Uniqueness of positive solutions for a class of one-dimensional $(p,q)$-Laplacian problems

We prove the uniqueness of positive solutions to the one-dimensional $(p,q)$-Laplacian problem cases -(φ (u))=λ h(t)f(u),\ 0< t< 1, \\ u(0)=u(1)=0, cases where φ (u)=|u|ᵖ⁻²u+|u|q-2u, $p>q>1$, h(t)~ t-γ for some γ ∈ 0,1), z⁻ʳf(z) is decreasing for z large for some r∈ (0,q-1), and λ is a large parameter. We allow both the infinite semipositone/positone cases i.e. f(0⁺)∈ \-∞ ,∞ \.