Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem

Abstract

In this article, we consider the following Schrödinger-Poisson problem: −ε2Δu+V(y)u+Φ(y)u=∣u∣p−1u,y∈R3,−ΔΦ(y)=u2,y∈R3,\left\{\begin{array}{ll}-{\varepsilon }^{2}\Delta u+V(y)u+\Phi (y)u={| u| }^{p-1}u,& y\in {{\mathbb{R}}}^{3},\\ -\Delta \Phi (y)={u}^{2},& y\in {{\mathbb{R}}}^{3},\end{array}\right. where ε>0\varepsilon \gt 0 is a small parameter, 1<p<51\lt p\lt 5, and V(y)V(y) is a potential function. We construct multi-peak solution concentrating at the critical points of V(y)V(y) through the Lyapunov-Schmidt reduction method. Moreover, by using blow-up analysis and local Pohozaev identities, we prove that the multi-peak solution we construct is non-degenerate. To our knowledge, it seems be the first non-degeneracy result on the Schödinger-Poisson system.

Keywords

• multi-peak solution
• non-degeneracy
• schrödinger-poisson system
• 35j15
• 35j20