In this paper, we prove existence of distributional solutions of a nonlinear elliptic system, related to the Keller-Segel model. Our starting point is the boundedness theorem (for solutions of elliptic equations) proved by Guido Stampacchia and Neil Trudinger.

In this paper we study the existence of solutions of the Dirichlet problem associated to the following nonlinear PDE

$$-\mathrm{d}\mathrm{i}\mathrm{v}(a(x)\mathrm{\nabla }u|\mathrm{\nabla }u{|}^{p-2})-\mathrm{d}\mathrm{i}\mathrm{v}(|u{|}^{(r-1)\lambda +1}\mathrm{\nabla }u|\mathrm{\nabla }u{|}^{\lambda -2})=f$$

where $1<\lambda \le p$, $r>1$ and $f\in L^1(\mathrm{\Omega })$.