We consider the elliptic equation −u+u = 0 with nonlinear boundary condition ∂u ∂n = λu + g(λ, x, u), where g(λ,x,s) s → 0, as |s|→∞ and g is oscillatory. We provide sufficient conditions on g for the existence of unbounded sequences of stable solutions, unstable solutions, and turning points, even in the absence of resonant solutions.
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Castro, A., Pardo, R. Infinitely Many Stability Switches in a Problem with Sublinear Oscillatory Boundary Conditions. J Dyn Diff Equat 29, 485–499 (2017).