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 { (-Delta)(s)u - lambda u = vertical bar u vertical bar (2)* (-) (2)u in Omega, u = 0 in R-n\Omega, where (-Delta)(s) is the fractional Laplace operator, s is an element of (0, 1), Omega is an open bounded set of R-n, n > 2s, with Lipschitz boundary, lambda > 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 {L(K)u + lambda u + vertical bar u vertical bar(2)* (-) (2)u + f(x, u) = 0 in Omega, u = 0 in R-n\Omega, where L-K is a general non-local integrodifferential operator of order s and f is a lower order perturbation of the critical power vertical bar u vertical bar(2)* (-) (2)u. 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 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.