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

Abstract

We study existence and multiplicity of positive ground states for the scalar curvature equation $Delta u+ K(|x|) u^{{n+2}{n-2}}=0$, x in R^n, $n geq 3$ when the function $K:R^+ to R^+$ is bounded above and below by two positive constants, i.e. $\underline{K} leq K(r) leq overline{K}$ for every positive r, it is decreasing in (0,R) and increasing in $(R,+infty)$ for a certain positive constant R. 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 $overline{K} / underline{K}$ which guarantees the existence of a large number of ground states with fast decay, i.e. such that $u(|x|) sim |x|^{2-n}$ as $|x| to +infty$, 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