Critical exponent and discontinuous nonlinearities
type: Item
instance of: scholarly article
title: Critical exponent and discontinuous nonlinearities
zbMATH Open document ID: 0821.35049
author: M. Badiale
published in: Differential and Integral Equations
publication date: 1995-10-03T00:00:00Z
review text: We prove existence of a positive solution to the problem \[ -\Delta u=| u|^{2^*- 2} u+bh (u-a), \quad u(x)>0 \;\;\text{in }\Omega, \quad u(x)=0\;\;\text{on } \partial\Omega, \tag{1} \] where \(\Omega\) is a bounded regular open set \(\subset \mathbb{R}^ N\), \(2^*= {{2N} \over {n-2}}\) is the critical Sobolev exponent, \(h\) is the Heaviside function and \(a\), \(b\) are strictly positive parameters. The nonlinearity in (1) has critical power-like behavior and a discontinuity jump of length \(b\) in the point \(u=a\). So in problem (1) there are two difficulties to be overcome: the presence of a discontinuity and the lack of compactness due to the critical exponent.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35R05
zbMATH DE Number: 459200
zbMATH Keywords: discontinuous nonlinearities
zbMATH Keywords: critical Sobolev exponent
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 459200