Multiplicity of Radial Ground States for the Scalar Curvature Equation Without Reciprocal Symmetry

We study existence and multiplicity of positive ground states for the scalar curvature equation **formula** when the function K:R+→R+ is bounded above and below by two positive constants, i.e. **formula** for every r>0, it is decreasing in (0,R) and increasing in (R,+∞) for a certain R>0. We recall that in this case ground states have to be radial, so the problem is reduced to an ODE and, then, to a dynamical system via Fowler transformation. We provide a smallness non perturbative (i.e. computable) condition on the ratio **formula** which guarantees the existence of a large number of ground states with fast decay, i.e. such that u(|x|)∼|x|2−n as |x|→+∞, which are of bubble-tower type. We emphasize that if K(r) has a unique critical point and it is a maximum the radial ground state with fast decay, if it exists, is unique.