Open AccessA Brezis-Nirenberg result for non-local critical equations in low dimension
Author(s) -
Raffaella Servadei,
Enrico Valdinoci
Publication year - 2013
Publication title -
communications on pure and applied analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.077
H-Index - 42
eISSN - 1553-5258
pISSN - 1534-0392
DOI - 10.3934/cpaa.2013.12.2445
Subject(s) - nirenberg and matthaei experiment , bounded function , mathematics , sobolev space , omega , lipschitz continuity , context (archaeology) , dimension (graph theory) , fractional laplacian , operator (biology) , laplace operator , lambda , boundary (topology) , pure mathematics , type (biology) , mathematical analysis , combinatorics , physics , paleontology , biochemistry , chemistry , repressor , quantum mechanics , gene , transcription factor , optics , biology , ecology
The present paper is devoted to the study of a nonlocal fractional equation involving critical nonlinearities. In the recent papers [14, 18, 19] we investigated the existence of non-trivial solutions for this problem when the set Omega is an open bounded subset of R-n with n >= 4s 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