相关论文: The Brezis--Nirenberg problem for the H\'{e}non eq…

The present paper studies the non-local fractional analogue of the famous paper of Brezis and Nirenberg in [4]. Namely, we focus on the following model, $$\begin{align*}\left(\mathcal{P}\right) \begin{cases} \left(-\Delta\right)^s u-\lambda…

This article investigates the multiplicity of solutions to the Brezis-Nirenberg problem on smooth bounded domains in the hyperbolic space $\mathbb{B}^N$ for $N \ge 4$. Specifically, we study the critical semilinear equation…

We consider the nonlinear Schr\"{o}dinger equation $-\Delta u+(\lambda a(x)+1)u=|u|^{p-1}u$ on a locally finite graph $G=(V,E)$. We prove via the Nehari method that if $a(x)$ satisfies certain assumptions, for any $\lambda>1$, the equation…

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}…

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 +…

We study a singularly perturbed Dirichlet problem for the $p$-Laplacian with competing superlinear terms, \[ -\varepsilon \Delta_p u = a(x)|u|^{q-2}u - b(x)|u|^{\gamma-2}u, \qquad u|_{\partial\Omega}=0, \] where $1<p<q<\gamma<p^*$, $a\geq…

In this paper we study existence of ground state solution to the following problem $$ (- \Delta)^{\alpha}u = g(u) \ \ \mbox{in} \ \ \mathbb{R}^{N}, \ \ u \in H^{\alpha}(\mathbb R^N) $$ where $(-\Delta)^{\alpha}$ is the fractional Laplacian,…

We prove the existence of a ground state for some variational problems in Hilbert spaces, following the approach of Berestycki and Lions. Next, we examine the problem of constructing ground state solutions…

In this paper, we apply the method of the Nehari manifold to study the Kirchhoff type equation \begin{equation*} -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u) \end{equation*} submitted to Dirichlet boundary conditions. Under a…

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 work we prove the existence of ground state solutions for the following class of problems \begin{equation*} \left\{ \begin{array}{ll} \displaystyle - \Delta_1 u + (1 + \lambda V(x))\frac{u}{|u|} & = f(u), \quad x \in \mathbb{R}^N,…

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*}…

We prove the existence of ground state solution to the following problem. \begin{align*} (-\Delta)^{s}u+u&=\lambda|u|^{-\gamma-1}u+P(x)|u|^{p-1}u,~\text{in}~\mathbb{R}^N\setminus\Omega\\ N_su(x)&=0,~\text{in}~\Omega \end{align*} where…

In this article, we are study the following Dirichlet problem with Choquard type non linearity \[ -\Delta_{\mathbb{H}} u = a u+ \left(\int_{\Omega}\frac{|u(\eta)|^{Q^*_\lambda}}{|\eta^{-1}\xi|^{\lambda}}d\eta\right)|u|^{Q^*_\lambda-2}u \;…

We show that the classical Brezis-Nirenberg problem $$ -\Delta u=u|u| + \lambda u\ \hbox{in}\ \Omega, u=0\ \hbox{on}\ \partial\Omega, $$ when $\Omega$ is a bounded domain in $\mathbb R^6$ has a sign-changing solution which blows-up at a…

he equation $-\Delta u = \lambda e^u$ posed in the unit ball $B \subseteq \R^N$, with homogeneous Dirichlet condition $u|_{\partial B} = 0$, has the singular solution $U=\log\frac1{|x|^2}$ when $\lambda = 2(N-2)$. If $N\ge 4$ we show that…

Let $\lambda^*>0$ denote the largest possible value of $\lambda$ such that \begin{align*} \left\{\begin{aligned} \Delta^2 u & = \la e^u && \text{in $B $} u &= \pd{u}{n} = 0 && \text{on $ \pa B $} \end{aligned} \right. \end{align*} has a…

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…

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…

We show that the elliptic equation with a non-Lipschitz right-hand side, $-\Delta u = \lambda |u|^{\beta-1}u - |u|^{\alpha-1}u$ with $\lambda>0$ and $0<\alpha<\beta<1$, considered on a smooth star-shaped domain $\Omega$ subject to zero…