Existence of solutions for a cantilever beam problem

We are concerned with the fourth-order nonuniform cantilever beam problem(I(x)WΔ∇(x))Δ∇=f(x,W(x)),(I(x)WΔ∇(x))Δ∇=f(x,W(x)),W(a)=WΔ(a)=0W(a)=WΔ(a)=0, I(b)WΔ∇(b)=(I(x)WΔ∇Δ(x))|x=b=0I(b)WΔ∇(b)=(I(x)WΔ∇(x))Δ|x=b=0. Under various assumptions on f we prove the existence of at least one, two, or three positive solutions for this measure chain boundary value problem using the Green function approach.