2024/10/29 by José M. Mazón, Mazon, J. M., A. Molino +3
Mathematics · Computer Science · #Stochastic processes and statistical mechanics #Fixed Point Theorems Analysis #Computational Geometry and Mesh Generation
paper · pdf · doi:10.48550/arxiv.2410.21927
This paper deals with Gelfand-type problems \- Δm u = λf(u), · amp;\hboxin Ω, λ · gt;0,
u =0, · amp;\hboxon ∂mΩ, . 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^* ∈ (0, ∞) such that, for 0 ≤ λ< λ^*, problem \eqrefGelfand10 admits a minimal bounded solution uλ and there are not solution for λ> λ^*. Moreover, assuming f is convex, we show that Problem \eqrefGelfand10 admits a minimal bounded solution for λ= λ^*. We also show that uλ 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.