Nonlinear distributions and quasilinear ellipticity
type: Item
instance of: scholarly article
title: Nonlinear distributions and quasilinear ellipticity
zbMATH Open document ID: 0629.35043
DOI: 10.1016/0022-0396(87)90195-1
author: Victor L. Shapiro
published in: Journal of Differential Equations
publication date: 1987-01-01T00:00:00Z
review text: Consider in a bounded domain \(\Omega \subset R^ N\) the equation \[ Q(u)\equiv -D_ i[a^{ij}(x,u)D_ ju+b^ i(x,u)u]+c^ i(x,u)D_ iu+d(x,u)u=f(x,u)+g_ 0(x,u)-D_ ig_ i(x,u) \] where functions \(a^{ij}\), \(b^ i\), \(c^ i\), d, \(g_ i\) satisfy a linear growth condition in u and uf(x,u) may be arbitrary quickly converging to - \(\infty\) as \(u\to \pm \infty\). It is proved the existence of a distribution solution \(u\in H_ 0^ 1(\Omega)\) of this equation with f(x,u), \(f(x,u)u\in L^ 1(\Omega)\). Further, by some additional conditions it is shown the existence of a distribution solution \(u\in H^ 1_ 0(\Omega)\) of the equation at resonance \[ Q(u)=\lambda^*_ 1u+f(x,u)+g_ 0(x,u)-D_ ig_ i(x,u) \] with f(x,u), \(uf(x,u)\in L^ 1(\Omega)\) where \(\lambda^*_ 1\) is the generalized first eigenvalue of Q.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 4023842
zbMATH Keywords: quasilinear
zbMATH Keywords: generalized first eigenvalue
zbMATH Keywords: growth condition
zbMATH Keywords: existence
zbMATH Keywords: distribution solution
zbMATH Keywords: equation at resonance
reviewed by: László Simon
MaRDI profile type: Publication
full work available at URL: 0022-0396(87)90195-1
OpenAlex ID: W1984918332
cites work: Q4167193
cites work: Variational inequalities and flow in porous media
cites work: Nonlinear perturbations of a linear elliptic problem near its first eigenvalue
cites work: Q4040640
cites work: Remarks on some quasilinear elliptic equations
cites work: Q5593890
cites work: Q4405392
cites work: Resonance and Quasilinear Ellipticity
description: scientific article; zbMATH DE number 4023842