Regularity results for an anisotropic nonlinear Dirichlet problem

Abstract

In this paper we consider the homogeneous Dirichlet problem associated to the model equation − div ⁡ ( a ( x ) | ∇u | p − 2 ∇u ) − div ⁡ ( | u | ( r − 1 ) q + 1 | ∇u | q − 2 ∇u ) = f in Ω , where Ω is a bounded open subset of 2, $ ]]> R N , N > 2 , 1<q<p<N, \\frac {1}{q'} $ ]]> r > 1 q ′ and f belongs to 1 $ ]]> L m ( Ω ) , m > 1 . We give some regularity results according to the values of the exponents ( r − 1 ) q + 1 and m in suitable ranges depending on N , p and q .