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

Abstract

We study asymptotic behaviors of positive solutions to the Yamabe equation and the k -Yamabe equation near isolated singular points and establish expansions up to arbitrary orders. Such results generalize an earlier pioneering work by Caffarelli, Gidas, and Spruck and a work by Korevaar, Mazzeo, Pacard, and Schoen on the Yamabe equation and a work by Han, Li, and Teixeira on the k -Yamabe equation. The study is based on a combination of classification of global singular solutions and an analysis of linearized operators at these global singular solutions. Such linearized equations are uniformly elliptic near singular points for 1 k n=2 and become degenerate for n=2 < k n. In a significant portion of the paper, we establish a degree 1 expansion for the k -Yamabe equation for n=2 < k < n, generalizing a similar result for k D 1 by Korevaar, Mazzeo, Pacard, and Schoen and for 2 k n=2 by Han, Li, and Teixeira.

Local estimates for conformal $Q$-curvature equations

“…Later, Korevaar, Mazzeo, Pacard and Schoen in [32] studied refined asymptotics and expanded such a singular solution u to the first order. Han, Li and Li [24] recently established the expansions up to arbitrary orders for such a singular solution u. Subsequent to [6], other secondorder Yamabe type equations with isolated singularities related to (1.2) have also been studied; see, for example, [9,24,25,33,34,40,52] and the references therein.…”

On the $σ_2$-Nirenberg problem on $\mathbb{S}^2$

“…where u * is some radially symmetric solution of f (λ(A u )) = 1 on R n \{0} and α is some positive number. See [32] for expansions to arbitrary orders. See also [70] for expansions of solutions of conformal quotient equations.…”

“…Recall that when dimension n ≥ 3, the existence of solutions of the σ k -Yamabe problem has been proved for k ≥ n/2, k = 2 or when (M, g) is locally conformally flat, the compactness of the set of solutions has been proved for k ≥ n/2 when the manifold is not conformally equivalent to the standard sphere − they were established in [11,22,27,31,45,54,65]. For more recent works on σ k -Yamabe type problems, see for example [1,3,4,7,8,9,19,20,21,23,30,32,33,38,39,40,41,42,55,56,60,66,67,68] and references therein. However, there are still many challenging open problems on general compact Riemannian manifolds -the compactness remains open for 2 ≤ k ≤ n/2 and the existence remains open for 2 < k < n/2.…”

Local pointwise second derivative estimates for strong solutions to the $$\sigma _k$$-Yamabe equation on Euclidean domains

“…(1.2) and their counterparts on Riemannian manifolds were first studied by Viaclovsky in [63]. Since then, these equations have been addressed by various authors -for a partial list of references, see [1][2][3][8][9][10][11][12]14,[16][17][18][19]24,27,28,[31][32][33]35,[39][40][41]43,44,46,47,53,54,56,64,65] in the positive case and [13,23,25,29,30,42,45,55] in the negative case. When k = 1, these equations reduce to the original Yamabe equation.…”