Multiple positive bound state solutions of a critical Choquard equation

Abstract

IIn this paper we consider the problem $$ \left\{ \begin{array}{rcl} -Δu+V_λ(x)u=(I_μ*|u|^{2^{*}_μ})|u|^{2^{*}_μ-2}u \ \ \mbox{in} \ \ \mathbb{R}^{N},\\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^{N}, \end{array} \right.\leqno{(P_λ)} $$ where $V_{\lambda }=\lambda +V_0$ with $\lambda \ge 0$, $V_0\in L^{N/2}({\mathbb{R}}^N)$, $I_{\mu }=\frac{1}{|x{|}^{\mu }}$ is the Riesz potential with $0<\mu <\min \{N,4\}$ and $2_{\mu }^{\ast }=\frac{2N-\mu }{N-2}$ with $N\ge 3$. Under some smallness assumption on $V_0$ and $\lambda$ we prove the existence of two positive solutions of $(P_{\lambda })$. In order to prove the main result, we used variational methods combined with degree theory.

In this new version, we change the title and prove two new main results with the collaboration of Professor Dr Riccardo Molle