BOUNDEDNESS OF SOLUTIONS TO SINGULAR ANISOTROPIC ELLIPTIC EQUATIONS

We prove the uniform boundedness of all solutions for a generalclass of Dirichlet anisotropic elliptic problems of the form-triangle--> pu+ Phi 0(u,del u) = Psi(u,del u) +fin ohm, u= 0 on partial derivative ohm,where ohm subset of RN(N >= 2) is a bounded open set, triangle--> pu=& sum;Nj=1 partial derivative j(|partial derivative ju|pj-2 partial derivative ju)and Phi 0(u,del u) =(a0+& sum;Nj=1aj|partial derivative ju|pj)|u|m-2u, witha0>0,m,pj>1,aj >= 0 for 1 <= j <= NandN/p=& sum;Nk=1(1/pk)>1. We assume thatf is an element of Lr(ohm)withr > N/p. The feature of this study is the inclusion of a possibly singular gradient-dependent term Psi(u,del u) =& sum;Nj=1|u|theta j-2u|partial derivative ju|qj, where theta j>0 and0 <= qj< pjfor 1 <= j <= N. The existence of such weak solutions is contained in a recent paper by the authors.