Optimal Lyapunov inequalities for disfocality and Neumann boundary conditions using L^p norms

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October 2008, 20(4): 877-888. doi: 10.3934/dcds.2008.20.877

Optimal Lyapunov inequalities for disfocality and Neumann boundary conditions using $L^p$ norms

Antonio Cañada ^1, and Salvador Villegas ^1,

1.	Departamento de Análisis Matemático, Universidad de Granada, 18071, Granada, Spain, Spain

Received January 2007 Revised August 2007 Published January 2008

Abstract
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Motivated by the applications to nonlinear resonant boundary value problems with Neumann boundary conditions, this paper is devoted to the study of $L^{p}$ Lyapunov-type inequalities ($1 \leq p \leq \infty$) with mixed boundary conditions. We carry out a complete treatment of the problem for any constant $p \geq 1.$ Our main result is derived from a detailed analysis of the relationship between the existence of nontrivial solutions of these two different boundary problems.

Keywords: existence and uniqueness., disfocality, Lyapunov inequality, mixed boundary conditions, Neumann boundary value problems, resonance.

Mathematics Subject Classification: Primary: 34B05; Secondary: 34B1.

Citation: Antonio Cañada, Salvador Villegas. Optimal Lyapunov inequalities for disfocality and Neumann boundary conditions using $L^p$ norms. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 877-888. doi: 10.3934/dcds.2008.20.877

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