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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[39]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[39]

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

[1]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[2]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[3]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[4]

Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065

[5]

Rafael G. L. D'Oliveira, Marcelo Firer. Minimum dimensional Hamming embeddings. Advances in Mathematics of Communications, 2017, 11 (2) : 359-366. doi: 10.3934/amc.2017029

[6]

Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637

[7]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

[8]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935

[9]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1

[10]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228

[11]

Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265

[12]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213

[13]

F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605

[14]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209

[15]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[16]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[17]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[18]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475

[19]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810

[20]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (79)
  • HTML views (64)
  • Cited by (0)

Other articles
by authors

[Back to Top]