This paper deals with Gelfand-type problems 0.1 \(\begin{aligned} \qquad \qquad \left\{ \begin{array}{ll} - \Delta _m u = \lambda f(u), \quad & \hbox {in} \ \Omega , \ \lambda >0, \\ u =0, \quad & \hbox {on} \ \partial _m\Omega , \end{array} \right. \end{aligned}\) in the framework of Random Walk Spaces, which includes as particular cases: Gelfand-type problems posed on locally finite weighted connected graphs and Gelfand-type problems driven by convolution integrable kernels. Under the same assumption on the nonlinearity f as in the local case, we show there exists an extremal parameter \(\lambda ^* \in (0, \infty )\) such that, for \(0 \le \lambda < \lambda ^*\) , problem (0.1) admits a minimal bounded solution \(u_\lambda \) and there are not solution for \(\lambda > \lambda ^*\) . Moreover, assuming f is convex, we show that Problem (0.1) admits a minimal bounded solution for \(\lambda = \lambda ^*\) . We also show that \(u_\lambda \) are stable, and, for f strictly convex, we show that they are the unique stable solutions. We give simple examples that illustrate the many situations that can occur when solving Gelfand-type problems on weighted graphs.
