Anisotropic degenerate elliptic problem with singular gradient lower order term

In this paper, we investigate the existence and regularity results of anisotropic elliptic equations with degenerate coercivity and a singular lower order term exhibiting natural growth with respect to the gradient. The model problem we consider is 0.1 - ∑ i = 1 N D i [ | D i u | p i - 2 D i u ( 1 + | u | ) γ ] + ∑ i = 1 N | D i u | p i | u | θ = f in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain in R N , γ > 0 , 0 ≤ θ < 1 and 2 ≤ p 1 ≤ p 2 ≤ ⋯ ≤ p N . The results of our study will rely on the summability of f in specific Lebesgue spaces and the value of γ .