Related papers: Uniqueness of solutions for a nonlocal elliptic ei…

This paper deals with Gelfand-type problems \begin{equation}\label{Gelfand10} \qquad\qquad\left\{\begin{array}{ll} - \Delta_m u = \lambda f(u), \quad&\hbox{in} \ \Omega, \ \lambda >0, \\[10pt] u =0, \quad&\hbox{on} \ \partial_m\Omega,…

We deal with existence, uniqueness and regularity of nonnegative solutions to a Dirichlet problem for equations as \begin{equation*} \displaystyle -\operatorname{div}\left(\frac{|\nabla u|^{p-2}\nabla u}{(1+u)^{\theta(p-1)}}\right) = h(u)f…

Let $\Omega \Subset \mathbb R^n$, $f \in C^1(\mathbb R^{N\times n})$ and $g\in C^1(\mathbb R^N)$, where $N,n \in \mathbb N$. We study the minimisation problem of finding $u \in W^{1,\infty}_0(\Omega;\mathbb R^N)$ that satisfies \[ \big\|…

We study the behavior of weak solutions to the singular quasilinear elliptic problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$, in a bounded domain with the Dirichlet boundary condition, where $p>1$, $\gamma>0$,…

Let $p$ and $q$ be locally H\"{o}lder functions in $\RR^N$, $p>0$ and $q\geq 0$. We study the Emden-Fowler equation $-\Delta u+ q(x)|\nabla u|^a=p(x)u^{-\gamma}$ in $\RR^N$, where $a$ and $\gamma$ are positive numbers. Our main result…

Using the sub-supersolution method we study the existence of positive solutions for the anisotropic problem \begin{equation} -\sum_{i=1}^N\frac{\partial}{\partial x_i}\left( \left|\frac{\partial u}{\partial x_i}\right|^{p_i-2}\frac{\partial…

We study the bifurcation of solutions of semilinear elliptic boundary value problems of the form \begin{align*} \begin{aligned} -\Delta u &= f_\lambda(|x|,u,|\nabla u|) &&\text{in }\Omega, u &= 0 &&\text{on }\partial\Omega, \end{aligned}…

In this paper we study the geometry and the topology of unbounded domains in the Hyperbolic Space $\mathbb{H} ^n$ supporting a bounded positive solution to an overdetermined elliptic problem. Under suitable conditions on the elliptic…

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 the existence of a maximal solution of $-\Gd u+g(u)=f(x)$ in a domain $\Gw\subset \BBR^N$ with compact boundary, assuming that $f\in (L^1_{loc}(\Gw))_+$ and that $g$ is nondecreasing, $g(0)\geq 0$ and $g$ satisfies the…

In this paper, we consider the following overdetermined eigenvalue problem on an unbounded domain $\Omega\subset\mathbb{R}^{N+1}$ with $N\geq1$ \begin{equation} \left\{ \begin{array}{ll} -\Delta u=\lambda u\,\, &\text{in}\,\, \Omega,\\ u=0…

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in…

In this paper, we study the existence and the summability of solutions to a Robin boundary value problem whose prototype is the following: $$ \begin{cases} -\text{div}(b(|u|)\nabla u)=f &\text{in }\Omega,\\[.2cm] \displaystyle\frac{\partial…

We study sharp conditions for the existence and nonexistence of infinitely many nonnegative solutions to the problem $-\Delta_p u = \lambda f(u)$ in a bounded domain with Dirichlet boundary conditions, where $f$ is a continuous function…

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…

In this paper we study the Dirichlet eigenvalue problem $$ -\Delta_p u-\Delta_{J,p}u =\lambda|u|^{p-2}u \quad \text{ in } \Omega,\quad u=0 \quad\text{ in } \Omega^c=\mathbb{R}^N\setminus\Omega. $$ Here $\Delta_p u$ is the standard local…

Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following anisotropic elliptic Neumann problem with Hardy-H\'{e}non weight $$ \begin{cases} -\nabla(a(x)\nabla u)+a(x)u=a(x)|x-q|^{2\alpha}u^p,\,\,\,\,…

In this article, we study the following fractional Laplacian equation with critical growth and singular nonlinearity $$\quad (-\Delta)^s u = \lambda a(x) u^{-q} + u^{2^*_s-1}, \quad u>0 \; \text{in}\; \Omega,\quad u = 0 \; \mbox{in}\;…

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{cases} \displaystyle - \Delta_{1} u = h(u)f & \text{in}\, \Omega,\newline…

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…