Localized nodal solutions of higher topological type for nonlinear Schrödinger–Poisson system

Abstract

In this paper, we focus our attention on following Schrödinger–Poisson system: −ε2Δu+V(x)u+λψu=|u|p−2u,x∈R3,−ε2Δψ=u2,u∈H1(R3).where 4<p<6 and ε,λ>0 are small positive parameters. Under a local condition imposed on the potential V, we study above system and obtain an infinite sequence of localized sign-changing solutions by applying the symmetric mountain pass theorem. Precisely, these solutions are constructed as higher topological type critical points of the energy functional and concentrated at a local minimum set of the potential V. Our method follows the same spirit of Chen and Wang’s method (Chen and Wang, 2017) which does not need any non-degeneracy condition of the limiting equations, but we cannot use it directly due to the presence of nonlocal term ψ(x)=14π∫R3u2(y)|x−y|dy. We employ some new analytical skills to overcome the obstacles caused by the nonlocal term, our results improve and extend some related ones in the literature.