Variational methods in the study of boundary value problems with discontinuous nonlinearity
type: Item
instance of: scholarly article
title: Variational methods in the study of boundary value problems with discontinuous nonlinearity
zbMATH Open document ID: 0557.35045
DOI: 10.1007/BF02848538
author: Giovanna Cerami
publication date: 1983-01-01T00:00:00Z
review text: One examines the problem: \((*)\quad Lu=\lambda f(x,u)+p(x)\) in \(\Omega\) ; \(u=0\) on \(\partial \Omega\), L is a uniformly elliptic operator, \(\lambda\in R\) and f(x,t) is a Hölder function in \({\bar \Omega}\times (R-\{a\}).\) Four sets of conditions are formulated to ensure the existence of a function u(x) for fixed \(\lambda\) or, in general, a couple (u(x),\(\lambda)\) satisfying (*) almost everywhere. The conditions either concern the behaviour of \(f(x,t)^ for\) \(t\to 0\), \(t\to \infty\) and in the vicinity of the discontinuity at \(t=a\) or they relate the properties of f with the spectrum of L, the coefficients defining L and the behaviour of p(x). The technique used to prove the basic theorems rely upon approximating f(x,t) by a sequence \(\{f_ n\}\) of continuous functionals, subsequent determination of critical points \(\{u_ n\}\) of suitable functions for the regularized problem and finally concluding about properties of the limit \(\lim_{n\to \infty} u_ n\).
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35J35
zbMATH DE Number: 3888164
zbMATH Keywords: discontinuous nonlinearity
zbMATH Keywords: variational formulation
zbMATH Keywords: existence
zbMATH Keywords: critical points
zbMATH Keywords: regularized problem
MaRDI profile type: Publication
cites work: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I
cites work: Dual variational methods in critical point theory and applications
cites work: Q4146134
cites work: Q4765147
cites work: Q3661974
cites work: Variational methods for non-differentiable functionals and their applications to partial differential equations
cites work: Q4200614
cites work: A global theory of steady vortex rings in an ideal fluid
cites work: Symmetry and related properties via the maximum principle
cites work: Q5736313
cites work: Q4175937
cites work: Elliptic eigenvalue problems with discontinuous nonlinearities
cites work: Q3918776
cites work: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus
cites work: Maximal and minimal solutions of elliptic differential equations with discontinuous non-linearities
cites work: A Variational Method for Boundary Value Problems with Discontinuous Nonlinearities
cites work: A duality principle for non-convex optimisation and the calculus of variations
cites work: Elliptic Partial Differential Equations of Second Order
cites work: On positive solutions of nonlinear elliptic eigenvalue problems
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 3888164