The functional $J_{\Omega }=\frac {1}{|\Omega |}\int _{\Omega }[(\Delta u)^2-b|\nabla |^2+\psi (u)]dxdy$ is considered on bounded domains $\Omega \subset R^2$ and configurations $u$, which are functions belonging to $H^2_{loc}(R^2)$. For such $u$ the value of $J_{\Omega }(u)$ is studied in the limit where $\Omega $ extends to the whole plane $R^2$, which we denote $J(u)$. The integrand defining $J_{\Omega }$ is radially symmetric, and we raise the question whether $J(u)$ has a minimizer $u^{\star }$ which is radially symmetric, namely has the form $u^{\star }(x,y)=v(\sqrt {x^2+y^2})$. It is shown that the minimal value of $J(u)$ over the radially symmetric configurations is equal to its minimal value over 1-dimensional configurations, which are functions of the form $u(x,y)=v(x)$. Restricting to 1-dimensional configurations yields the 1-dimensional model, hence $\lambda _2\leq \lambda _1$, where $\lambda _1$ and $\lambda _2$ are the values of the 1 and 2-dimensional models respectively. Then the question which we address is whether strict inequality holds in $\lambda _2\leq \lambda _1$. We establish a sufficient condition for a configuration $u$ to satisfy $J(u)\geq \lambda _1$, and use it to conclude that the minimal value of $J(u)$ on the class of quasi-convex configurations is $\lambda _1$.

We study the stability of standing waves $e^{i \omega t}\phi_{\omega}(x)$ for a nonlinear Schrödinger equation with critical power nonlinearity $|u|^{4/n}u$ and a potential $V(x)$ in $\mathbb R^n$. Here, $\omega\in \mathbb R$ and $\phi_{\omega}(x)$ is a ground state of the stationary problem. Under suitable assumptions on $V(x)$, we show that $e^{i \omega t}\phi_{\omega}(x)$ is stable for sufficiently large $\omega$. This result gives a different phenomenon from the case $V(x)\equiv 0$ where the strong instability was proved by M.I. Weinstein [25].

Motivated by the study of the Cauchy problem with bore-like initial data, we show the "well-posedness" for Korteweg-de Vries and Benjamin-Ono equations with initial data in Zhidkov spaces $X^s$, with respectively $s>1$ and $s>5/4$. Here, "well-posedness" includes local (global in some cases) existence, uniqueness under a supplementary assumption, and continuity with respect to the initial data.

We obtain solutions of the nonlinear degenerate parabolic equation \[ \frac{\partial \rho}{\partial t}= \mbox{div} \Big\{ \rho \nabla c^\star \left[ \nabla \left(F^\prime(\rho)+V\right) \right] \Big\} \] as a steepest descent of an energy with respect to a convex cost functional. The method used here is variational. It requires fewer uniform convexity assumptions than those imposed by Alt and Luckhaus in their pioneering work [4]. In fact, their assumptions may fail in our equation. This class of equations includes the Fokker-Planck equation, the porous-medium equation, the fast diffusion equation and the parabolic p-Laplacian equation.