# American Institute of Mathematical Sciences

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

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.
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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