Conformal metrics and Ricci tensors in the pseudo-Euclidean space

We consider constant symmetric tensors $T$ on $R^n$, $n\geq 3$, and we study the problem of finding metrics $\bar{g}$ conformal to the pseudo-Euclidean metric $g$ such that $\mbox{Ric}\,\bar{g} = T$. We show that such tensors are determined by the diagonal elements and we obtain explicitly the metrics $\bar{g}$. As a consequence of these results we get solutions globally defined on $R^n$ for the equation $- \varphi \Delta_g \varphi +n \vert\vert\nabla_g \varphi\vert\vert^2/2 + \lambda \varphi^2 = 0.$Moreover, we show that for certain unbounded functions $\overline{K}$ defined on $R^n$, there are metrics conformal to the pseudo-Euclidean metric with scalar curvature $\overline{K}$.