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In the paper, we consider the fractional elliptic system \begin{equation*}\left\{\begin{array}{ll} (- \Delta)^{\frac{\alpha_1}{2}}u(x)+\sum\limits^n_{i=1}b_i(x)\frac{\partial u}{\partial x_i}+B(x)u(x)=f(x,u,v),& \mbox { in } \Omega,\\ (-…

We consider the supercritical problem -\Delta u = |u|^{p-2}u in \Omega, u=0 on \partial\Omega, where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N},$ $N\geq3,$ and $p\geq2^{*}:= 2N/(N-2).$ Bahri and Coron showed that if $\Omega$ has…

We study the following problem \[ \begin{cases} -\Delta u = \lambda u + u^{2^*-2} v & \hbox{in} \Omega,\\ -\Delta v= \mu v^{2^*-1} + u^{2^*-1} & \hbox{in} \Omega,\\ u> 0,v> 0 & \hbox{in} \Omega,\\ u=v=0 & \hbox{on} \partial \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 $\Omega \subset \R^N, N \geq 3$…

We consider a second-order nonlocal parabolic MEMS equation with Dirichlet boundary conditions: \[ u_t-\Delta u=\frac{\lambda}{(1-u)^2\bigl(1+\int_\Omega\frac{1}{1-u}\,dx\bigr)^2},\quad x\in\Omega,\ t>0, \] where…

We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient, whose model is $$ \begin{cases} -\Delta u +…

We consider the nonlinear equation $$-u'' = f(u) + h , \quad \text{on} \quad (-1,1),$$ where $f : {\mathbb R} \to {\mathbb R}$ and $h : [-1,1] \to {\mathbb R}$ are continuous, together with general Sturm-Liouville type, multi-point boundary…

Given $\Omega(\subseteq\;R^{1+m})$, a smooth bounded domain and a nonnegative measurable function $f$ defined on $\Omega$ with suitable summability. In this paper, we will study the existence and regularity of solutions to the quasilinear…

In this paper, we study the following class of weighted Choquard equations \begin{align*} -\Delta u =\lambda u + \Bigg(\displaystyle\int\limits_\Omega \frac{Q(|y|)F(u(y))}{|x-y|^\mu}dy\Bigg) Q(|x|)f(u) ~~\textrm{in}~~ \Omega~~ \text{and}~~…

This work is devoted to the Dirichlet problem for the equation (-\Delta u = \lambda u + |x|^\alpha |u|^{2^*-2} u) in the unit ball of $\mathbb{R}^N$. We assume that $\lambda$ is bigger than the first eigenvalues of the laplacian, and we…

We consider the semilinear Dirichlet problem \[ \Delta u+kg(u)=\mu _1 \varphi _1+\cdots +\mu _n \varphi _n+e(x) \;\; \mbox{for $x \in \Omega$}, \;\; u=0 \;\; \mbox{on $\partial \Omega$}, \] where $\varphi _k$ is the $k$-th eigenfunction of…

We consider singularly perturbed nonlinear Schr\"odinger equations \be \label{eq:0.1} - \varepsilon^2 \Delta u + V(x)u = f(u), \ \ u > 0, \ \ v \in H^1(\R^N) \ee where $V \in C(\R^N, \R)$ and $f$ is a nonlinear term which satisfies the…

In this paper we analyze the asymptotic behaviour as $p\to 1^+$ of solutions $u_p$ to $$ \left\{ \begin{array}{rclr} -\Delta_p u_p&=&\frac{\lambda}{|x|^p}|u_p|^{p-2}u_p+f&\quad \mbox{ in } \Omega,\\ u_p&=&0 &\quad \mbox{ on }\partial\Omega,…

We consider nonlinear second order elliptic problems of the type \[ -\Delta u=f(u) \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega, \] where $\Omega$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some general…

Consider the Dirichlet problem with respect to an elliptic operator \[ A = - \sum_{k,l=1}^d \partial_k \, a_{kl} \, \partial_l - \sum_{k=1}^d \partial_k \, b_k + \sum_{k=1}^d c_k \, \partial_k + c_0 \] on a bounded Wiener regular open set…

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,…

The aim of this paper is to extend previous results regarding the multiplicity of solutions for quasilinear elliptic problems with critical growth to the variable exponent case. We prove, in the spirit of \cite{DPFBS}, the existence of at…

In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in \cite{HS1} to $\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace…

We consider semilinear elliptic problems of the form \[ -\Delta u + \lambda u = f(x,u), \quad u\in H^1_0(A), \] where $A\subset\mathbb{R}^N$, $N\geq3$, is either a bounded or unbounded annulus, and $\lambda \geq0$. We study a broad class of…

In this paper, we study the following singular nonlinear elliptic problem \begin{equation}\label{eq:1} \left\{ \begin{array}{ll} \displaystyle (-\Delta)^{\frac \alpha 2} u=\lambda |u|^{r-2}u+\mu\frac{|u|^{q-2}u}{|x|^{s}}\quad &{\rm in…