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.
CitationRos, X., Serra, J. Nonexistence results for nonlocal equations with critical and supercritical nonlinearities. "Communications in partial differential equations", 02 Gener 2015, vol. 40, núm. 1, p. 115-133.
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