Existence and blow up behavior of prescribed mass solutions on large smooth domains to the Kirchhoff equation with combined nonlinearities

Abstract: In this paper, we consider the existence, multiplicity and the asymptotic behavior of prescribed mass solutions to the following nonlinear Kirchhoff equation with mixed nonlinearities: [ \begin{cases} -(a+b\int|\nabla u|^2\mathrm{d}x)\Delta u+V(x)u+\lambda u=|u|^{q-2}u+\beta|u|^{p-2}u \quad&\text{in}\ \Omega, \int_{\Omega}|u|^2\mathrm{d}x=\alpha, \end{cases} ] both on large bounded smooth star-shaped domain $\Omega\subset\mathbb{R}^{3}$ and on $\mathbb{R}^{3}$, where $2<p<\frac{14}{3}<q<6$ and $V(x)$ is the potential. The standard approach based on the Pohozaev identity to obtain normalized solutions is invalid due to the presence of potential $V(x)$.