Uniqueness and nondegeneracy for some nonlinear elliptic problems in a ball.
type: Item
instance of: scholarly article
title: Uniqueness and nondegeneracy for some nonlinear elliptic problems in a ball.
zbMATH Open document ID: 1109.35039
DOI: 10.1016/S0022-0396(02)00194-8
author: Amandine Aftalion
author: Filomena Pacella
published in: Journal of Differential Equations
publication date: 2004-01-26T00:00:00Z
review text: This paper is devoted to the uniqueness of non-degeneracy of solutions of nonlinear problems of the following type \[ \begin{cases} \Delta_p u+ f(r,u)= 0\quad &\text{in }B,\\ u> 0\quad &\text{in }B,\\ u= 0\quad &\text{on }\partial B,\end{cases} \] where \(B\) is the unit ball in \(\mathbb R^N\) centered at the origin, \(N\geq 2\), \(r=| x|\) and \(\Delta_p\) denotes the \(p\)-Laplace operator \[ \Delta_p u= \text{div}(|\nabla u|^{p-2}\nabla u),\quad p> 1. \] To this end the authors use the maximum principle and an implicit function theorem that they derive in a suitable weighted space.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35B50
zbMATH DE Number: 2032994
zbMATH Keywords: implicit function theorem
zbMATH Keywords: nonlinear elliptic problem
zbMATH Keywords: maximum principle
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
cites work: An elementary proof of the uniqueness of positive radial solutions of a quasilinear Dirichlet problem
cites work: Dual variational methods in critical point theory and applications
cites work: Q4227364
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, perturbation of simple eigenvalues, and linearized stability
cites work: Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle
cites work: Q4237649
cites work: Monotonicity and symmetry results for \(p\)-Laplace equations and applications
cites work: Uniqueness theorems for positive radial solutions of quasilinear elliptic equations in a ball
cites work: Existence and uniqueness of nonnegative solutions of quasilinear equations in \(\mathbb{R}^ n\)
cites work: Symmetry and related properties via the maximum principle
cites work: Global and local behavior of positive solutions of nonlinear elliptic equations
cites work: Symmetry of positive solutions of a quasilinear elliptic equation via isoperimetric inequalities
cites work: Uniqueness of positive solutions of \(\Delta u-u+u^ p=0\) in \(R^ n\)
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: Uniqueness of positive radial solutions of \(\Delta u+f(u)=0\) in \({\mathbb{R}}^ n\)
cites work: Exact multiplicity of positive solutions for a class of semilinear problems
cites work: Linear and nonlinear aspects of vortices. The Ginzburg-Landau model
cites work: Uniqueness of positive solutions of semilinear equations in \(R^ n\).
cites work: Q4221424
cites work: On bifurcation from infinity
cites work: Uniqueness of ground states for quasilinear elliptic equations
cites work: Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities.
cites work: Uniqueness of solutions of nonlinear Dirichlet problems
cites work: A strong maximum principle for some quasilinear elliptic equations
full work available at URL: s0022-0396(02)00194-8
OpenAlex ID: W2031202697
description: scientific article; zbMATH DE number 2032994