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Title: Equations involving fractional Laplacian operator: Compactness and application
Contributor(s): Yan, Shusen  (author); Yang, Jianfu (author); Yu, Xiaohui (author)
Publication Date: 2015
Open Access: Yes
DOI: 10.1016/j.jfa.2015.04.012Open Access Link
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Abstract: 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.
Publication Type: Journal Article
Grant Details: ARC/DP130102773
Source of Publication: Journal of Functional Analysis, 269(1), p. 47-79
Publisher: Academic Press
Place of Publication: Maryland Heights, United States of America
ISSN: 1096-0783
Field of Research (FOR): 010110 Partial Differential Equations
Peer Reviewed: Yes
HERDC Category Description: C1 Refereed Article in a Scholarly Journal
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