Existence and asymptotic behaviors of normalized solutions for Kirchhoff equations with critical Sobolev exponent

Abstract

In this paper, we consider the existence and asymptotic behaviors of normalized ground state to Kirchhoff equation with Sobolev critical exponent and mixed nonlinearities$\{\begin{array}{cc}-(a+b\underset{R^3}{\int }|\nabla u{|}^2)\Delta u=\lambda u+\mu |u{|}^{q-2}u+|u{|}^{4}u,& x\in R^3,\\ \underset{R^3}{\int }{u}^2=c^2,\end{array}$ where $a,b,c>0$ are constants, $\lambda \in R,\mu >0$ and $2<q<6$. When $10/3⩽q<6$, we show that the problem has a normalized ground state solution under suitable assumptions on μ and c which is a mountain pass solution. Furthermore, we prove precise asymptotic behaviors of ground states as $\mu \to 0$ and $\mu \to \infty$ for $2<q<6$. After scaling, the ground state converges to Aubin-Talanti babbles (minimizers of Sobolev inequality) as $\mu \to 0$ for $10/3⩽q<6$. However, the ground state converges to minimizers of Gagliardo-Nirenberg inequality as $\mu \to 0$ for $2<q<10/3$ or as $\mu \to \infty$ for $14/3⩽q<6$.