Equations involving fractional Laplacian operator: Compactness and application

In this paper, we consider the following problem involving fractional Laplacian operator: (-Delta)(alpha)u = vertical bar u vertical bar(2)*(alpha-2-epsilon) u + lambda u in Omega, u = 0 on partial derivative Omega, (1) where Omega is a smooth bounded domain in R-N, epsilon is an element of [0, 2(alpha)* - 2), 0 < alpha < 1, 2(alpha)* = 2N/N - 2 alpha, and (-Delta)(alpha) is either the spectral fractional Laplacian or the restricted fractional Laplacian. We show for problem (1) with the spectral fractional Laplacian that for any sequence of solutions u(n) of (1) corresponding to epsilon(n) is an element of [0, 2(alpha)* - 2), satisfying parallel to lu(n)parallel to(H) <= C in the Sobolev space H defined in (1.2), u(n) converges strongly in H provided that N > 6 alpha and lambda > 0. The same argument can also be used to obtain the same result for the restricted fractional Laplacian. An application of this compactness result is that problem (1) possesses infinitely many solutions under the same assumptions. (C) 2015 Elsevier Inc. All rights reserved.