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.
Origine : Fichiers produits par l'(les) auteur(s)
Loading...