M.Sc Thesis

M.Sc StudentPoliakovsky Arkady
SubjectOn Nonlinear Eigenvalue Problems Involving the
DepartmentDepartment of Mathematics
Supervisor PROF. Itai Shafrir


This thesis is devoted to the study of positive eigenfunctions for the singular eigenvalue problem,

where is  a bounded smooth domain in containing 0, with zero boundary condition on . Here , λ and μ are real parameters and η is a nonnegative function on Ω,  continuous on ,  such that . One is led to the study

of the problem  ( P) from the minimization  problem,


since a positive minimizer for (0.0.1) is an eigenfunction for (P) with . We prove that a minimizer for (0.0.1) exists if and only if , and that the later occurs if and only if  for some critical value  (under certain hypo-

theses on η). In the case  it follows that there is no positive eigenfunction for (P) with in  Nevertheless, we prove that in this case a positive eigenfunction  does exists. This solution is

obtained as the limit,  of the (suitably normalized) minimizers  for the minimization problems,

We show further that  is the unique positive eigenfunction, up to a multiplicative factor.