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

In this paper, we consider the existence and asymptotic behaviors of normalized ground state to Kirchhoff equation with Sobolev critical exponent and mixed nonlinearities {-( a+b integral (R)3|del u|(2))Delta u=lambda u+mu|u|(q-2)u+|u|(4)u,x is an element of R-3, integral(3 )(R)u(2) = c(2,) where a,b,c>0 are constants, lambda is an element of 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 mu and c which is a mountain pass solution. Furthermore, we prove precise asymptotic behaviors of ground states as mu -> 0 and mu ->infinity for 2<q<6. After scaling, the ground state converges to Aubin-Talanti babbles (minimizers of Sobolev inequality) as mu -> 0 for 10/3 <= q<6. However, the ground state converges to minimizers of Gagliardo-Nirenberg inequality as mu -> 0 for 2 < q<10/3 or as mu ->infinity for 14/3 <= q < 6.