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

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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<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. © 2013 Elsevier Inc.

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Ferone, V., & Murat, F. (2014). Nonlinear elliptic equations with natural growth in the gradient and source terms in Lorentz spaces. Journal of Differential Equations, 256(2), 577–608. https://doi.org/10.1016/j.jde.2013.09.013

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