Exact multiplicity of solutions to superlinear and sublinear problems
type: Item
instance of: scholarly article
title: Exact multiplicity of solutions to superlinear and sublinear problems
zbMATH Open document ID: 1004.35043
DOI: 10.1016/S0362-546X(01)00775-1
author: Junping Shi
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-08-25T00:00:00Z
review text: The author considers the equation \(\Delta u+\lambda f(u)=0\) in a bounded smooth domain \(\Omega\subset\mathbb R^n\) with Dirichlet boundary conditions. It is required that all Dirichlet eigenvalues \(\lambda_1<\lambda_2<\lambda_3<\dots\) of \(-\Delta\) on \(\Omega\) are simple. The nonlinearity \(f\in{\mathcal{C}}^1(\mathbb R)\) satisfies \(f(0)=0\), \(f'>0\), and \(f(u)/|u|\) is either decreasing or increasing for \(u\in\mathbb R\setminus\{0\}\). Moreover, the limits \(f(u)/u\rightarrow f\pm>0\) as \(u\rightarrow \pm\infty\) are assumed to exist. The main results of the paper state the existence of precisely \(0,1,2\) or \(4\) solutions for \(\lambda\) lying in certain parameter ranges depending on the quotients \(\lambda_k/f'(0)\), \(\lambda_k/f_+\), \(\lambda_k/f_-\). There are also informations on the Morse indices and the signs of the solutions. The proofs are based on bifurcation theorems and on degree arguments. The conditions on \(f\) imply that solutions are nondegenerate and that their Morse indices are determined by \(\lambda\).
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B32
zbMATH DE Number: 1788034
zbMATH Keywords: semlinear elliptic equations
zbMATH Keywords: exact multiplicity
zbMATH Keywords: bifurcation
zbMATH Keywords: asymptotically linear nonlinearities
reviewed by: Thomas J. Bartsch
MaRDI profile type: Publication
cites work: Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces
cites work: Q3899074
cites work: Sharp nonuniqueness results for some nonlinear problems
cites work: Solutions of superlinear elliptic equations and their morse indices
cites work: On the Morse indices of sign changing solutions of nonlinear elliptic problems
cites work: On the existence of sign changing solutions for semilinear Dirichlet problems
cites work: On nontrivial solutions of an asymptotically linear Dirichlet problem
cites work: Sharp multiplicity results for a semilinear dirichlet problem
cites work: Multiple Solutions for a Nonlinear Dirichlet Problem
cites work: Q4379844
cites work: Critical point theory and the number of solutions of a nonlinear Dirichlet problem
cites work: Infinite dimensional Morse theory and multiple solution problems
cites work: Bifurcation from simple eigenvalues
cites work: Bifurcation, perturbation of simple eigenvalues, and linearized stability
cites work: The effect of domain shape on the number of positive solutions of certain nonlinear equations
cites work: A NOTE ON BIFURCATION FROM INFINITY
cites work: Multiple solutions of some semilinear elliptic equations via the generalized Conley index
cites work: A note on multiple solutions of some semilinear elliptic problems
cites work: Q3900559
cites work: Résultats d'existence et de non-existence pour certains problèmes demi-linéaires à l'infini
cites work: Multiple solutions for asymptotically linear boundary value problems in which the nonlinearity crosses at least one eigenvalue
cites work: On the Positive Solutions of Semilinear Equations Δu + λu - hu p = 0 on the Compact Manifolds
cites work: Some global results for nonlinear eigenvalue problems
cites work: On bifurcation from infinity
cites work: On nonlinear elliptic problems with jumping nonlinearities
cites work: Q3927745
cites work: Q4850189
cites work: Blow up points of solution curves for a semilinear problem
cites work: Morse indices and exact multiplicity of solutions to semilinear elliptic problems
description: scientific article; zbMATH DE number 1788034