Elliptic systems with nonlinear diffusion and a convection term
Lucio Boccardo, Luigi Orsina · Discrete and Continuous Dynamical Systems · 2022
In this paper we prove existence (and summability properties) of solutions for the following elliptic system$ \left\{ \begin{array}{cl} -{\rm{div}}(A(x)\,{ abla} u) + u^{{\lambda}} = -{\rm{div}}( u^{{\lambda}} \, M(x)\,{ abla}\psi) + f(x)\,, & {\rm{in }}\; \Omega , \\ -{\rm{div}}(M(x)\,{ abla}\psi) = u^{\rho}\,, & {\rm{in }}\; \Omega , \\ u = 0 = \psi & \;{\rm{on}}\; \partial\Omega , \end{array} \right. $under some assumptions on $ {\lambda} > 0 $, $ \rho > 0 $ and $ f(x) $ in $ L^{{m}}(\Omega) $, $ m \geq 1 $. 'Return of the Patriarca' (see [3])