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Abstract

We prove the existence and multiplicity of positive radial solution to the problem \[ \left\{ \begin{array}{@{}l@{}} -\varDelta u=\lambda K(|x|)f(u)\quad \text{in}\ \varOmega, \\ \frac{\partial u}{\partial n}+\tilde{c}(u)u=0\ \text{on}\ |x|=r_{0},\quad u(x)\rightarrow 0\text{ as }|x|\rightarrow \infty , \end{array} \right. \] where $ \varOmega =\{x\in \mathbb{R}^{N}:|x| \gt r_{0} \gt 0\}$, $N \gt 2$, $f:(0,\infty )\rightarrow (0,\infty )$ is continuous, superlinear at $ \infty ,$ and is allowed to be singular at $ 0$.