Some elliptic problems involving the gradient on general bounded and exterior domains

In this article we consider the existence of positive singular solutions on bounded domains and also classical solutions on exterior domains. First we consider positive singular solutions of the following problems: \begin{equation} \label{eq_abst_1}-\Delta u = (1+g(x)) | \nabla u|^p \qquad \mbox{ in } B_1, \qquad u = 0 \mbox{ on } \;\; \partial B_1, \qquad \mbox{ and} \end{equation} \begin{equation} \label{eq_abst_2} -\Delta u = | \nabla u|^p \qquad \mbox{ in } \Omega, \qquad u = 0 \mbox{ on } \;\; \partial \Omega. \end{equation} In the first problem $B_1$ is the unit ball in $\mathbb{R}^N$ and in the second $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$. In both cases we assume $N \ge 3$, $\frac{N}{N-1}<p<2$ and in the first problem we assume $g \ge 0$ is a H\"older continuous function with $g(0)=0$. We obtain positive singular solutions in both cases. \\ For the second equation we also consider the case of $\Omega$ an exterior domain $\mathbb{R}^N$ where $N \ge 3$ and $p >\frac{N}{N-1}$. We prove the existence of a bounded positive classical solution with the additional property that $\nabla u(x) \cdot x>0$ for large $|x|$.