Raffaella Servadei and Enrico Valdinoci
A Brezis-Nirenberg result 
for non-local critical equations 
in low dimension
(68K, LaTeX)

ABSTRACT.  The present paper is devoted to the study of the following non-local fractional equation involving critical nonlinearities 
$$\left\{ 
egin{array}{ll} 
(-\Delta)^s u-\lambda u=|u|^{2^*-2}u & {\mbox{ in }} \Omega\ 
u=0 & {\mbox{ in }} \RR^n\setminus \Omega\,, 
\end{array} 
ight.$$ 
where $s\in (0,1)$ is fixed, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda$ is a positive parameter, $2^*$ is the fractional critical Sobolev exponent and $\Omega$ is an open bounded subset of $\RR^n$, $n\in(2s,4s)$\,, with Lipschitz boundary.