Gelfand-Type problems in Random Walk Spaces
This paper deals with Gelfand-type problems \begin{equation}\label{Gelfand10} \qquad\qquad\left\{\begin{array}{ll} - Δ_m u = λf(u), \quad&\hbox{in} \ Ω, \ λ>0, \\[10pt] u =0, \quad&\hbox{on} \ \partial_mΩ, \end{array} \right. \end{equation} 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 $λ^* \in (0, \infty)$ such that, for $0 \leq λ< λ^*$, problem \eqref{Gelfand10} admits a minimal bounded solution $u_λ$ and there are not solution for $λ> λ^*$. Moreover, assuming $f$ is convex, we show that Problem \eqref{Gelfand10} admits a minimal bounded solution for $λ= λ^*$. We also show that $u_λ$ 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.