Normalized ground states for Kirchhoff type equations with general nonlinearities

Abstract

In this paper, we consider the following Kirchhoff type equation\begin{equation*} \begin{cases}-\displaystyle\Big ( a+b\int_{\mathbb{R}^3}|\nabla u|^2 \Big ) \Delta u-\lambda u=f(u), & \textrm{in}\,\,\mathbb{R}^3,\\[3mm]u\in H^1(\mathbb{R}^3),\end{cases}\end{equation*}with an $L^2$ constraint $\int_{\mathbb{R}^3}|u|^2dx=m$, where $a,b,m > 0$, $\lambda\in\mathbb{R}$ will arise as a Lagrange multiplier and the nonlinearity $f$ is merely continuous and satisfies general mass supercritical conditions. Both in the Sobolev subcritical and critical cases, we establish the existence of ground states to this problem and derive some basic behavior of the ground state energy $E_m$ when $m > 0$ varies. Our results generalize and improve the ones in [L. Jeanjean, S.-S. Lu, Calc. Var. 59, 174 (2020)] and other related literature.