Nonexistence results for nonlocal equations with critical and supercritical nonlinearities

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