2025/11/27 by Tapia, José Carmona, Aparicio, Antonio J. Martínez, Martínez-Aparicio, Pedro J.
Computer Science · Engineering · Mathematics · #35B09 #35B40 #35J61 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · doi:10.48550/arxiv.2511.22733
openalex publication_date 2025/11/27 · openalex created_date 2025/12/03 · openalex updated_date 2026/07/28
In this article, we investigate the existence and multiplicity of solutions to the Robin problem \begincases -Δu = λf(u) amp; in Ω, (∂ u)/(∂ ν) + γu=0 amp; on ∂Ω, \endcases where Ω⊂ ℝN (N≥ 1) is a smooth bounded domain, and λ, γ>0. Our main assumption is that f\colon ℝ→ ℝ is a locally Lipschitz function, possibly sign-changing, such that f(s)>0 for every s∈ (α,β), where 0<α<β are two zeros of f. Without any further conditions, we establish the existence of two nonnegative solutions whose maximum lies in (α,β) for sufficiently large λ. Moreover, we analyse the limiting behaviour of the solution set of this Robin problem, showing that it degenerates into that of the associated Neumann problem as γ→ 0 and into that of the associated Dirichlet problem as γ→∞.