  We study a semilinear elliptic problem with a singular nonlinear term of the
type $g(u)=-u^{-1}$, using a variational approach. Note that the minus sign is
important since the corresponding term in the Euler-Lagrange functional is
concave. Contrary to the convex case there are no solutions for the Dirichlet
problem, due to the power being $-1$. We therefore study the Neumann problem
and prove a local existence result for solutions bifurcating from constant
solutions. In the radial case we show that one of the two bifurcation branches
is global and unbounded, and we find its asympotic behaviour.
