u ) , V is a continuous and periodic potential function with positive infimum, f ( x , t ) ∈ C ( R 3 × R , 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 λ small enough and ε fixed by a combination of variational methods and truncation technique.