Multi-bump solutions for Choquard equation with deepening potential well

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

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