January  2021, 20(1): 159-191. doi: 10.3934/cpaa.2020262

Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains

Dipartimento di Matematica, Università di Roma "Tor Vergata", via della Ricerca Scientifica 1, 00133 Roma, ITALY

Received  June 2020 Revised  August 2020 Published  January 2021 Early access  October 2020

Fund Project: The research of T. D'Aprile is partially supported by the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome "Tor Vergata", CUP E83C18000100006, and by the group GNAMPA of INdAM Istituto Nazionale di Alta Matematica

We are concerned with the existence of blowing-up solutions to the following boundary value problem
$ -\Delta u = \lambda V(x) e^u-4\pi N {\mathit{\boldsymbol{\delta}}}_0\;\hbox{ in } \Omega, \quad u = 0 \;\hbox{ on }\partial \Omega, $
where
$ \Omega $
is a smooth and bounded domain in
$ \mathbb{R}^2 $
such that
$ 0\in\Omega $
,
$ V $
is a positive smooth potential,
$ N $
is a positive integer and
$ \lambda>0 $
is a small parameter. Here
$ {\mathit{\boldsymbol{\delta}}}_0 $
defines the Dirac measure with pole at
$ 0 $
. We assume that
$ \Omega $
is
$ (N+1) $
-symmetric and we find conditions on the potential
$ V $
and the domain
$ \Omega $
under which there exists a solution blowing up at
$ N+1 $
points located at the vertices of a regular polygon with center
$ 0 $
.
Citation: Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure and Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262
References:
[1]

S. Baraket and F. Pacard, Construction of singular limits for a semilinear elliptic equation in dimension 2, Calc. Var. Partial Differ. Equ., 6 (1998), 1-38.  doi: 10.1007/s005260050080.

[2]

D. BartolucciC. C. ChenC. S. Lin and G. Tarantello, Profile of blow-up solutions to mean field equations with singular data, Commun. Partial Differ. Equ., 29 (2004), 1241-1265.  doi: 10.1081/PDE-200033739.

[3]

D. Bartolucci and G. Tarantello, Liouville type equations with singular data and their application to periodic multivortices for the electroweak theory, Commun. Math. Phys., 229 (2002), 3-47.  doi: 10.1007/s002200200664.

[4]

T. BartschA. Pistoia and T. Weth, $N$-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the $\sinh$-Poisson and the Lane-Emden-Fowler equations, Commun. Math. Phys., 297 (2010), 653-686.  doi: 10.1007/s00220-010-1053-4.

[5]

H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of $-\Delta u = V (x)e^u$ in two dimensions, Commun. Partial Differ. Equ., 16 (1991), 1223-1253.  doi: 10.1080/03605309108820797.

[6]

X. Chen, Remarks on the existence of branch bubbles on the blowup analysis of equation $-\Delta u = e^2u$ in dimension two, Commun. Anal. Geom., 7 (1999), 295-302.  doi: 10.4310/CAG.1999.v7.n2.a4.

[7]

C. C. Chen and C. C. Lin, Mean field equation of Liouville type with singular data: topological degree, Commun. Pure Appl. Math., 68 (2015), 887-947.  doi: 10.1002/cpa.21532.

[8]

C. C. Chen and C. C. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Commun. Pure Appl. Math., 55 (2002), 728-771.  doi: 10.1002/cpa.3014.

[9]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-623.  doi: 10.1215/S0012-7094-91-06325-8.

[10]

T. D'Aprile, Blow-up phenomena for the Liouville equation with a singular source of integer multiplicity, J. Differ. Equ., 266 (2019), 7379-7415.  doi: 10.1016/j.jde.2018.12.005.

[11]

T. D'Aprile, Multiple blow-up solutions for the Liouville equation with singular data, Commun. Partial Differ. Equ., 38 (2013), 1409-1436.  doi: 10.1080/03605302.2013.799487.

[12]

T. D'Aprile and J. Wei, Bubbling solutions for the Liouville equation with singular sources: non-simple blow-up, J. Funct. Anal., 279 (2020), Art. 108605. doi: 10.1016/j.jfa.2020.108605.

[13]

M. Del PinoP. Esposito and M. Musso, Nondegeneracy of entire solutions of a singular Liouville equation, Proc. Am. Math. Soc., 140 (2012), 581-588.  doi: 10.1090/S0002-9939-2011-11134-1.

[14]

M. Del PinoP. Esposito and M. Musso, Two dimensional Euler flows with concentrated vorticities, Trans. Am. Math. Soc., 362 (2010), 6381-6395.  doi: 10.1090/S0002-9947-2010-04983-9.

[15]

M. Del PinoM. Kowalczyk and M. Musso, Singular limits in Liouville-type equation, Calc. Var. Partial Differ. Equ., 24 (2005), 47-81.  doi: 10.1007/s00526-004-0314-5.

[16]

P. Esposito, Blow up solutions for a Liouville equation with singular data, SIAM J. Math. Anal., 36 (2005), 1310-1345.  doi: 10.1137/S0036141003430548.

[17]

P. EspositoM. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 227-257.  doi: 10.1016/j.anihpc.2004.12.001.

[18]

P. EspositoM. Musso and A. Pistoia, On the existence and profile of nodal solutions for a two-dimensional elliptic problem with large exponent in nonlinearity, Proc. Lond. Math. Soc., 94 (2007), 497-519.  doi: 10.1112/plms/pdl020.

[19]

P. EspositoM. Musso and A. Pistoia, Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent, J. Differ. Equ., 227 (2006), 29-68.  doi: 10.1016/j.jde.2006.01.023.

[20]

P. EspositoA. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in $ \mathbb{R}^2$, J. Anal. Math., 100 (2006), 249-280.  doi: 10.1007/BF02916763.

[21]

M. Grossi and A. Pistoia, Multiple blow-up phenomena for the $\sinh$-Poisson equation, Arch. Ration. Mech. Anal., 209 (2013), 287-320.  doi: 10.1007/s00205-013-0625-9.

[22]

T. J. Kuo and C. S. Lin, Estimates of the mean field equations with integer singular sources: non-simple blowup, J. Differ. Geom., 103 (2016), 377-424. 

[23]

Y. Y. Li, On a singularly perturbed elliptic equation, Adv. Differ. Equ., 2 (1997), 955-980. 

[24]

Y. Y. Li and I. Shafrir, Blow-up analysis for solutions of $-\Delta u = V e^u$ in dimension two, Indiana Univ. Math. J., 43 (1994), 1255-1270.  doi: 10.1512/iumj.1994.43.43054.

[25]

L. Ma and J. Wei, Convergence for a Liouville equation, Comment. Math. Helv., 76 (2001), 506-514.  doi: 10.1007/PL00013216.

[26]

C. S. Lin and C. L. Wang, Elliptic functions, Green functions and the mean field equation on tori, Ann. of Math., 172 (2010), 911-954.  doi: 10.4007/annals.2010.172.911.

[27]

C. S. Lin and S. Yan, Bubbling solutions for relativistic abelian Chern-Simons model on a torus, Commun. Math. Phys., 297 (2010), 733-758.  doi: 10.1007/s00220-010-1056-1.

[28]

J. Moser, A sharp form of an inequality by N.Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.

[29]

K. Nagasaki and T. Suzuki, Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities, Asymptotic Anal., 3 (1990), 173-188. 

[30]

J. Prajapat and G. Tarantello., On a class of elliptic problem in $ \mathbb{R}^2$: Symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 967-985.  doi: 10.1017/S0308210500001219.

[31]

T. Suzuki., Two-dimensional Emden-Fowler equation with exponential nonlinearity, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), 493-512. Progr. Nonlinear Differential Equations Appl., 7, Birkhäuser Boston, Boston, MA, 1992.

[32]

G. Tarantello, Analytical Aspects of Liouville-Type Equations with Singular Sources, North-Holland, Amsterdam, 2004. doi: 10.1016/S1874-5733(04)80009-3.

[33]

G. Tarantello, A quantization property for blow up solutions of singular Liouville-type equations, J. Funct. Anal., 219 (2005), 368-399.  doi: 10.1016/j.jfa.2004.07.006.

[34]

G. Tarantello, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4608-0.

[35]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  doi: 10.1512/iumj.1968.17.17028.

[36]

J. WeiD. Ye and F. Zhou, Bubbling solutions for an anisotropic Emden-Fowler equation, Calc. Var. Partial Differ. Equ., 28 (2007), 217-247.  doi: 10.1007/s00526-006-0044-y.

[37]

J. Wei and L. Zhang, Estimates for Liouville equation with quantized singularities, arXiv: 1905.04123.

[38]

V. H. Weston, On the asymptotic solution of a partial differential equation with an exponential nonlinearity, SIAM J. Math. Anal., 9 (1978), 1030-1053.  doi: 10.1137/0509083.

[39]

Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-6548-9.

show all references

References:
[1]

S. Baraket and F. Pacard, Construction of singular limits for a semilinear elliptic equation in dimension 2, Calc. Var. Partial Differ. Equ., 6 (1998), 1-38.  doi: 10.1007/s005260050080.

[2]

D. BartolucciC. C. ChenC. S. Lin and G. Tarantello, Profile of blow-up solutions to mean field equations with singular data, Commun. Partial Differ. Equ., 29 (2004), 1241-1265.  doi: 10.1081/PDE-200033739.

[3]

D. Bartolucci and G. Tarantello, Liouville type equations with singular data and their application to periodic multivortices for the electroweak theory, Commun. Math. Phys., 229 (2002), 3-47.  doi: 10.1007/s002200200664.

[4]

T. BartschA. Pistoia and T. Weth, $N$-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the $\sinh$-Poisson and the Lane-Emden-Fowler equations, Commun. Math. Phys., 297 (2010), 653-686.  doi: 10.1007/s00220-010-1053-4.

[5]

H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of $-\Delta u = V (x)e^u$ in two dimensions, Commun. Partial Differ. Equ., 16 (1991), 1223-1253.  doi: 10.1080/03605309108820797.

[6]

X. Chen, Remarks on the existence of branch bubbles on the blowup analysis of equation $-\Delta u = e^2u$ in dimension two, Commun. Anal. Geom., 7 (1999), 295-302.  doi: 10.4310/CAG.1999.v7.n2.a4.

[7]

C. C. Chen and C. C. Lin, Mean field equation of Liouville type with singular data: topological degree, Commun. Pure Appl. Math., 68 (2015), 887-947.  doi: 10.1002/cpa.21532.

[8]

C. C. Chen and C. C. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Commun. Pure Appl. Math., 55 (2002), 728-771.  doi: 10.1002/cpa.3014.

[9]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-623.  doi: 10.1215/S0012-7094-91-06325-8.

[10]

T. D'Aprile, Blow-up phenomena for the Liouville equation with a singular source of integer multiplicity, J. Differ. Equ., 266 (2019), 7379-7415.  doi: 10.1016/j.jde.2018.12.005.

[11]

T. D'Aprile, Multiple blow-up solutions for the Liouville equation with singular data, Commun. Partial Differ. Equ., 38 (2013), 1409-1436.  doi: 10.1080/03605302.2013.799487.

[12]

T. D'Aprile and J. Wei, Bubbling solutions for the Liouville equation with singular sources: non-simple blow-up, J. Funct. Anal., 279 (2020), Art. 108605. doi: 10.1016/j.jfa.2020.108605.

[13]

M. Del PinoP. Esposito and M. Musso, Nondegeneracy of entire solutions of a singular Liouville equation, Proc. Am. Math. Soc., 140 (2012), 581-588.  doi: 10.1090/S0002-9939-2011-11134-1.

[14]

M. Del PinoP. Esposito and M. Musso, Two dimensional Euler flows with concentrated vorticities, Trans. Am. Math. Soc., 362 (2010), 6381-6395.  doi: 10.1090/S0002-9947-2010-04983-9.

[15]

M. Del PinoM. Kowalczyk and M. Musso, Singular limits in Liouville-type equation, Calc. Var. Partial Differ. Equ., 24 (2005), 47-81.  doi: 10.1007/s00526-004-0314-5.

[16]

P. Esposito, Blow up solutions for a Liouville equation with singular data, SIAM J. Math. Anal., 36 (2005), 1310-1345.  doi: 10.1137/S0036141003430548.

[17]

P. EspositoM. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 227-257.  doi: 10.1016/j.anihpc.2004.12.001.

[18]

P. EspositoM. Musso and A. Pistoia, On the existence and profile of nodal solutions for a two-dimensional elliptic problem with large exponent in nonlinearity, Proc. Lond. Math. Soc., 94 (2007), 497-519.  doi: 10.1112/plms/pdl020.

[19]

P. EspositoM. Musso and A. Pistoia, Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent, J. Differ. Equ., 227 (2006), 29-68.  doi: 10.1016/j.jde.2006.01.023.

[20]

P. EspositoA. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in $ \mathbb{R}^2$, J. Anal. Math., 100 (2006), 249-280.  doi: 10.1007/BF02916763.

[21]

M. Grossi and A. Pistoia, Multiple blow-up phenomena for the $\sinh$-Poisson equation, Arch. Ration. Mech. Anal., 209 (2013), 287-320.  doi: 10.1007/s00205-013-0625-9.

[22]

T. J. Kuo and C. S. Lin, Estimates of the mean field equations with integer singular sources: non-simple blowup, J. Differ. Geom., 103 (2016), 377-424. 

[23]

Y. Y. Li, On a singularly perturbed elliptic equation, Adv. Differ. Equ., 2 (1997), 955-980. 

[24]

Y. Y. Li and I. Shafrir, Blow-up analysis for solutions of $-\Delta u = V e^u$ in dimension two, Indiana Univ. Math. J., 43 (1994), 1255-1270.  doi: 10.1512/iumj.1994.43.43054.

[25]

L. Ma and J. Wei, Convergence for a Liouville equation, Comment. Math. Helv., 76 (2001), 506-514.  doi: 10.1007/PL00013216.

[26]

C. S. Lin and C. L. Wang, Elliptic functions, Green functions and the mean field equation on tori, Ann. of Math., 172 (2010), 911-954.  doi: 10.4007/annals.2010.172.911.

[27]

C. S. Lin and S. Yan, Bubbling solutions for relativistic abelian Chern-Simons model on a torus, Commun. Math. Phys., 297 (2010), 733-758.  doi: 10.1007/s00220-010-1056-1.

[28]

J. Moser, A sharp form of an inequality by N.Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.

[29]

K. Nagasaki and T. Suzuki, Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities, Asymptotic Anal., 3 (1990), 173-188. 

[30]

J. Prajapat and G. Tarantello., On a class of elliptic problem in $ \mathbb{R}^2$: Symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 967-985.  doi: 10.1017/S0308210500001219.

[31]

T. Suzuki., Two-dimensional Emden-Fowler equation with exponential nonlinearity, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), 493-512. Progr. Nonlinear Differential Equations Appl., 7, Birkhäuser Boston, Boston, MA, 1992.

[32]

G. Tarantello, Analytical Aspects of Liouville-Type Equations with Singular Sources, North-Holland, Amsterdam, 2004. doi: 10.1016/S1874-5733(04)80009-3.

[33]

G. Tarantello, A quantization property for blow up solutions of singular Liouville-type equations, J. Funct. Anal., 219 (2005), 368-399.  doi: 10.1016/j.jfa.2004.07.006.

[34]

G. Tarantello, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4608-0.

[35]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.  doi: 10.1512/iumj.1968.17.17028.

[36]

J. WeiD. Ye and F. Zhou, Bubbling solutions for an anisotropic Emden-Fowler equation, Calc. Var. Partial Differ. Equ., 28 (2007), 217-247.  doi: 10.1007/s00526-006-0044-y.

[37]

J. Wei and L. Zhang, Estimates for Liouville equation with quantized singularities, arXiv: 1905.04123.

[38]

V. H. Weston, On the asymptotic solution of a partial differential equation with an exponential nonlinearity, SIAM J. Math. Anal., 9 (1978), 1030-1053.  doi: 10.1137/0509083.

[39]

Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-6548-9.

[1]

Xing Wang, Chang-Qi Tao, Guo-Ji Tang. Differential optimization in finite-dimensional spaces. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1495-1505. doi: 10.3934/jimo.2016.12.1495

[2]

Paolo Maria Mariano. Line defect evolution in finite-dimensional manifolds. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 575-596. doi: 10.3934/dcdsb.2012.17.575

[3]

A. Jiménez-Casas, Mario Castro, Justine Yassapan. Finite-dimensional behavior in a thermosyphon with a viscoelastic fluid. Conference Publications, 2013, 2013 (special) : 375-384. doi: 10.3934/proc.2013.2013.375

[4]

Barbara Panicucci, Massimo Pappalardo, Mauro Passacantando. On finite-dimensional generalized variational inequalities. Journal of Industrial and Management Optimization, 2006, 2 (1) : 43-53. doi: 10.3934/jimo.2006.2.43

[5]

Laurence Cherfils, Hussein Fakih, Alain Miranville. Finite-dimensional attractors for the Bertozzi--Esedoglu--Gillette--Cahn--Hilliard equation in image inpainting. Inverse Problems and Imaging, 2015, 9 (1) : 105-125. doi: 10.3934/ipi.2015.9.105

[6]

Sergey Popov, Volker Reitmann. Frequency domain conditions for finite-dimensional projectors and determining observations for the set of amenable solutions. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 249-267. doi: 10.3934/dcds.2014.34.249

[7]

Duy Phan, Lassi Paunonen. Finite-dimensional controllers for robust regulation of boundary control systems. Mathematical Control and Related Fields, 2021, 11 (1) : 95-117. doi: 10.3934/mcrf.2020029

[8]

Michael L. Frankel, Victor Roytburd. A Finite-dimensional attractor for a nonequilibrium Stefan problem with heat losses. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 35-62. doi: 10.3934/dcds.2005.13.35

[9]

Sami Baraket, Soumaya Sâanouni, Nihed Trabelsi. Singular limit solutions for a 2-dimensional semilinear elliptic system of Liouville type in some general case. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1013-1063. doi: 10.3934/dcds.2020069

[10]

Igor Chueshov, Alexander V. Rezounenko. Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1685-1704. doi: 10.3934/cpaa.2015.14.1685

[11]

Futoshi Takahashi. Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data. Conference Publications, 2013, 2013 (special) : 729-736. doi: 10.3934/proc.2013.2013.729

[12]

Chiara Zanini. Singular perturbations of finite dimensional gradient flows. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 657-675. doi: 10.3934/dcds.2007.18.657

[13]

Futoshi Takahashi. Singular extremal solutions to a Liouville-Gelfand type problem with exponential nonlinearity. Conference Publications, 2015, 2015 (special) : 1025-1033. doi: 10.3934/proc.2015.1025

[14]

Haitao Yang, Yibin Zhang. Boundary bubbling solutions for a planar elliptic problem with exponential Neumann data. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5467-5502. doi: 10.3934/dcds.2017238

[15]

Weiwei Ao. Sharp estimates for fully bubbling solutions of $B_2$ Toda system. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1759-1788. doi: 10.3934/dcds.2016.36.1759

[16]

Jean-François Babadjian, Francesca Prinari, Elvira Zappale. Dimensional reduction for supremal functionals. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1503-1535. doi: 10.3934/dcds.2012.32.1503

[17]

Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032

[18]

Ovidiu Savin. A Liouville theorem for solutions to the linearized Monge-Ampere equation. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 865-873. doi: 10.3934/dcds.2010.28.865

[19]

Zongming Guo, Juncheng Wei. Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2561-2580. doi: 10.3934/dcds.2014.34.2561

[20]

Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, 2021, 29 (5) : 2829-2839. doi: 10.3934/era.2021016

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (249)
  • HTML views (76)
  • Cited by (0)

Other articles
by authors

[Back to Top]