A nonsymmetric asymptotically linear elliptic problem
type: Item
instance of: scholarly article
title: A nonsymmetric asymptotically linear elliptic problem
zbMATH Open document ID: 0844.35035
DOI: 10.12775/TMNA.1994.031
author: A. Marino
author: Anna Maria Micheletti
author: Angela Pistoia
published in: Topological Methods in Nonlinear Analysis
publication date: 1996-08-29T00:00:00Z
review text: Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\). The paper is concerned with the semilinear elliptic problem \[ \Delta u+ g(x, u)= te_1 \quad \text{in }\Omega, \qquad u= 0 \quad \text{on }\partial\Omega,\tag \(*\) \] where \(g(x, u)= \alpha u^++ \beta u^-+ g_0(x, u)\), \(g_0(x, u)/u\to 0\) as \(|u|\to \infty\), \(e_1\) is the positive eigenvalue of the Laplacian and \(\alpha, \beta, t\in \mathbb{R}\). To \((*)\) there corresponds a functional \[ f_t(u)= \int_\Omega (\textstyle{{1\over 2}} |\nabla u|^2- G(x, u)+ te_1 u)dx \] in \(H^1_0(\Omega)\) and critical points of \(f_t\) are solutions of \((*)\). It is shown that for \((\alpha, \beta)\) in certain regions of \(\mathbb{R}^2\), if \(t\) is large enough, then \((*)\) has at least one, two, three, respectively four solutions. Existence of one solution is shown by using a variant of the saddle point theorem of Rabinowitz. Two and three solutions are obtained by linking-type arguments where careful estimates are needed in order to show that certain linking levels are different. An additional argument gives a fourth critical point. It should also be noted that a rather general sufficient condition for \(f_t\) to satisfy the Palais-Smale condition is given in this paper.
reviewed by: Andrzej Szulkin
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 828286
zbMATH Keywords: jumping nonlinearity
zbMATH Keywords: semilinear elliptic equation
zbMATH Keywords: linking
zbMATH Keywords: multiple solutions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 828286