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2022 Gelfand-type problems involving the 1-Laplacian operator
Alexis Molino, Sergio Segura de León
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Publ. Mat. 66(1): 269-304 (2022). DOI: 10.5565/PUBLMAT6612211

Abstract

In this paper, the theory of Gelfand problems is adapted to the 1-Laplacian setting. Concretely, we deal with the following problem:

Δ1u=λf(u)in Ω,u=0on Ω,

where ΩN (N1) is a domain, λ0, and f:[0,+[]0,+[ is any continuous increasing and unbounded function with f(0)>0.

We prove the existence of a threshold λ*=h(Ω)f(0) (h(Ω) being the Cheeger constant of Ω) such that there exists no solution when λ>λ* and the trivial function is always a solution when λλ*. The radial case is analyzed in more detail, showing the existence of multiple (even singular) solutions as well as the behavior of solutions to problems involving the p-Laplacian as p tends to 1, which allows us to identify proper solutions through an extra condition.

Citation

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Alexis Molino. Sergio Segura de León. "Gelfand-type problems involving the 1-Laplacian operator." Publ. Mat. 66 (1) 269 - 304, 2022. https://doi.org/10.5565/PUBLMAT6612211

Information

Received: 25 May 2020; Accepted: 22 December 2020; Published: 2022
First available in Project Euclid: 4 January 2022

Digital Object Identifier: 10.5565/PUBLMAT6612211

Subjects:
Primary: 35J20 , 35J75 , 35J92

Keywords: 1-Laplacian operator , Gelfand problem , nonlinear elliptic equations

Rights: Copyright © 2022 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.66 • No. 1 • 2022
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