elliptic problems with nonlinear terms depending on the gradient and singular on the boundary

{−δu = u^qα|∇u |^q +λf (x) in ω

u = 0 on ∂ω, (p)

where ω ⊂ rⁿ is a bounded domain, 1 < q ≤ 2, α ∈ r and f ≥ 0. we prove that:

(1) if qα < −1, then problem (p) has a distributional solution for all f ∈ l1(ω), and all

λ > 0.

(2) if −1 ≤ qα < 0, then problem (p) has a solution for all f ∈ l^s(ω), where s > n/q

if n ≥ 2, and without any restriction on λ.

(3) if q = 2 and −1 ≤ qα < 0 then problem (p) has infinitely many solutions under

suitable hypotheses on f .

(4) if 0 ≤ qα and f ∈ l¹(ω) satisfies