The Brezis-Nirenberg result for the fractional Laplacian

The aim of this paper is to deal with the non-local fractional counterpart of the Laplace equation involving critical non-linearities studied in the famous paper of Brezis and Nirenberg (1983). Namely, our model is the equation (equations found) where (−Δ)sis the fractional Laplace operator, s ∈ (0, 1), Ω is an open bounded set of Rn, n > 2s, with Lipschitz boundary, λ > 0 is a real parameter and 2∗ = 2n/(n − 2s) is a fractional critical Sobolev exponent. In this paper we first study the problem in a general framework; indeed we consider the equation (equations found), where LKis a general non-local integrodifferential operator of order s and f is a lower order perturbation of the critical power |u|2 ∗ − 2u. In this setting we prove an existence result through variational techniques. Then, as a concrete example, we derive a Brezis-Nirenberg type result for our model equation; that is, we show that if λ1,sis the first eigenvalue of the non-local operator (−Δ)swith homogeneous Dirichlet boundary datum, then for any λ ∈ (0, λ1,s) there exists a non-trivial solution of the above model equation, provided n≥4s. In this sense the present work may be seen as the extension of the classical Brezis-Nirenberg result to the case of non-local fractional operators.