Singular extremal solutions for supercritical elliptic equations in a ball

We study positive singular solutions to the Dirichlet problem for the semilinear elliptic equation Δu+λf(u)=0 in the unit ball on RN. We assume that f has the form f(u)=up+g(u), where p>(N+2)/(N−2) and g(u) is a lower order term. We first show the uniqueness of the singular solution to the problem, and then study the existence of the singular extremal solution. In particular, we show a necessary and sufficient condition for the existence of the singular extremal solution in terms of the weak eigenvalue of the linearized problem.