  We consider the curvature of a family of warped products of two
pseduo-Riemannian manifolds $(B,g_B)$ and $(F,g_F)$ furnished with metrics of
the form $c^{2}g_B \oplus w^2 g_F$ and, in particular, of the type $w^{2
\mu}g_B \oplus w^2 g_F$, where $c, w \colon B \to (0,\infty)$ are smooth
functions and $\mu$ is a real parameter. We obtain suitable expressions for the
Ricci tensor and scalar curvature of such products that allow us to establish
results about the existence of Einstein or constant scalar curvature structures
in these categories. If $(B,g_B)$ is Riemannian, the latter question involves
nonlinear elliptic partial differential equations with concave-convex
nonlinearities and singular partial differential equations of the
Lichnerowicz-York type among others.
