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


  • 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


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


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.





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.