Nonexistence results for nonlocal equations with critical and supercritical nonlinearities

TL;DR

This paper establishes nonexistence of bounded solutions for certain nonlocal nonlinear equations with critical and supercritical nonlinearities, extending classical results to operators associated with Levy processes and fractional Laplacians.

Contribution

It provides new nonexistence results for solutions to nonlocal equations involving Levy-type operators and fractional Laplacians, using a generalized variational inequality approach.

Findings

Nonexistence of solutions for nonlocal operators with specific kernel conditions.

Results apply to Dirichlet problems in star-shaped domains with critical and supercritical nonlinearities.

Extension of classical nonexistence results to fractional and Levy process-based operators.

Abstract

We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form \[Lu(x)=\sum a_{ij}\partial_{ij}u+{\rm PV}\int_{\R^n}(u(x)-u(x+y))K(y)dy.\] These operators are infinitesimal generators of symmetric L\'evy processes. Our results apply to even kernels $K$ satisfying that $K(y)|y|^{n+\sigma}$ is nondecreasing along rays from the origin, for some $\sigma\in(0,2)$ in case $a_{ij}\equiv0$ and for $\sigma=2$ in case that $(a_{ij})$ is a positive definite symmetric matrix. Our nonexistence results concern Dirichlet problems for $L$ in star-shaped domains with critical and supercritical nonlinearities (where the criticality condition is in relation to $n$ and $\sigma$). We also establish nonexistence of bounded solutions to semilinear equations involving other nonlocal operators such as the higher order fractional Laplacian $(-\Delta)^s$ (here $s>1$) or the fractional $p$-Laplacian. All these nonexistence results follow from a general variational inequality in the spirit of a classical identity by Pucci and Serrin.