Electron. J. Differential Equations, Vol. 2018 (2018), No. 123, pp. 1-25.

Existence of nontrivial solutions for a perturbation of Choquard equation with Hardy-Littlewood-Sobolev upper critical exponent

Yu Su, Haibo Chen

Abstract:
In this article, we consider the problem
$$ -\Delta u =\Big(\int_{\mathbb{R}^{N}} \frac{|u|^{2^{*}_{\mu}}}{|x-y|^{\mu}} \,dy\Big) |u|^{2^{*}_{\mu}-2}u + f(x,u) \quad\text{in }\mathbb{R}^{N}, $$
where $N\geqslant3$, $\mu\in(0,N)$ and $2^{*}_{\mu}=\frac{2N-\mu}{N-2}$. Under suitable assumptions on f(x,u), we establish the existence and non-existence of nontrivial solutions via the variational method.

Submitted November 8, 2017. Published June 15, 2018.
Math Subject Classifications: 35J20, 35J60.
Key Words: Hardy-Littlewood-Sobolev upper critical exponent; Choquard equation.

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Yu Su
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: yizai52@qq.com
Haibo Chen
School of Mathematics and Statistics
Central South University
Changsha, 410083 Hunan, China
email: math_chb@163.com

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