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We study the large time behavior of non-negative solutions to the nonlinear diffusion equation with critical gradient absorption $$\partial\_t u - \Delta\_{p}u + |\nabla u|^{q\_*} = 0 \quad \hbox{in} (0,\infty)\times\mathbb{R}^N\ ,$$ for…

We consider the equation $- \e^2 \D u + u= u^p$ in $\Omega \subseteq \R^N$, where $\Omega$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of $\partial \O$, for $N \geq 3$ and…

Given $\mu > 0$, we study the elliptic problem: \begin{align*} \text{ find } (u,\lambda) \in H_0^1(\Omega) \times \mathbb{R} \text{ such that } -\Delta u + \lambda u = |u|^{p-2}u \text{ in } \Omega \text{ and } \int_\Omega|u|^2dx = \mu,…

We consider the following elliptic system with Neumann boundary: \begin{equation} \begin{cases} -\Delta u + \mu u=v^p, &\hbox{in } \Omega, \\-\Delta v + \mu v=u^q, &\hbox{in } \Omega, \\\frac{\partial u}{\partial n} = \frac{\partial…

We consider weak positive solutions to the critical $p$-Laplace equation with Hardy potential in $\mathbb R^N$ $$-\Delta_p u -\frac{\gamma}{|x|^p} u^{p-1}=u^{p^*-1}$$ where $1<p<N$, $0\le \gamma <\left(\frac{N-p}{p}\right)^p$ and…

We study the fractional laplacian problem (-\Delta)^s u &=& u^p -\epsilon u^q \quad\text{in }\quad \Omega, u &\in& H^s(\Omega)\cap L^{q+1}(\Omega),u &>&0 \quad\text{in }\quad \Omega, u&=&0 \quad\text{in}\quad \mathbb{R}^N\setminus\Omega,…

We review the indefinite sublinear elliptic equation $-\Delta u=a(x)u^{q}$ in a smooth bounded domain $\Omega\subset\mathbb{R}^{N}$, with Dirichlet or Neumann homogeneous boundary conditions. Here $0<q<1$ and $a$ is continuous and changes…

We study positive solutions of equation (E1) $-\Delta u + u^p|\nabla u|^q= 0$ ($0\leq p$, $0\leq q\leq 2$, $p+q>1$) and (E2) $-\Delta u + u^p + |\nabla u|^q =0$ ($p>1$, $1<q\leq 2$) in a smooth bounded domain $\Omega \subset \mathbb{R}^N$.…

In this article, we study the existence and multiplicity of solutions of the following $(p,q)$-Laplace equation with singular nonlinearity: \begin{equation*} \left\{\begin{array}{rllll} -\Delta_{p}u-\ba\Delta_{q}u & = \la u^{-\de}+ u^{r-1},…

Let $\Omega \subset \mathbb{R}^N$ be a bounded domain and $\delta(x)$ be the distance of a point $x\in \Omega$ to the boundary. We study the positive solutions of the problem $\Delta u +\frac{\mu}{\delta(x)^2}u=u^p$ in $\Omega$, where $p>0,…

In this paper, we are concerned with the well-known Brezis-Nirenberg problem \begin{equation*} \begin{cases} -\Delta u= u^{2^*-1}+\varepsilon u^{q-1},\quad u>0, &{\text{in}~\Omega},\\ \quad \ \ u=0, &{\text{on}~\partial \Omega}, \end{cases}…

We establish a lower bound for the number of sign changing solutions with precisely two nodal domains to the singularly perturbed nonlinear elliptic equation -{\epsilon}^{2}{\Delta}_{g}u+u=|u|^{p-2}u on an n-dimensional Riemannian manifold…

The aim of this paper is to prove the existence of weak solutions to the equation $\Delta u + u^p = 0$ which are positive in a domain $\Omega \subset {\Bbb R}^N$, vanish at the boundary, and have prescribed isolated singularities. The…

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 consider the problem \[-\Delta u + W(x)u = ((1/{|x|^{\alpha}} * |u|^{p}) |u|^{p-2}u, u \in H_{0}^{1}(\Omega)\], where $\Omega$ is an exterior domain in $\mathbb{R}^{N}$, $N\geq3,$ $\alpha \in(0,N)$, $p\in[2,(2N-\alpha)/(N-2)$, $W$ is…

We study the problem $-\Delta u=\lambda u-u^{-1}$ with a Neumann boundary condition; the peculiarity being the presence of the singular term $-u^{-1}$. We point out that the minus sign in front of the negative power of $u$ is particularly…

We establish the existence of multiple sign-changing solutions to the quasilinear critical problem $$-\Delta_{p} u=|u|^{p^*-2}u, \qquad u\in D^{1,p}(\mathbb{R}^{N}),$$ for $N\geq4$, where $\Delta_{p}u:=\mathrm{div}(|\nabla u|^{p-2}\nabla…

We discuss recent advances in the theory of quasilinear equations of the type $ -\Delta_{p} u = \sigma u^{q} \; \; \text{in} \;\; \mathbb{R}^n, $ in the case $0<q< p-1$, where $\sigma$ is a nonnegative measurable function, or measure, for…

We study the problem \begin{equation*} (I_{\epsilon}) \left\{\begin{aligned} -\Delta u- \frac{\mu u}{|x|^2}&=u^p -\epsilon u^q \quad\text{in }\quad \Omega, \\ u&>0 \quad\text{in }\quad \Omega, \\ u &\in H^1_0(\Omega)\cap L^{q+1}(\Omega),…

In this paper, we mainly consider nonnegative weak solutions $u\in D^{1,p}(\R^{N})$ to the doubly $D^{1,p}(\R^{N})$-critical nonlocal quasi-linear Schr\"{o}dinger-Hartree equation: \begin{align*} -\Delta_p u- \mu…