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). In this paper we first study the problem in a general framework; indeed we consider an equation driven by a general non-local integrodifferential operator and in presence of a lower order perturbation of the critical power. In this setting we prove an existence result through variational techniques. Then, as a concrete example, we derive a Brezis-Nirenberg type result for an equation driven by the fractional Laplacian (-Delta)(s); that is, we show that if lambda(1,s) is the first eigenvalue of the non-local operator (-Delta)(s) with homogeneous Dirichlet boundary datum, then for any lambda is an element of (0, lambda(1,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.