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This paper is devoted to the study of Lyapunov-type inequality for Neumannboundary conditions at higher eigenvalues. Our main result is derived from adetailed analysis about the number and distribution of zeros of nontrivialsolutions and their first derivatives, together with the use of suitableminimization problems. This method of proof allows to obtain new information onLyapunov constants. For instance, we prove that as in the classical result byLyapunov, the best constant is not attained. Additionally, we exploit therelation between Neumann boundary conditions and disfocality to provide newnonresonance conditions at higher eigenvalues.

Lyapunov inequalities for Neumann boundary conditions at higher eigenvalues