Estimates of the scalar curvature equation via the method of moving planes III

Isolated singularities of solutions to the Yamabe equation

“…The equation (4) can be seen as a perturbation of the equation (3), and so one can ask under what conditions on K the Fowler solutions still serve as asymptotic models. This question has been studied by C. C. Chen and C. S. Lin (see [4][5][6]9]), whose work has inspired some of the techniques employed in our paper. The main motivation of the present work was to determine whether these asymptotic results could be extended to a more general setting, namely, for an arbitrary background metric.…”

Arbitrarily large solutions of the conformal scalar curvature problem at an isolated singularity

“…In the case of non-constant K, Chen and Lin [2], [5] and Zhang [9] give conditions on K such that every C 2 positive solution u(x) of (1.7) satisfies (1.6).…”

Isolated singularities of nonlinear elliptic inequalities

“…4) was left open in that paper (see [11, open question at the bottom of p. 1887 and conjecture on p. 1889]). Many authors (see for example [1], [2], [3], [4], [5], [6], [7]) have studied the asymptotic behavior at an isolated singularity of solutions of the differential equation −∆u = f (u) under various conditions on the positive function f . Of particular relevance to our results is a result of Lions [8] which states that every C 2 positive solution of −∆u = u p in a punctured neighborhood of the origin in R R R n is asymptotically harmonic as |x| → 0 + provided p < n/(n − 2) (if n = 2, p < ∞).…”