Multi-bump solutions for Choquard equation with deepening potential well

Abstract

We study the existence of multi-bump solutions to Choquard equation $$ \begin{array}{ll} -\Delta u + (\lambda a(x)+1)u=\displaystyle\big(\frac{1}{|x|^{\mu}}\ast |u|^p\big)|u|^{p-2}u \mbox{ in } \,\,\, \R^3, \end{array} $$ where $\mu \in (0,3), p\in(2, 6-\mu)$, $\lambda$ is a positive parameter and the nonnegative function $a(x)$ has a potential well $ \Omega:=int (a^{-1}(0))$ consisting of $k$ disjoint bounded components $ \Omega:=\cup_{j=1}^{k}\Omega_j$. We prove that if the parameter $\lambda$ is large enough then the equation has at least $2^{k}-1$ multi-bump solutions.