The Gelfand problem for the Infinity Laplacian

Fernando Charro; Byungjae Son; Peiyong Wang; Fernando Charro; Byungjae Son; Peiyong Wang

doi:10.3934/mine.2023022

Mathematics in Engineering

2023, Volume 5, Issue 2: 1-28. doi: 10.3934/mine.2023022

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The Gelfand problem for the Infinity Laplacian

1.
Department of Mathematics, Wayne State University, 656 W. Kirby St., Detroit, MI 48202, USA
2.
Department of Mathematics and Statistics, University of Maine, Orono, ME 04469, USA

Received: 10 December 2021 Revised: 21 February 2022 Accepted: 22 February 2022 Published: 11 April 2022

We study the asymptotic behavior as $ p\to\infty $ of the Gelfand problem

$ \begin{equation*} \left\{ \begin{aligned} -&\Delta_{p} u = \lambda\,e^{u}&& \text{in}\ \Omega\subset \mathbb{R}^n\\ &u = 0 && \text{on}\ \partial\Omega. \end{aligned} \right. \end{equation*} $

Under an appropriate rescaling on $ u $ and $ \lambda $, we prove uniform convergence of solutions of the Gelfand problem to solutions of

$ \left\{ \begin{aligned} &\min\left\{|\nabla{}u|-\Lambda\,e^{u}, -\Delta_{\infty}u\right\} = 0&& \text{in}\ \Omega,\\ &u = 0\ &&\text{on}\ \partial\Omega. \end{aligned} \right. $

We discuss existence, non-existence, and multiplicity of solutions of the limit problem in terms of $ \Lambda $.
- fully nonlinear,
- infinity Laplacian,
- Gelfand problem
Citation: Fernando Charro, Byungjae Son, Peiyong Wang. The Gelfand problem for the Infinity Laplacian[J]. Mathematics in Engineering, 2023, 5(2): 1-28. doi: 10.3934/mine.2023022

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Abstract

We study the asymptotic behavior as $ p\to\infty $ of the Gelfand problem

$ \begin{equation*} \left\{ \begin{aligned} -&\Delta_{p} u = \lambda\,e^{u}&& \text{in}\ \Omega\subset \mathbb{R}^n\\ &u = 0 && \text{on}\ \partial\Omega. \end{aligned} \right. \end{equation*} $

Under an appropriate rescaling on $ u $ and $ \lambda $, we prove uniform convergence of solutions of the Gelfand problem to solutions of

$ \left\{ \begin{aligned} &\min\left\{|\nabla{}u|-\Lambda\,e^{u}, -\Delta_{\infty}u\right\} = 0&& \text{in}\ \Omega,\\ &u = 0\ &&\text{on}\ \partial\Omega. \end{aligned} \right. $

We discuss existence, non-existence, and multiplicity of solutions of the limit problem in terms of $ \Lambda $.

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