Qualitative analysis of singular solutions for nonlinear elliptic equations with potentials

We consider the structure of radially symmetric singular solutions for elliptic equations with the Hardy term and power nonlinearity. In the critical case, it is shown that there exists a unique non-oscillatory singular solution, around which infinitely many singular solutions are oscillating. We also study the subcritical and supercritical cases and make clear the difference of structure from the critical case. Our results can be applied to various problems such as the minimizing problem related to the Caffarelli–Kohn–Nirenberg inequality, the scalar field equation and a self-replication model.