A uniqueness result for a class of singular p-Laplacian Dirichlet problem with non-monotone forcing term

Abstract:

We prove uniqueness of positive solutions for the problem \begin{equation*} -\Delta _{p}u=\lambda \frac {f(u)}{u^{\beta }}\text { in }\Omega , u=0\text { on }\partial \Omega , \end{equation*} where $\beta \in (0,1)$, $1<p\leq 2$, $\Omega$ is bounded domain in $\mathbb {R}^{n}$ with smooth boundary $\partial \Omega$, $f:[0,\infty )\rightarrow (0,\infty )$ is of class $C^{1}$ with $f(z)/z^{\beta }$ decreasing for $z$ large, and $\lambda$ is a large parameter. Here the forcing term $f(z)$ is not required to be increasing even for large $z$.