Asymptotic blow-up analysis for singular Liouville type equations with applications

Abstract

We generalize the pointwise estimates obtained in [2,19] and [34] concerning blow-up solutions of the Liouville type equation:−Δu=|x|2αW(x)euinΩ, with Ω⊂R2 open and bounded, α∈(−1,+∞) and W any Lipschitz continuous function which satisfies 0<a≤W≤b<∞. We focus to the case (left open in [2] and [34]) where the parameter α∈N, whose analysis is much more involved as we need to resolve the difficulty of a genuinely non radial behaviour of blow-up solutions. In the worst situation there is no chance (in general) to resolve the profile in the form of a solution of a Liouville equation in R2, instead we need to adopt iterated blow-up arguments.Next, we refine our blow up analysis to cover a class of planar Liouville type problems (see (1.27)–(1.28) below) arising from the study of Cosmic Strings (cfr. [28,35]). In this context, we are able to distinguish between a single blow-up radial profile and the case of multiple blow-up profiles, typical of non radial solutions. As a consequence we obtain a (radial) symmetry result which is interesting in itself but also contributes towards the “sharp” solvability issue for the planar problem (1.27)–(1.28).