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

Francesca Dalbono, Andrea Sfecci, Matteo Franca

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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
Lingua originale English 1-20 20 Journal of Dynamics and Differential Equations Published - 2020

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