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L estimate for some nonlinear elliptic partial differential equations and application to an existence result. (English) Zbl 0785.35033

Summary: Consider the nonlinear elliptic equation


where 𝒜(u)=-div(a(x,u,Du))+a 0 (x,u,Du) is a Leray-Lions operator defined on W 0 1,p (Ω) with a 0 (x,s,ξ)sα 0 |s| p , α 0 >0, and where H is a first-order term satisfying |H(x,s,ξ)|C 0 +C 1 |ξ| p . The main goal of this paper is to prove an L estimate for the bounded solutions of (E) when f belongs to L q (Ω) and g belongs to (L r (Ω)) N with r=p ' q and max(1,N/p)<q+. In view of the method and results developed in the author’s previous work, this implies the existence of a solution for equation (E).

35J65Nonlinear boundary value problems for linear elliptic equations
35B45A priori estimates for solutions of PDE
35D05Existence of generalized solutions of PDE (MSC2000)
35B35Stability of solutions of PDE