In this paper, we are concerned with the following type of fractional problems: $$ \begin{ cases}\dis (-\Delta)^{ s} u-\mu\frac{ u}{ |x|^{ 2s}}-\lambda u=|u|^{ 2^*_{ s}-2}u+f(x,u), &\text{ in} \Omega,\ \ \, u=0\,&\text{ in} \R^N\backslash\Omega \end{ cases} \eqno { (*)} $$ where $s\in (0,1)$, $2^*_{ s}=2N/(N-2s)$ is the critical Sobolev exponent, $f(x,u)$ is a lower order perturbation of critical Sobolev nonlinearity. We obtain the existence of the solution for (*) through variational methods. In particular we derive a Br\'ezis-Nirenberg type result when $f(x,u)=0$.
Categories: math.AP