In this paper we prove the existence of nonnegative nontrivial solutions of the system $$\left\{\begin{array}{rcll} \Delta u & = & u & \mbox{in } \Omega,\\ \Delta v & = & v, & \end{array}\right.$$ with nonlinear coupling through the boundary given by $$ \left\{\begin{array}{rcll} \frac{\partial u}{\partial n} & = & f(x,u,v) & \mbox{on } \partial \Omega, \\ \frac{\partial v}{\partial n} & = & g(x,u,v), \end{array}\right. $$ under suitable assumptions on the nonlinear terms $f$ and $g$. For the proof we use a fixed-point argument and the key ingredient is a Liouville type theorem for a system of Laplace equations with nonlinear coupling through the boundary of power type in the half space.

We study the blowup mechanism for a simplified system of chemotaxis. First, Moser's iteration scheme is applied and the blowup point of the solution is characterized by the behavior of the local Zygmund norm. Then, Gagliardo-Nirenberg's inequality gives $\varepsilon_0>0$ satisfying $\limsup_{t\uparrow T_{\max}}\left\Vert u(t)\right\Vert_{L^1\left(B_R(x_0)\cap \Omega\right)}\geq \varepsilon_0$ for any blowup point $x_0\in \overline{\Omega}$ and $R>0$. On the other hand, from the study of the Green's function it appears that $t\mapsto\left\Vert u(t)\right\Vert_{L^1\left(B_R(x_0)\cap\Omega\right)}$ has a bounded variation. Those facts imply the finiteness of blowup points, and then, the chemotactic collapse at each blowup point and an estimate of the number of blowup points follow.

We consider here some differential operators arising from the so called Carnot-Carathéodory metric spaces associated with a family of vector fields $X = (X_1, \ldots, X_k)$, which include the Hörmander type as a special case. We prove some weak and strong comparison results for solutions of the relevant differential $inequalities$. We then use these results to get some symmetry and monotonicity properties of solutions of the relevant partial differential equations.