Classification of positive solutions to the critical fractional Choquard equation in

ABSTRACT

The first aim of this paper is to classify the positive solutions of the fractional Choquard equation $(-\mathrm{\Delta }{)}^{s/2}u=\left(I_{\alpha }\ast u^{2_{s,\alpha }^{\ast }}\right)u^{2_{s,\alpha }^{\ast }-1},x\in {\mathbb{R}}^N,$ where $2_{s,\alpha }^{\ast }=(N+\alpha )/(N-s)$ is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality.

Moreover, based on the uniqueness and non-degeneracy of the solution of the above equation, we then study the perturbed Choquard equation $(-\mathrm{\Delta }{)}^{s/2}u=\left(I_{\alpha }\ast u^{2_{s,\alpha }^{\ast }}\right)u^{2_{s,\alpha }^{\ast }-1}+\epsilon k(x)u^q,x\in {\mathbb{R}}^N.$ By using the finite-dimensional reduction, we obtain the existence of at least one positive solution if $|\epsilon |$ is suitable small.

Keywords:

• Fractional Choquard equation
• Uniqueness
• Lyapunov–Schmidt method

MSC(2010):

• 35J50
• 58E30

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Huxiao Luo http://orcid.org/0000-0001-7715-8699

Additional information

Funding

This work is supported by National Natural Science Foundation of China (Grant No. 11901532).