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

The present paper is devoted to the study of the following non-local fractional equation involving critical nonlinearities{eqnarray}(-Δ)^s u-λ u=|u|2^*-2u, in Ω \=0, in R^n Ω,{eqnarray} 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 Rⁿ, $n>2s$, 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 Rⁿ 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 extensionof some classical results for the Laplacian to the case of non-local fractional operators.