Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result becomes false for some arbitrarily small, smooth perturbations of the potential.
"Examples of non-isolated blow-up for perturbations of the scalar curvature equation on non-locally conformally flat manifolds." J. Differential Geom. 98 (2) 349 - 356, October 2014. https://doi.org/10.4310/jdg/1406552253