Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions

We study positive solutions to (singular) boundary value problems of the form: \begin{document}$\left\{ \begin{align} & -\left( {{\varphi }_{p}}(u') \right)'=\lambda h(t)\frac{f(u)}{{{u}^{\alpha }}},~\ \ t\in (0,1),~~ \\ & u'(1)+c(u(1))u(1)=0,~ \\ & u(0)=0, \\ \end{align} \right.$ \end{document} where \begin{document}$\varphi_p(u): = |u|^{p-2}u$\end{document} with \begin{document}$p>1$\end{document} is the \begin{document}$p$\end{document} -Laplacian operator of \begin{document}$u$\end{document} , \begin{document}$λ>0$\end{document} , \begin{document}$0≤α , \begin{document}$c:[0,∞)\rightarrow (0,∞)$\end{document} is continuous and \begin{document}$h:(0,1)\rightarrow (0,∞)$\end{document} is continuous and integrable. We assume that \begin{document}$f∈ C[0,∞)$\end{document} is such that \begin{document}$f(0) , \begin{document}$\lim_{s\rightarrow ∞}f(s) = ∞$\end{document} and \begin{document}$\frac{f(s)}{s^{α}}$\end{document} has a \begin{document}$p$\end{document} -sublinear growth at infinity, namely, \begin{document}$\lim_{s \rightarrow ∞}\frac{f(s)}{s^{p-1+α}} = 0$\end{document} . We will discuss nonexistence results for \begin{document}$λ≈ 0$\end{document} , and existence and uniqueness results for \begin{document}$λ \gg 1$\end{document} . We establish the existence result by a method of sub-supersolutions and the uniqueness result by establishing growth estimates for solutions.