Related papers: The critical dimension for a fourth order elliptic…

This paper studies the existence of positive normalized solutions to the singular elliptic equation \[ -\Delta u + \lambda u = u^{-r} + u^{p-1} \quad \text{in } \Omega, \] with the Dirichlet boundary condition $u=0$ on $\partial\Omega$ and…

We analyze the nonlinear elliptic problem $\Delta u=\frac{\lambda f(x)}{(1+u)^2}$ on a bounded domain $\Omega$ of $\R^N$ with Dirichlet boundary conditions. This equation models a simple electrostatic Micro-Electromechanical System (MEMS)…

We investigate stable solutions of elliptic equations of the type \begin{equation*} \left \{ \begin{aligned} (-\Delta)^s u&=\lambda f(u) \qquad {\mbox{ in $B_1 \subset \R^{n}$}} \\ u&= 0 \qquad{\mbox{ on $\partial B_1$,}}\end{aligned}\right…

We consider the regularity of the extremal solution of the nonlinear eigenvalue problem (S)_\lambda \qquad {rcr} -\Delta u + c(x) \cdot \nabla u &=& \frac{\lambda}{(1-u)^2} \qquad {in $ \Omega$}, u &=& 0 \qquad {on $ \pOm$}, where $ \Omega…

We are interested in regularity properties of semi-stable solutions for a class of singular semilinear elliptic problems with advection term defined on a smooth bounded domain of a complete Riemannian manifold with zero Dirichlet boundary…

We study the extremal solution for the problem $(-\Delta)^s u=\lambda f(u)$ in $\Omega$, $u\equiv0$ in $\R^n\setminus\Omega$, where $\lambda>0$ is a parameter and $s\in(0,1)$. We extend some well known results for the extremal solution when…

We consider the reaction-diffusion problem $-\Delta_g u = f(u)$ in $\mathcal{B}_R$ with zero Dirichlet boundary condition, posed in a geodesic ball $\mathcal{B}_R$ with radius $R$ of a Riemannian model $(M,g)$. This class of Riemannian…

We study the effect of the parameter $\lambda$, the dimension $N$, the profile $f$ and the geometry of the domain $\Omega \subset\mathbb{R}^N$, on the question of uniqueness of the solutions to the following elliptic boundary value problem…

This paper deals with solutions to the equation \begin{equation*} -\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $\lambda_+,\lambda_- > 0$, $q \in (0,1)$,…

We study the regularity of the extremal solution $u^*$ to the singular reaction-diffusion problem $-\Delta_p u = \lambda f(u)$ in $\Omega$, $u =0$ on $\partial \Omega$, where $1<p<2$, $0 < \lambda < \lambda^*$, $\Omega \subset \mathbb{R}^n$…

We consider the class of semi-stable solutions to semilinear equations $-\Delta u=f(u)$ in a bounded smooth domain $\Omega$ of $R^n$ (with $\Omega$ convex in some results). This class includes all local minimizers, minimal, and extremal…

We analyze nonnegative solutions of the nonlinear elliptic problem $\Delta u=\frac{\lambda f(x)}{u^2}+P$, where $\lambda>0$ and $P\geq0$, on a bounded domain $\Omega$ of $\mathbb{R}^N$ ($N\geq 1$) with a Dirichlet boundary condition. This…

We examine equations of the form {eqnarray*} \{{array}{lcl} \hfill \HA u &=& \lambda g(x) f(u) \qquad \text{in}\ \Omega \hfill u&=& 0 \qquad \qquad \qquad \text{on}\ \pOm, {array}. {eqnarray*} where $ \lambda >0$ is a parameter and $…

We study the branch of semi-stable and unstable solutions (i.e., those whose Morse index is at most one) of the Dirichlet boundary value problem $-\Delta u=\frac{\lambda f(x)}{(1-u)^2}$ on a bounded domain $\Omega \subset \R^N$, which…

We study the regularity of stable solutions to the problem $$ \left\{ \begin{array}{rcll} (-\Delta)^s u &=& f(u) & \text{in} \quad B_1\,, u &\equiv&0 & \text{in} \quad \mathbb R^n\setminus B_1\,, \end{array} \right. $$ where $s\in(0,1)$.…

We study the singular semilinear equation $-Pu = \frac{f}{u^\gamma}$ on a bounded domain $\Omega$ with Dirichlet condition $u \equiv 0$ on $\partial \Omega$ , where $P$ is a second-order elliptic differential operator in nondivergence form.…

The parabolic problem $u_t-\Delta u=\frac{\lambda f(x)}{(1-u)^2}+P$ on a bounded domain $\Omega$ of $R^n$ with Dirichlet boundary condition models the microelectromechanical systems(MEMS) device with an external pressure term. In this…

We consider the supercritical elliptic problem -\Delta u = \lambda e^u, \lambda > 0, in an exterior domain $\Omega = \mathbb{R}^N \setminus D$ under zero Dirichlet condition, where D is smooth and bounded in \mathbb{R}^N, N greater or equal…

We study the existence/nonexistence of positive solution to the problem of the type: \begin{equation}\tag{$P_{\lambda}$} \begin{cases} \Delta^2u-\mu a(x)u=f(u)+\lambda b(x)\quad\textrm{in $\Omega$,}\\ u>0 \quad\textrm{in $\Omega$,}\\…

In this paper, we are concerned with the following type of fractional problems: $$ \begin{cases}\dis (-\Delta)^{s} u-\mu\frac{u}{|x|^{2s}}-\lambda u=|u|^{2^*_{s}-2}u+f(x,u), &\text{in} \Omega,\ \ \, u=0\,&\text{in} \R^N\backslash\Omega…