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 \begin{equation}\label{1}\nonumber - \Bigl(a+b\int_{\R^3} {{{\left| {\nabla u} \right|}^2}}\Bigl) Δu =λu+ {| u |^{p - 2}}u+μ{| u |^{q - 2}}u \text { in } \mathbb{R}^{3} \end{equation} 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 $\frac{14}{3}<q<p\le 6$, $\hat{I}¼>0$ and $\hat{I}»\in \mathbb{R}$ 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 N. 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 $({\int }_{{\mathbb{R}}^3}{\left|\nabla u\right|}^2)\hat{I}”u$ 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.