  We study the problem $-\Delta u=\lambda u-u^{-1}$ with a Neumann boundary
condition; the peculiarity being the presence of the singular term $-u^{-1}$.
We point out that the minus sign in front of the negative power of $u$ is
particularly challenging, since no convexity argument can be invoked. Using
bifurcation techniques we are able to prove the existence of solution
$(u_\lambda,\lambda)$ with $u_\lambda$ approaching the trivial constant
solution $u=\lambda^{-1/2}$ and $\lambda$ close to an eigenvalue of a suitable
linearized problem. To achieve this we also need to prove a generalization of a
classical two-branch bifurcation result for potential operators. Next we study
the radial case and show that in this case one of the bifurcation branches is
global and we find the asymptotical behavior of such a branch. This results
allows to derive the existence of multiple solutions $u$ with $\lambda$ fixed.
