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

Abstract

Abstract In this article, we consider the following Schrödinger-Poisson problem: − ε 2 Δ u + V ( y ) u + Φ ( y ) u = ∣ u ∣ p − 1 u , y ∈ R 3 , − Δ Φ ( y ) = u 2 , y ∈ R 3 , \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 < 5 1\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.