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 Δu+K(|x|)un+2n-2=0,x∈Rn,n>2,when the function K: R+→ R+ is bounded above and below by two positive constants, i.e. 0 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 K¯/K̲ 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.