Gelfand type problems involving the 1-Laplacian operator

2020/05/27 by Molino, Alexis, de León, Sergio Segura
#35J20 #35J75 #35J92 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2005.13657

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) · amp;\hboxin Ω ;
u=0 · amp;\hboxon ∂Ω ; . where Ω⊂ℝ^N (N≥1) is a domain, λ≥ 0 and f>:>[0,+∞[→]0,+∞[ is any continuous increasing and unbounded function with f(0)>0. It is proved the existence of a threshold λ^*=(h(Ω))/(f(0)) (being h(Ω) the Cheeger constant of Ω) such that there exists no solution when λ>λ^* and the trivial function is always a solution when λ≤λ^*. The radial case is analyzed in more detail showing the existence of multiple solutions (even singular) as well as the behaviour of solutions to problems involving the p--Laplacian as p tends to 1, which allows us to identify proper solutions through an extra condition.

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