Regularizing effect in some Mingione’s double phase problems with very singular data

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 $f\in L^1(\mathrm{\Omega })$.