Lyapunov inequalities for Partial Differential Equations at radial higher eigenvalues

TL;DR

This paper investigates $L_p$ Lyapunov inequalities for radial higher eigenvalues of linear PDEs with Neumann boundary conditions, revealing critical relations between $p$ and $N/2$ that influence inequality optimality.

Contribution

It establishes new optimal Lyapunov inequalities at radial higher eigenvalues, highlighting the role of the relation between $p$ and $N/2$ and analyzing zero distribution of solutions.

Findings

Relation between $p$ and $N/2$ determines inequality nature

Significant differences in solutions based on subcritical, supercritical, or critical cases

Use of minimizing sequences and zero distribution analysis

Abstract

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.