This 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.