Normalized solutions for the general Kirchhoff type equations

In the present paper, we prove the existence, non-existence and multiplicity of positive normalized solutions ( λ c , u c ) ∈ ℝ × H 1 (ℝ N ) to the general Kirchhoff problem − M ( ∫ R N | ∇ u | 2 d x ) Δ u + λ u = g ( u ) in R N , u ∈ H 1 ( R N ) , N ≥ 1 , satisfying the normalization constraint ∫ R...

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Bibliographic Details
Published in:Acta mathematica scientia 2024-09, Vol.44 (5), p.1886-1902
Main Authors: Liu, Wenmin, Zhong, Xuexiu, Zhou, Jinfang
Format: Article
Language:English
Subjects:
Analysis
Mathematics
Mathematics and Statistics
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Summary:In the present paper, we prove the existence, non-existence and multiplicity of positive normalized solutions ( λ c , u c ) ∈ ℝ × H 1 (ℝ N ) to the general Kirchhoff problem − M ( ∫ R N | ∇ u | 2 d x ) Δ u + λ u = g ( u ) in R N , u ∈ H 1 ( R N ) , N ≥ 1 , satisfying the normalization constraint ∫ R N u 2 d x = c , where M ∈ C ([0, ∞)) is a given function satisfying some suitable assumptions. Our argument is not by the classical variational method, but by a global branch approach developed by Jeanjean et al. [J Math Pures Appl, 2024, 183: 44–75] and a direct correspondence, so we can handle in a unified way the nonlinearities g ( s ), which are either mass subcritical, mass critical or mass supercritical.
ISSN:0252-9602
1572-9087
DOI:10.1007/s10473-024-0514-3