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We consider the following supercritical problem for the Lane-Emden system: \begin{equation}\label{eq00} \begin{cases} -\Delta u_1=|u_2|^{p-1}u_2\ &in\ D,\\ -\Delta u_2=|u_1|^{q-1}u_1 \ &in\ D,\\ u_1=u_2=0\ &on\ \partial D, \end{cases}…

We consider the following slightly supercritical problem for the Lane-Emden system with Neumann boundary conditions: \begin{equation*} \begin{cases} -\Delta u_1=|u_2|^{p_\epsilon-1}u_2,\ &in\ \Omega,\\ -\Delta u_2=|u_1|^{q_\epsilon-1}u_1, \…

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…

We study the pure Neumann Lane-Emden problem in a bounded domain \[ -\Delta u = |u|^{p-1} u \text{ in }\Omega, \qquad \partial_\nu u=0 \text{ on }\partial \Omega, \] in the subcritical, critical, and supercritical regimes. We show existence…

We consider the supercritical Lane-Emden problem $$(P_\eps)\qquad -\Delta v= |v|^{p_\eps-1} v \ \hbox{in}\ \mathcal{A} ,\quad u=0\ \hbox{on}\ \partial\mathcal{A} $$ where $\mathcal A$ is an annulus in $\rr^{2m},$ $m\ge2$ and…

We establish the existence of finitely many sign-changing solutions to the Lane-Emden system $$-\Delta u=|v|^{q-2}v,\quad -\Delta v=|u|^{p-2}u \quad \text{ in }\mathbb{R}^N, \ \ N\geq 4,$$ where the exponents $p$ and $q$ lie on the critical…

In this paper, we address the solvability of the critical Lane-Emden system \[\begin{cases} -\Delta u=|v|^{p-1}v &\mbox{in } \Omega_\epsilon,\\ -\Delta v=|u|^{q-1}u &\mbox{in } \Omega_\epsilon,\\ u=v=0 &\mbox{on } \partial \Omega_\epsilon,…

We concern a family $\{(u_{\varepsilon},v_{\varepsilon})\}_{\varepsilon > 0}$ of solutions of the Lane-Emden system on a smooth bounded convex domain $\Omega$ in $\mathbb{R}^N$ \[\begin{cases} -\Delta u_{\varepsilon} = v_{\varepsilon}^p…

We consider the Neumann problem $$(P)\qquad - \Delta v + v= v^{q-1} \ \text{in }\ \mathcal{D}, \ v > 0 \ \text{in } \ \mathcal{D},\ \partial_\nu v = 0 \ \text{on } \partial\mathcal{D} ,$$ where $\mathcal{D} $ is an open bounded domain in…

We study the following Lane-Emden system \[ -\Delta u=|v|^{q-1}v \quad \text{ in } \Omega, \qquad -\Delta v=|u|^{p-1}u \quad \text{ in } \Omega, \qquad u_\nu=v_\nu=0 \quad \text{ on } \partial \Omega, \] with $\Omega$ a bounded regular…

We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair $\boldsymbol\varepsilon = (\varepsilon_1, \varepsilon_2 )$ of positive parameters, we consider a perforated domain…

We consider the Lane-Emden Dirichlet problem \begin{equation}\tag{1} \left\{\begin{array}{lr}-\Delta u= |u|^{p-1}u\qquad \mbox{ in }\Omega u=0\qquad\qquad\qquad\mbox{ on }\partial \Omega \end{array}\right. \end{equation} when $p>1$ and…

Our purpose of this paper is to study the nonexistence of nonnegative very weak solutions of \begin{equation}\label{eq 0.1} \displaystyle (-\Delta)^\alpha u = u^p+\nu\quad {\rm in}\quad \Omega,\qquad\ u=g\quad {\rm in}\quad \mathbb{…

We study the existence of sign changing solutions to the following problem $$ (P) \quad \quad \quad \left\{ \begin{array}{ll} \Delta u+|u|^{p-1}u=0 \quad & {\rm in} \quad \Omega_\epsilon; u=0 \quad & {\rm on} \quad\partial \Omega_\epsilon,…

We consider the supercritical problem {equation*} -\Delta u=|u| ^{p-2}u\text{\in}\Omega,\quad u=0\text{\on}\partial\Omega, {equation*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$ and $p$ smaller than the critical exponent…

We prove the existence, uniqueness, and sharp bilateral pointwise estimates for positive bounded solutions to the Lane--Emden type problem \[ \begin{cases} L u = \sum\limits_{i=1}^{m}\sigma_{i} u^{q_{i}}+\sigma_0, \quad u\geq0 & \text{in }…

We examine the H\'enon equation $ -\Delta u =|x|^\alpha u^p$ in $ \Omega \subset \mathbb{R}^N$ with $u=0$ on $ \partial \Omega$ where $ 0 < \alpha$. We show there exists a sequence $ \{p_k\}_k \subset [ \frac{N+2}{N-2}, p_{\alpha}(N)]$ with…

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 investigate a Dirichlet problem for the Laplace equation in a domain of $\mathbb{R}^2$ with two small close holes. The domain is obtained by making in a bounded open set two perforations at distance $|\epsilon_1|$ one from the other and…

In this paper we study the asymptotic behavior of minimal energy solutions to the Lane-Emden system $-\Delta u = v^p$ and $-\Delta v = u^q$ on bounded domains as the index $(p,q)$ approaches to the critical hyperbola from below. Precisely,…