Nonlinear elliptic equations with natural growth in the gradient and source terms in Lorentz spaces

In this paper we consider the problem{u∈W01,p(Ω),−diva(x,u,Du)+b(x,u,Du)=h(x,u,Du)in D′(Ω), where −diva(x,u,Du) is a Leray-Lions operator which is defined on W01,p(Ω) with coercivity α, where the growth with respect to Du of h(x,u,Du) is controlled by αγ|Du|p, and where b(x,u,Du) satisfies a similar growth condition but “has the good sign”. The main feature of the problem is that the source terms belong to the Lorentz space LNp,∞(Ω). When two smallness conditions are satisfied (the second one depends on the behavior of b(x,u,Du) when |u| tends to infinity), we prove the existence of a solution which further satisfies eδp−1|u|−1∈W01,p(Ω) for every δ with γ⩽δ<δ0, for some threshold δ0. The key ingredient in the proof of the existence result is an a priori estimate which holds true for every solution to the problem which satisfies the above mentioned exponential regularity condition.