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.

In this paper, we consider the following coupled Schr\”{o}dinger system with doubly critical exponents, which can be seen as a counterpart of the Brezis-Nirenberg problem $$\left\{% \begin{array}{ll} -\Delta u+\lambda_1 u=\mu_1 u^5+ \beta u^2v^3, & \hbox{$x\in \Omega$}, \\ -\Delta v+\lambda_2 v=\mu_2 v^5+ \beta v^2u^3, & \hbox{$x\in \Omega$}, \\ u=v=0,& \hbox{$x\in \partial\Omega$}, \\ \end{array}% \right.$$ where $\Omega$ is a ball in $\R^3,$ $-\lambda_1(\Omega)<\lambda_1,\lambda_2<-\frac14\lambda_1(\Omega)$, $\mu_1,\mu_2>0$ and $\beta>0$. Here $\lambda_1(\Omega)$ is the first eigenvalue of $-\Delta$ with Dirichlet boundary condition in $\Omega$. We show that the problem has at least one nontrivial solution for all $\beta>0$.

In this paper, we study the existence of critical points for the following functional $$I(u)=\frac{1}{2}\ds\int_{\R^N}|\nabla u|^2+\ds\int_{\R^N}|u|^2|\nabla u|^2-\frac{N}{4(N+1)}\ds\int_{\R^N}|u|^{\frac{4(N+1)}{N}},$$ constrained on $S_c=\{u\in H^1(\R^N)|~\int_{\R^N}|u|^2|\nabla u|^2<+\infty,~|u|_2=c,c>0\}$, where $N\geq1$. The constraint problem is $L^2$-critical. We prove that the minimization problem $i_c=\inf\limits_{u\in S_c}I(u)$ has no minimizer for all $c>0$. We also obtain a threshold value of $c$ separating the existence and nonexistence of critical points for $I(u)$ restricted to $S_c$.