Uniqueness of nonnegative weak solution to $u^p\le (-\Delta {)}^{\frac{\alpha }{2}}u$ on ${\mathbb{R}}^N$

Abstract

This note shows that under $(p,\alpha ,N)\in (1,\infty )\times (0,2)\times {\mathbb{Z}}_{+}$ the fractional order differential inequality $$(†)u^p\le (-\Delta {)}^{\frac{\alpha }{2}}uin{\mathbb{R}}^N$$ has the property that if $N\le \alpha$ then a nonnegative solution to $(†)$ is unique, and if $N>\alpha$ then the uniqueness of a nonnegative weak solution to $(†)$ occurs when and only when $p\le N/(N-\alpha )$, thereby innovatively generalizing Gidas-Spruck's result for $u^p+\Delta u\le 0$ in ${\mathbb{R}}^N$ discovered in \cite{GS}.

14 pages, 0 fingure