Solutions of some nonlinear elliptic problems, which are singular on a submanifold
type: Item
instance of: scholarly article
title: Solutions of some nonlinear elliptic problems, which are singular on a submanifold
zbMATH Open document ID: 0897.35027
DOI: 10.1016/S0021-7824(97)89968-0
author: Michèle Grillot
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1998-06-23T00:00:00Z
review text: Let \((M,g)\) be a \(C^2\) complete connected Riemannian manifold of dimension \(N\geq 3\) and \(\Sigma\) a compact submanifold of \(M\) of dimension \(d\geq 0\) without boundary. First we prove a removability result and secondly, if \(M\) is compact, we construct solutions of \[ -\Delta_g u(\cdot) +h(.,u(\cdot)) =0\text{ in }M \setminus \Sigma \] which blow-up on \(\Sigma\). More precisely: if \(h(x,r)= | r|^{q-1}r\), \(q>1\) and \((N<d+2\) and \(1<q<(N-d)/(N-d-2))\) or \((N=d+2\) and \(q>1)\), then for all \(\gamma\in [0,\infty]\), there exists a unique function \(u_\gamma\in C^1(M \setminus \Sigma)\), solution of the above equation and satisfying \(\lim_{\text{dist} (x,\Sigma)\to 0} u_\gamma(x)/ \mu(x)= \gamma\) where \(\mu\) is the fundamental solution of the Laplacian in \(\mathbb{R}^{N-d}\). Moreover if \(\gamma= \infty\), then \[ \lim_{\text{dist} (x,\Sigma) \to 0} \text{dist} (x,\Sigma)^{2/(q-1)} u_\infty(x) =\biggl[ \bigl(2/(q-1)\bigr) \bigl( 2q/(q-1)-N +d\bigr) \biggr]^{1/(q-1)}. \] We have a similar result when \(h(x,r) =e^r\) and \(N=d+2\). Those solutions play a role in Riemannian geometry in the study of conformal deformation of hyperbolic space: their existence implies the existence of complete Riemannian metrics conformally related to the classical metric of the hyperbolic space and with scalar curvature equal to \(-1\).
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 1146376
zbMATH Keywords: fundamental solution
zbMATH Keywords: conformal deformation of hyperbolic space
MaRDI profile type: Publication
cites work: `Large' solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behaviour
cites work: Singularités éliminables pour des équations semi-linéaires
cites work: Le spectre d'une variété riemannienne. (The spectrum of a Riemannian manifold)
cites work: Removable singularities for some nonlinear elliptic equations
cites work: Anisotropic singularities of solutions of nonlinear elliptic equations in \({\mathbb{R}}^ 2\)
cites work: A probabilistic approach to one class of nonlinear differential equations
cites work: FURTHER STUDIES OF EMDEN'S AND SIMILAR DIFFERENTIAL EQUATIONS
cites work: Q3993500
cites work: Q4864646
cites work: On solutions of δu=f(u)
cites work: Q4324333
cites work: Q4052566
cites work: Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations
cites work: On the inequality \(\Delta u \geqq f (u)\)
cites work: Scalar curvature and conformal deformation of hyperbolic space
cites work: Singularities of elliptic equations with an exponential nonlinearity
cites work: Singularités éliminables d'équations elliptiques non linéaires
cites work: Singular solutions of some nonlinear elliptic equations
cites work: Q4726693
cites work: Semilinear elliptic equations with uniform blow-up on the boundary
full work available at URL: s0021-7824(97)89968-0
OpenAlex ID: W2009175826
Wikidata QID: Q127152181
description: scientific article; zbMATH DE number 1146376