Related papers: Coron's problem for the critical Lane-Emden system

Let $1<p<+\infty$ and let $\Omega\subset\mathbb R^N$ be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the…

We prove that, for any real $\lambda$, the system $-\Delta u +\lambda u = u^3-\beta uv^2$, $ -\Delta v+\lambda v =v^3-\beta vu^2$, $ u,v\in H^1_0(\Omega),$ where $\Omega$ is a bounded smooth domain of $R^3$, admits a bounded family of…

We use variational methods to study the existence of nontrivial and radially symmetric solutions to the H\`enon-Lane-Emden system with weights, when the exponents involved lie on the "critical hyperbola". We also discuss qualitative…

We study the following problem \[ \begin{cases} -\Delta u = \lambda u + u^{2^*-2} v & \hbox{in} \Omega,\\ -\Delta v= \mu v^{2^*-1} + u^{2^*-1} & \hbox{in} \Omega,\\ u> 0,v> 0 & \hbox{in} \Omega,\\ u=v=0 & \hbox{on} \partial \Omega,…

We are concerned with the Lane-Emden problem \begin{equation*} \begin{cases} -\Delta u=u^{p} &{\text{in}~\Omega},\\[0.5mm] u>0 &{\text{in}~\Omega},\\[0.5mm] u=0 &{\text{on}~\partial \Omega}, \end{cases} \end{equation*} where $\Omega\subset…

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…

We study the problem of the existence and nonexistence of positive solutions to a superlinear second-order divergence type elliptic equation with measurable coefficients $(*)$: $-\nabla\cdot a\cdot\nabla u=u^p$ in an unbounded cone--like…

Let $\Omega$ be a bounded domain in $\mathbb R^{N}$, $N\geq3$ with smooth boundary, $a>0, \lambda>0$ and $0<\delta<3$ be real numbers. Define $2^*:=\displaystyle\frac{2N}{N-2}$ and the characteristic function of a set $A$ by $\chi_A$. We…

In this paper, we are devoted to studying the positive weak, punctured or distributional solutions to the biharmonic Lane-Emden equation \begin{equation*} \Delta^{2} u=u^{p} \quad \quad \text{in} \ \mathbb{R}^{N}\setminus Z, \end{equation*}…

In this paper we are proving the existence of a nontrivial solution of the ${p}(x)$- Laplacian equation with Dirichlet boundary condition. We will use the variational method and concentration compactness principle involving positive radon…

We consider the Hamiltonian system with Neumann boundary conditions: \[ -\Delta u + \mu u=v^{q }, \quad -\Delta v+ \mu v=u^{p} \quad \text{ in $\Omega$}, \qquad u, v >0 \quad \text{ in $\Omega$,} \qquad \partial_\nu u= \partial_\nu v=0…

We consider the supercritical elliptic problem -\Delta u = \lambda e^u, \lambda > 0, in an exterior domain $\Omega = \mathbb{R}^N \setminus D$ under zero Dirichlet condition, where D is smooth and bounded in \mathbb{R}^N, N greater or equal…

We compute the Morse index of $1$-spike solutions of the semilinear elliptic problem \begin{equation}\label{abstr} \tag{$\mathcal P_p$} \begin{cases} -\Delta u= u^p & \text{in $\Omega$} \\ u=0 & \text{on $\partial\Omega$} \\ u>0 & \text{in…

Let $ \ti \Om $ be a bounded convex domain in Euclidean $ n $ space, $ \hat x \in \ar \ti \Om, $ and $ r > 0. $ Let $ \ti u = (\ti u^1, \ti u^2, \dots, \ti u^N) $ be a weak solution to \[\nabla \cdot \left (|\nabla \ti u |^{p-2} \nabla \ti…

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$ and $\delta(x)=\text{dist}\,(x,\partial \Omega)$. Assume $\mu>0$, $\nu$ is a nonnegative finite measure on $\partial \Omega$ and $g \in C(\Omega \times \mathbb{R}_+)$. We study…

This paper deals with existence and multiplicity of positive solutions for a quasilinear problem with Neumann boundary conditions, set in a ball. The problem admits at least one constant non-zero solution and it involves a nonlinearity that…

In this paper, we provide an affirmative answer to the {\it conjecture A} for bounded simple rotationally symmetric domains $\Omega\subset \mathbb{R}^n(n\geq 3)$ along $x_n$ axis. Precisely, we use a new simple argument to study the…

Our main result shows that the mass $2\pi$ is critical for the minimal Keller-Segel system \begin{align}\label{prob:abstract}\tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v), \\ v_t = \Delta v - v + u, \end{cases}…

This paper investigates the existence of positive solutions of a singular boundary value problem with negative exponent similar to standard Emden--Fowler equation. A necessary and sufficient condition for the existence of $C[0,1]$ positive…

In this paper, we are concerned with the critical order Lane-Emden-Hardy equations \begin{equation*} (-\Delta)^{\frac{n}{2}}u(x)=\frac{u^{p}(x)}{|x|^{a}} \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n} \end{equation*} with $n\geq4$…