Positive singular solutions of a certain elliptic PDE

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Date
2023-07-17
Authors
Mohammadnejad, Negar
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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.
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partial differential equations, positive singular solutions of PDE, linearization, continuity method, Banach's fixed point theorem, PDE
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