On the critical Choquard equation with potential well

Abstract

In this paper we are interested in the following nonlinear Choquard equation $$ -Δu+(λV(x)-β)u =\big(|x|^{-μ}\ast |u|^{2_μ^{\ast}}\big)|u|^{2_μ^{\ast}-2}u\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$ where $λ,β\in\mathbb{R}^+$, $0<μ 0$ is a constant such that the operator $-Δ+λV(x)-β$ is non-degenerate, we prove the existence of ground state solutions which localize near the potential well int $V^{-1}(0)$ for $λ$ large enough and also characterize the asymptotic behavior of the solutions as the parameter $λ$ goes to infinity. Furthermore, for any $0<β<β_{1}$, we are able to find the existence of multiple solutions by the Lusternik-Schnirelmann category theory, where $β_{1}$ is the first eigenvalue of $-Δ$ on $Ω$ with Dirichlet boundary condition.