Remarks on variational methods and lower-upper solutions
type: Item
instance of: scholarly article
title: Remarks on variational methods and lower-upper solutions
zbMATH Open document ID: 0941.35031
author: Luca Merizzi
author: Susanna Terracini
author: Monica Conti
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2000-08-13T00:00:00Z
review text: The authors consider the following class of equations \[ \begin{cases} -\Delta u(x)= f\bigl(x,u(x) \bigr),\quad x\in\Omega,\\ u=0\text{ on }\partial \Omega,\end{cases} \tag{1} \] where \(\Omega\subset\mathbb{R}^N\) is a bounded smooth domain. Letting \(F(x,t)=\int^t_0 f(x,s)ds\), the energy functional associated to (1) is defined by the formula \[ J(u)=\int_\Omega \Bigl(\textstyle {1\over 2}|\nabla u|^2-F(x,u) \Bigr)dx,\quad u\in \overset\circ W^1_2 (\Omega). \] Under suitable structural assumptions imposed on \(f,J\) turns out to be of class \(C^1\), and weak solutions of (1) are sought as critical points of \(J\). Assuming the existence of lower and upper solutions with appropriate ordering relations, the authors' exploit the variational structure of problem (1) to show the existence of local \(J\)-minimizers.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B38
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 1406550
zbMATH Keywords: semilinear elliptic equations
zbMATH Keywords: existence and multiplicity of solutions
zbMATH Keywords: lower and upper solutions
reviewed by: M. Fuchs
MaRDI profile type: Publication
full work available at URL: s000300050009
OpenAlex ID: W1996546405
DOI: 10.1007/S000300050009
description: scientific article; zbMATH DE number 1406550