The Complementarity of Normalized Solutions for Kirchhoff Equations with Mixed Nonlinearity 
Received:October 16, 2023 Revised:March 08, 2024 
Key Words:
normalized solutions Kirchhoff type equation mixed nonlinearty

Fund Project:Supported by the Basic and Applied Basic Research Foundation of Guangdong Province (Grant No.2022A1515010644). 

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Abstract: 
In this paper, we study the existence of solutions for Kirchhoff equation $$\Big(a+b \int_{\mathbb{R}^{3}}\nabla u^{2} \text{d} x\Big)\Delta u=\lambda u+\muu^{q2} u+u^{p2} u,~~x\in \mathbb{R}^{3}$$ with mass constraint condition $$S_{c}:=\Big\{u \in H^{1}(\mathbb{R}^{3}):\int_{\mathbb{R}^{3}}u^{2} \text{d} x=c\Big\},$$ where $a$, $b$, $c>0$, $\mu\in \mathbb{R}$, $20$ when $(p, q)$ belongs to a certain domain in $\mathbb{R}^{2}$. We prove the existence of normalized solutions by using constraint minimization, concentration compactness principle and Minimax methods. We partially extend the results which have been studied. 
Citation: 
DOI:10.3770/j.issn:20952651.2024.04.007 
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