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

\(- \left(a+b\int_{{\mathbb{R}^3}} {{{\left| {\nabla u} \right|}^2}}\right) \Delta u=\lambda u+ {| u |^{p - 2}}u+\mu {| u |^{q - 2}}u\) 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.