| Abstract: |
| The first goal of this paper is to establish the existence of a positive solution for the singular boundary value problem (1.1), where $\mathcal{B}$ is a general boundary operator of Dirichlet, Neumann or Robin type, either classical or non-classical, not necessarily of Sturm--Liouville type. Since $f(u):=au^p -\lambda u$, $u\geq 0$, is not increasing if $\lambda>0$, the uniqueness of the positive solution is not obvious, even when $a(x)$ is a positive constant. The second goal of this paper is to establish the uniqueness of the positive solution of (1.1) in that case. At a later stage, denoting by $L_\lambda$ the unique positive solution of (1.1) when $a(x)$ is a positive constant, we will characterize the point-wise behavior of $L_\lambda$ as $\lambda \to \pm \infty$. Finally, we establish the uniqueness of the positive solution of (1.1) when $a(x)$ is non-increasing in $[0,R]$, $\lambda \geq 0$, and $\beta < 0$ if $-u`(0)+\beta u(0)=0$. This is a joint work with Juli\`{a}n L\`{o}pez-G\`{o}mez and Andrea Tellini. |
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