Regularity results to a class of elliptic equations with explicit $u$-dependence and Orlicz growth

We study the regularity properties of the weak solutions $u:\Omega \subseteq \mathbb R ^n\to\mathbb R $ to problems of the type \begin{equation*} \begin{cases} -{\rm div}\, a(x,D u)+b(x)\phi'(|u|) {\frac{u}{|u|}} =f & \text{in}\,\,\Omega \\ u=0 & \text{on}\,\,\partial\Omega \end{cases} \end{equation*} with $\Omega\subset\mathbb R ^n$ a bounded open set and where the function $a(x,\xi)$ satisfies growth conditions with respect to the second variable expressed through an N-function $\phi$. We prove that, under a suitable interplay between the lower order terms and the datum $f$, which is assumed only to belong to $L^1(\Omega)$, the solutions are bounded in $\Omega$. Next, if $a(x,\xi)$ depends on $x$ through a Hölder continuous function, we take advantage from the boundedness of the solution $u$ to prove the higher differentiability and the higher integrability of its gradient, under mild assumptions on the data.