Positive solutions for a nonhomogeneous Schrödinger-Poisson system

Abstract

In this article, we consider the following Schrödinger-Poisson system: −Δu+u+k(x)ϕ(x)u=f(x)∣u∣p−1u+g(x),x∈R3,−Δϕ=k(x)u2,x∈R3,\left\{\begin{array}{ll}-\Delta u+u+k\left(x)\phi \left(x)u=f\left(x)| u{| }^{p-1}u+g\left(x),& x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi =k\left(x){u}^{2},& x\in {{\mathbb{R}}}^{3},\end{array}\right. with p∈(3,5)p\in \left(3,5). Under suitable assumptions on potentials f(x)f\left(x), g(x)g\left(x) and k(x)k\left(x), then at least four positive solutions for the above system can be obtained for sufficiently small ‖g‖H−1(R3)\Vert g{\Vert }_{{H}^{-1}\left({{\mathbb{R}}}^{3})} by taking advantage of variational methods and Lusternik-Schnirelman category.

Keywords

• schrödinger-poisson system
• lusternik-schnirelman category
• multiple solutions
• 35a15
• 35j10
• 35b09