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This paper is devoted to the study of $L_{p}$ Lyapunov-type inequalities ($ \1 \leq p \leq +\infty$) for linear partial differential equations at radialhigher eigenvalues. More precisely, we treat the case of Neumann boundaryconditions on balls in $\real^{N}$. It is proved that the relation between thequantities $p$ and $N/2$ plays a crucial role to obtain nontrivial and optimalLyapunov inequalities. By using appropriate minimizing sequences and a detailedanalysis about the number and distribution of zeros of radial nontrivialsolutions, we show significant qualitative differences according to the studiedcase is subcritical, supercritical or critical.

Lyapunov inequalities for Partial Differential Equations at radial higher eigenvalues