Lyapunov inequalities for Partial Differential Equations at radial higher eigenvalues

Abstract

This paper is devoted to the study of $L_p$ Lyapunov-type inequalities ($1\le p\le +\infty$) for linear partial differential equations at radial higher eigenvalues. More precisely, we treat the case of Neumann boundary conditions on balls in ${\Re }^N$. 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.

15 pages