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 $$ \textstyle\begin{cases} -(\phi (u^{\prime }))^{\prime }=\lambda h(t)f(u),\ 0< t< 1, \\ u(0)=u(1)=0, \end{cases} $$ where $\phi (u^{\prime })=|u^{\prime }|^{p-2}u^{\prime }+|u^{\prime }|^{q-2}u^{ \prime }$, $p>q>1$, $h(t)\sim t^{-\gamma }$ for some $\gamma \in \lbrack 0,1)$, $z^{-r}f(z)$ is decreasing for z large for some $r\in (0,q-1)$, and λ is a large parameter. We allow both the infinite semipositone/positone cases i.e. $f(0^{+})\in \{-\infty ,\infty \}$.