Related papers: Multiple solutions to the Bahri-Coron problem in s…

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 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…

Let $\Omega$ be a bounded domain in $\R^n$, $n\ge 3$ with smooth boundary $\partial\Omega$ and a small hole. We give the first example of sign-changing {\it bubbling} solutions to the nonlinear elliptic problem $$ -\Delta u=|u|^{{n+2\over…

In this paper we construct nontrivial exterior domains $\Omega \subset \mathbb{R}^N$, for all $N\geq 2$, such that the problem $$\left\{ {ll} -\Delta u +u -u^p=0,\ u >0 & \mbox{in }\; \Omega, {1mm] \ u= 0 & \mbox{on }\; \partial \Omega,…

We investigate the overdetermined torsion problem $\begin{cases} -\Delta u = 1 & \text{in}\ \Omega\\ u=0 & \text{on}\ \partial \Omega\\ \frac{\partial u}{\partial \nu}=\text{const.} & \text{on}\ \partial \Omega, \end{cases}$ where $\Omega$…

We study the existence of nontrivial unbounded domains $\Omega$ in $\mathbb{R}^N$ such that the overdetermined problem $$ -\Delta u = 1 \quad \text{in $\Omega$}, \qquad u=0, \quad \partial_\nu u=\textrm{const} \qquad \text{on $\partial…

In this note we construct smooth bounded domains $\Omega \subset \mathbb R^2$, other than disks, for which the overdetermined problem $$ \left\{ \begin{alignedat}{2} \Delta u + \lambda u &= 0 &\qquad& \text{ in } \Omega, \newline u &= b…

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…

In this paper, we study the Brezis-Nirenberg problem on bounded smooth domains of R3. Using the algebraic topological argument of Bahri-Coron[2] as implemented in [6] combined with the Brendle[4]- Schoen[8]'s bubble construction, we solve…

We prove the existence of infinitely many solutions to a class of non-symmetric Dirichlet problems with exponential nonlinearities. Here the domain $\Omega \subset\subset \mathbb{R}^{2l}$ where $2l$ is the order of the equation. Considered…

We establish the existence of positive solutions to a general class of overdetermined semilinear elliptic boundary problems on suitable bounded open sets $\Omega\subset\mathbb{R}^n$. Specifically, for $n\leq 4$ and under mild technical…

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}…

In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of complex problems $$ (-i\nabla - A(\mu x))^{2}u= \mu |u|^{q-2}u + |u|^{2^{*}-2}u \ \mbox{in} \ \Omega, \ \ \ \ u \in…

Let $N \geq 3$ and $\Omega \subset \mathbb{R}^N$ be $C^2$ bounded domain. We study the existence of positive solution $u \in H^1(\Omega)$ of \begin{align*} \left\{ \begin{array}{l} -\Delta u + \lambda u = \frac{|u|^{2^*(s)-2}u}{|x-x_1|^s} +…

We construct nontrivial smooth bounded domains $\Omega \subseteq \mathbb{R}^n$ of the form $\Omega_0 \setminus \overline{\Omega}_1$, bifurcating from annuli, for which there exists a positive solution to the overdetermined boundary value…

We construct nontrivial unbounded domains $\Omega$ in the hyperbolic space $\mathbb{H}^N$, $N \in \{2,3,4\}$, bifurcating from the complement of a ball, such that the overdetermined elliptic problem \begin{equation} -\Delta_{\mathbb{H}^N}…

We study the critical Neumann problem \begin{equation*} \begin{cases} -\Delta u = |u|^{2^*-2}u &\text{in }\Sigma_\omega,\\ \quad\frac{\partial u}{\partial\nu}=0 &\text{on }\partial\Sigma_\omega, \end{cases} \end{equation*} in the unbounded…

In this paper, we investigate the existence of multiple solutions to the following multi-critical elliptic problem \begin{equation}\label{eq:0.1} \left\{\begin{aligned} -\Delta u & =\lambda |u|^{p-2}u…

We prove the existence of two smooth families of unbounded domains in $\mathbb{R}^{N+1}$ with $N\geq1$ such that \begin{equation} -\Delta u=\lambda u\,\, \text{in}\,\,\Omega, \,\, u=0,\,\,\partial_\nu…

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}…