Existence of nontrivial solutions for a quasilinear Schrödinger–Poisson system in R3 with periodic potentials

Wei, Chongqing and Li, Anran and Zhao, Leiga (2023) Existence of nontrivial solutions for a quasilinear Schrödinger–Poisson system in R3 with periodic potentials. ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS, 2023 (48). pp. 1-15. ISSN 1417-3875

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Abstract

In this paper, we study the following quasilinear Schrödinger–Poisson system in R3 −∆u+V(x)u+λϕu = f(x,u), x ∈ R3, −∆ϕ−ε4∆4ϕ = λu2, x ∈R3, where λ and ε are positive parameters, ∆4u = div(|∇u|2∇u), V is a continuous and periodic potential function with positive infimum, f(x,t) ∈ C(R3 × 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.

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