Normalized solutions of the autonomous Kirchhoff equation with Sobolev critical exponent: Sub- and super-critical cases

Abstract

In the present paper, we investigate the existence and multiplicity properties of the normalized solutions to the following Kirchhoff-type equation with Sobolev critical growth ( P ) { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + λ u = μ | u | p − 2 u + | u | 4 u , a m p ; in  R 3 , u > 0 ,   ∫ R 3 | u | 2 d x = c 2 , a m p ; in  R 3 , \begin{equation*} \begin {cases} -\left (a+b\int _{\mathbb {R}^3}|abla u|^2dx\right )\Delta u+\lambda u=\mu |u|^{p-2}u+|u|^{4}u, \quad &\text {in } \mathbb {R}^3,\\ u>0, \ \int _{\mathbb {R}^3}|u|^2dx=c^2, \quad &\text {in }\mathbb {R}^3, \end{cases} \tag {$P$}\end{equation*} where a ,   b ,   c ,   μ > 0 a, \ b, \ c, \ \mu >0 and 4 > p > 6 4>p>6 . We consider both the L 2 L^2 -subcritical and the L 2 L^2 -supercritical cases. Precisely, in the L 2 L^2 -subcritical case, by combining the truncation method, the concentration-compactness principle and genus theory, we obtain the multiplicity of the normalized solutions for problem ( P ) (P) . In the L 2 L^2 -supercritical case, by using a fiber map and the concentration-compactness principle, we obtain a couple of normalized solutions for problem ( P ) (P) , as well as their asymptotic behavior. These results extend and complement the existing results from Sobolev subcritical growth to the critical Sobolev setting.