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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

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 and Continuous Dynamical Systems, 2008, 20 (4) : 877-888. doi: 10.3934/dcds.2008.20.877
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