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

Abstract: 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 $|x|$.