On the solvability of singular Liouville equations on compact surfaces of arbitrary genus
Author:
Alessandro Carlotto
Journal:
Trans. Amer. Math. Soc. 366 (2014), 12371256
MSC (2010):
Primary 35A01, 35J15, 35J20, 35J61, 35J75, 35R01
Published electronically:
September 19, 2013
MathSciNet review:
3145730
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Abstract: In the first part of this article, we complete the program announced in the preliminary note by the author and Malchiodi by proving a conjecture presented in a previous paper that states the equivalence of contractibility and stability for generalized spaces of formal barycenters and hence we get purely algebraic conditions for the solvability of the singular Liouville equation on Riemann surfaces. This relies on a structure decomposition theorem for in terms of maximal strata, and on elementary combinatorial arguments based on the selection rules that define such spaces. Moreover, we also show that these solvability conditions on the parameters are not only sufficient, but also necessary, at least when for some the value approaches . This disproves a conjecture made in Section 3 of a paper by Tarantello and gives the first nonexistence result for this class of PDEs without any genus restriction. The argument we present is based on a combined use of the maximum/comparison principle and of a Pohozaev type identity and applies for an arbitrary choice both of the underlying metric of and of the datum .
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 A. AMBROSETTI, A. MALCHIODI, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Univ. Press, Cambridge Studies in Advanced Mathematics 104, 2007. MR 2292344 (2008k:35129)
 2.
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 3.
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 5.
 D. BARTOLUCCI, F. DE MARCHIS, A. MALCHIODI, Supercritical conformal metrics on surfaces with conical singularities, I. M. R. N. 2011 (2011), no. 24, 56255643. MR 2863376
 6.
 D. BARTOLUCCI, C. S. LIN, G. TARANTELLO, Uniqueness and symmetry results for solutions of a mean field equation on via a new bubbling phenomenon, Comm. Pure Appl. Math. 64 (2011), 16771730. MR 2838340
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 D. BARTOLUCCI, E. MONTEFUSCO, Blowup analysis, existence and qualitative properties of solutions of the twodimensional EmdenFowler equation with singular potential, Math. Meth. Appl. Sci. 30 (2007), 23092327. MR 2362955 (2008k:35131)
 8.
 D. BARTOLUCCI, G. TARANTELLO, Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory, Comm. Math. Phys. 229 (2002), 347. MR 1917672 (2003e:58026)
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 A. CARLOTTO, A. MALCHIODI, Weighted barycentric sets and singular Liouville equations on compact surfaces, Journal of Functional Analysis 262 (2012), no. 2, 409450. MR 2854708
 12.
 C. C. CHEN, C. S. LIN, Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math. 56 (2003), no. 12, 16671727. MR 2001443 (2004h:35065)
 13.
 F. DE MARCHIS, Multiplicity result for a scalar field equation on compact surfaces, Comm. in Part. Diff. Eq. 33 (2008), no. 12, 22082224. MR 2475336 (2010b:53060)
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 Z. DJADLI, A. MALCHIODI, Existence of conformal metrics with constant curvature, Ann. Math. 168 (2008), pp. 813858. MR 2456884 (2009h:53074)
 17.
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Additional Information
Alessandro Carlotto
Affiliation:
Department of Mathematics, Stanford University, Stanford, California 94305
Email:
carlotto@stanford.edu
DOI:
http://dx.doi.org/10.1090/S000299472013058473
Keywords:
Singular Liouville equations,
formal barycenters,
Pohozaev identity.
Received by editor(s):
November 21, 2011
Published electronically:
September 19, 2013
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
