Lyapunov inequalities for partial differential equations atradial higher eigenvalues

Antonio Cañada; Salvador Villegas

doi:10.3934/dcds.2013.33.111

Article Contents

2013, Volume 33, Issue 1: 111-122. Doi: 10.3934/dcds.2013.33.111

This issue Previous Article Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions Next Article Existence and qualitative properties of solutions for nonlinear Dirichlet problems

Lyapunov inequalities for partial differential equations at radial higher eigenvalues

Antonio Cañada^1, and
Salvador Villegas^1,

1.
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, Campus Fuentenueva. 18071 Granada

Received: August 2011

Revised: November 2011

Published: January 2013

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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 $\Bbb{R}^{N}$. 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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Mathematics Subject Classification: Primary: 35J25, 35B07; Secondary: 35J20.

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References

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