A priori estimates and existence for elliptic equations with gradient dependent terms

International audienceWe consider, in a bounded domain Ω of R N , a class of nonlinear elliptic equations in divergence form asa_0 u - div(a(x, u, Du)) = H (x, u, Du) in Ω, u = 0 on ∂Ω,where a_0 ≥ 0, the second order part is a coercive, pseudomonotone operator of Leray-Lions type in the Sobolev space W^{1,p}_0 (Ω), p > 1, and the function H grows at most like |Du|^q + f (x), with p-1 < q < p. Assuming f (x) to belong to an (optimal) Lebesgue class Lmm , with m < N/p , we prove a priori estimates and existence of solutions, discussing several ranges of the exponents m, q and p which include cases of singular data (L^1 data or measures). The obtention of a priori estimates is not straightforward because of the "superlinear" character of the first order terms. To this purpose we use a new approach, generalizing the method introduced in our note [29]. We complete the results known in the previous literature where either q ≤ p-1 or m ≥ N/p. This paper has been published in Ann. Sc. Norm. Sup. Pisa, 13, (2014), pp. 137-205