Existence and nonexistence of positive singular solutions for semilinear elliptic problems with applications in astrophysics
type: Item
instance of: scholarly article
title: Existence and nonexistence of positive singular solutions for semilinear elliptic problems with applications in astrophysics
zbMATH Open document ID: 0833.35042
author: Yi Li
author: Jairo Santanilla
published in: Differential and Integral Equations
publication date: 1996-03-18T00:00:00Z
review text: The first main theorem establishes the existence of infinitely many positive solutions \(u\in C^2(\mathbb{R}^n\backslash O)\) of the semilinear elliptic problem \[ -\Delta u= f(|x|) u^p\text{ in }\mathbb{R}^n\backslash O,\;u(x)\sim|x|^{2-n}\text{ as }|x|\to 0\text{ and as }|x|\to \infty,\tag{1} \] where \(f\) is a nonnegative locally Hölder continuous function in \((0, \infty)\) such that \(\int^\infty_0 r^a f(r)dr< \infty\), \(a= n-1- p(n- 2)\), \(n\geq 3\), \(p> 1\). Such a solution \(u\) automatically has ``finite total mass'', i.e., \(\int_{\mathbb{R}^n} |\Delta u|dx< \infty\), but infinite energy. The necessity of the integrability condition on \(f\) follows from an oscillation criterion of \textit{E. S. Noussair} and the reviewer [Proc. R. Soc. Edinburgh, Sect. A 75, 67-81 (1976; Zbl 0372.35004)]. Additional theorems yield existence or nonexistence of positive singular solutions to Dirichlet problems for Hénon type equations in a deleted neighborhood of the origin. A typical such equation has the singularity \(f(r)= 0(r^{- b})\) as \(r\to O+\) for \(b< 2\). Previous results of similar type are described by the first author and \textit{W.-M. Ni} [Arch. Ration. Mech. Anal. 108, 175-194 (1989; Zbl 0705.35039)]. The problems studied here correspond to the Matukuma and Hénon models in astrophysics for a globular stellar cluster with a black hole at the origin.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 85A15
Mathematics Subject Classification ID: 35Q75
zbMATH DE Number: 763472
zbMATH Keywords: finite total mass solution
zbMATH Keywords: Hénon models
zbMATH Keywords: oscillation criterion
reviewed by: Charles A. Swanson
MaRDI profile type: Publication
Recommended article: Existence of positive entire solutions of some semilinear elliptic equations
Recommended article: Structure of positive radial solutions of Matukuma's equation
Recommended article: Q3819412
Recommended article: Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign‐changing non‐linearity
Recommended article: Bubble solutions for Hénon type equation with nearly critical exponent in \(\mathbb{R}^N\)
Recommended article: Dirichlet problem for a semilinear elliptic equation with singular coefficient
Recommended article: A class of elliptic equations with singular and critical nonlinearities
Recommended article: Semilinear elliptic equations of Matukuma-type and related topics
Recommended article: Solutions for a class of singular semilinear elliptic equations
Recommended article: Multi-peak solutions for the Hénon equation with slightly subcritical growth
description: scientific article; zbMATH DE number 763472