Dual variational methods in critical point theory and applications
type: Item
instance of: scholarly article
title: Dual variational methods in critical point theory and applications
zbMATH Open document ID: 0273.49063
DOI: 10.1016/0022-1236(73)90051-7
author: Paul H. Rabinowitz
author: A. Ambrosetti
published in: Journal of Functional Analysis
publication date: 1973-01-01T00:00:00Z
review text: Consider the nonlinear elliptic partial differential equation \[ L(u) \equiv -\sum_{i,j=1}^n (a_{ij}(x)u_{x_i})_{x_j} + c(x)u = p(x,u),\quad x\in\Omega,\ u = 0,\ x \in\partial\Omega, \tag{*}\] where \(\Omega\subset\mathbb R^n\) is a smooth bounded domain. Formally, the critical points of the functional \[ I(u) = \int_\Omega \left[ \frac12 \sum_{i,j=1}^n (a_{ij}(x)u_{x_i})_{x_j} + c(x)u^2 - P(x,u(x))\right] \,dx, \] where \(P(x,u)\) is a primitive of \(p(x,u)\), are solutions of (*). The authors construct dual variational methods to enable them to prove the existence and estimate the number of critical points possessed by a real continuously differentiable functional on a real Banach space, and then apply their results to various existence problems for equations of type (*). They also apply them to problems with linear term added, i.e. \[ L(u) = a(x)u + p(x,u),\quad x\in\Omega;\ u=0,\ x \in\partial\Omega, \] as well as to nonlinear integral equations of the form \[ v(x) = \int_\Omega g(x,y)q(y,v(y))\,dy. \]
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 3429675
Wikidata QID: Q56519801
MaRDI profile type: Publication
full work available at URL: 0022-1236(73)90051-7
OpenAlex ID: W1981288703
cites work: Q4760461
cites work: Q3847203
cites work: Generalizing the Lusternik‐Schnirelman theory of critical points
cites work: Lusternik-Schnirelman theory on Banach manifolds
cites work: Infinite dimensional manifolds and nonlinear elliptic eigenvalue problems
cites work: Ljusternik-Schnirelman theory and non-linear eigenvalue problems
cites work: Q5636090
cites work: A minimum-maximum principle for a class of non-linear integral equations
cites work: Q5590275
cites work: Q5631509
cites work: Q5662797
cites work: On the existence of multiple solutions for a class of nonlinear boundary value problems
cites work: Q5680495
cites work: Q4764973
cites work: On a class of non-linear integral equations
cites work: Some aspects of nonlinear eigenvalue problems
cites work: A generalized Morse theory
cites work: Nonlinear sturm‐lionville problems for second order ordinary differential equations
cites work: Superlinear Sturm-Liouville problems
cites work: Q3269389
cites work: Q5537946
cites work: ON THE EIGENFUNCTIONS OF QUASILINEAR ELLIPTIC PROBLEMS
cites work: Existence theorems for equations of hammerstein type
description: scientific article