The Complementarity of Normalized Solutions for Kirchhoff Equations with Mixed Nonlinearity

Abstract:

      In this paper, we study the existence of solutions for Kirchhoff equation $$-\Big(a+b \int_{\mathbb{R}^{3}}|\nabla u|^{2} \text{d} x\Big)\Delta u=\lambda u+\mu|u|^{q-2} u+|u|^{p-2} u,~~x\in \mathbb{R}^{3}$$ 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.