Existence of solutions of semilinear elliptic problems on unbounded domains
type: Item
instance of: scholarly article
title: Existence of solutions of semilinear elliptic problems on unbounded domains
zbMATH Open document ID: 0837.35051
author: Wen-Ching Lien
author: Hwai-Chiuan Wang
author: Shyuh-Yaur Tzeng
published in: Differential and Integral Equations
publication date: 1993-12-08T00:00:00Z
review text: This paper is devoted to the study of the existence of positive solutions \(u\in H^1_0(\Omega)\) of the semilinear elliptic equation \[ - \Delta u+ \lambda u= |u|^{p- 2} u\quad\text{in }\Omega,\tag{\(*\)} \] where \(\Omega\) is an unbounded domain in \(\mathbb{R}^N\), \(\lambda\in \mathbb{R}\), \(N\geq 2\) and \(p\in (2, 2^*)\), where \(2^*= 2N/(N- 2)\) for \(N\geq 3\) and \(2^*= \infty\) for \(N= 2\). Let \[ f(u)= {1\over 2} \int_\Omega (|\nabla u|^2+ \lambda u^2)dx,\;\alpha(\Omega):= \inf\Biggl\{f(u): u\in H^1_0(\Omega), \int_\Omega |u|^p dx= 1\Biggr\}. \] Section 2 provides various tools for the study of this problem, like the following result: if \(\Omega_1\subset \Omega_2\), then \(\alpha(\Omega_2)\leq \alpha(\Omega_1)\), and if \(\Omega_1\subsetneqq a_1\) and \(\alpha(\Omega_1)\) admits a minimizer, i.e. there exists a solution \(u\in H^1_0(\Omega)\) of \((*)\) (with \(\Omega= \Omega_1\)) such that \(f(u)= \alpha(\Omega_1)\) and \(\int_\Omega |u|^p dx= 1\), then \(\alpha(\Omega_2)< \alpha(\Omega_1)\). The authors prove that \(\alpha(\Omega)= \alpha\) if \(\Omega\subsetneqq \mathbb{R}^N\) is a ball up domain (Sections 2,3) i.e. if for any \(r> 0\) there exists \(x\in \Omega\) such that \(B(x; r)\subset \Omega\). Using an adaptation of a compactness result by P. L. Lions, they also prove that if \(\Omega\) is a periodic domain (Section 4) which means that there exists a partition \(\{Q_m\}\) of \(\Omega\) and points \(\{y_m\}\) in \(\mathbb{R}^N\) such that \(\{y_m\}\) forms a subgroup of \(\mathbb{R}^N\), \(Q_0\) is bounded and \(Q_m= Q_0 y_m\), then \(\alpha(\Omega)\) admits a minimizer and therefore \(\alpha(\Omega)> \alpha(\mathbb{R}^N)\). This situation covers several cases already known in the literature and extends them. Section 5 is devoted to finite perturbed periodic domains, a case which offers various interesting examples and is especially interesting in view of the non-existence results by Esteban-Lions. The main tool of this section is the followin…
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 459209
zbMATH Keywords: existence of positive solutions
zbMATH Keywords: periodic domain
reviewed by: Jean Dolbeault
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 459209