The Gelfand problem for the 1-homogeneous p-Laplacian Carmona Tapia, José Molino Salas, Alexis Rossi, Julio D. Gelfand problem Elliptic equations Viscosity solutions In this paper, we study the existence of viscosity solutions to the Gelfand problem for the 1-homogeneous p-Laplacian in a bounded domain Ω ⊂ ℝN, that is, we deal with − 1 p − 1|∇u|2−p div(|∇u|p−2∇u) = λeu in Ω with u = 0 on ∂Ω. For this problem we show that, for p ∈ [2, ∞], there exists a positive critical value λ∗ = λ∗(Ω, N, p) such that the following holds: ∙ If λ < λ∗, the problem admits a minimal positive solution wλ. ∙ If λ > λ∗, the problem admits no solution. Moreover, the branch of minimal solutions {wλ} is increasing with λ. In addition, using degree theory, for fixed p we show that there exists an unbounded continuum of solutions that emanates from the trivial solution u = 0 with λ = 0, and for a small fixed λ we also obtain a continuum of solutions with p ∈ [2, ∞]. 2020-03-02T11:54:34Z 2020-03-02T11:54:34Z 2019 info:eu-repo/semantics/article Tapia, J. C., Salas, A. M., & Rossi, J. D. (2019). The Gelfand problem for the 1-homogeneous p-Laplacian. Advances in Nonlinear Analysis, 8(1), 545-558. http://hdl.handle.net/10481/59925 10.1515/anona-2016-0233 eng http://creativecommons.org/licenses/by/3.0/es/ info:eu-repo/semantics/openAccess Atribución 3.0 España Walter de Gruyter GmbH