We consider a class of semilinear elliptic equations of the form \begin{equation}\label{eq:abs} -\Delta u(x,y)+W'(u(x,y))=0,\quad (x,y)\in\mathbb{R}^{2} \end{equation} where $W:\mathbb{R}\to\mathbb{R}$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-1)^{2}$. We show, via variational methods, that for any $j\geq 2$, the equation (\ref{eq:abs}) has a solution $v_{j}\in C^{2}(\mathbb{R}^{2})$ with $|v_{j}(x,y)|\leq 1$ for any $(x,y)\in\mathbb{R}^{2}$ satisfying the following symmetric and asymptotic conditions: setting $\tilde v_{j}(\rho,\theta)=v_{j}(\rho\cos( \theta),\rho\sin(\theta))$, there results \begin{eqnarray*} &\hbox{$\tilde v_{j}(\rho,\frac{\pi}{2}+\theta)=-\tilde v_{j}(\rho,\frac{\pi}{2}-\theta)$ and $\tilde v_{j}(\rho,\theta+\frac{\pi}{j})=-\tilde v_{j}(\rho,\theta)$, $\forall (\rho,\theta)\in \mathbb{R}^{+}\times\mathbb{R}$}&\\ & \hbox{and $\tilde v_{j}(\rho,\theta)\to 1$ as $\rho\to+\infty$ for any $\theta\in [\frac{\pi}2-\frac{\pi}{2j},\frac{\pi}2)$.}& \end{eqnarray*}

We study the fourth-order nonlinear critical problem $\Delta^2 u= u^{2^*-1}$ in a smooth, bounded domain $\Omega \subset \mathbb{R}^n$, $n \ge 5$, subject to the boundary conditions $u=\Delta u-d u_\nu=0$ on $\partial \Omega$. We provide estimates for the range of parameters $d \in \mathbb{R}$ for which this problem admits a positive solution. If the domain is the unit ball, we obtain an almost complete description.

Given any domain $\Omega \subseteq \mathbb R^N$, $w \in L^1_{loc} (\Omega)$ and a differentiable function $A: \mathbb R^N \to \mathbb [0,\infty)$ which is $p$-homogeneous and strictly convex, we consider the minimization problem $$ \inf \Big \{ \frac{\int_{\Omega} A( \nabla u ) } { \left( \int_{\Omega} w (x) |u|^q \right)^{\frac{p}{q} }} \, \colon \, u \in \mathcal{D}^{1,\mathcal{p}}_0 (\Omega) \quad 0< \int_{\Omega} w (x) |u|^q < \infty \Big \}. $$ If the infimum is achieved and $q =p >1$, without additional regularity assumptions on $\Omega$ or the weight function $w$, we show that the minimizer is unique up to a constant factor. The same conclusion holds when $A$ is allowed to depend on $x \in \Omega$ and satisfies natural growth assumptions. Some of our results also hold when $q < p$.

For $p\in (1,+\infty)$ we derive a weighted $L^p$ estimate for the (spatial) gradient of the solution $u$ of a degenerate parabolic differential equation. Here the underlying domain $\Omega\subset\mathbf{R}^n$, $n\ge 2$, is unbounded and the equation may degenerate only at infinity along some unbounded branch of $\Omega$. Our estimate is strictly related with the still-open problem of giving a concrete characterization of the interpolation space between $W^{2,p}(\Omega)$ and $L^p(\Omega)$ to which the (spatial) gradient of $u$ belongs.

In this paper we prove the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, for rough initial data.