Electron. J. Diff. Equ., Vol. 2016 (2016), No. 03, pp. 1-17.

Existence and nonexistence of nontrivial solutions for Choquard type equations

Tao Wang

In this article, we consider the nonlocal problem
 -\Delta u+u=q(x)\Big(\int_{\mathbb{R}^N}\frac{q(y)|u(y)|^p}{|x-y|^{N-\alpha}}dy
 \Big)|u|^{p-2}u,\quad x\in \mathbb{R}^N,
where $N\geq 3$, $\alpha\in (0,N)$, $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-2}$ and q(x) is a given potential. Under suitable assumptions on q(x), we prove the existence and nonexistence of nontrivial solutions.

Submitted November 8, 2015. Published January 4, 2016.
Math Subject Classifications: 35A15, 35J20, 35J60.
Key Words: Choquard equation; nonlocal nonlinearities; variational methods.

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Tao Wang
School of Mathematics and Statistics
Central South University
Changsha, Hunan 410083, China
email: wt_61003@163.com

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