Symmetry and monotonicity of positive solutions for a Choquard equation involving the logarithmic Laplacian operator

In this paper, we study a Schrödinger–Choquard equation involving the logarithmic Laplacian operator in R n : L ▵ u ( x ) + ω u ( x ) = C n , s ( | x | 2 s - n ∗ u p ) u r , x ∈ R n , where 0 < s < 1 , p > 1 , r > 0 , n ≥ 2 , ω > 0 . Using the direct method of moving planes, we prove...

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Bibliographic Details
Published in:Journal of fixed point theory and applications 2024-09, Vol.26 (3), Article 36
Main Authors: Cao, Linfen, Kang, Xianwen, Dai, Zhaohui
Format: Article
Language:English
Subjects:
Analysis
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
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Summary:In this paper, we study a Schrödinger–Choquard equation involving the logarithmic Laplacian operator in R n : L ▵ u ( x ) + ω u ( x ) = C n , s ( | x | 2 s - n ∗ u p ) u r , x ∈ R n , where 0 < s < 1 , p > 1 , r > 0 , n ≥ 2 , ω > 0 . Using the direct method of moving planes, we prove that if u satisfies some suitable asymptotic properties, then u must be radially symmetric and monotone decreasing about some point in the whole space. The key ingredients of the proofs are the narrow region principle and decay at infinity theorem; the ideas can be applied to problems involving more general nonlocal operators.
ISSN:1661-7738
1661-7746
DOI:10.1007/s11784-024-01121-y