Summary: The present paper is devoted to the study of the following non-local fractional equation involving critical nonlinearities
where is fixed, is the fractional Laplace operator, is a positive parameter, is the fractional critical Sobolev exponent and is an open bounded subset of , , with Lipschitz boundary. In the recent papers [14, 18, 19] we investigated the existence of non-trivial solutions for this problem when is an open bounded subset of with 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 . 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 (and consequently ) our result is the classical result obtained by Brezis and Nirenberg in the famous paper . 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.