We study the existence of $L^2$ normalized solutions for nonlinear Schrödinger equations and systems. Under new Palais-Smale type conditions, we develop new deformation arguments for the constrained functional on $S_m=\{ u; \, \int_{\mathbb R^N} |u^2 | =m\}$ or $S_{m_1}\times S_{m_2}$. As applications, we give other proofs to the results of [5,8, 22]. As to the results of [5, 22], our deformation result enables us to apply the genus theory directly to the corresponding functional to obtain infinitely many solutions. As to the result [8], via our deformation result, we can show the existence of vector solution without using constraint related to the Pohozaev identity.

The asymptotic stability of two-dimensional stationary flows in a non-symmetric exterior domain is considered. Under the smallness condition on initial perturbations, we show the stability of the small stationary flow whose leading profile at spatial infinity is given by the rotating flow decaying in the scale-critical order $O(|x|^{-1})$. Especially, we prove the $L^p$-$L^q$ estimates to the semigroup associated with the linearized equations.

In this paper, our goal is to study the following class of Hénon type problems \begin{equation*} \left\{\begin{array}{rclcl}\displaystyle -\Delta u & & = \lambda u+|x|^{\alpha}k(u_+)+ f(x) &\mbox{in}&B_1, \\ u & & = 0 & \mbox{on} & \partial B_1, \end{array}\right. \end{equation*} where $B_1$ is the unit ball in $\mathbb R^N$, $k(t)$ is a $C^1$ function in $[0,+\infty)$ which is assumed to be in the critical growth range with subcritical perturbation, $f$ is radially symmetric and belongs to $L^{\mu}(B_1)$ for suitable $\mu$ depending on $N\geq 3$. Under appropriate hypotheses on the constant $\lambda$, we prove existence of at least two radial solutions for this problem using variational methods.