Global weak solutions of a parabolic-elliptic Keller-Segel system with gradient dependent chemotactic coefficients

We consider the following Keller-Segel system with gradient dependent chemotactic coefficient: \begin{document}$ \begin{equation*} \begin{cases} u_{t} = \Delta u- \chi \nabla\cdot (uf(|\nabla v|)\nabla v),\\ 0 = \Delta v -v+g(u), \end{cases} \end{equation*} $\end{document} in smooth bounded domains $ \Omega \subset \mathbb{R}^{n}, \,n\geq 1 $ with $ f(\xi) = \big(\xi^{p-2}\big(1+\xi^{p}\big)^{\frac{q-p}{p}}\big), \,1