Regularizing effect in some double phase problems

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

\begin{document}$ \begin{equation*} { } -{{{\rm{\;div}}}}\big(a(x)\,\nabla u|\nabla u|^{p-2}\big) -{{{\rm{\;div}}}}\big( |u|^{(r-1)\lambda+1}\nabla u|\nabla u|^{\lambda-2}\big) = f \end{equation*} $\end{document}

where $ 1 < \lambda \leq p $, $ r > 1 $ and $ f \in L^1(\Omega) $.