Electron. J. Differential Equations, Vol. 2016 (2016), No. 306, pp. 1-11.

Multiplicity of solutions to a nonlocal Choquard equation involving fractional magnetic operators and critical exponent

Fuliang Wang, Mingqi Xiang

Abstract:
In this article, we study the multiplicity of solutions to a nonlocal fractional Choquard equation involving an external magnetic potential and critical exponent, namely,
$$\displaylines{ (a+b[u]_{s,A}^2)(-\Delta)_A^su+V(x)u =\int_{\mathbb{R}^N}\frac{|u(y)|^{2_{\mu,s}^*}}{|x-y|^{\mu}}dy|u|^{2_{\mu,s}^*-2}u +\lambda h(x)|u|^{p-2}u\quad \text{in }\mathbb{R}^N, \cr [u]_{s,A}=\Big(\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^N} \frac{|u(x)-e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}\,dx\,dy\Big) ^{1/2} }$$
where $a\geq 0$, b>0, $0<s<\min\{1,N/4\}$, $4s\leq \mu<N$, $V:\mathbb{R}^N\to \mathbb{R}$ is a sign-changing scalar potential, $A:\mathbb{R}^N\to \mathbb{R}^N$ is the magnetic potential, $(-\Delta )_A^s$ is the fractional magnetic operator, $\lambda>0$ is a parameter, $2_{\mu,s}^*=\frac{2N-\mu}{N-2s}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality and $2<p<2_s^*$. Under suitable assumptions on a,b and $\lambda$, we obtain multiplicity of nontrivial solutions by using variational methods. In particular, we obtain the existence of infinitely many nontrivial solutions for the degenerate Kirchhoff case, that is, a=0, b>0.

Submitted October 27, 2016. Published November 30, 2016.
Math Subject Classifications: 49A50, 26A33, 35J60, 47G20.
Key Words: Choquard equation; fractional magnetic operator; variational method; critical exponent.

Show me the PDF file (277 KB), TEX file for this article.

Fuliang Wang
College of Science
Civil Aviation University of China
Tianjin 300300, China
email: flwang@cauc.edu.cn
Mingqi Xiang
College of Science
Civil Aviation University of China
Tianjin 300300, China
email: xiangmingqi_hit@163.com

Return to the EJDE web page