The Brezis-Nirenberg result for the fractional Laplacian

Abstract

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 \[ { ( − Δ ) s u − λ u = | u | 2 ∗ − 2 u a m p ; in Ω , u = 0 a m p ; in R n ∖ Ω , \left \{ \begin {array}{ll} (-\Delta )^s u-\lambda u=|u|^{2^*-2}u & {\mbox { in }} \Omega ,\\ u=0 & {\mbox { in }} \mathbb {R}^n\setminus \Omega \,, \end {array} \right . \] where ( − Δ ) s (-\Delta )^s is the fractional Laplace operator, s ∈ ( 0 , 1 ) s\in (0,1) , Ω \Omega is an open bounded set of R n \mathbb {R}^n , n > 2 s n>2s , with Lipschitz boundary, λ > 0 \lambda >0 is a real parameter and 2 ∗ = 2 n / ( n − 2 s ) 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 \[ { L K u + λ u + | u | 2 ∗ − 2 u + f ( x , u ) = 0 a m p ; in Ω , u = 0 a m p ; in R n ∖ Ω , \left \{ \begin {array}{ll} \mathcal L_K u+\lambda u+|u|^{2^*-2}u+f(x, u)=0 & \mbox {in } \Omega ,\\ u=0 & \mbox {in } \mathbb {R}^n\setminus \Omega \,, \end {array}\right . \] where L K \mathcal L_K is a general non-local integrodifferential operator of order s s and f f is a lower order perturbation of the critical power | u | 2 ∗ − 2 u |u|^{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 λ 1 , s \lambda _{1,s} is the first eigenvalue of the non-local operator ( − Δ ) s (-\Delta )^s with homogeneous Dirichlet boundary datum, then for any λ ∈ ( 0 , λ 1 , s ) \lambda \in (0, \lambda _{1,s}) there exists a non-trivial solution of the above model equation, provided n ⩾ 4 s n\geqslant 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.