  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.
