Lyapunov inequalities for partial differential equations at radial higher eigenvalues
Antonio Cañada Salvador Villegas
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.
keywords: Neumann boundary value problems Lyapunov inequalities radial eigenvalues. partial differential equations
Some new qualittative properties on the solvability set of pendulum-type equations
Antonio Cañada Antonio J. Ureña
Please refer to Full Text.
keywords: Pendulum-type equations asymptotic results linear damping Baire category. Dirichlet boundary conditions solvability set Riemann-Lebesgue lemma
Optimal Lyapunov inequalities for disfocality and Neumann boundary conditions using $L^p$ norms
Antonio Cañada Salvador Villegas
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

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