Equations involving fractional Laplacian operator: Compactness and application

In this paper, we consider the following problem involving fractional Laplacian operator:equation(1)(−Δ)αu=|u|2α⁎−2−εu+λuinΩ,u=0on ∂Ω, where Ω is a smooth bounded domain in RNRN, ε∈[0,2α⁎−2), 0<α<10<α<1, 2α⁎=2NN−2α, and (−Δ)α(−Δ)α 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 unun of (1) corresponding to εn∈[0,2α⁎−2), satisfying ‖un‖H≤C‖un‖H≤C in the Sobolev space H defined in (1.2), unun converges strongly in H   provided that N>6αN>6α and λ>0λ>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.