Year: 2024
Author: Xingdong Tang, Guixiang Xu, Chunyan Zhang, Jihui Zhang
Annals of Applied Mathematics, Vol. 40 (2024), Iss. 3 : pp. 219–248
Abstract
In this paper, we consider the fractional critical Schrödinger equation (FCSE) $$(-\Delta)^su-|u|^{2^*_s-2}u=0,$$ where $u∈\dot{H}^s(\mathbb{R}^N),$ $N≥4,$ $0<s<1$ and $2^∗_s=\frac{2N}{N−2s}$ is the critical Sobolev exponent of order $s.$ By virtue of the variational method and the concentration compactness principle with the equivariant group action, we obtain some new type of nonradial, sign-changing solutions of (FCSE) in the energy space $\dot{H}^s(\mathbb{R}^N)$. The key component is that we take the equivariant group action to construct several subspace of $\dot{H}^s(\mathbb{R}^N)$ with trivial intersection, then combine the concentration compactness argument in the Sobolev space with fractional order to show the compactness property of Palais-Smale sequences in each subspace and obtain the multiple solutions of (FCSE) in $\dot{H}^s(\mathbb{R}^N).$
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aam.OA-2024-0006
Annals of Applied Mathematics, Vol. 40 (2024), Iss. 3 : pp. 219–248
Published online: 2024-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 30
Keywords: Fractional critical Schrödinger equation sign-changing solution the concentration-compactness principle the equivariant group action the mountain pass theorem.