Principal Eigenvalues For Some Nonlocal Problems And Their Applications

In practical applications, reaction-diffusion equations are not able to describe some natural phenomena more accurately in many cases, such as in population ecol-ogy, image processing, material science and so on. In the last ten years, nonlocal dispersal equation which is generated by integral operators is established to de-scribe nonlocal effects. Recently, nonlocal dispersal equations have been considered extensively and there present many results. However, less is known about the steady states of nonlocal equations. This is mainly due to the deficiency of the compactness of nonlocal operators or their inverse, and the regularity of solutions associated with nonlocal equations, which do not only bring many technical difficulties but also lead to the essential change of dynamics. We know that the principal eigenvalue plays a central role in the study of nonlocal equations. Thus, it is of significance to study the existence and properties of the principal eigenvalues of nonlocal operators.Firstly, we study nonlocal weighted Dirichlet eigenvalue problem. By consid-ering the asymptotic behavior of principal eigenvalues of auxiliary eigenvalue prob-lems, we obtain the existence of principal eigenvalues for the weighted eigenvalue problem. Then, combining the properties of nonlocal operators, their variational formulations are obtained as well. However, comparing to the local eigenvalue, it is not enough to ensure the existence of principal eigenvalues only when the weight functions are bounded. That is, the existence of principal eigenvalues depends on the properties of weight functions. Additionally, we discuss the monotonicity and continuous dependence of principal eigenvalues with respect to the weight function by their variational formulations. Moreover, we can extend the existence result to the weighted eigenvalue problem associated with general kernel function.Secondly, we investigate the existence and properties of principal eigenvalues of nonlocal weighted mixed boundary eigenvalue problems and their applications. As there may not always admit principal eigenvalues, the mixed boundary case will be more complicated and the existence of principal eigenvalues depends on the properties of both weight function and kernel function. Here, the existence can be obtained by an auxiliary eigenvalue problem. Then, we discuss the continuous de-pendence of principal eigenvalues on the mixed boundary condition, and the relations between the principal eigenvalues of mixed boundary weighted eigenvalue problem and Dirichlet boundary weighted eigenvalue problems as well as Neumann boundary weighted eigenvalue problems. Furthermore, as an application of our main results, we consider the existence and stability of positive solutions of a nonlocal problem.Finally, we concern with the limit of certain sequences of principal eigenval-ues associated with some nonlocal eigenvalue problems and positive solutions of a nonlocal equation. Here, the existence, uniqueness, stability and the dependence of nontrivial bounded positive solutions on the parameter A are discussed. After establishing the limit of a sequence of principal eigenvalues corresponding to some nonlocal eigenvalue problems, the existence of positive solutions can be obtained. Additionally, we can get the uniqueness and stability of positive solutions by the analysis method. Particularly, when there exists no nontrivial bounded positive solutions, the longtime behavior of solutions corresponding to nonlocal dispersal equations will be different from the results associated with the local version. More-over, the asymptotic behavior of positive solutions of nonlocal stationary problem on the parameter λ is also distinct from the corresponding results associated with some nonlinear elliptic equations.