On the critical Choquard equation with potential well

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In this paper we are interested in the following nonlinear Choquard equation -Δu+(λV(x)-β)u =(|x|⁻μ |u|2μ)|u|2μ-2u4.14mmin1.14mm Rⁿ, where λ,β+, 0<μ<N, N≥4, 2μ=(2N-μ)/(N-2) is the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality and the nonnegative potential function V∈ C(Rⁿ,R) such that Ω:=int V⁻¹(0) is a nonempty bounded set with smooth boundary. If β>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<β₁, we are able to find the existence of multiple solutions by the Lusternik-Schnirelmann category theory, where β₁ is the first eigenvalue of -Δ on Ω with Dirichlet boundary condition.