Related papers: Convexity for free boundaries with singular term (…

Given a smooth domain $\Omega\subset\RR^N$ such that $0 \in \partial\Omega$ and given a nonnegative smooth function $\zeta$ on $\partial\Omega$, we study the behavior near 0 of positive solutions of $-\Delta u=u^q$ in $\Omega$ such that $u…

This paper is concerned with power concavity properties of the solution to the parabolic boundary value problem \begin{equation} \tag{$P$} \left\{\begin{array}{ll} \partial_t u=\Delta u +f(x,t,u,\nabla u) &…

In this paper we construct nontrivial exterior domains $\Omega \subset \mathbb{R}^N$, for all $N\geq 2$, such that the problem $$\left\{ {ll} -\Delta u +u -u^p=0,\ u >0 & \mbox{in }\; \Omega, {1mm] \ u= 0 & \mbox{on }\; \partial \Omega,…

We study the generalized boundary value problem for nonnegative solutions of of $-\Delta u+g(u)=0$ in a bounded Lipschitz domain $\Omega$, when $g$ is continuous and nondecreasing. Using the harmonic measure of $\Omega$, we define a trace…

We consider the Dirichlet problem for positive solutions of the equation $-\Delta_p (u) = f(u)$ in a convex, bounded, smooth domain $\Omega \subset\R^N$, with $f$ locally Lipschitz continuous. \par We provide sufficient conditions…

In this survey we provide an overview of nonlinear elliptic homogeneous boundary value problems featuring singular zero-order terms with respect to the unknown variable whose prototype equation is $$ -\Delta u = {u^{-\gamma}} \ \text{in}\…

We investigate the problem $$-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \mbox{ in } \Omega, \quad \frac{\partial u}{\partial \mathbf{n}} = 0 \mbox{ on } \partial \Omega, \leqno{(P_\lambda)} $$ where $\Omega$ is a bounded smooth…

We study an inhomogeneous minimization problems associated to the $p(x)$-Laplacian. We make a thorough analysis of the essential properties of their minimizers and we establish a relationship with a suitable free boundary problem. On the…

We consider the quasi-linear eigenvalue problem $-\Delta_p u = \lambda g(u)$ subject to Dirichlet boundary conditions on a bounded open set $\Omega$, where $g$ is a locally Lipschitz continuous functions. Imposing no further conditions on…

In this manuscript we deal with regularity issues and the asymptotic behaviour (as $p \to \infty$) of solutions for elliptic free boundary problems of $p-$Laplacian type ($2 \leq p< \infty$): \begin{equation*} -\Delta_p u(x) +…

Let $2 \le N\in\mathbb{N}$, $\Omega$ be a bounded open in $\mathbb{R}^{N}$, $T\in (0,\infty)$, $Q=\Omega\times (0,T)$, $u$ be a weak solution of parabolic equation $\displaystyle \frac{\partial u}{\partial t} -Lu= f$, where $L$ is an…

This paper investigates the existence and regularity of strong solutions to the incompressible Navier-Stokes equations within a bounded domain $\Omega \subset \mathbb{R}^3$, subject to the boundary condition $(u\cdot \vec{n})|_{\partial…

We consider the optimization problem of minimizing $\int_{\Omega}|\nabla u|^p dx$ with a constrain on the volume of $\{u>0\}$. We consider a penalization problem, and we prove that for small values of the penalization parameter, the…

Let $\Omega\subset\mathbb{R}^{d}$ be an open set. Given a boundary datum $g$ on $\partial\Omega$ and a function $K:\bar {\Omega} \to\mathcal{K}$, the family of all compact convex subsets of $\mathbb{R}^{d}$, we prove the existence of…

We investigate the properties of the Cheeger sets of rotationally invariant, bounded domains $\Omega \subset \mathbb{R}^n$. For a rotationally invariant Cheeger set $C$, the free boundary $\partial C \cap \Omega$ consists of pieces of…

We consider the supercritical problem {equation*} -\Delta u=|u| ^{p-2}u\text{\in}\Omega,\quad u=0\text{\on}\partial\Omega, {equation*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$ and $p$ smaller than the critical exponent…

In this paper we consider the existence of solution for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-\Delta)^su + u &= Q(x) |u|^{p-1}u\;\;\mbox{in}\;\;\R^N \setminus \Omega\\…

We consider the boundary value problem \begin{equation} - \Delta u = \lambda c(x)u+ \mu(x) |\nabla u|^2 + h(x), \qquad u \in H^1_0(\Omega) \cap L^{\infty}(\Omega), \leqno{(P_{\lambda})} \end{equation} where $\Omega \subset \R^N, N \geq 3$…

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities: $$({-}{ \Delta})^{\frac{\alpha}{2}}u- \gamma \frac{u}{|x|^{\alpha}}=…

We study the following elliptic problem $-A(u) = \lambda u^q$ with Dirichlet boundary conditions, where $A(u) (x) = \Delta u (x) \chi_{D_1} (x)+ \Delta_p u(x) \chi_{D_2}(x)$ is the Laplacian in one part of the domain, $D_1$, and the…