Related papers: Supercritical elliptic problems on nonradial domai…

By a combination of variational and topological techniques in the presence of invariant cones, we detect a new type of positive axially symmetric solutions of the Dirichlet problem for the elliptic equation $$ -\Delta u + u = a(x)|u|^{p-2}u…

The paper concerns with positive solutions of problems of the type $-\Delta u+a(x)\, u=u^{p-1}+\varepsilon u^{2^*-1}$ in $\Omega\subseteq\mathbb{R}^N$, $N\ge 3$, $2^*={2N\over N-2}$, $2<p<2^*$. Here $\Omega$ can be an exterior domain, i.e.…

In this article we consider the existence of positive singular solutions on bounded domains and also classical solutions on exterior domains. First we consider positive singular solutions of the following problems: \begin{equation}…

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$ and let $m$ be a possibly discontinuous and unbounded function that changes sign in $\Omega$. Let $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ be a continuous…

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,…

We prove new results on the existence of positive radial solutions of the elliptic equation $-\Delta u= \lambda h(|x|,u)$ in an annular domain in $\mathbb{R}^{N}, N\geq 2$. Existence of positive radial solutions are determined under 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…

In this paper we consider nonlinear elliptic PDEs of the type $$-\Delta_p u+a(x)|u|^{p-2}u=|u|^{p^*-2}u \qquad \mbox{ in }\Omega,$$ where $1<p<N$ and $p^*=Np/(N-p)$ is the critical Sobolev exponent, and allowing the asymptotic behavior of…

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 shall prove a multiplicity result for semilinear elliptic problems with a super-critical nonlinearity of the form, \begin{equation}\label{con-c} \left \{ \begin{array}{ll} -\Delta u =|u|^{p-2} u+\mu |u|^{q-2}u, & x \in \Omega\\ u=0, & x…

We deal with the following semilinear equation in exterior domains \[-\Delta u + u = a(x)|u|^{p-2}u,\qquad u\in H^1_0({A_R}), \] where ${A_R} := \{x\in\mathbb{R}^N:\, |x|>{R}\}$, $N\ge 3$, $R>0$. Assuming that the weight $a$ is positive and…

In this paper, we study the positive solutions to the following singular and non local elliptic problem posed in a bounded and smooth domain $\Omega\subset \R^N$, $N> 2s$: % \begin{eqnarray*} (P_\lambda)\left\{\begin{array}{lll}…

Let us consider a semilinear boundary value problem $ - \Delta u= f(x,u),$ in $\Omega,$ with Dirichlet boundary conditions, where $ \Omega \subset \mathbb{R}^N $, $N> 2,$ is a bounded smooth domain. We provide sufficient conditions…

We consider the supercritical problem \[ -\Delta v=|v|^{p-2}v in \Theta_{\epsilon}, v=0 on \partial\Theta_{\epsilon}, \] where $\Theta$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq3$, $p>2^{\ast}:=2N/(N-2)$, and…

In this article, we study the existence of positive solutions to elliptic equation (E1) $$(-\Delta)^\alpha u=g(u)+\sigma\nu \quad{\rm in}\quad \Omega,$$ subject to the condition (E2) $$u=\varrho\mu\quad {\rm on}\quad \partial\Omega\ \ {\rm…

We consider positive solutions of the following elliptic Hamiltonian systems \begin{equation} \left\{ \begin{aligned} -\Delta u+u&=a(x)v^{p-1}~~~\text{in}~~A_R\\ -\Delta v+v&=b(x)u^{q-1}~~~\text{in}~~A_R~~~~~~~~~~~~~~~~~(0.1)\\ u,…

We deals with nonlinear elliptic Dirichlet problems of the form $${\rm div}(|D u|^{p-2}D u )+f(u)=0\quad\mbox{ in }\Omega,\qquad u\in H^{1,p}_0(\Omega) $$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n\ge 2$, $p> 1$ and $f$ has…

This paper is devoted to the study of the existence of positive solutions for a problem related to a higher order fractional differential equation involving a nonlinear term depending on a fractional differential operator,…

In this work we analyze the existence of solutions to the nonlinear elliptic system: \begin{equation*} \left\{ \begin{array}{rcll} -\Delta u & = & v^q+\a g & \text{in }\Omega , \\ -\Delta v& = &|\nabla u|^{p}+\l f &\text{in }\Omega , \\…

We are concerned with positive solutions of equation (E) $(-\Delta)^s u=f(u)$ in a domain $\Omega \subset \mathbb{R}^N$ ($N>2s$), where $s \in (\frac{1}{2},1)$ and $f\in C^{\alpha}_{loc}(\mathbb{R})$ for some $\alpha \in(0,1)$. We establish…