Uniqueness of positive radial solutions for a class of infinite semipositone $p$-Laplacian problems in a ball

Abstract:

We prove uniqueness of positive radial solutions to the $p$-Laplacian problem \begin{equation*} \left \{ \begin {array}{c} -\Delta _{p}u=\lambda f(u)\text { in }\Omega , \\ u=0\text { on }\partial \Omega , \end{array} \right . \end{equation*} where $\Delta _{p}u=\operatorname {div}(|\nabla u|^{p-2}\nabla u),p\geq 2,\ \Omega$ is the open unit ball in $R^{N}, N>1,\ f:(0,\infty )\rightarrow \mathbb {R}$ is concave, $p$-sublinear at $\infty$ with infinite semipositone structure at $0$, and $\lambda$ is a large parameter.