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

DOI: 10.2140/pjm.2012.257.75

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