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

摘要

In this paper we consider the problem where?div a(x,u,Du) is a Leray-Lions operator which is defined on W_0~(1,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 L~(N/p,∞) (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 e~(δ/p?1|u|)?1 ∈ W_0~(1,p) (Ω) for every δ with γ≤δ<δ_0, for some threshold δ_0. 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.