The effect of the shape of the domain on the existence of solutions of an equation involving the critical Sobolev exponent
type: Item
instance of: scholarly article
title: The effect of the shape of the domain on the existence of solutions of an equation involving the critical Sobolev exponent
zbMATH Open document ID: 0838.35042
author: Pablo Padilla
published in: Journal of Differential Equations
publication date: 1996-05-30T00:00:00Z
review text: In a bounded regular domain \(\Omega\) of \(\mathbb{R}^n\) the semilinear elliptic problem \[ - \Delta u= u^p,\;u> 0\quad\text{in }\Omega,\;u= 0\quad\text{on }\partial\Omega\tag{\(*\)} \] is studied, where \(p= (n+ 2)/(n- 2)\) is the critical Sobolev exponent. In a previous paper, \textit{W. W. Ding} [J. Partial Differ. Equations 2, No. 4, 83-88 (1989; Zbl 0694.35067)] constructed a nontrivial solution of \((*)\) even in a contractible domain. The present paper gives a generalization of this result. The main theorem states that, if \(\Omega\) is a domain such that there exists a ball \(B_R(x)\), \(x\in \Omega\), and a subset \(\Sigma\) of \(\mathbb{R}^n\) with \(\partial\Sigma\) disjoint from \(\partial B_R(x)\), satisfying \(\Omega\cap B_R(x)= B_R(x)\backslash \Sigma\), then there exists a solution of \((*)\), provided \(\text{meas}(\Sigma)\) is small and \(\text{dist}(\partial\Sigma, \partial B_R(x))\) is large enough. This theorem applies for example to domains, where \(\Omega\) is a ball in \(\mathbb{R}^n\) with a small hole. In the last section, also a multiplicity result is proved.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 842915
zbMATH Keywords: existence of solutions in topologically trivial domains
zbMATH Keywords: critical Sobolev exponent
zbMATH Keywords: multiplicity
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0019
OpenAlex ID: W2037474632
DOI: 10.1006/JDEQ.1996.0019
Recommended article: Q4033115
Recommended article: An answer to the problem posed by H. Brezis
Recommended article: Q3978165
Recommended article: Q4863664
Recommended article: On the uniqueness of positive solution of an elliptic equation
Recommended article: On a class of semilinear elliptic problems in ℝN with critical growth
Recommended article: On some semilinear elliptic problems with singular potentials involving symmetry
Recommended article: Q4407817
Recommended article: Positive solutions for semilinear elliptic equations with critical weighted Hardy-Sobolev exponents
Recommended article: Multiplicity of solutions for singular semilinear elliptic equations with critical Hardy-Sobolev exponents
description: scientific article; zbMATH DE number 842915