Related papers: A supercritical elliptic problem in a cylindrical …

In this paper, we consider the logarithmic elliptic equations with critical exponent \begin{equation} \begin{cases} -\Delta u=\lambda u+ |u|^{2^*-2}u+\theta u\log u^2, \\ u \in H_0^1(\Omega), \quad \Omega \subset \R^N. \end{cases}…

We study the elliptic system \begin{equation*} \begin{cases} -\Delta u_1 - \kappa_1u_1 = \mu_1|u_1|^{p-2}u_1 + \lambda\alpha|u_1|^{\alpha-2}|u_2|^\beta u_1, \\ -\Delta u_2 - \kappa_2u_2 = \mu_2|u_2|^{p-2}u_2 +…

Let $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\delta$ be the distance to $\partial \Omega$. We study positive solutions of equation (E) $-L_\mu u+ g(|\nabla u|) = 0$ in $\Omega$ where $L_\mu=\Delta +…

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.…

We study the existence and multiplicity of sign changing solutions of the following equation $ \begin{cases} -\Delta u = \mu |u|^{2^{\star}-2}u+\frac{|u|^{2^{*}(t)-2}u}{|x|^t}+a(x)u \quad\text{in}\quad \Omega, u=0…

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…

Semilinear elliptic equations of the form $-\Delta u =\lambda|u|^{p-2}u- |u|^{q-2}u$ in bounded and unbounded domains are considered. In the plane of exponents $p\times q$, the so-called critical exponents curve is introduced which…

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,…

Let $1<p<N$, $p^{*}=Np/(N-p)$, $0<s<p$, $p^{*}(s)=(N-s)p/(N-p)$, and $\Om\in C^{1}$ be a bounded domain in $\R^{N}$ with $0\in\bar{\Om}.$ In this paper, we study the following problem \[ \begin{cases}…

For $ p \in (1,N)$ and a domain $\Omega$ in $\mathbb{R}^N$, we study the following quasi-linear problem involving the critical growth: \begin{eqnarray*} -\Delta_p u - \mu g|u|^{p-2}u = |u|^{p^{*}-2}u \ \mbox{ in } \mathcal{D}_p(\Omega),…

The aim of this paper is to study the critical elliptic equations with Stein-Weiss type convolution parts $$ \displaystyle-\Delta u =\frac{1}{|x|^{\alpha}}\left(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{2_{\alpha,…

This work deals with the existence of at least two positive solutions for the class of quasilinear elliptic equations with cylindrical singularities and multiple critical nonlinearities that can be written in the form \begin{align*}…

In this paper, we prove that there exists at most one positive radial weak solution to the following quasilinear elliptic equation with singular critical growth \[ \begin{cases} -\Delta_{p}u-{\displaystyle…

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…

Let $N \ge 4$, $\Omega$ be a bounded domain in $\mathbb{R}^N$, and let $\Sigma \subset \Omega$ be a smooth closed submanifold of dimension $k$ with $2 \le k \le N-2$. We study the existence of positive solutions $u \in H_0^1(\Omega)$ to the…

We study the following class of quasilinear degenerate elliptic equations with critical nonlinearity \begin{align*} \begin{cases}-\Delta_{\gamma,p} u= \lambda |u|^{q-2}u+|u|^{p_{\gamma}^{*}-2}u & \text{ in } \Omega\subset \mathbb{R}^N, \\…

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,…

Let $\Omega\subset \mathbb{R}^N$ be a bounded regular domain, $0<s<1$ and $N>2s$. We consider $$ (P)\left\{ \begin{array}{rcll} (-\Delta)^s u &= & \frac{u^{q}}{d^{2s}} & \text{ in }\Omega , \\ u &> & 0 & \text{in }\Omega , \\ u & = & 0 &…

We consider the following nonlinear elliptic system of Hamiltonian type with critical exponents: \begin{equation*} \begin{cases} -\Delta u + V(|y'|,y'')\, u = |v|^{p-1}v, & \text{in } \mathbb{R}^N,\newline -\Delta v + V(|y'|,y'')\, v =…

In the present paper we investigate the following semilinear singular elliptic problem: \begin{equation*} (\rm P)\qquad \left \{\begin{array}{l} -\Delta u = \dfrac{p(x)}{u^{\alpha}}\quad \text{in} \Omega \\ u = 0\ \text{on} \Omega,\ u>0…