Positive normalized solution to the Kirchhoff equation with general nonlinearities of mass super-critical

In present paper, we study the normalized solutions $(\lambda_c, u_c)\in \R\times H^1(\R^N)$tothefollowingKirchhoffproblem$$ -\left(a+b\int_{\R^N}|\nabla u|^2dx\right)\Delta u+\lambda u=g(u)~\hbox{in}~\R^N,;1\leq N\leq 3 $$ satisfying the normalization constraint ${\displaystyle {\int }_{{\mathbb{R}}^N}{u}^2=c,}$ which appears in free vibrations of elastic strings. The parameters $a,b>0$ are prescribed as is the mass $c>0$. The nonlinearities $g(s)$ considered here are very general and of mass super-critical. Under some suitable assumptions, we can prove the existence of ground state normalized solutions for any given $c>0$. After a detailed analysis via the blow up method, we also make clear the asymptotic behavior of these solutions as $c\to 0^{+}$ as well as $c\to +\mathrm{\infty }$.