Liouville-type theorems for Lane--Emden inequalities involving nonlocal operators

We establish a Liouville-type theorem for nonnegative weak supersolutions to ${\mathcal{L}}_{K}u=u^q$ in ${\mathbb{R}}^n$, where ${\mathcal{L}}_K$ is a translation-invariant integro-differential operator of order $2s$ with $s\in (0,1)$. The kernel $K$ is assumed to be even and satisfy uniform ellipticity bounds. We prove that the only nonnegative supersolution is the trivial one $u\equiv 0$ in the range $1<q\le \frac{n}{n-2s}$ for $n>2s$ (and for all $q>1$ when $n\le 2s$). Our proof is elementary and relies on a test function method combined with a dyadic decomposition of the nonlocal tail. Notably, our argument does not rely on the maximum principle or the fundamental solution.