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

In this paper we consider the problem equation(P){−Δu=uqα|∇u|q+λf(x)in Ωu=0on ∂Ω, where Ω⊂RNΩ⊂RN is a bounded domain, 10λ>0.(2)If −1≤qα<0−1≤qα<0, then problem (P) has a solution for all f∈Ls(Ω)f∈Ls(Ω), where s>Nq if N≥2N≥2, and without any restriction on λλ.(3)If q=2q=2 and −1≤qα<0−1≤qα<0 then problem (P) has infinitely many solutions under suitable hypotheses on ff.(4)If 0≤qα0≤qα and f∈L1(Ω)f∈L1(Ω) satisfies λ1(f)=infϕ∈W01,2(Ω)∫Ω|∇ϕ|2dx∫Ωfϕ2dx>0, then problem (P) has a positive solution if 0<λ<λ1(f)0<λ<λ1(f) and no positive solution for large λλ.