Uniqueness of non-negative solutions to an integral equation of the Choquard type

ABSTRACT

Let $u\in L_{\mathrm{loc}}^{\frac{2n(p-1)}{\alpha +\beta }}({\mathbb{R}}^n)$ be a non-negative solution of the integral equation $u(x)={\int }_{{\mathbb{R}}^n}\frac{u^{p-1}(y)}{|x-y{|}^{n-\alpha }}{\int }_{{\mathbb{R}}^n}\frac{u^p(z)}{|y-z{|}^{n-\beta }}\mathrm{d}z\mathrm{d}y,x\in {\mathbb{R}}^n,$where $0<\alpha ,\beta <n$ and $p\ge 2$. We prove that $u\equiv 0$ if $\frac{2n}{2n-\alpha -\beta }<p<\frac{n+\beta }{n-\alpha }$ and u must assume an explicit form if $p=\frac{n+\beta }{n-\alpha }$. As an application, we obtain a similar result for non-negative distributional solutions of the corresponding static Choquard-type equation. The main tool we use is the method of moving planes in integral forms.

KEYWORDS:

• Integral equation
• Choquard equation
• Liouville theorem
• classification of solutions
• uniqueness of solutions

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 45K05
• 35R11
• 35A02
• 35B53

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is funded by University of Economics and Law, VNU-HCM