Lyapunov inequalities for partial differential equations

This paper is devoted to the study of Lp Lyapunov-type inequalities (1⩽p⩽+∞) for linear partial differential equations. More precisely, we treat the case of Neumann boundary conditions on bounded and regular domains in RN. It is proved that the relation between the quantities p and N/2 plays a crucial role. This fact shows a deep difference with respect to the ordinary case. The linear study is combined with Schauder fixed point theorem to provide new conditions about the existence and uniqueness of solutions for resonant nonlinear problems.