Strong stability results for nonlinear elliptic equations with respect to very singular perturbation of the data
type: Item
instance of: scholarly article
zbMATH Open document ID: 1162.35371
DOI: 10.1142/S0219199701000378
author: Luigi Orsina
author: Alessio Porretta
published in: Communications in Contemporary Mathematics
publication date: 2001-07-29T00:00:00Z
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 1624594
MaRDI profile type: Publication
cites work: On a nonlinear partial differential equation having natural growth terms and unbounded solution
cites work: Existence and nonexistence of solutions for some nonlinear elliptic equations
cites work: Existence of bounded solutions for non linear elliptic unilateral problems
title: Strong stability results for nonlinear elliptic equations with respect to very singular perturbation of the data
review text: The motivation of the present work is the study of nonlinear elliptic equations of the form \(-\Delta u+h(u)|\nabla u|^2=f\), in \(\Omega \setminus K\), where \(f\) belongs to \(L^\infty (\Omega)\), \(K\) is a compact subset of \(\Omega\) with zero capacity and \(h\) is a continuous function such that: \(h(s)s\geq \gamma >1\), for every \(s\geq s_0>0\).NEWLINENEWLINE\textit{H. Brézis} and \textit{L. Nirenberg} [Topol. Methods Nonlinear Anal. 9, No. 2, 201--219 (1997; Zbl 0905.35027)] proved that if \(u\) is a smooth solution to this problem in \(\Omega \subset K\), then it has to be a smooth function in the whole \(\Omega \). This leads to a non-existence result, when replacing \(f\) by \(f+\mu\), where \(\mu\) is a measure with support in a compact set with zero capacity. Then, Brézis raised the conjecture concerning the behavior of the solution \(u_n\) of the preceding equation, when the second member is \(f_n\) (\(f_n\) in \(L^\infty(\Omega)\)) and \((f_n)_n\) converges to \(f\) in a weak sense, namely, NEWLINE\[NEWLINE\int_{\Omega \setminus I(K)}|f_n-f|\,dxNEWLINE\]NEWLINE tends to 0, for every neighborhood \(I(K)\) of \(K\).NEWLINENEWLINEThe authors solve this conjecture in a generalized way, considering the equation NEWLINE\[NEWLINE-\text{div}\,a(x,u,\nabla u)+H(x,u,\nabla u)=f, \quad \text{in}\;\Omega,NEWLINE\]NEWLINE with homogeneous Dirichlet boundary conditions on \(\partial \Omega\). The Carathéodory function \(a\) is assumed to have a \((p-1)\)-growth with respect to its last variable: \(|a(x,s,\xi)|\leq\beta|\xi|^{p-1}\) and \(a(x,s,\xi )\cdot\xi\geq \alpha|\xi|^p\), for \(\alpha \) and \(\beta \) positive; \(a\) is also assumed to be monotone with respect to this last variable. The Carathéodory function \(H\) is assumed to have a \(p\)-growth with respect to its last variable: \(g(s)|\xi|^p\leq H(x,s,\xi )\leq \theta (x)+b(s)|\xi|^p\), where \(g\) satisfies some growth conditions.NEWLINENEWLINEThe main result asserts that if \((f_n)_n\) (with \(f_n\) n…
reviewed by: Alain Brillard
description: scientific article; zbMATH DE number 1624594