An analytic construction of singular solutions related to a critical Yamabe problem

We answer affirmatively a question posed by Aviles in 1983, concerning the construction of singular solutions of semilinear equations without using phase-plane analysis. Fully exploiting the semilinearity and the stability of the linearized operator in any dimension, our techniques involve a careful gluing in weighted L∞ spaces that handles multiple occurrences of criticality, without the need of derivative estimates. The above solution constitutes an Ansatz for the Yamabe problem with a prescribed singular set of maximal dimension (n−2)/2, for which, using the same machinery, we provide an alternative construction to the one given by Pacard. His linear theory uses Lp-theory on manifolds, while our strategy relies solely on asymptotic analysis and is suitable for generalization to non-local problems. Indeed, in a forthcoming paper, we will prove analogous results in the fractional setting.