On a Parabolic-Elliptic system with gradient dependent chemotactic coefficient

We consider a second order PDEs system of Parabolic-Elliptic type with chemotactic terms. The system describes the evolution of a biological species "$u$" moving towards a higher concentration of a chemical stimuli "$v$" in a bounded and open domain of $\mathcal{R}^N$. In the system considered, the chemotaxis sensitivity depends on the gradient of $v$, i.e., the chemotaxis term has the following expression $$- div \left(\chi u |\nabla v|^{p-2}\nabla v \right),$$ where $\chi$ is a positive constant and $p$ satisfies $$p \in (1, \infty), \quad \mbox{ if } N=1 \quad \mbox{ and } \quad p\in \left(1, \frac{N}{N-1}\right), \quad \mbox{ if } N\geq 2.$$ We obtain uniform bounds in time in $L^{\infty}(\Omega)$ of the solutions. For the one-dimensional case we prove the existence of infinitely many non-constant steady-states for $p\in (1,2)$ for any $\chi$ positive and a given positive mass.