Lyapunov-type Inequalities for Differential Equations

Let us consider the linear boundary value problem u′′(x) + a(x)u(x) = 0, x ∈ (0, L), u′(0) = u′(L) = 0, (0.1) where a ∈ Λ0 and Λ0 is defined by Λ0 = {a ∈ L∞(0, L) \ {0} : Z L 0 a(x) dx ≥ 0, (0.1) has nontrivial solutions}. Classical Lyapunov inequality states that Z L 0 a(x) dx > 4/L for any function a ∈ Λ0, where a(x) = max{a(x), 0}. The constant 4/L is optimal. Let us note that Lyapunov inequality is given in terms of ‖a‖1, the usual norm in the space L(0, L). In this paper we review some recent results on Lp Lyapunovtype inequalities, 1 < p ≤ +∞, for ordinary and partial differential equations on a bounded and regular domain in R . In the last case, it is showed that the relation between the quantities p and N/2 plays a crucial role, pointing out a deep difference with respect to the ordinary case. In the proof, the best constants are obtained by using a related variational problem and Lagrange multiplier theorem. Finally, the linear results are combined with Schauder fixed point theorem in the study of resonant nonlinear problems. Mathematics Subject Classification (2000). Primary 34B05; Secondary 35J25.