Related papers: Nodal solutions of the Brezis-Nirenberg problem in…

We consider the Brezis-Nirenberg problem: $$-\Delta u =\lambda u + |u|^{p-1}u\qquad \mbox{in}\,\, \Omega,\quad u=0\,\, \mbox{on}\,\,\ \partial\Omega,$$ where $\Omega$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 3$,…

We show that the classical Brezis-Nirenberg problem $$\Delta u + |u|^{4 \over N-2} u + \varepsilon u = 0 ,\quad {\mbox {in}} \quad \Omega, \quad u= 0 , \quad {\mbox {on}} \quad \partial \Omega$$ admits nodal solutions clustering around a…

In this paper, we consider the Brezis-Nirenberg problem \begin{equation*} \left\{\begin{aligned} &-\Delta u = \lambda u+|u|^{2^*-2}u, \quad &\mbox{in}\,\Omega,\\ &u=0,\quad &\mbox{on}\, \partial\Omega, \end{aligned}\right. \end{equation*}…

In this paper, we prove that the Brezis-Nirenberg problem -\Delta u = |u|^{p-1}u+\epsilon u in \Omega; u=0 on \partial \Omega where \Omega is a symmetric bounded smooth domain in R^N, N\geq 7 and p = (N+2)/(N-2), has a solution with the…

The problem \begin{equation} \label{bn} -\Delta u=|u|^{4\over n-2}u+\lambda V u\ \hbox{in}\ \Omega,\ u=0\ \hbox{on}\ \partial\Omega \end{equation} where $\Omega$ is a bounded regular domain in $\mathbb R^n$, $\lambda\in \mathbb R$ and $V\in…

We consider the Brezis-Nirenberg problem: \begin{equation*} \begin{cases} -\Delta u = \lambda u + |u|^{2^* -2}u & \hbox{in}\ \Omega\\ u=0 & \hbox{on}\ \partial \Omega, \end{cases} \end{equation*} where $\Omega$ is a smooth bounded domain in…

In this paper, we are interested in the existence of solutions for the following Choquard type Brezis-Nirenberg problem \begin{align*} \left\{ \begin{array}{ll} -\Delta…

In this paper, we consider the Brezis-Nirenberg problem $$ -\Delta u=\lambda u+|u|^{\frac{4}{N-2}}u,\quad\mbox{in}\,\, \Omega,\quad u=0,\quad\mbox{on}\,\, \partial\Omega, $$ where $\lambda\in\mathbb{R}$, $\Omega\subset\mathbb R^N$ is a…

In this work we study the existence of nodal solutions for the problem $$ -\Delta u = \lambda u e^{u^2+|u|^p} \text{ in }\Omega, \; u = 0 \text{ on }\partial \Omega, $$ where $\Omega\subseteq \mathbb R^2$ is a bounded smooth domain and…

In this paper we consider nodal radial solutions of the problem $$ \begin{cases} -\Delta u=|u|^{2^*-2}u+\lambda u&\text{ in }B,\\ u=0&\text{ on }\partial B \end{cases} $$ where $2^*=\frac{2N}{N-2}$ with $3\le N\le6$ and $B$ is the unit ball…

We consider the problem $\Delta u + \lambda u +u^5 = 0$, $u>0$, in a smooth bounded domain $\Omega$ in ${\mathbb R}^3$, under zero Dirichlet boundary conditions. We obtain solutions to this problem exhibiting multiple bubbling behavior at…

We study the following Brezis-Nirenberg problem of Kirchhoff type $$ \left\{\aligned &-(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u = \lambda|u|^{q-2}u + \delta |u|^{2}u, &\quad \text{in}\ \Omega, \\ &u=0,& \text{on}\ \partial\Omega,…

We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation $$-\Delta u =\left(\int_{\Omega}\frac{|u|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dy\right)|u|^{2_{\mu}^{\ast}-2}u+\lambda…

In this work we study the existence of solutions to the critical Brezis-Nirenberg problem when one deals with the spectral fractional Laplace operator and mixed Dirichlet-Neumann boundary conditions, i.e., $$ \left\{\begin{array}{rcl}…

In this paper we are concerned with the well-known Brezis-Nirenberg problem \begin{equation*} \begin{cases} -\Delta u= u^{\frac{N+2}{N-2}}+\varepsilon u, &{\text{in}~\Omega},\\ u>0, &{\text{in}~\Omega},\\ u=0, &{\text{on}~\partial \Omega}.…

In this paper, we study the higher order Brezis-Nirenberg problem under the Navier boundary condition \be\label{eq} \begin{cases} (-\Delta)^m u=\varepsilon u+u^{p} & \text { in }\, \Omega, \\ u>0 & \text { in }\, \Omega, \\ u=-\Delta…

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^N$ with $N\geq 3$, $1<\alpha$, $2^{\ast}=\frac{2N}{N-2}$ and $\{u_\alpha\}\subset H_{0}^{1,2\alpha}(\Omega)$ be a critical point of the functional \begin{equation*}…

We consider the positive solutions of the nonlinear eigenvalue problem $-\Delta_{\mathbb{H}^n} u = \lambda u + u^p, $ with $p=\frac{n+2}{n-2}$ and $u \in H_0^1(\Omega),$ where $\Omega$ is a geodesic ball of radius $\theta_1$ on…

In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation with Neumann boundary condition \begin{equation*} \begin{aligned} -\Delta u &= \lambda \alpha(x)u +…

This work is devoted to the Dirichlet problem for the equation (-\Delta u = \lambda u + |x|^\alpha |u|^{2^*-2} u) in the unit ball of $\mathbb{R}^N$. We assume that $\lambda$ is bigger than the first eigenvalues of the laplacian, and we…