Nonexistence Results for Nonlocal Equations with Critical and Supercritical Nonlinearities

We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form [GRAPHICS] These operators are infinitesimal generators of symmetric Levy 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 is an element of (0, 2) in case a ( ij ) equivalent to 0 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.