Some Existence Results for a Singular Elliptic Problem via Bifurcation Theory

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.