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A Brezis-Nirenberg result for non-local critical equations in low dimension. (English) Zbl 06171054

Summary: The present paper is devoted to the study of the following non-local fractional equation involving critical nonlinearities

(-Δ) s u-λu=|u| 2 * -2 u,inΩu=0,in n Ω,

where s(0,1) is fixed, (-Δ) s is the fractional Laplace operator, λ is a positive parameter, 2 * is the fractional critical Sobolev exponent and Ω is an open bounded subset of n , n>2s, with Lipschitz boundary. In the recent papers [14, 18, 19] we investigated the existence of non-trivial solutions for this problem when Ω is an open bounded subset of n with n4s and, in this framework, we prove some existence results.

Aim of this paper is to complete the investigation carried on in [14, 18, 19], by considering the case when 2s<n<4s. In this context, we prove an existence theorem for our problem, which may be seen as a Brezis-Nirenberg type result in low dimension. In particular when s=1 (and consequently n=3) our result is the classical result obtained by Brezis and Nirenberg in the famous paper [4]. In this sense the present work may be considered as the extension of some classical results for the Laplacian to the case of non-local fractional operators.

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