On a second order eigenvalue problem and its application

Abstract

A linear eigenvalue problem governed by a second order differential equation with separate and general boundary conditions is considered and a new monotonicity result on the principal eigenvalue with respect to the coefficient of the advection term is established. The main approach is based on the functional proposed by Liu and Lou [16] and a key finding lies in the nice properties of the associated Fréchet operator when confined at suitable points and function spaces. As an application, this monotonicity result is used to study a class of competitive parabolic systems and the so-called “competitive exclusion principle” is observed in a larger parameter region than several existing works, which is a nontrivial improvement.