Liouville theorems for nonlocal Lane--Emden inequalities with degenerate kernels

Abstract

We establish a Liouville-type theorem for the inequality ${\mathcal{L}}_{K}u\ge |u{|}^q$ in ${\mathbb{R}}^n$, where ${\mathcal{L}}_K$ is a translation-invariant, degenerate elliptic integro-differential operator whose kernel is even and satisfies $0\le K(z)\le \Lambda |z{|}^{-n-2s}$. No lower ellipticity bound is imposed on the kernel, and no sign condition is imposed on the solution. We prove that every such solution is trivial when $1\le q\le \frac{n}{n-2s}$ if $n>2s$, and for every $q\ge 1$ if $n\le 2s$. The proof combines a test function method with a dyadic decomposition of the nonlocal tail and uses neither the maximum principle nor a fundamental solution.

13 pages. v2: title changed; kernel assumption weakened to a one-sided upper bound; new corollary on mixed local-nonlocal operators