Normalized solutions to a class of Kirchhoff equations with Sobolev critical exponent

Gongbao Li¹ ; Xiao Luo² ; Tao Yang³

¹Central China Normal University, School of Mathematics and Statistics
²Hefei University of Technology, School of Mathematics
³Zhejiang Normal University, Department of Mathematics

Annales Fennici Mathematici, Tome 47 (2022) no. 2, pp. 895-925

Résumé

In this paper, we consider the existence and asymptotic properties of solutions to the following Kirchhoff equation $- \left(a+b\int_{{\mathbb{R}^3}} {{{\left| {\nabla u} \right|}^2}}\right) \Delta u=\lambda u+ {| u |^{p - 2}}u+\mu {| u |^{q - 2}}u$ in $\mathbb{R}^{3}$ under the normalized constraint $\int_{{\mathbb{R}^3}} {{u}^2}=c^2$, where $a>0$, $b>0$, $c>0$, $2<q<\frac{14}{3}<p\leq 6$ or $\frac{14}{3}<q< p\leq 6$, $\mu>0$ and $\lambda\in\mathbb{R}$ appears as a Lagrange multiplier. In both cases for the range of $p$ and $q$, the Sobolev critical exponent $p=6$ is involved and the corresponding energy functional is unbounded from below on $S_c=\{ u \in H^{1}({\mathbb{R}^3})\colon \int_{{\mathbb{R}^3}} {{u}^2}=c^2 \}$. If $2<q<\frac{10}{3}$ and $\frac{14}{3}<p<6$, we obtain a multiplicity result to the equation. If $2<q<\frac{10}{3}<p=6$ or $\frac{14}{3}<q< p\leq 6$, we get a ground state solution to the equation. Furthermore, we derive several asymptotic results on the obtained normalized solutions. Our results extend the results of Soave (J. Differential Equations 2020 & J. Funct. Anal. 2020), which studied the nonlinear Schrödinger equations with combined nonlinearities, to the Kirchhoff equations. To deal with the special difficulties created by the nonlocal term $({\int_{{\mathbb{R}^3}} {\left| {\nabla u} \right|} ^2}) \Delta u$ appearing in Kirchhoff type equations, we develop a perturbed Pohozaev constraint approach and we find a way to get a clear picture of the profile of the fiber map via careful analysis. In the meantime, we need some subtle energy estimates under the $L^2$-constraint to recover compactness in the Sobolev critical case.  

Détail

DOI : 10.54330/afm.120247

Keywords: Kirchhoff equation, Sobolev critical exponent, normalized solutions, asymptotic property, variational methods