Vol. 257, No. 1, 2012

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Resonant solutions and turning points in an elliptic problem with oscillatory boundary conditions

Alfonso Castro and Rosa Pardo

Vol. 257 (2012), No. 1, 75–90
Abstract

We consider the elliptic equation Δu + u = 0 with nonlinear boundary conditions ∂u∕∂n = λu + g(λ,x,u), where the nonlinear term g is oscillatory and satisfies g(λ,x,s)∕s 0 as |s|→ 0. We provide sufficient conditions on g for the existence of sequences of resonant solutions and turning points accumulating to zero.

Keywords
turning points, resonance, stability, instability, multiplicity, Steklov eigenvalues, bifurcation, sublinear oscillating boundary conditions
Mathematical Subject Classification 2010
Primary: 35B32, 35B34, 35B35, 35J25, 58J55
Secondary: 35J60, 35J65
Milestones
Received: 3 June 2011
Revised: 22 November 2011
Accepted: 3 May 2012
Published: 19 June 2012
Authors
Alfonso Castro
Department of Mathematics
Harvey Mudd College
301 Platt Boulevard
Claremont, CA 91711
United States
Rosa Pardo
Departamento de Matemática Aplicada
Universidad Complutense de Madrid
Avenida Complutense s/n
28040 Madrid
Spain