Regularizing effect for some p-Laplacian systems

We study existence and regularity of weak solutions for the following $p$-Laplacian system \begin{cases} -\Delta_p u+A\varphi^{\theta+1}|u|^{r-2}u=f, \ &u\in W_0^{1,p}(\Omega),\-\Delta_p \varphi=|u|^r\varphi^\theta,
&\varphi\in W_0^{1,p}(\Omega), \end{cases} where $\mathrm{\Omega }$ is an open bounded subset of ${\mathbb{R}}^N$ $(N\ge 2)$, $\Delta_p v :=\operatorname{div}(|\nabla v|^{p-2}\nabla v)$isthe$p$-Laplacianoperator,for$1<p<N$,$A>0$,$r>1$,$0\leq\theta<p-1$and$f$ belongs to a suitable Lebesgue space. In particular, we show how the coupling between the equations in the system gives rise to a regularizing effect producing the existence of finite energy solutions.