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We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, ∆u + n(n−2) 4 u n+2 n−2 = 0, in the neighbourhood of isolated singularities in the standard Euclidean ball. Although asymptotic radial symmetry for such solutions was proved some time ago, [2], we present a much simpler and more geometric derivation of this fact. We also discuss a refinement, showing that any such solution is asymptotic to one of the deformed radial singular solutions. Finally we give some applications of these refined asymptotics, first to computing the global Pohoẑaev invariants of solutions on the sphere with isolated singularities, and then to the regularity of the moduli space of all such solutions.

Given a nondegenerate minimal hypersurface Σ in a Riemannian manifold, we prove that, for all ε small enough there exists uε, a critical point of the Allen-Cahn energy Eε(u) = ε 2 |∇u| 2 + (1 − u 2 ) 2 , whose nodal set converges to Σ as ε tends to 0. Moreover, if Σ is a volume nondegenerate constant mean curvature hypersurface, then the same conclusion holds with the function uε being a critical point of Eε under some volume constraint.

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The aim of this paper is to prove the existence of weak solutions to the equation ∆u+u p = 0 which are positive in a domain Ω ⊂ R N , vanish at the boundary, and have prescribed isolated singularities. The exponent p is required to lie in the interval (N/(N − 2), (N + 2)/(N − 2)). We also prove the existence of solutions to the equation ∆u + u p = 0 which are positive in a domain Ω ⊂ R n and which are singular along arbitrary smooth k-dimensional submanifolds in the interior of these domains provided p lie in the interval ((n − k)/(n − k − 2), (n − k + 2)/(n − k − 2)). A particular case is when p = (n + 2)/(n − 2), in which case solutions correspond to solutions of the singular Yamabe problem. The method used here is a mixture of different ingredients used by both authors in their separate constructions of solutions to the singular Yamabe problem, along with a new set of scaling techniques.

We extend the results and methods of [6] to prove the existence of constant positive scalar curvature metrics g which are complete and conformal to the standard metric on S N \ Λ, where Λ is a disjoint union of submanifolds of dimensions between 0 and (N − 2)/2. The existence of solutions with isolated singularities occupies the majority of the paper; their existence was previously established by Schoen [12], but the proof we give here, based on the techniques of [6], is more direct, and provides more information about their geometry. When Λ is discrete we also establish that these solutions are smooth points in the moduli spaces of all such solutions introduced and studied in [7] and [8]