A Brezis-Nirenberg result for non-local critical equations in low dimension

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