Electron. J. Differential Equations, Vol. 2017 (2017), No. 52, pp. 1-14.

Ground state solutions for Choquard type equations with a singular potential

Tao Wang

Abstract:
This article concerns the Choquard type equation
$$ -\Delta u+V(x)u=\Big(\int_{\mathbb{R}^N}\frac{|u(y)|^p}{|x-y|^{N-\alpha}}dy\Big) |u|^{p-2}u,\quad x\in \mathbb{R}^N, $$
where $N\geq3$, $\alpha\in ((N-4)_+,N)$, $2\leq p<(N+\alpha)/(N-2)$ and V(x) is a possibly singular potential and may be unbounded below. Applying a variant of the Lions' concentration-compactness principle, we prove the existence of ground state solution of the above equations.

Submitted June 6, 2016. Published February 21, 2017.
Math Subject Classifications: 35A15, 35A20, 35J20.
Key Words: Choquard equation; singular potential; ground state solution; Lions' concentration-compactness principle.

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Tao Wang
College of Mathematics and Computing Science
Hunan University of Science and Technology
Xiangtan, Hunan 411201, China
email: wt_61003@163.com

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