We study existence and multiplicity of radial 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, 1) and increasing in (1 , + ∞). Chen and Lin (Commun Partial Differ Equ 24:785–799, 1999) had shown the existence of a large number of bubble tower solutions if K is a sufficiently small perturbation of a positive constant. Our main purpose is to improve such a result by considering a non-perturbative situation: we are able to prove multiplicity assuming that the ratio K¯/K̲ is smaller than some computable values.
Francesca Dalbono, M.F. (2020). Multiplicity of ground states for the scalar curvature equation. ANNALI DI MATEMATICA PURA ED APPLICATA, 199(1), 273-298 [10.1007/s10231-019-00877-2].
Multiplicity of ground states for the scalar curvature equation
Francesca Dalbono;
2020-01-01
Abstract
We study existence and multiplicity of radial 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, 1) and increasing in (1 , + ∞). Chen and Lin (Commun Partial Differ Equ 24:785–799, 1999) had shown the existence of a large number of bubble tower solutions if K is a sufficiently small perturbation of a positive constant. Our main purpose is to improve such a result by considering a non-perturbative situation: we are able to prove multiplicity assuming that the ratio K¯/K̲ is smaller than some computable values.
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