Principal eigenvalues of elliptic BVPs with glued Dirichlet-Robin mixed boundary conditions. Large potentials on the boundary conditions

Abstract

In this paper we perform a complete study about the existence, uniqueness and simplicity of the principal eigenvalue of a class of elliptic eigenvalue problems with glued Dirichlet-Robin boundary conditions on a component of the boundary of the domain and Dirichlet boundary conditions on the other component of the boundary. Moreover, this principal eigenvalue and its normalized principal eigenfunction are approached in R×H1(Ω) by principal eigen-pairs of boundary eigenvalue problems with classical mixed boundary conditions. In some sense, the glued mixed boundary conditions analyzed in this work may be regarded as classical mixed boundary conditions with large potentials in the Robin boundary condition. The principal eigenvalues analyzed in this paper, play a crucial role in analyzing the existence, uniqueness and asymptotic behavior of the positive solutions of certain kinds of semilinear elliptic boundary value problems with nonlinear boundary conditions and spatial heterogeneities. The main technical tools used to carry out the mathematical analysis of this work are variational and monotonicity techniques. The results in [2], [3] and [4] play a crucial role to obtain some of the results of this work.