Positive singular solutions of a certain elliptic PDE

dc.contributor.authorMohammadnejad, Negar
dc.contributor.examiningcommitteeLui, Shiu Hong
dc.contributor.examiningcommitteeSlevinsky, Richard Mikaël
dc.contributor.supervisorCowan, Craig
dc.date.accessioned2023-07-25T19:36:33Z
dc.date.available2023-07-25T19:36:33Z
dc.date.issued2023-07-17
dc.date.submitted2023-07-17T17:02:38Zen_US
dc.degree.disciplineMathematicsen_US
dc.degree.levelMaster of Science (M.Sc.)
dc.description.abstractIn 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.noteOctober 2023
dc.identifier.urihttp://hdl.handle.net/1993/37429
dc.language.isoeng
dc.rightsopen accessen_US
dc.subjectpartial differential equations
dc.subjectpositive singular solutions of PDE
dc.subjectlinearization
dc.subjectcontinuity method
dc.subjectBanach's fixed point theorem
dc.subjectPDE
dc.titlePositive singular solutions of a certain elliptic PDE
dc.typemaster thesisen_US
local.subject.manitobano
project.funder.identifierhttps://doi.org/10.13039/100010318
project.funder.nameUniversity of Manitoba
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