Structural stability for variable exponent elliptic problems. II. The $p(u)$-laplacian and coupled problems. - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Nonlinear Analysis: Theory, Methods and Applications Année : 2010

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

Résumé

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.
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Dates et versions

hal-00402869 , version 1 (08-07-2009)

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Paternité - Pas d'utilisation commerciale

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Boris Andreianov, Mostafa Bendahmane, Stanislas Ouaro. Structural stability for variable exponent elliptic problems. II. The $p(u)$-laplacian and coupled problems.. Nonlinear Analysis: Theory, Methods and Applications, 2010, 72 (12), pp. 4649-4660. ⟨10.1016/j.na.2010.02.044⟩. ⟨hal-00402869⟩
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