Variational approach for a class of cooperative systems
type: Item
instance of: scholarly article
title: Variational approach for a class of cooperative systems
zbMATH Open document ID: 1260.35031
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2012-09-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1407.5556
reviewed by: Luisa Consiglieri
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J57
zbMATH DE Number: 6082080
zbMATH Keywords: coexistence states
zbMATH Keywords: variational methods
zbMATH Keywords: spectral bound
author: Pablo Álvarez-Caudevilla
MaRDI profile type: Publication
OpenAlex ID: W1991191249
arXiv ID: 1407.5556
cites work: Global attractivity and singular perturbation for a class of nonlinear cooperative systems
cites work: Existence and uniqueness of coexistence states for some nonlinear elliptic systems
cites work: Uniqueness and existence of positive solutions for, weakly coupled general sublinear systems
cites work: Metasolutions in cooperative systems
cites work: The dynamics of a class of cooperative systems
cites work: Some remarks on classical problems and fine properties of Sobolev spaces
cites work: Positive solutions of a semilinear elliptic equation on a compact manifold
cites work: Periodic Solutions of Nonlinear Hyperbolie Partial Differential Equations
cites work: The maximum principle for cooperative weakly coupled elliptic systems and some applications
cites work: Q5692606
cites work: Q4695147
cites work: Strong positivity in \(C(\bar\Omega)\) for elliptic systems
cites work: The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems
cites work: Steady States, Global Existence and Blow-up for Fourth-Order Semilinear Parabolic Equations of Cahn–Hilliard Type
cites work: A Maximum Principle for an Elliptic System and Applications to Semilinear Problems
cites work: Standing Wave Solutions for a System Derived from the Fitzhugh–Nagumo Equations for Nerve Conduction
cites work: Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equations
cites work: Q4328224
cites work: Remarks on sublinear elliptic equations
cites work: On the Positive Solutions of Semilinear Equations Δu + λu - hu p = 0 on the Compact Manifolds
cites work: Asymptotic behaviour of principal eigenvalues for a class of cooperative systems
cites work: Spectral theory and nonlinear functional analysis
cites work: Bifurcation from simple eigenvalues
cites work: Bifurcation, perturbation of simple eigenvalues, and linearized stability
DOI: 10.1016/J.NA.2012.05.010
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review text: The main result of the paper establishes the existence and uniqueness of positive minimizers \(u\in H^1_0(\Omega)\) of the Euler-Lagrange equation \(D_u\mathcal E(u)v=0\), with NEWLINE\[NEWLINE\mathcal E(u)=\int_\Omega \left( |\nabla u|^2-\lambda u^2-\gamma|(-\Delta-\lambda)^{-1/2}u|^2\right)/2\mathrm{dx} +\int_\Omega a\int_0^u f(\cdot,\xi)\xi d\xi\mathrm{dx}.NEWLINE\]NEWLINE The result is proved through classical variational methods under the assumptions: \(\gamma>0\); the domain \(\Omega\subset\mathbb R^n\) (\(n\geq 1\)) is bounded and of class \(C^{2+\mu}\) for some \(\mu\in ]0,1[\); the function \(a\in C^{\mu}(\bar\Omega)\) is such that \(a^{-1}(\{0\})=\bar\Omega\setminus\Omega_+\) and \(\bar\Omega_+\subset\Omega\), where \(\Omega_+:=\{x\in\Omega:\;a(x)>0\}\) is of class \(C^{2+\mu}\); the function \(f\in C^{\mu, 1+\mu}(\bar\Omega \times [0,\infty[)\) obeys some conditions that imply that \(f(\cdot,u)\rightarrow \infty\) as \(u\rightarrow \infty\) uniformly in \(\bar\Omega\); and \(\lambda\in\mathbb R\) is strictly lesser than the principal eigenvalue of the Laplace operator in \(\Omega\) under homogeneous Dirichlet boundary conditions. The limiting behavior of the positive minimizers is also analyzed as the parameter \(\gamma\) approaches the limiting values for which the existence of positive solutions is held. Furthermore, the map \(\gamma\mapsto u\) is shown that it is increasing and of class \(C^1\).
description: scientific article; zbMATH DE number 6082080