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hal-00326563, version 2

## Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient

Haydar Abdelhamid 1, Marie-Françoise Bidaut-Véron () 1

Comptes Rendus de l Académie des Sciences - Series I - Mathematics 346 (2008) 1251-1256

Abstract: Thanks to a change of unknown we compare two elliptic quasilinear problems with Dirichlet data in a bounded domain of $\mathbb{R}^{N}.$ The first one, of the form $-\Delta_{p}u=\beta(u)\left\vert \nabla u\right\vert ^{p}+\lambda f(x),$ where $\beta$ is nonnegative, involves a gradient term with natural growth. The second one, of the form $-\Delta_{p}v=\lambda f(x)(1+g(v))^{p-1}$ where $g$ is nondecreasing, presents a source term of order $0$. The correlation gives new results of existence, nonexistence and multiplicity for the two problems.

• 1:  Laboratoire de Mathématiques et Physique Théorique (LMPT)
• CNRS : UMR6083 – Université François Rabelais - Tours
• Domain : Mathematics/Analysis of PDEs
• Keywords : Quasilinear elliptic equations – renormalized solutions – extremal solutions – measure data
• Available versions :  v1 (2008-10-06) v2 (2008-11-20)

• hal-00326563, version 2
• oai:hal.archives-ouvertes.fr:hal-00326563
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• Submitted on: Wednesday, 19 November 2008 11:36:59
• Updated on: Thursday, 12 February 2009 17:47:48