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:Ω ⊆ R ⁿ→ R to problems of the type {equation*} {cases} -{ div}\, a(x,D u)+b(x)φ'(|u|) {u/|u|} =f & in\,\,Ω \\ u=0 & on\,\,∂Ω {cases} {equation*} with Ω⊂ R ⁿ a bounded open set and where the function a(x,ξ) satisfies growth conditions with respect to the second variable expressed through an N-function φ. We prove that, under a suitable interplay between the lower order terms and the datum f, which is assumed only to belong to L¹(Ω), the solutions are bounded in Ω. Next, if a(x,ξ) 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.