On some critical problems for the fractional Laplacian operator

We study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional Laplacian: { (-Delta)alpha/2 = lambda = lambda U-Q + U N+alpha/N-alpha, u > 0 in Omega, u = 0 on (sic)Omega where Omega subset of R-N is a smooth bounded domain, N >= 1, lambda > 0, 0 < q < N+alpha/N-alpha, 0 0, at least one if lambda = Lambda, no solution if lambda > Lambda. For q = 1 we show existence of at least one solution for 0 < lambda < lambda(1) and nonexistence for lambda >= lambda(1). When q > 1 the existence is shown for every lambda > 0. Also we prove that the solutions are bounded and regular. (C) 2012 Elsevier Inc. All rights reserved.