M.Sc Thesis | |

M.Sc Student | Poliakovsky Arkady |
---|---|

Subject | On Nonlinear Eigenvalue Problems Involving the p-Laplacian |

Department | Department 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,

(0.0.1)

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.