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