The Brezis-Nirenberg problem for fractional elliptic operators

Let L= div (A(x)delta) be a uniformly elliptic operator in divergence form in a bounded open subset of Rn. We study the effect of the operator L on the existence and nonexistence of positive solutions of the nonlocal Brezis-Nirenberg problem where (-L)s denotes the fractional power of -L with zero Dirichlet boundary values on , 0<s<1, n>2s and is a real parameter. By assuming A(x)A(x0) for all x and A(x)A(x0)+|x-x0|sigma In near some point x0, we prove existence theorems for any (0,1,s(-L)), where 1,s(-L) denotes the first Dirichlet eigenvalue of (-L)s. Our existence result holds true for sigma>2s and n4s in the interior case (x0) and for sigma>2s(n-2s)n-4s and n>4s in the boundary case (x0). Nonexistence for star-shaped domains is obtained for any 0.