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The yamabe equation in a non-local setting. (English) Zbl 1273.49011

Summary: Aim of this paper is to study the following elliptic equation driven by a general non-local integro-differential operator K such that K u+λu+|u| 2 * -2 u=0 in Ω,u=0 in n Ω, where s(0,1),Ω is an open bounded subset of n ,n>2s, with Lipschitz boundary, λ is a positive real parameter, 2 * =2n/(n-2s) is a fractional critical Sobolev exponent, while K is the non-local integrodifferential operator

K u(x)= n u(x+y)+u(x-y)-2u(x)K(y)dy,x n ·

As a concrete example, we consider the case when K(x)=|x| -(n+2s) , which gives rise to the fractional Laplace operator -(-Δ) s . We show that our problem admits a nontrivial solution for any λ>0, provided n4s and λ is different from the eigenvalues of (-Δ) s . This result may be read as the non-local fractional counterpart of the one obtained by Capozzi, Fortunato and Palmieri and by Gazzola and Ruf for the classical Laplace equation with critical nonlinearities.

In this sense the present work may be seen as the extension of some classical results for the Laplacian to the case of non-local fractional operators.

MSC:
49J35Minimax problems (existence)
35A15Variational methods (PDE)
35S15Boundary value problems for pseudodifferential operators
47G20Integro-differential operators
45G05Singular nonlinear integral equations