Yan, Shusen

Yang, Jianfu

Yu, Xiaohui

2015

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

Journal of Functional Analysis, 269(1), p. 47-79

1096-0783

0022-1236

https://hdl.handle.net/1959.11/18795

Elsevier Inc

Equations involving fractional Laplacian operator: Compactness and application

Journal Article

Publication