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

TL;DR

This paper proves nonexistence (Liouville theorems) for nonnegative weak solutions of fractional Lane-Emden equations in unbounded domains, establishing sharp conditions on the exponent p and domain geometry.

Contribution

It establishes new nonexistence results for fractional Lane-Emden equations in unbounded domains, including sharp thresholds for the exponent p in half-space and exterior domain settings.

Findings

No weak solutions for p below critical exponents in certain unbounded domains.

Sharp nonexistence threshold p = (N+α)/(N−α) in half-space.

Application to classical solutions in specific exterior and half-space domains.

Abstract

Our purpose of this paper is to study the nonexistence of nonnegative very weak solutions of \begin{equation}\label{eq 0.1} \displaystyle (-\Delta)^\alpha u = u^p+\nu\quad {\rm in}\quad \Omega,\qquad\ u=g\quad {\rm in}\quad \mathbb{ R}^N\setminus \Omega, \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 $$ (-\Delta)^\alpha u = u^p\quad {\rm in}\quad \Omega, \qquad u\ge 0\quad {\rm in}\quad \mathbb{ R}^N\setminus \Omega, $$ where $\Omega=\mathbb{R}^N\setminus B_{r_0}(0)$ with $r_0>0$ or $\Omega=\mathbb{R}^{N-1}\times(0,+\infty)$.