Combined effects of concave and convex nonlinearities in some elliptic problems
type: Item
instance of: scholarly article
title: Combined effects of concave and convex nonlinearities in some elliptic problems
zbMATH Open document ID: 0805.35028
author: Giovanna Cerami
author: A. Ambrosetti
author: Haïm Brezis
published in: Journal of Functional Analysis
publication date: 1994-10-30T00:00:00Z
review text: Let \(\Omega\) be a bounded domain in \(\mathbb{R}^ N\) and consider the semilinear elliptic problem \[ -\Delta u = f_ \lambda (x,u) \quad \text{ in } \Omega, \quad u = 0 \quad \text{ on } \partial \Omega \tag{1} \] where \(f_ \lambda : \Omega \times \mathbb{R} \to \mathbb{R}\) and \(\lambda\) is a real parameter. When \(f_ \lambda\) is sublinear, for example, \(f_ \lambda = \lambda u^ q\), \(0<q<1\), it is known that (1) has a unique positive solution for all \(\lambda>0\). On the other hand, if \(f_ \lambda = \lambda | u |^{q-1}u\), the problem (1) admits infinitely many solutions. The purpose of the present note is to study (1) when \(f_ \lambda\) is the sum of a sublinear and superlinear term. It is shown that the combined effects of these two nonlinearities change considerably the structure of the solution set.
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 599811
zbMATH Keywords: concave and convex nonlinearities
zbMATH Keywords: semilinear elliptic problem
reviewed by: Vesa Mustonen
MaRDI profile type: Publication
full work available at URL: jfan.1994.1078
OpenAlex ID: W1988167079
DOI: 10.1006/JFAN.1994.1078
description: scientific article; zbMATH DE number 599811