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 ( | D u | p − 2 D u ( 1 + | u | ) θ ( p − 1 ) φ ) = g φ in Ω , u = 0 on ∂ Ω , where Ω is an open subset of R N with γ ( Ω ) < 1 , N ⩾ 2 , 1 < p < + ∞ , θ ⩾ 0 and φ is the density of Gauss measure.