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 $\Omega$ is an open bounded subset of $\mathbb{R}^N$ $(N\geq 2)$, $\Delta_p v :=\operatorname{div}(|\nabla v|^{p-2}\nabla v)$ is the $p$-Laplacian operator, 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.