On a chemorepulsion elliptic system with nonlinear production and weak data

Let $ \Omega $ be a bounded Lipschitz domain in $ {\mathbb{R}}^N $, with $ N \geq 3 $ and $ \chi > 0 $. Let $ M(x) $ be a symmetric positive definite matrix with coefficients $ m_{ij} \in L^{\infty} (\Omega) $. We study the problem:

$ \left\{ \begin{array}{ll} - \mbox{div}(M(x) \nabla u) + u = \chi \mbox{div}(uM(x) \, \nabla \psi ) + f(x), \\ - \mbox{div}(M(x) \nabla \psi) +\psi = u^{\theta}+g(x). \end{array} \right. $

Under the assumptions

$ 1. $ $ \theta < \frac{N+2}{N-2} $;

$ 2. $ $ f \geq 0 $, $ f \in L^{m}(\Omega) $, for $ m > \frac{2N}{N+2} $;

$ 3. $ $ g \geq 0 $, $ g \in L^{q}(\Omega) $, for $ q > \frac{2N}{N+2} $,

we prove the existence of at least one weak solution in $ W^{1, \frac{N}{N-1}} (\Omega) \times W^{1, 2} (\Omega) $. Moreover, under the assumptions

$ 4. $ $ \theta < \frac{2}{N-2} $;

$ 5. $ $ g \in L^{q}(\Omega) \mbox{ for some }q > \frac{N}{2}, $

we have

$ u \in W^{1, 2} (\Omega), \quad \psi \in W^{1, 2} (\Omega) \cap L^{\infty}(\Omega). $