Gelfand-Type problems in Random Walk Spaces

TL;DR

This paper investigates Gelfand-type nonlinear problems within Random Walk Spaces, establishing existence, stability, and uniqueness of solutions depending on a critical parameter, and illustrating diverse behaviors on weighted graphs.

Contribution

It extends Gelfand problems to Random Walk Spaces, proving existence of a critical parameter and solution properties, including stability and uniqueness, in a broad nonlocal framework.

Findings

Existence of a critical parameter mbda* for solutions.

Solutions are stable and unique when the nonlinearity is strictly convex.

Examples demonstrate diverse solution behaviors on weighted graphs.

Abstract

This paper deals with Gelfand-type problems \begin{equation}\label{Gelfand10} \qquad\qquad\left\{\begin{array}{ll} - \Delta_m u = \lambda f(u), \quad&\hbox{in} \ \Omega, \ \lambda >0, \\[10pt] u =0, \quad&\hbox{on} \ \partial_m\Omega, \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 $\lambda^* \in (0, \infty)$ such that, for $0 \leq \lambda < \lambda^*$, problem \eqref{Gelfand10} admits a minimal bounded solution $u_\lambda$ and there are not solution for $\lambda > \lambda^*$. Moreover, assuming $f$ is convex, we show that Problem \eqref{Gelfand10} 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.