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

Abstract

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