A class of nonlinear degenerate elliptic equations related to the Gauss measure

We prove a priori estimates and existence results for Dirichlet problems whose model case is:{−div(|Du|p−2Du(1+|u|)θ(p−1)φ)=gφin Ω,u=0on ∂Ω, where Ω   is an open subset of RNRN with γ(Ω)<1γ(Ω)<1, N⩾2N⩾2, 1