  Eigenvector-dependent nonlinear eigenvalue problems are considered which
arise from the finite difference discretizations of the Gross-Pitaevskii
equation. Existence and uniqueness of positive eigenvector for both one and two
dimensional cases and existence of antisymmetric eigenvector for one
dimensional case are proved. In order to compute eigenpairs corresponding to
excited states as well as ground state, homotopies for both one and two
dimensional problems are constructed respectively and the homotopy paths are
proved to be regular and bounded. Numerical results are presented to verify the
theories derived for both one and two dimensional problems.
