Gelfand-Type problems in Random Walk Spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916459346657280 |
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| author | Mazon, J. M. Molino, A. Toledo, J. |
| author_facet | Mazon, J. M. Molino, A. Toledo, J. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_21927 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gelfand-Type problems in Random Walk Spaces Mazon, J. M. Molino, A. Toledo, J. Analysis of PDEs 35R02, 05C81, 05C81, 35A15, 53J61 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. |
| title | Gelfand-Type problems in Random Walk Spaces |
| topic | Analysis of PDEs 35R02, 05C81, 05C81, 35A15, 53J61 |
| url | https://arxiv.org/abs/2410.21927 |