Related papers: On the Lane-Emden conjecture
In this short note, we prove a sharp quantization for positive solutions of Lane-Emden problems in a bounded planar domain. This result has been conjectured by De Marchis, Ianni and Pacella [6, Remark 1.2].
We prove that suitably generic pairs of linear equations on an even number of variables are uncommon. This verifies a conjecture of Kam\v{c}ev, Morrison and the second author. Moreover, we prove that any large system containing such a…
We consider the Lane-Emden problem on planar domains. When the exponent is large, the existence and multiplicity of solutions strongly depend on the geometric properties of the domain, which also deeply affect their qualitative behavior.…
We study positive solutions to the fractional Lane-Emden system \begin{equation*} \tag{S}\label{S} \left\{ \begin{aligned} (-\Delta)^s u &= v^p+\mu \quad &&\text{in } \Omega \\ (-\Delta)^s v &= u^q+\nu \quad &&\text{in } \Omega\\ u = v &= 0…
We study positive solutions of the superlinear Lane-Emden inequality \(-\Delta u\ge \sigma u^q\), \(q>1\), on infinite locally finite weighted graphs and connected domains of such graphs. We first prove that solvability is equivalent to the…
We prove uniqueness of least-energy solutions to the fractional Lane-Emden equation, under homogeneous Dirichlet exterior conditions, when the underlying domain is a ball $B \subset \mathbb{R}^N$. The equation is characterized by a…
We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solutions which grow at most like the distance to the boundary to a power given by the natural scaling exponent of the equation; in other words,…
We deal with very weak positive supersolutions to the H\'enon-Lane-Emden system on neighborhoods of the origin. In our main theorem we prove a sharp nonexistence result.
In this paper we study the Lane-Emden-Fowler equation $$(P)_\epsilon\ \{\Delta u+|u|^{q-2}u=0 \ \hbox{in}\ \mathcal D_\epsilon, u=0 \ \hbox{on}\ \partial\mathcal D_\epsilon.$$ Here $\mathcal D_\epsilon = \mathcal D \setminus \{x \in…
We consider the Lane-Emden Dirichlet problem \begin{equation}\tag{1} \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. \end{equation} when $p>1$ and…
A nonlinear fourth-order parabolic equation in one space dimension with periodic boundary conditions is studied. This equation arises in the context of fluctuations of a stationary nonequilibrium interface and in the modeling of quantum…
In this paper, we establish a Liouville theorem for solutions to the Lane Emden equation involving Baouendi Grushin operators. We focus on solutions that are stable outside a compact set. Specifically, we prove that when p is smaller than…
The Lane-Emden equation has been used to model several phenomenas in theoretical physics, mathematical physics and astrophysics such as the theory of stellar structure. This study is an attempt to utilize the collocation method with the…
We establish existence of positive non-decreasing radial solutions for a nonlocal nonlinear Neumann problem both in the ball and in the annulus. The nonlinearity that we consider is rather general, allowing for supercritical growth (in the…
We present a counter-example to the recent claim that supermultiplets of N-extended supersymmetry with no central charge and in 1-dimension are specified unambiguously by providing the numbers of component fields in all available…
We consider ground state solutions of the critical Lane-Emden system \[\begin{cases} -\Delta u = v^p &\text{in } \mathbb{R}^n,\\ -\Delta v = u^q &\text{in } \mathbb{R}^n,\\ u,v >0\ &\text{in } \mathbb{R}^n, \end{cases}\] where $n \ge 3$ and…
We show that Connes' embedding conjecture on von Neumann algebras is equivalent to the existence of certain algebraic certificates for a polynomial in noncommuting variables to satisfy the following nonnegativity condition: The trace is…
We give partial affirmative answers to Landis conjecture in all dimensions for two different types of linear, second order, elliptic operators in a domain $\Omega\subset \mathbb{R}^N$. In particular, we provide a sharp decay criterion that…
We investigate a coupled system of elliptic equations of Lane--Emden--Fowler type on a bounded domain $\Omega \subset \mathbb{R}^n$ ($n \geq 1$) with homogeneous Dirichlet boundary conditions. The system is characterized by sublinear…
For a cocycle of invertible real $n$-by-$n$ matrices, the Multiplicative Ergodic Theorem gives an Oseledets subspace decomposition of $\mathbb{R}^n$; that is, above each point in the base space, $\mathbb{R}^n$ is written as a direct sum of…