Related papers: Bubble concentration on spheres for supercritical …

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 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…

We consider the semilinear Lane-Emden problem in a smooth bounded domain of the plane. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions as the exponent p of the nonlinearity goes to infinity. Among other…

We study the existence of sign-changing solutions with multiple bubbles to the slightly subcritical problem $$-\Delta u=|u|^{2^*-2-\e}u \hbox{in}\Omega, \quad u=0 \hbox{on}\partial \Omega,$$ where $\Omega$ is a smooth bounded domain in…

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…

In this paper, we investigate the following critical elliptic equation $$ -\Delta u+V(y)u=u^{\frac{N+2}{N-2}},\,\,u>0,\,\,\text{in}\,\R^{N},\,\,u\in H^{1}(\R^{N}), $$ where $V(y)$ is a bounded non-negative function in $\R^{N}.$ Assuming…

We study the following elliptic problem involving slightly subcritical non-power nonlinearity $$\left\{\begin{array}{lll} -\Delta u =\frac{|u|^{2^*-2}u}{[\ln(e+|u|)]^\epsilon}\ \ &{\rm in}\ \Omega, \\[2mm] u= 0 \ \ & {\rm on}\…

We consider the semilinear Lane-Emden problem \begin{equation}\label{problemAbstract}\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.\tag{$\mathcal…

For a domain $\Omega\subset\dR^N$ we consider the equation $ -\Delta u + V(x)u = Q_n(x)\abs{u}^{p-2}u$ with zero Dirichlet boundary conditions and $p\in(2,2^*)$. Here $V\ge 0$ and $Q_n$ are bounded functions that are positive in a region…

We study the existence and the profile of sign-changing solutions to the slightly subcritical problem $$ -\De u=|u|^{2^*-2-\eps}u \hbox{in} \cB, \quad u=0 \hbox{on}\partial \cB, $$ where $\cB$ is the unit ball in $\rr^N$, $N\geq 3$,…

In this paper, we extend the analysis of the subcritical approximation of the Nirenberg problem on spheres recently conducted in \cite{MM19, MM}. Specifically, we delve into the scenario where the sequence of blowing up solutions exhibits a…

We consider the following critical fractional Schr\"{o}dinger equation \begin{equation*} (-\Delta)^s u+V(|y'|,y'')u = u^{2_s^*-1},\quad u>0,\quad y =(y',y'') \in \mathbb{R}^3\times\mathbb{R}^{N-3}, \end{equation*} where $N\geq 3,s\in(0,1)$,…

For a bounded set $\Omega \subset \mathbb R^N$ and a perturbation $V \in C^1(\overline{\Omega})$, we analyze the concentration behavior of a blow-up sequence of positive solutions to \[ -\Delta u_\epsilon + \epsilon V = N(N-2)…

We consider the non linear focusing wave equation $\partial_{tt}u-\Delta u-u|u|^{p-1}=0$ in large dimensions and for radially symmetric data, in the energy supercritical zone for p large enough. We construct finite time blow up solutions…

Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold without boundary and $\Gamma$ be a non degenerate closed geodesic of $(M,g)$. We prove that the supercritical problem $$-\Delta_gu+h u=u^{\frac{n+1}{n-3}\pm\epsilon},\ u>0,\…

Given an isoparametric function $f$ on the $n$-dimensional round sphere, we consider functions of the form $u=w\circ f$ to reduce the semilinear elliptic problem \[ -\Delta_{g_0}u+\lambda u=\lambda\ | u\ | ^{p-1}u\qquad\text{ on…

Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following elliptic Dirichlet problem $$ \begin{cases} -\Delta\upsilon= e^{\upsilon}-s\phi_1-4\pi\alpha\delta_p-h(x)\,\,\,\,…

In this paper, we deal with the boundary value problem $-\Delta u= |u|^{4/(n-2)}u/[\ln (e+|u|)]^\varepsilon$ in a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, $n\geq 3$ with homogenous Dirichlet boundary condition. Here…

In this survey we report on some recent results related to various singular phenomena arising in the study of some classes of nonlinear elliptic equations. We establish qualitative results on the existence, nonexistence or the uniqueness of…

For the prescribed scalar curvature equation on $S^n$ ($n \ge 6$), we consider the situation where the number of bubbles tends to infinity in the Lyapunov-Schmidt (finite dimension) reduction method. In an outstanding paper by Wei and Yan,…