The Gelfand problem for the 1-homogeneous p-Laplacian
| العنوان: | The Gelfand problem for the 1-homogeneous p-Laplacian |
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| المؤلفون: | Julio D. Rossi, José Carmona Tapia, Alexis Molino Salas |
| المصدر: | Advances in Nonlinear Analysis, Vol 8, Iss 1, Pp 545-558 (2017) Digibug. Repositorio Institucional de la Universidad de Granada instname |
| بيانات النشر: | Walter de Gruyter GmbH, 2017. |
| سنة النشر: | 2017 |
| مصطلحات موضوعية: | Pure mathematics, viscosity solutions, elliptic equations, Gelfand problem, 01 natural sciences, 35j60, Trivial solution, 35j15, 0103 physical sciences, Continuum (set theory), 0101 mathematics, Physics, QA299.6-433, Degree (graph theory), 010102 general mathematics, Elliptic equations, Critical value, 35j70, Homogeneous, Bounded function, Viscosity solutions, Domain (ring theory), p-Laplacian, 010307 mathematical physics, gelfand problem, Analysis |
| الوصف: | 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, ∞]. The first author was partially supported by MINECO–FEDER Grant MTM2015-68210-P (Spain) and Junta de Andalucía FQM-194 (Spain). The second author was partially supported by MINECO–FEDER Grant MTM2015-68210-P (Spain), Junta de Andalucía FQM-116 (Spain) and MINECO Grant BES-2013- 066595 (Spain). The third author was partially supported by CONICET (Argentina) and MINECO–FEDER Grant MTM2015-70227-P (Spain). |
| تدمد: | 2191-950X 2191-9496 |
| URL الوصول: | https://explore.openaire.eu/search/publication?articleId=doi_dedup___::3818c43001ce516143b1dbc9375c48ec https://doi.org/10.1515/anona-2016-0233 |
| حقوق: | OPEN |
| رقم الأكسشن: | edsair.doi.dedup.....3818c43001ce516143b1dbc9375c48ec |
| قاعدة البيانات: | OpenAIRE |
| تدمد: | 2191950X 21919496 |
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