A Class Of Second-order Differential Operator Eigenvalue Problems And Their Applications

This thesis is mainly concerned with a linear eigenvalue problem governed by a second order operator.On the one hand,by applying the approach in functional analysis as well as some skills in nonlinear analysis,a monotonicity result of the principal eigenvalue with respect to the coefficient of the first order term is established.On the other hand,as an application,such a theoretical result is then applied to a competition system and the global dynamics is revealed.The main body of this thesis is divided into the following four chapters:In the first chapter,an introduction on the background and recent development is given.Also,the main problem that is under consideration and the main results we obtained are presented.In the second chapter,a preliminary including the theory of the smoothness of the principal eigenvalue and the theory of monotone dynamical systems is exhibited.The third chapter is devoted to the main proofs.Specifically,in subsection 3.1,focusing on a linear functional and its Fréchet operator,many fundamental and important properties are shown,and based on these,the monotonicity result of the principal eigenvalue is established by considering four cases;in subsection 3.2,a non-existence result of positive steady states is firstly proved by the monotonicity result,and then by appealing to the monotone theory,a picture on the global dynamics of the competitive system is obtained.A short discussion is displayed in Chapter 4,where a comparison between our results and the existing results in related works is included.