Normalized solutions to a class of Kirchhoff equations with Sobolev critical exponent (2103.08106v1)

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) \Delta u =\lambda u+ {| u |^{p - 2}}u+\mu {| u |^{q - 2}}u \text { in } \mathbb{R}^{3} \end{equation} under the normalized constraint $\int_{{\mathbb{R}^3}} {{u}^2}=c^2$, where $a!>!0$, $b!>!0$, $c!>!0$, $2!<!q!<!\frac{14}{3}!<! p!\leq!6$ or $\frac{14}{3}!<!q!<! p!\leq! 6$, $\mu!>!0$ and $\lambda!\in!\R$ 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 $S_c=\Big{ u \in H^{1}({\mathbb{R}^3}): \int_{{\mathbb{R}^3}} {{u}^2}=c^2 \Big}$. 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 N. Soave (J. Differential Equations 2020 $&$ J. Funct. Anal. 2020), which studied the nonlinear Schr\"{o}dinger equations with combined nonlinearities, to the Kirchhoff equations. To deal with the special difficulties created by the nonlocal term $({\int_{{\R^3}} {\left| {\nabla u} \right|} ^2}) \Delta u$ 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.