Liouville results for semilinear integral equations with conical diffusion

TL;DR

This paper establishes sharp nonexistence results for positive supersolutions of certain semilinear integral equations with conical diffusion, identifying the critical exponent and demonstrating the existence of solutions beyond it.

Contribution

It provides new Liouville-type nonexistence results for a broad class of nonlocal operators with conical diffusion, extending previous work to more general operators and regimes.

Findings

Nonexistence of positive supersolutions for subcritical exponents

Existence of solutions for supercritical exponents

Results are sharp and apply to a wide class of operators

Abstract

Nonexistence results for positive supersolutions of the equation $$-Lu=u^p\quad\text{in $\mathbb R^N_+$}$$ are obtained, $-L$ being any symmetric and stable linear operator, positively homogeneous of degree $2s$, $s\in(0,1)$, whose spectral measure is absolutely continuous and positive only in a relative open set of the unit sphere of $\mathbb R^N$. The results are sharp: $u\equiv 0$ is the only nonnegative supersolution in the subcritical regime $1\leq p\leq\frac{N+s}{N-s}\,$, while nontrivial supersolutions exist, at least for some specific $-L$, as soon as $p>\frac{N+s}{N-s}$. \\ The arguments used rely on a rescaled test function's method, suitably adapted to such nonlocal setting with weak diffusion; they are quite general and also employed to obtain Liouville type results in the whole space.