Singular solutions and a critical Yamabe problem revisited

Hardy Chan, Azahara DelaTorre

Published 2019-12-21Version 1

We answer affirmatively a question of Aviles posed in 1983, concerning the construction of singular solutions of semilinear equations without using phase-plane analysis. Our techniques involve a careful gluing in weighted $L^\infty$ spaces, fully exploiting the semilinearity and the stability of the linearized operator in any dimension. By the same machinery, we also provide an alternative proof for the existence of singular solutions for the Yamabe problem whose singular set has maximal dimension $(n-2)/2$. This is already known to Pacard, whose proof involves H\"{o}lder spaces and $L^p$-theory on manifolds. Our approach, however, study the equations in the ambient space and is therefore suitable for generalization to nonlocal problems. In a forthcoming paper, we will prove analogous results in the fractional setting.