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: -(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, 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).