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arXiv · 1712.08264

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

Fashun Gao·Edcarlos D. da Silva·Minbo Yang·Jiazheng Zhou
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Abstract

In this paper we consider the nonlinear Choquard equation $$ -\Delta u+V(x)u =\left(\int_{\mathbb{R}^N}\frac{G(y,u)}{|x-y|^{\mu}}dy\right)g(x,u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$ where $0<\mu<N$, $N\geq3$, $g(x,u)$ is of critical growth due to the Hardy--Littlewood--Sobolev inequality and $G(x,u)=\displaystyle\int^u_0g(x,s)ds$. Firstly, by assuming that the potential $V(x)$ might be sign-changing, we study the existence of Mountain-Pass solution via a concentration-compactness principle for the Choquard equation. Secondly, under the conditions introduced by Benci and Cerami \cite{BC1}, we also study the existence of high energy solution by using a global compactness lemma for the nonlocal Choquard equation.

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Fashun Gao, Edcarlos D. da Silva, Minbo Yang, Jiazheng Zhou. 2017-12-22. Existence of solutions for critical Choquard equations via the concentration compactness method. https://arxiv.org/abs/1712.08264

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