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 ⊆ Rn → R to elliptic problems −div a(x,Du) + b(x)ϕ′(|u|) u |u| = f in Ω , u = 0 on ∂Ω , with Ω ⊂ Rn 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 H¨older 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.