The Gelfand problem for the 1-homogeneous p-Laplacian
Metadatos
Mostrar el registro completo del ítemEditorial
Walter de Gruyter GmbH
Materia
Gelfand problem Elliptic equations Viscosity solutions
Fecha
2019Referencia bibliográfica
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.
Patrocinador
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).Resumen
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, ∞].