In this paper we study Brezis-Nirenberg type results for radial solutions of a quasilinear elliptic equation of the form $$ \begin{cases}-\Delta_pu= \lambda C(|x|)|u|^{p-2}u+ B(|x|) |u|^{q-2}u, \ a.e.\ x\in B_R(0)\subset\mathbb R^N,\ R>0,\\ u=0,\quad \mbox{on }\partial B_R(0), \end{cases} $$ where $\lambda\in\mathbb R$, $q\ge p>1$, $\Delta_pu=\mbox{div}(|\nabla u|^{p-2}\nabla u)$, $B_R(0)$ denotes the ball of radius $R>0$ centered at $0$ in $\mathbb R^N$, and the weights $ B,\ C$ are appropriate positive measurable radially symmetric functions.

We study a fourth-order geometric evolution problem on a network of curves in a bounded domain $\Omega$. The flow decreases a weighted total length of the curves and preserves the enclosed volumes. Stationary solutions of the flow are critical points of a partition problem in $\Omega$. In this paper we study the linearized stability of stationary solutions using the $H^{-1}$-gradient flow structure of the problem. Important issues are the development of an appropriate PDE formulation of the geometric problem and Poincaré type estimate on a network of curves.

We consider an equation of the type $A(u+k*u)=f$, where $A$ is a linear second-order elliptic operator, $k$ is a scalar function depending on time only and $k*u$ denotes the standard time convolution of functions defined in $(-\infty,T)$ with their supports in $[0,T]$. The previous equation is endowed with second-order dynamical boundary conditions. Assuming that the kernel $k$ is unknown and a supplementary condition is given, $k$ can be recovered and global existence, uniqueness and continuous dependence results can be shown.

We study the Dirichlet problem associated to the equation for self-similar surfaces under mean curvature flow for graphs over the Euclidean plane with a disk removed. We show the existence of a solution provided the boundary conditions on the boundary circle are small enough and satisfy some symmetries.

We obtain a pointwise, a priori bound for the vorticity of axially symmetric solutions to the three-dimensional Navier-Stokes equations. The bound is in the form of a reciprocal of a power of the distance to the axis of symmetry. This seems to be the first general pointwise upper bound established for the axially symmetric Navier-Stokes equations.

In this paper we study the existence of multiple solutions to the equation \begin{align*} -{\Delta_p} u = f(x,u)-|u|^{p-2}u \end{align*} with the nonlinear boundary condition \begin{align*} |\nabla u|^{p-2} \frac{\partial u}{ \partial \nu} = \lambda |u|^{p-2}u+g(x,u). \end{align*} We establish the existence of a smallest positive solution, a greatest negative solution, and a nontrivial sign-changing solution when the parameter $\lambda$ is greater than the second eigenvalue of the Steklov eigenvalue problem. Our approach is based on truncation techniques and comparison principles for nonlinear elliptic differential inequalities. In particular, we make use of variational and topological tools, such as critical point theory, the mountain-pass theorem, the second deformation lemma and variational characterizations of the second eigenvalue of the Steklov eigenvalue problem.