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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data

Authors: Olivier Guibé and Anna Mercaldo
Journal: Trans. Amer. Math. Soc. 360 (2008), 643-669
MSC (2000): Primary 35J60; Secondary 35A35, 35J25, 35R10
Published electronically: June 25, 2007
MathSciNet review: 2346466
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Abstract: In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is

$ (P)$ $ \displaystyle \left\{\begin{array}{ll} - \bigtriangleup _p u -\hbox{div }(c(x)... ... & \hbox{in} ~ \Omega,\\ u=0 & \hbox{on} ~ \partial\Omega, \end{array}\right. $

where $ \Omega $ is a bounded open subset of $ \mathbb{R}^N$, $ N\geq 2$, $ \bigtriangleup _p$ is the so-called $ p-$Laplace operator, $ 1< p< N$, $ \mu$ is a Radon measure with bounded variation on $ \Omega $, $ 0\le\gamma\le p-1$, $ 0\le\lambda\le p-1$, and $ \vert c\vert$ and $ b$ belong to the Lorentz spaces $ L^{\frac{N}{p-1},r}(\Omega) $, $ \frac{N}{p-1}\leq r \leq +\infty$, and $ L^{N,1}(\Omega)$, respectively. In particular we prove the existence under the assumptions that $ \gamma=\lambda=p-1$, $ \vert c\vert$ belongs to the Lorentz space $ L^{\frac{N}{p-1},r}(\Omega)$, $ \frac{N}{p-1}\leq r<+\infty$, and $ \Vert c\Vert _{ L^{\frac{N}{p-1},r}(\Omega)}$ is small enough.

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Additional Information

Olivier Guibé
Affiliation: Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS, Université de Rouen, Avenue de l’Université BP.12, 76801 Saint Etienne du Rouvray, France

Anna Mercaldo
Affiliation: Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Complesso Monte S. Angelo, via Cintia, 80126 Napoli, Italy

Keywords: Existence, nonlinear elliptic equations, noncoercive problems, measures data.
Received by editor(s): December 16, 2003
Received by editor(s) in revised form: May 23, 2005, and August 2, 2005
Published electronically: June 25, 2007
Article copyright: © Copyright 2007 American Mathematical Society