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 v -v+g(u), \end{cases} \end{equation*} $\end{document} in smooth bounded domains $ \Omega \subset \mathbb{R}^{n}, \,n\geq 1 f(\xi) \big(\xi^{p-2}\big(1+\xi^{p}\big)^{\frac{q-p}{p}}\big), \,1<q\leq p<\infty and g(\xi) \frac{\xi}{(1+\xi)^{1-\beta}}, \,\xi \geq 0, \,\beta \in [0, 1]. show existence of a global weak solution, L^{\infty} $-norm, if$ 1<q\leq p<\infty,\; \; n 1,\\ 1<q<\frac{n}{n-1},\; \quad n\geq 2.