Normalized ground states for Kirchhoff equations in ${\mathbb{R}}^3$ with a critical nonlinearity

Abstract

This paper is concerned with the existence of ground states for a class of Kirchhoff type equation with combined power nonlinearities \begin{equation*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u(x)|^{2}\right) Δu =λu+|u|^{p-2}u+u^{5}\quad \ \text{for some} \ λ\in\mathbb{R},\quad x\in\mathbb{R}^{3}, \end{equation*} with prescribed $L^2$-norm mass \begin{equation*} \int_{\mathbb{R}^{3}}u^{2}=c^{2} \end{equation*} in Sobolev critical case and proves that the equation has a couple of solutions $(u_c,{\lambda }_c)\in S(c)\times \mathbb{R}$ for any $c>0$, $a,b>0$ and $\frac{14}{3}\le p<6,$ where $S(c)=\{u\in H^1({\mathbb{R}}^3):{\int }_{{\mathbb{R}}^3}{u}^2=c^2\}.$ \textbf{Keywords:} Kirchhoff type equation; Critical nonlinearity; Normalized ground states \noindent{AMS Subject Classification:\, 37L05; 35B40; 35B41.}

18 pages

Cited by in corpus (2)

• On the Kirchhoff equation with prescribed mass and general nonlinearities
• Positive solutions with prescribed mass for a planar Choquard equation with critical growth