Lyapunov inequalities for Partial Differential Equations at radial higher eigenvalues

Details Lyapunov inequalities for Partial Differential Equations at radial higher eigenvalues Lyapunov inequalities for Partial Differential Equations at radial higher eigenvalues

2011-09-23

arxiv.org/abs/1109.5020v1

Mathematics

Analysis of PDEs

15 pages

Scientific paper

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

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