𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Multiplicity results near the principal eigenvalue for boundary-value problems with periodic nonlinearity

✍ Scribed by A. Cañada

Publisher
John Wiley and Sons
Year
2007
Tongue
English
Weight
119 KB
Volume
280
Category
Article
ISSN
0025-584X

No coin nor oath required. For personal study only.

✦ Synopsis

Abstract

Let us consider the boundary‐value problem

equation image

where g: ℝ → ℝ is a continuous and T ‐periodic function with zero mean value, not identically zero, (λ, a) ∈ ℝ^2^ and $ \tilde h $ ∈ C [0, π ] with ∫^π^ ~0~ $ \tilde h $(x) sin x dx = 0. If λ ~1~ denotes the first eigenvalue of the associated eigenvalue problem, we prove that if (λ, a) → (λ ~1~, 0), then the number of solutions increases to infinity. The proof combines Liapunov–Schmidt reduction together with a careful analysis of the oscillatory behavior of the bifurcation equation. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

📜 SIMILAR VOLUMES

A Multiplicity Result for a Nonlinear Bo
A Multiplicity Result for a Nonlinear Boundary Value Problem
✍ Kim Goudreau 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 196 KB

In this paper, we establish existence and multiplicity results for the problem Ž . Ž. w x Ž . Ž . xЉ q x q f t, x, xЈ s s t a.e. in 0, ; x 0 s x s ␥ , under the Ambrosetti᎐Prodi type condition, with f being a Caratheodory function.

Multiplicity Results for Boundary Value
Multiplicity Results for Boundary Value Problems with Potentials Oscillating around Resonance
✍ Patrick Habets; Enrico Serra; Massimo Tarallo 📂 Article 📅 1997 🏛 Elsevier Science 🌐 English ⚖ 440 KB

## Consider the Dirichlet boundary value problem ) u=0, on 0, where 0 is a bounded domain R N and \* 1 is the first eigenvalue of &2 in 0, under Dirichlet boundary conditions. Let . 1 be the corresponding eigenfunction. Such a resonance problem is easy to deal with if the potential G(x, u)= | u 0