Exact multiplicity of positive solutions for a class of semilinear problem. II
type: Item
instance of: scholarly article
title: Exact multiplicity of positive solutions for a class of semilinear problem. II
zbMATH Open document ID: 0947.35067
author: Tiancheng Ouyang
author: Junping Shi
published in: Journal of Differential Equations
publication date: 2000-10-30T00:00:00Z
review text: The authors study positive solutions to the semilinear problem: \[ \begin{cases} \Delta u + \lambda f(u) = 0,& \quad \text{ in }B^n, \\ u = 0,& \quad \text{ on }B^n, \end{cases} \tag{1} \] on the unit ball \(B^n\) in \({\mathbb R}^n\) with \(n\geq 1\), where \(\lambda\) is a positive parameter and \(f\) is a smooth function. This problem has a very long history and many contributions have been made in the last several decades on the existence and multiplicity of positive solutions. But a complete description on the global structure of the positive solution set of (1) is very hard to obtain even for certain typical classes of nonlinearities. The authors have obtained interesting results on this problem. Using the result of Gidas-Ni-Nirenberg on radial symmetry of positive solutions of (1), this problem is reduced to positive solutions of the following ordinary differential equation: \[ \begin{cases} u'' + {{n-1}\over r}u' + \lambda f(u) = 0, &\quad r\in (0,1), \\ u'(0) = u(1) = 0. &\quad \end{cases} \tag{2} \] Then using bifurcation analysis methods together with a Morse index estimate, the authors developed a unified approach to obtain the exact multiplicity of the positive solutions for several classes of nonlinear functions \(f(u)\), and the precise shape of the global bifurcation diagrams. The authors discovered that the shape of the bifurcation curve depends on the shape of the graph of the function \(f(u)/u\) and the growth rate of \(f\). Typical examples of the nonlinear function \(f(u)\) treated in this paper includes (A) \(f(u) = u^p\) with \(0<p<1\) or \(1<p<(n+2)/(n-2)\), (B) \(f(u) = \sqrt{u^2+2u}\), (C) \(f(u) = -u+u^p-c\) with \(c\geq 0\) and \(1<p<(n+2)/(n-2)\), (D) \(f(u) = u(u-b)(c-u)\) with \(0<2b<c\), (E) \(f(u) = u^p - u^q\) with \(0<p<q\), (F) \(f(u) = u^p + u^q\) with \(0<q<1<p\leq n/(n-2)\). The authors also give a partial answer to a conjecture of \textit{P. L. Lions} proposed [SIAM Rev. 24, 441-467 (1982; Zbl 0511.35033)], which claims that t…
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B32
zbMATH DE Number: 1384461
zbMATH Keywords: semilinear elliptic equation
zbMATH Keywords: global positive solution structure
zbMATH Keywords: bifurcation
zbMATH Keywords: Morse index
reviewed by: Yiming Long
MaRDI profile type: Publication
cites work: Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces
cites work: Nonlinear eigenvalue problems having precisely two solutions
cites work: Positive solutions for some semi-positone problems via bifurcation theory
cites work: Combined effects of concave and convex nonlinearities in some elliptic problems
cites work: Positive solutions of asymptotically linear elliptic eigenvalue problems
cites work: Nonlinear scalar field equations. I: Existence of a ground state
cites work: Q3672366
cites work: Uniqueness of stable and unstable positive solutions for semipositone problems
cites work: Branches of radial solutions for semipositone problems
cites work: Existence results for classes of sublinear semipositone problems
cites work: Non-Negative Solutions for a Class of Radially Symmetric Non-Positone Problems
cites work: Uniqueness of the ground state solutions of △u+f(u)=0 in Rn, n≥3
cites work: Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball.
cites work: A Geometric Proof of the Kwong-McLeod Uniqueness Result
cites work: Uniqueness of the ground state solution for \(\Delta u - u + u^3=0\) and a variational characterization of other solutions
cites work: Bifurcation from simple eigenvalues
cites work: Bifurcation, perturbation of simple eigenvalues, and linearized stability
cites work: Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems
cites work: On positive solutions of some singularly perturbed problems where the nonlinearity changes sign
cites work: On Positive Solutions of Semilinear Elliptic Equations
cites work: Q5185568
cites work: The set of positive solutions of semilinear equations in large balls
cites work: Symmetry and related properties via the maximum principle
cites work: Q3933274
cites work: Quasilinear Dirichlet problems driven by positive sources
cites work: Heavy rotating string—a nonlinear eigenvalue problem
cites work: Solution curves for semilinear equations on a ball
cites work: Exact multiplicity results for boundary value problems with nonlinearities generalising cubic
cites work: An Exact Multiplicity Result for a class of Semilinear Equations
cites work: On the Kolodner-Coffman method for the uniqueness problem of Emden-Fowler BVP
cites work: Uniqueness of Radial Solutions of Semilinear Elliptic Equations
cites work: On ground state solutions of \(-\Delta{} u=u^ p-u^ q\)
cites work: Uniqueness of the positive solution of \(\Delta u+f(u)=0\) in an annulus
cites work: A Counterexample to the Nodal Domain Conjecture and a Related Semilinear Equation
cites work: On the Existence of Positive Solutions of Semilinear Elliptic Equations
cites work: Uniqueness of Positive Radial Solutions of Δu + f(u) = 0 in ℝ n , II
cites work: Uniqueness of positive radial solutions of \(\Delta u+f(u)=0\) in \({\mathbb{R}}^ n\)
cites work: Uniqueness of solutions of nonlinear Dirichlet problems
cites work: Uniqueness and nonuniqueness for positive radial solutions of Δu + f(u, r) = 0
cites work: Exact multiplicity of positive solutions for a class of semilinear problems
cites work: Uniqueness of positive solutions of semilinear equations in \(R^ n\).
cites work: Uniqueness of non-negative solutions of semilinear equations in \({\mathbb{R}}^ n\)
cites work: Some global results for nonlinear eigenvalue problems
cites work: On bifurcation from infinity
cites work: Global solution branches of two point boundary value problems
cites work: Morse indices and exact multiplicity of solutions to semilinear elliptic problems
cites work: On a singular nonlinear semilinear elliptic problem
cites work: Global bifurcation of steady-state solutions
cites work: Existence, uniqueness, and nondegeneracy of positive solutions of semilinear elliptic equations
Recommended article: Exact multiplicity of positive solutions for a class of semilinear problems
Recommended article: Q5322250
Recommended article: Exact multiplicity of positive solutions for a class of singular semilinear equations
Recommended article: Exact multiplicity of positive solutions of a semipositone problem with concave-convex nonlinearity
Recommended article: An Exact Multiplicity Result for a class of Semilinear Equations
Recommended article: Exact multiplicity of solutions for some semilinear Dirichlet problems
Recommended article: Exact multiplicity of positive solutions in semipositone problems with concave-convex type nonlinearities
Recommended article: Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity
Recommended article: Exact multiplicity of positive solutions for a class of semilinear equations on a ball
description: scientific article; zbMATH DE number 1384461