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We study the problem $-\Delta u=\lambda u-u^{-1}$ with a Neumann boundarycondition; 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 $u$ isparticularly challenging, since no convexity argument can be invoked. Usingbifurcation techniques we are able to prove the existence of solution$(u_\lambda,\lambda)$ with $u_\lambda$ approaching the trivial constantsolution $u=\lambda^{-1/2}$ and $\lambda$ close to an eigenvalue of a suitablelinearized problem. To achieve this we also need to prove a generalization of aclassical two-branch bifurcation result for potential operators. Next we studythe radial case and show that in this case one of the bifurcation branches isglobal and we find the asymptotical behavior of such a branch. This resultsallows to derive the existence of multiple solutions $u$ with $\lambda$ fixed.

Radial Solutions and a Local Bifurcation Result for a Singular Elliptic Problem with Neumann Condition