HAL : hal-00402869, version 1

DOI : 10.1016/j.na.2010.02.044

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Nonlinear Analysis: Theory, Methods and Applications A 72, 12 (2010) 4649-4660

Structural stability for variable exponent elliptic problems. II. The $p(u)$-laplacian and coupled problems.

Boris Andreianov 1, Mostafa Bendahmane 2, Stanislas Ouaro 3

(06/2010)

We study well-posedness for elliptic problems under the form $$b(u)-\div \mathfrak{a}(x,u,\Grad u)=f,$$ where $\mathfrak{a}$ satisfies the classical Leray-Lions assumptionswith an exponent $p$ that may depend both on the space variable $x$ and on the unknown solution $u$. A prototype case is the equation $u-\div \Bigl( |\grad u|^{p(u)-2}\grad u \Bigr)=f$. We have to assume that $\inf_{x\in\overline{\Om},\,z\in\R} p(x,z)$ is greater than the space dimension $N$. Then, under mild regularity assumptions on $\Om$ and on the nonlinearities, we show that the associated solution operator is an order-preserving contraction in $L^1(\Om)$. In addition, existence analysis for a sample coupled system for unknowns $(u,v)$ involving the $p(v)$-laplacian of $u$ is carried out. Coupled elliptic systems with similar structure appear in applications, e.g. in modelling of stationary thermo-rheological fluids.

1 :	Laboratoire de Mathématiques (LM-Besançon)
	CNRS : UMR6623 – Université de Franche-Comté
2 :	Centro de Investigación en Ingeniería Matemática [Concepción] (CI²MA)
	Universidad de Concepción
3 :	LAME Ouagadougou, Burkina-Faso (LAME)
	Université de Ouagadougou

Domaine

:

Mathématiques/Equations aux dérivées partielles

Mots Clés : variable exponent – $p(u)$-laplacian – thermo-rheological fluids – well-posedness – Young measures

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Contributeur : Boris Andreianov <>

Soumis le : Mercredi 8 Juillet 2009, 15:14:33

Dernière modification le : Vendredi 23 Avril 2010, 11:59:17