Nonexistence results for nonlocal equations with critical and supercritical nonlinearities

2013/09/20 by Ros-Oton, Xavier, Serra, Joaquim · 6 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1309.5407

Abstract

We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form Lu(x)=∑ a_ij∂_iju+\rm PV∫_\Rⁿ(u(x)-u(x+y))K(y)dy. 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.

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