Asymptotic behavior of solutions to the σ k -Yamabe equation near isolated singularities

Abstract

Abstract. σ k -Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In [38] YanYan Li proved that an admissible solution with an isolated singularity at 0 ∈ R n to the σ k -Yamabe equation is asymptotically radially symmetric. In this work we prove that an admissible solution with an isolated singularity at 0 ∈ R n to the σ k -Yamabe equation is asymptotic to a radial solution to the same equation on R n \ {0}. These results generalize earlier pioneering work in this direction on the classical Yamabe equation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelli et al, we formulate and prove a general asymptotic approximation result for solutions to certain ODEs which include the case for scalar curvature and σ k curvature cases. An alternative proof is also provided using analysis of the linearized operators at the radial solutions, along the lines of approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.

Asymptotic Expansions of Solutions of the Yamabe Equation and the σk‐Yamabe Equation near Isolated Singular Points

“…There have been many results related to the theme of estimates (1.5) and (1.6) for other equations. Han, Li, and Teixeira [9] studied the k -Yamabe equation near isolated singularities and derived similar estimates for its solutions. Caffarelli, Jin, Sire, and Xiong [2] studied fractional semilinear elliptic equations with isolated singularities.…”

Semilinear elliptic system with boundary singularity

“…Especially, for the internal isolated singularity of the scalar case, see [3,8,16,20,22]. See also Li [21] and Han-Li-Teixeira [19] for conformally invariant fully nonlinear elliptic equations. The Sobolev critical exponent case p = n+2 n−2 is of particular interest, because the equation connects to the Yamabe problem and the conformal invariance, which leads to a richer isolated singularity structure.…”

Local Analysis of Solutions of Fractional Semi-Linear Elliptic Equations with Isolated Singularities

“…When σ = 1, Theorem 1.1 and Theorem 1.2 were proved in [3] by Caffarelli, Gidas and Spruck. We may also see [16] for this classical case, and [18,12] for some conformally invariant fully nonlinear equations with isolated singularities. A similar upper bound in (3) was obtained in [9] under additional assumptions that the equations are globally satisfied on the whole space and the conformal metric is complete.…”