Singular solutions for semilinear elliptic equations with general supercritical growth

A positive radial singular solution for Δ u+f(u)=0 Δ u + f ( u ) = 0 with a general supercritical growth is constructed. An exact asymptotic expansion as well as its uniqueness in the space of radial functions are also established. These results can be applied to the bifurcation problem Δ u+λ f(u)=0 Δ u + λ f ( u ) = 0 on a ball. Our method can treat a wide class of nonlinearities in a unified way, e.g., uᵖlog u u p log u , exp (uᵖ) exp ( u p ) and exp (⋯ exp (u)⋯ ) exp ( ⋯ exp ( u ) ⋯ ) as well as uᵖ u p and eᵘ e u . Main technical tools are intrinsic transformations for semilinear elliptic equations and ODE techniques.