The existence and nonexistence of global $L^2$-constrained
minimizers for Kirchhoff equations with $L^2$-subcritical general
nonlinearity
Abstract
In this paper, we study the existence of global $L^2$-constrained
minimizers related to the following Kirchhoff type equation: $$
-\left(a+b\ds\int_{\R^N}|\nabla
u|^2\right)\Delta
u-f(u)=\lambda
u,~~~x\in
\R^N,~\lambda\in\R,$$
where $N\leq3$, $a,$ $b>0$ are
constants, $f(u)$ is a general $L^2$-subcritical nonlinearity. By
using the concentration compactness principle, we prove the sharp
existence and nonexistence of global $L^2$-constraint minimizers.
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