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

Abstract

In this paper, we consider the existence and asymptotic properties of solutions to the following Kirchhoff equation

$-(a+b{\int }_{{\mathbb{R}}^3}{\left|\mathrm{\nabla }u\right|}^2)\mathrm{\Delta }u=\lambda u+|u{|}^{p-2}u+\mu |u{|}^{q-2}u$ in ${\mathbb{R}}^3$

under the normalized constraint ${\int }_{{\mathbb{R}}^3}{u}^2=c^2$, where $a>0$, $b>0$, $c>0$, $2<q<\frac{14}{3}<p\le 6$ or $\frac{14}{3}<q<p\le 6$, $\mu >0$ 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 $S_c=\{u\in H^1({\mathbb{R}}^3):{\int }_{{\mathbb{R}}^3}{u}^2=c^2\}$. 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 $\frac{14}{3}<q<p\le 6$, 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 $({\int }_{{\mathbb{R}}^3}{\left|\mathrm{\nabla }u\right|}^2)\mathrm{\Delta }u$ 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.