Positive singular solutions of a certain elliptic PDE
| dc.contributor.author | Mohammadnejad, Negar | |
| dc.contributor.examiningcommittee | Lui, Shiu Hong | |
| dc.contributor.examiningcommittee | Slevinsky, Richard Mikaël | |
| dc.contributor.supervisor | Cowan, Craig | |
| dc.date.accessioned | 2023-07-25T19:36:33Z | |
| dc.date.available | 2023-07-25T19:36:33Z | |
| dc.date.issued | 2023-07-17 | |
| dc.date.submitted | 2023-07-17T17:02:38Z | en_US |
| dc.degree.discipline | Mathematics | en_US |
| dc.degree.level | Master of Science (M.Sc.) | |
| dc.description.abstract | In this thesis, we study the existence of positive singular solutions of a system of partial differential equations on a bounded domain. We first consider the solution of the following problem:\begin{equation} \label{base equation} \left\{ \begin{array}{lr} -\Delta w= | \nabla w|^p& \text{in}~~ B_1 \backslash \{0\},\\ w=0 & \text{on}~~ \partial B_1. \end{array} \right. \end{equation} Then we use its well-known solution to study the positive singular solutions of its perturbations on $B_1$ which is a unit ball centered at the origin in $\mathbb{R}^N$ and where we assume $N\ge 3$ and $\frac{N}{N-1}<p<2$ \begin{equation} \label{main equation od the thesis} \left\{ \begin{array}{lr} -\Delta u= (1+\kappa_1(x)) | \nabla v|^p& \text{in}~~ B_1 \backslash \{0\},\\ -\Delta v= (1+\kappa_2(x)) | \nabla u|^p& \text{in}~~ B_1 \backslash \{0\},\\ u=v=0 & \text{on}~~ \partial B_1. \end{array} \right. \end{equation} In this equation, $\kappa_1$ and $ \kappa_2$ are both non-negative, continuous functions such that $\kappa_1(0)=\kappa_2(0)=0$. We want to show the existence of positive singular solutions of equation \eqref{main equation od the thesis} on the domain. | |
| dc.description.note | October 2023 | |
| dc.identifier.uri | http://hdl.handle.net/1993/37429 | |
| dc.language.iso | eng | |
| dc.rights | open access | en_US |
| dc.subject | partial differential equations | |
| dc.subject | positive singular solutions of PDE | |
| dc.subject | linearization | |
| dc.subject | continuity method | |
| dc.subject | Banach's fixed point theorem | |
| dc.subject | PDE | |
| dc.title | Positive singular solutions of a certain elliptic PDE | |
| dc.type | master thesis | en_US |
| local.subject.manitoba | no | |
| project.funder.identifier | https://doi.org/10.13039/100010318 | |
| project.funder.name | University of Manitoba |