Existence of positive solution of the Choquard equation $-Δ u+a(x)u=(I_μ*|u|^{2^{*}_μ})|u|^{2^{*}_μ-2}u$ in $\mathbb{R}^N$

Claudianor O. Alves, Giovany M. Figueiredo

Published 2018-12-12Version 1

In this paper we show existence of a positive solution to the problem $$ \left\{ \begin{array}{rcl} -\Delta u+a(x)u=(I_{\mu}*|u|^{2^{*}_{\mu}})|u|^{2^{*}_{\mu}-2}u,\\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^{N},\\ u \in D^{1,2}(\mathbb{R}^{N}), \end{array} \right.\leqno{(P)} $$ where $I_\mu=\frac{1}{|x|^\mu}$ is the Riesz potential, $0<\mu<\min\{N,4\}$ and $2^{*}_{\mu}=\frac{(2N-\mu)}{N-2}$ with $N\geq 3$. In order to prove the main result, we used variational methods combined with a splitting theorem.