Jannat, Farzaneh2024-08-292024-08-292024-08-272024-08-27http://hdl.handle.net/1993/38463This thesis explores a class of nonlinear elliptic partial differential equations known as Hénon-type equations, employed in various scientific domains including physics, biology, and applied mathematics. These equations are particularly useful in modeling pattern formation, reaction-diffusion processes, and population dynamics. The central focus is on determining solutions to equations of the form: \[ \left\{ \begin{array}{lr} Lu = f, & \text{in}~ \Omega,\\ u = 0, & \text{on}~ \partial \Omega. \end{array} \right. \] \noindent We investigate the existence of positive singular solutions for Hénon-type equations involving nonlinear gradient terms. Specifically, we study equations of the form: \[ \left\{ \begin{array}{lr} -\Delta u = (1 + g(x))|x|^\alpha |\nabla u|^p, & \text{in}~ B_1 \backslash \{0\},\\ u = 0, & \text{on}~ \partial B_1, \end{array} \right. \] where \( p > 1 \), \( \alpha > 0 \), \( B_1 \) is the unit ball centered at the origin in \( \mathbb{R}^N \), and \( g(x) \) is a Hölder continuous function with \( g(0) = 0 \). We prove the existence of positive singular solutions under various parameters under various conditions. Moreover, we extend the analysis to exterior domains in \( \mathbb{R}^N \), exploring the existence of positive classical solutions to equations of the form: \[ \left\{ \begin{array}{lr} -\Delta u = |x|^\alpha |\nabla u|^p, & \text{in}~ \Omega,\\ u = 0, & \text{on}~ \partial \Omega, \end{array} \right. \] where \( \Omega \) is an exterior domain that does not contain the origin in its closure. \noindent We also consider the case where the solution has two singular points in $\Omega$. In particular, we consider: \[ \left\{ \begin{array}{lr} -\Delta u = |\nabla u|^p, & \text{in}~ \Omega \backslash \{\xi_1, \xi_2\},\\ u = 0, & \text{on}~ \partial \Omega, \end{array} \right. \] where $\Omega$ is a bounded domain in $\IR^N$ with $ \xi_1,\xi_2 \in \Omega$. \noindent This work contributes to understanding singular solutions in H\'enon-type equations, offering new insights into their behavior across various domains and parameter ranges.engHénon EquationSingular SolutionDirichlet Boundary ConditionLaplace-Beltrami OperatorBanach Fixed Point TheoremInner-Outer Gluing TechniqueMethod of ContinuityPartial Differential Equations (PDEs)doctoral thesis