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Nonexistence Results for Nonlocal Equations with Critical and Supercritical Nonlinearities

2014; Taylor & Francis; Volume: 40; Issue: 1 Linguagem: Inglês

10.1080/03605302.2014.918144

1532-4133

Xavier Ros‐Oton, Joaquim Serra,

Nonlinear Differential Equations Analysis

Abstract We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form These operators are infinitesimal generators of symmetric Lévy processes. Our results apply to even kernels K satisfying that K(y)|y| n+σ is nondecreasing along rays from the origin, for some σ ∈ (0, 2) in case a ij ≡ 0 and for σ = 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 σ). We also establish nonexistence of bounded solutions to semilinear equations involving other nonlocal operators such as the higher order fractional Laplacian (− Δ) 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. Keywords: Fractional LaplacianIntegro-differential operatorsNonexistenceSupercritical nonlinearities2010 Mathematics Subject Classfication: 35J6045K05 Acknowledgments The authors thank Xavier Cabré for his guidance and useful discussions on the topic of this paper.