Sign-changing multi-bump solutions for Choquard equation with deepening potential well

Abstract

In this paper, we are concerned with the existence of sign-changing multi-bump solutions for the following nonlinear Choquard equation\begin{equation}\tag{0.1}-\Delta u+(\lambda V(x)+1)u=(I_{\alpha}\ast|u|^p)|u|^{p-2}u \quad \text{in } \mathbb{R}^N,\end{equation}where $I_\alpha$ is the Riesz potential, $\lambda \in \mathbb{R}^{+}$, $ (N-4)^{+}< \alpha< N$,$2\le p < ({N+\alpha})/({N-2})$, and $V(x)$ is a nonnegative continuous function with a potential well $\Omega:= {\rm int}(V^{-1}(0))$ which possesses $k$ disjoint bounded components $\Omega_1, \ldots, \Omega_k$. We prove the existence of sign-changing multi-bump solutions for (0.1) if $\lambda$ is large enough.