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 $\lambda ,\beta \in {\mathbb{R}}^{+}$, $0<\mu <N$, $N\ge 4$, $2_{\mu }^{\ast }=(2N-\mu )/(N-2)$ is the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality and the nonnegative potential function $V\in \mathcal{C}({\mathbb{R}}^N,\mathbb{R})$ such that $Ω:=\mbox{int} V^{-1}(0)$ is a nonempty bounded set with smooth boundary. If $\beta >0$ is a constant such that the operator $-\Delta +\lambda V(x)-\beta$ is non-degenerate, we prove the existence of ground state solutions which localize near the potential well int $V^{-1}(0)$ for $\lambda$ large enough and also characterize the asymptotic behavior of the solutions as the parameter $\lambda$ goes to infinity. Furthermore, for any $0<\beta <{\beta }_1$, we are able to find the existence of multiple solutions by the Lusternik-Schnirelmann category theory, where ${\beta }_1$ is the first eigenvalue of $-\Delta$ on $\Omega$ with Dirichlet boundary condition.

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Cited by in corpus (1)

• Existence of solutions for critical Choquard equations via the concentration compactness method