We generalize the pointwise estimates obtained in [2,19] and [34] concerning blow-up solutions of the Liouville type equation: with open and bounded, and W any Lipschitz continuous function which satisfies . We focus to the case (left open in [2] and [34]) where the parameter , 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 , 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).

Bartolucci, D., & Tarantello, G. (2017). Asymptotic blow-up analysis for singular Liouville type equations with applications. JOURNAL OF DIFFERENTIAL EQUATIONS, 262, 3887-3931 [10.1016/j.jde.2016.12.003].

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

Bartolucci D;Tarantello G
2017-04-05

Abstract

We generalize the pointwise estimates obtained in [2,19] and [34] concerning blow-up solutions of the Liouville type equation: with open and bounded, and W any Lipschitz continuous function which satisfies . We focus to the case (left open in [2] and [34]) where the parameter , 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 , 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).
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - Analisi Matematica
English
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Bartolucci, D., & Tarantello, G. (2017). Asymptotic blow-up analysis for singular Liouville type equations with applications. JOURNAL OF DIFFERENTIAL EQUATIONS, 262, 3887-3931 [10.1016/j.jde.2016.12.003].
Bartolucci, D; Tarantello, G
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/198720
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