Normalized solutions to Kirchhoff equation with the Sobolev critical exponent in high dimensional spaces

Abstract

The following well-known Kirchhoff equation with the Sobolev critical exponent has been extensively studied, \begin{equation*} -\Big(a+b\int_{\mathbb R^N} | \nabla u|^2dx\Big) Δu+λu=μ|u|^{q-2}u+|u|^{2^*-2}u \ \ {\rm in}\ \ \mathbb{R}^N, \ \ N\geq4, \end{equation*} having prescribed mass ${\int }_{{\mathbb{R}}^N}|u{|}^{2}dx=c$, where $a$, $c$ are two positive constants, $b,\hat{I}¼$ are two parameters, $\hat{I}»$ appears as a real Lagrange multiplier and $2<q<2^{\ast }$, $2^{\ast }$ is the Sobolev critical exponent. Firstly, for the special case $\hat{I}¼=0$ and $N\ge 4$, the above equation reduces to a pure critical Kirchhoff equation, we obtain a complete conclusion including the existence, nonexistence and multiplicity of the normalized solutions by the variational methods. Secondly, when $\hat{I}¼>0$, $N\ge 5$ and $2<q<2+\frac{4}{N}$, we investigate the existence of the positive normalize solution under suitable assumptions on parameter $b$ and mass $c$. To the best of our knowledge, it is the first time to consider the above case, which is a more complicated case not only the difficulties on checking the Palais-Smale condition, but also the constraint functional requesting the intricate concave-convex structure. Lastly, when $\hat{I}¼>0$ and $N=4$, we obtain a local minimizer solution and a mountain pass solution under explicit conditions on $b$ and $c$. It is worth noting that the second solution is obtained by introducing a new functional to establish a threshold for the mountain pass level, which is the key step for the fulfillment of the Palais-Smale condition. This paper provides a refinement and extension of the results of the normalized solutions for Kirchhoff type problem in high-dimensional spaces.