Some classes of singular nonlinear elliptic equations

This thesis investigates the Lane-Emden type problem\[ \mathcal{L}u = \sum\limits_{i=1}^{m}\sigma_{i} u^{q_{i}}+\sigma_0, \quad u\geq0 \qquad \text{in} \quad \Omega, \quad\lim \limits_{x \rightarrow y} u(x) = f(y), \qquad y \in \partial^{\infty} \Omega,\]where $ \mathcal{L}u = - \text{div}(\mathcal{A} \nabla u)$ is a uniformly elliptic operator with bounded coefficients, $0 < q_{i} < 1$ with $i= 1,2,\dots,m$, each $\sigma_{i}$ is a nonnegative locally finite Borel measure on a regular domain $\Omega \subset \mathbb{R}^n$ possessing a positive Green function $G$ associated with the operator $\mathcal{L}$ and the boundary function $f \in \mathcal{C}^{+}({\partial^{\infty} \Omega})$. We establish the necessary and sufficient conditions for the existence of positive bounded solutions to the Lane-Emden type problem in the domain $\Omega$ involving several subnatural growth terms with the elliptic operator $\mathcal{L}$ and the fractional Laplacian $(- \Delta)^{\alpha}$ where $0 < \alpha < \frac{n}{2}$ on $\R^n$.As an application, we also find the same results for the continuous solutions to the sublinear elliptic equation\[ - \Delta u = \sum\limits_{i=1}^{m}\sigma_{i} u^{q_{i}}+\sigma_0, \quad u\geq0 \qquad \text{in} \quad \Omega, \quad \lim \limits_{x \rightarrow y} u(x) = f(y) \qquad y \in \partial^{\infty} \Omega.\]