Let \(\Omega \subset \mathbb {R}^N\; (N \ge 3)\) be a bounded smooth star-shaped domain with \(0 \in \Omega \) . Positive singular solutions with an isolated singular point at \(x=0\) of the Robin problems 0.1 \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=|x|^\alpha u^p \quad & \text { in } \Omega \backslash \{0\}, \\ \frac{\partial u}{\partial \nu }+d u=0 \quad & \hbox { on } \partial \Omega , \end{array} \right. \end{aligned}\) 0.2 \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=|x|^\alpha K(x) u^p \quad & \text { in } \Omega \backslash \{0\}, \\ \frac{\partial u}{\partial \nu }+d u=0 \quad & \hbox { on } \partial \Omega , \end{array} \right. \end{aligned}\) with \(\alpha >-2\) , \(\frac{N+\alpha }{N-2}<p \le p_c (N, \alpha )\) and some \(d>0\) are constructed, where \(K(x) \in C^1( \overline{\Omega })\) is a non-constant function satisfying \(0<a\le K(x) \le b\) with two positive constants a, b, and \(\begin{aligned}p_c (N,\alpha )&:=\frac{(N-2)^2-2(\alpha +2)(\alpha +N)-2 \sqrt{(\alpha +2)^3(\alpha +2N-2)}}{(N-2)(N-10-4\alpha )}\\ &<\frac{N+2+2\alpha }{N-2}.\end{aligned}\) It is seen that the equation in (0.1) or the equation \(\begin{aligned} -\Delta u=|x|^\alpha K(0) u^p \end{aligned}\) with \(p> \frac{N+\alpha }{N-2}\) admits a trivial positive radial singular solution in \(\mathbb {R}^N \backslash \{0\}\) each. The radial singular solution has stable properties for \(\frac{N+\alpha }{N-2}<p \le p_c (N, \alpha )\) . We first construct a family of positive radial singular solutions for homogeneous Dirichlet problems in \(B_R \backslash \{0\}\) with equations in (0.1) and (0.2) respectively, which can be seen as “sub-solutions" to the corresponding Robin problems. Taking the trivial radial singular solution in \(\mathbb {R}^N \backslash \{0\}\) as “super-solutions" to each of the problem, we can construct a family of positive singular solutions for the Robin problems via sub- and super-solution arguments, which extends results of Chiun-Chuan Chen and Chang-Shou Lin (J. Geom. Anal. 9:221-246,1999) to Robin problems with Hénon-Hardy equations and isolated singular points for \(p \in ( \frac{N+\alpha }{N-2}, p_c (N, \alpha )]\) . Our results can be used to obtain a family of positive slow-decay solutions for Steklov boundary value problems in exterior domains.
