On the solvability of singular Liouville equations on compact surfaces of arbitrary genus

Author:
Alessandro Carlotto

Journal:
Trans. Amer. Math. Soc. **366** (2014), 1237-1256

MSC (2010):
Primary 35A01, 35J15, 35J20, 35J61, 35J75, 35R01

Published electronically:
September 19, 2013

MathSciNet review:
3145730

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 non-existence 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 .

**1.**Antonio Ambrosetti and Andrea Malchiodi,*Nonlinear analysis and semilinear elliptic problems*, Cambridge Studies in Advanced Mathematics, vol. 104, Cambridge University Press, Cambridge, 2007. MR**2292344****2.**Thierry Aubin,*Some nonlinear problems in Riemannian geometry*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR**1636569****3.**A. Bahri and J.-M. Coron,*On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain*, Comm. Pure Appl. Math.**41**(1988), no. 3, 253–294. MR**929280**, 10.1002/cpa.3160410302**4.**Daniele Bartolucci, Chiun-Chuan Chen, Chang-Shou Lin, and Gabriella Tarantello,*Profile of blow-up solutions to mean field equations with singular data*, Comm. Partial Differential Equations**29**(2004), no. 7-8, 1241–1265. MR**2097983**, 10.1081/PDE-200033739**5.**Daniele Bartolucci, Francesca De Marchis, and Andrea Malchiodi,*Supercritical conformal metrics on surfaces with conical singularities*, Int. Math. Res. Not. IMRN**24**(2011), 5625–5643. MR**2863376**, 10.1093/imrn/rnq285**6.**Daniele Bartolucci, Chang-Shou Lin, and Gabriella Tarantello,*Uniqueness and symmetry results for solutions of a mean field equation on 𝕊² via a new bubbling phenomenon*, Comm. Pure Appl. Math.**64**(2011), no. 12, 1677–1730. MR**2838340**, 10.1002/cpa.20385**7.**Daniele Bartolucci and Eugenio Montefusco,*Blow-up analysis, existence and qualitative properties of solutions for the two-dimensional Emden-Fowler equation with singular potential*, Math. Methods Appl. Sci.**30**(2007), no. 18, 2309–2327. MR**2362955**, 10.1002/mma.887**8.**D. Bartolucci and G. Tarantello,*Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory*, Comm. Math. Phys.**229**(2002), no. 1, 3–47. MR**1917672**, 10.1007/s002200200664**9.**Haïm Brezis and Frank Merle,*Uniform estimates and blow-up behavior for solutions of -Δ𝑢=𝑉(𝑥)𝑒^{𝑢} in two dimensions*, Comm. Partial Differential Equations**16**(1991), no. 8-9, 1223–1253. MR**1132783**, 10.1080/03605309108820797**10.**Alessandro Carlotto and Andrea Malchiodi,*A class of existence results for the singular Liouville equation*, C. R. Math. Acad. Sci. Paris**349**(2011), no. 3-4, 161–166 (English, with English and French summaries). MR**2769900**, 10.1016/j.crma.2010.12.016**11.**Alessandro Carlotto and Andrea Malchiodi,*Weighted barycentric sets and singular Liouville equations on compact surfaces*, J. Funct. Anal.**262**(2012), no. 2, 409–450. MR**2854708**, 10.1016/j.jfa.2011.09.012**12.**Chiun-Chuan Chen and Chang-Shou Lin,*Topological degree for a mean field equation on Riemann surfaces*, Comm. Pure Appl. Math.**56**(2003), no. 12, 1667–1727. MR**2001443**, 10.1002/cpa.10107**13.**Francesca De Marchis,*Multiplicity result for a scalar field equation on compact surfaces*, Comm. Partial Differential Equations**33**(2008), no. 10-12, 2208–2224. MR**2475336**, 10.1080/03605300802523446**14.**Weiyue Ding, Jürgen Jost, Jiayu Li, and Guofang Wang,*The differential equation Δ𝑢=8𝜋-8𝜋ℎ𝑒^{𝑢} on a compact Riemann surface*, Asian J. Math.**1**(1997), no. 2, 230–248. MR**1491984**, 10.4310/AJM.1997.v1.n2.a3**15.**Zindine Djadli,*Existence result for the mean field problem on Riemann surfaces of all genuses*, Commun. Contemp. Math.**10**(2008), no. 2, 205–220. MR**2409366**, 10.1142/S0219199708002776**16.**Zindine Djadli and Andrea Malchiodi,*Existence of conformal metrics with constant 𝑄-curvature*, Ann. of Math. (2)**168**(2008), no. 3, 813–858. MR**2456884**, 10.4007/annals.2008.168.813**17.**Jerry L. Kazdan and F. W. Warner,*Curvature functions for compact 2-manifolds*, Ann. of Math. (2)**99**(1974), 14–47. MR**0343205****18.**Yan Yan Li,*Harnack type inequality: the method of moving planes*, Comm. Math. Phys.**200**(1999), no. 2, 421–444. MR**1673972**, 10.1007/s002200050536**19.**Yan Yan Li and Itai Shafrir,*Blow-up analysis for solutions of -Δ𝑢=𝑉𝑒^{𝑢} in dimension two*, Indiana Univ. Math. J.**43**(1994), no. 4, 1255–1270. MR**1322618**, 10.1512/iumj.1994.43.43054**20.**Andrea Malchiodi and David Ruiz,*New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces*, Geom. Funct. Anal.**21**(2011), no. 5, 1196–1217. MR**2846387**, 10.1007/s00039-011-0134-7**21.**M. SPIVAK,*A Comprehensive Introduction to Differential Geometry*, Publish or Perish Ed., Berkeley, 1979.**22.**Gabriella Tarantello,*Analytical, geometrical and topological aspects of a class of mean field equations on surfaces*, Discrete Contin. Dyn. Syst.**28**(2010), no. 3, 931–973. MR**2644774**, 10.3934/dcds.2010.28.931**23.**Marc Troyanov,*Prescribing curvature on compact surfaces with conical singularities*, Trans. Amer. Math. Soc.**324**(1991), no. 2, 793–821. MR**1005085**, 10.1090/S0002-9947-1991-1005085-9

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
35A01,
35J15,
35J20,
35J61,
35J75,
35R01

Retrieve articles in all journals with MSC (2010): 35A01, 35J15, 35J20, 35J61, 35J75, 35R01

Additional Information

**Alessandro Carlotto**

Affiliation:
Department of Mathematics, Stanford University, Stanford, California 94305

Email:
carlotto@stanford.edu

DOI:
https://doi.org/10.1090/S0002-9947-2013-05847-3

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.