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