Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data
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- by Olivier Guibé and Anna Mercaldo PDF
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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 \begin{equation}\tag {P} \begin {cases} - \bigtriangleup _p u -\operatorname {div}(c(x)|u|^{\gamma })+b(x)|\nabla u|^{\lambda } =\mu & \text {in $\Omega $},\\ u=0 & \text {in $\partial \Omega $}, \end{cases} \end{equation} 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 $|c|$ 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 $\gamma =\lambda =p-1$, $|c|$ belongs to the Lorentz space $L^{\frac {N}{p-1},r}(\Omega )$, $\frac {N}{p-1}\leq r<+\infty$, and $\|c\|_{ L^{\frac {N}{p-1},r}(\Omega )}$ is small enough.References
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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
- Email: Olivier.Guibe@univ-rouen.fr
- 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
- Email: mercaldo@unina.it
- 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
- © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 643-669
- MSC (2000): Primary 35J60; Secondary 35A35, 35J25, 35R10
- DOI: https://doi.org/10.1090/S0002-9947-07-04139-6
- MathSciNet review: 2346466