Liouville theorem for the fractional Lane-Emden Equation in unbounded domain

Abstract

Our purpose of this paper is to study the nonexistence of nonnegative very weak solutions of \begin{equation}\label{eq 0.1} \displaystyle (-Δ)^αu = u^p+ν\quad {\rm in}\quad Ω,\qquad\ u=g\quad {\rm in}\quad \mathbb{ R}^N\setminus Ω, \end{equation} where $\alpha \in (0,1)$, $p>0$, $\Omega$ is a unbounded $C^2$ domain in ${\mathbb{R}}^N$ with $N>2\alpha$, $g\in L^1({\mathbb{R}}^N\setminus \Omega ,\frac{dx}{1+|x{|}^{N+2\alpha }})$ nonnegative and $\nu$ is a nonnegative Radon measure. We obtain that\smallskip $(i)$ if $\Omega \supseteq \left({\mathbb{R}}^N\setminus \overline{B_{r_0}(0)}\right)$ for some $r_0>0$ and $p<\frac{N}{N-2\alpha }$, then fractional Lane-Emden equation has no weak solutions. $(ii)$ if $\Omega \supseteq \left\{x\in {\mathbb{R}}^N:x\cdot a>r_0\right\}$ for some $r_0\ge 0$, $a\in {\mathbb{R}}^N$ and $p<\frac{N+\alpha }{N-\alpha }$, then fractional Lane-Emden equation has no weak solutions. Here $\frac{N+\alpha }{N-\alpha }$ is sharp for the nonexistence in the half space. \smallskip The above Liouville theorem could be applied to obtain nonexistence of classical solution of the fractional Lane-Emden equations $$(-\Delta {)}^{\alpha }u=u^p\mathrm{in}\Omega ,u\ge 0\mathrm{in}{\mathbb{R}}^N\setminus \Omega ,$$ where $\Omega ={\mathbb{R}}^N\setminus B_{r_0}(0)$ with $r_0>0$ or $\Omega ={\mathbb{R}}^{N-1}\times (0,+\infty )$.

page 23. accepted by JMPA

References in corpus (1)

• A Liouville Theorem for the Fractional Laplacian