Infinitely many geometrically distinct solutions for periodic Schrödinger–Poisson systems

Abstract

Abstract This paper is dedicated to studying the following Schrödinger–Poisson system: {−△u+V(x)u+K(x)ϕ(x)u=f(x,u),x∈R3,−△ϕ=K(x)u2,x∈R3, $$ \textstyle\begin{cases} -\triangle u+V(x)u+K(x)\phi (x)u=f(x, u), \quad x\in {\mathbb {R}}^{3}, \\ -\triangle \phi =K(x)u^{2}, \quad x\in {\mathbb {R}}^{3}, \end{cases} $$ where V(x) $V(x)$, K(x) $K(x)$, and f(x,u) $f(x, u)$ are periodic in x. By using the non-Nehari manifold method, we establish the existence of ground state solutions for the above problem under some weak assumptions. Moreover, when f is odd in u, we prove that the above problem admits infinitely many geometrically distinct solutions. Our results improve and complement some related literature.

Keywords

• Schrödinger–Poisson system
• Nehari manifold
• Ground state
• Geometrically distinct solutions