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Gelfand-Type problems in Random Walk Spaces

J. M. Mazon, A. Molino, J. Toledo

Abstract

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.

Gelfand-Type problems in Random Walk Spaces

Abstract

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 as in the local case, we show there exists an extremal parameter such that, for , problem \eqref{Gelfand10} admits a minimal bounded solution and there are not solution for . Moreover, assuming is convex, we show that Problem \eqref{Gelfand10} admits a minimal bounded solution for . We also show that are stable, and, for 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.

Paper Structure

This paper contains 12 sections, 20 theorems, 290 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Preliminaires
  3. Random walk spaces
  4. Nonlocal operators
  5. The Lambert $W$ function
  6. Gelfand-Type Problems in Random Walk Spaces
  7. Assumptions on the spaces and previous results
  8. Gelfand-type problems
  9. Stability
  10. The exponential function and power-like functions. Examples
  11. The Gelfand problem
  12. The problem for power-like functions

Key Result

Proposition 2.6

(MST0, MSTBook) Let $[X,\mathcal{B},m,\nu]$ be a random walk space. The following statements are equivalent: (i) $[X,\mathcal{B},m,\nu]$ is $m$-connected. (ii) If $A,B\in\mathcal{B}$ satisfy $A\cup B=X$ and $L_m(A,B)= 0$, then either $\nu(A)=0$ or $\nu(B)=0$.

Figures (22)

  • Figure 1: $-W_0(-\lambda)$ and $-W_{-1}(-\lambda)$.
  • Figure 2: Linear weighted graph in Example \ref{['example4pt']}.
  • Figure 3: Solutions of \ref{['problem3Examxy']}. In dark--gray the graph $x=2(y-\lambda e^y)$, in light--gray $y=2(x-\lambda e^{x})$.
  • Figure 4: Bifurcation diagram for $\lambda\mapsto u(x)$, $x=2,3$, in Example \ref{['example4pt']}. $\lambda^*=\frac{1}{2e}\approx 0.184$.
  • Figure 5: Different weights (symmetric).
  • ...and 17 more figures

Theorems & Definitions (69)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Example 2.8
  • Example 2.9
  • Example 2.10
  • ...and 59 more