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

## Authors

• Gongbao Li Central China Normal University, School of Mathematics and Statistics
• Xiao Luo Hefei University of Technology, School of Mathematics
• Tao Yang Zhejiang Normal University, Department of Mathematics

## Keywords:

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

### Abstract

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.

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Section
Articles

2022-06-20

## How to Cite

Li, G., Luo, X., & Yang, T. (2022). Normalized solutions to a class of Kirchhoff equations with Sobolev critical exponent. Annales Fennici Mathematici, 47(2), 895–925. https://doi.org/10.54330/afm.120247