American Institute of Mathematical Sciences

Advanced
July  2016, 15(4): 1309-1333. doi: 10.3934/cpaa.2016.15.1309

Soliton solutions for a quasilinear Schrödinger equation with critical exponent

Wentao Huang 1, and  Jianlin Xiang 2,

1. 

Department of Mathematics, Central China Normal University, Wuhan, 430079, China
2. 

Department of Mathematics, Wuhan University of Technology, Wuhan, 430070, China

Received  October 2015 Revised  January 2016 Published  April 2016

This paper is concerned with the existence of soliton solutions for a quasilinear Schrödinger equation in $R^N$ with critical exponent, which appears from modelling the self-channeling of a high-power ultrashort laser in matter. By working with a perturbation approach which was initially proposed in [26], we prove that the given problem has a positive ground state solution.
Keywords: Ground state, quasilinear Schrödinger equation, positive solution, perturbation method., critical exponent.
Mathematics Subject Classification: Primary: 35J62; Secondary: 35B09, 35B3.
Citation: 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
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Func. Anal.}, 14 (1973), 349.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 437.  doi: 10.1002/cpa.3160360405.  Google Scholar

[3]

João M. Bezerra do Ó, Olímpio H. Miyagaki and Sérgio H.M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth,, \emph{J. Differential Equations}, 248 (2010), 722.  doi: 10.1016/j.jde.2009.11.030.  Google Scholar

[4]

H. Brandi, C. Manus, G. Mainfray, T. Lehner and G. Bonnaud, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma,, \emph{Phys. Fluids B}, 5 (1993), 3539.  doi: 10.1063/1.860828.  Google Scholar

[5]

F.G. Bass and N.N. Nasanov, Nonlinear electromagnetic-spin waves,, \emph{Phys. Rep.}, 189 (1990), 165.  doi: 10.1016/0370-1573(90)90093-H.  Google Scholar

[6]

L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations,, \emph{Exposition. Math.}, 4 (1986), 279.   Google Scholar

[7]

X.L. Chen and R.N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma,, \emph{Phys. Rev. Lett.}, 70 (1993), 2082.  doi: 10.1103/PhysRevLett.70.2082.  Google Scholar

[8]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach,, \emph{Nonlinear Anal. TMA.}, 56 (2004), 213.  doi: 10.1016/j.na.2003.09.008.  Google Scholar

[9]

S. Cuccagna, On instability of excited states of the nonlinear Schrödinger equation,, \emph{Physica D}, 238 (2009), 38.  doi: 10.1016/j.physd.2008.08.010.  Google Scholar

[10]

A. De Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation,, \emph{Commun. Math. Phys.}, 189 (1997), 73.  doi: 10.1007/s002200050191.  Google Scholar

[11]

Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent,, \emph{J. Math. Phys.}, 54 (2013).  doi: 10.1063/1.4774153.  Google Scholar

[12]

Y. Deng, S. Peng and J. Wang, Infinitely many sign-changing solutions for quasilinear Schrödinger equations in $R^N$,, \emph{Commun. Math. Sci.}, 9 (2011), 859.  doi: 10.4310/CMS.2011.v9.n3.a9.  Google Scholar

[13]

Y. Deng, S. Peng and S. Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth,, \emph{J. Differential Equations}, 258 (2015), 115.  doi: 10.1016/j.jde.2014.09.006.  Google Scholar

[14]

Y. Deng, S. Peng and S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations,, \emph{J. Differential Equations}, 260 (2016), 1228.  doi: 10.1016/j.jde.2015.09.021.  Google Scholar

[15]

Q. Han and F. Lin, Elliptic Partial Differential Equations,, Courant Lecture Notes in Mathematics, (1997).   Google Scholar

[16]

R.W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations,, \emph{Z. Phys. B}, 37 (1980), 83.   Google Scholar

[17]

A.M. Kosevich, B.A. Ivanov and A.S. Kovalev, Magnetic solitons,, \emph{Phys. Rep.}, 194 (1990), 117.  doi: doi:10.1016/0370-1573(90)90130-T.  Google Scholar

[18]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films,, \emph{J. Phys. Soc. Japan}, 50 (1981), 3262.  doi: 10.1143/JPSJ.50.3262.  Google Scholar

[19]

E. Laedke, K. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions,, \emph{J. Math. Phys.}, 24 (1983), 2764.  doi: 10.1063/1.525675.  Google Scholar

[20]

P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part I},, \emph{Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire}, 1 (1984), 109.   Google Scholar

[21]

P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part II,, \emph{Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire}, 1 (1984), 223.   Google Scholar

[22]

H. Lange, M. Poppenberg and H. Teismann, Nash-Moser methods for the solution of quasilinear Schrödinger equations,, \emph{Comm. Partial Differential Equations}, 24 (1999), 1399.  doi: 10.1080/03605309908821469.  Google Scholar

[23]

J. Liu, Y. Wang and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. II,, \emph{J. Differential Equations}, 187 (2003), 473.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[24]

J. Liu, Y. Wang and Z. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method,, \emph{Comm. Partial Differential Equations}, 29 (2004), 879.  doi: 10.1081/PDE-120037335.  Google Scholar

[25]

J. Liu and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. I,, \emph{Proc. Amer. Math. Soc.}, 131 (2003), 441.  doi: 10.1090/S0002-9939-02-06783-7 .  Google Scholar

[26]

X. Liu, J. Liu and Z. Wang, Quasilinear elliptic equations via perturbation method,, \emph{Proc. Amer. Math. Soc.}, 141 (2013), 253.  doi: 10.1090/S0002-9939-2012-11293-6 .  Google Scholar

[27]

X. Liu, J. Liu and Z. Wang, Quasilinear elliptic equations with critical growth via perturbation method,, \emph{J. Differential Equations}, 254 (2013), 102.  doi: 10.1016/j.jde.2012.09.006.  Google Scholar

[28]

X. Liu, J. Liu and Z. Wang, Ground states for quasilinear Schrödinger equations with critical growth,, \emph{Calc. Var. Partial Differential Equations}, 46 (2013), 641.  doi: 10.1007/s00526-012-0497-0.  Google Scholar

[29]

A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $R^N$,, \emph{J. Differential Equations}, 229 (2006), 570.  doi: 10.1016/j.jde.2006.07.001.  Google Scholar

[30]

V.G. Makhankov and V.K. Fedyanin, Nonlinear effects in quasi-one-dimensional models and condensed matter theory,, \emph{Phys. Rep.}, 104 (1984), 1.  doi: 10.1016/0370-1573(84)90106-6.  Google Scholar

[31]

P. Pucci and J. Serrin, A general variational idnetity,, \emph{Indiana Univ. Math. J.}, 35 (1986), 681.  doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[32]

M. Poppenberg, K. Schmitt and Z. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations,, \emph{Calc. Var. Partial Differential Equations}, 14 (2002), 329.  doi: 10.1007/s005260100105.  Google Scholar

[33]

G.R.W. Quispel and H.W. Capel, Equation of motion for the Heisenberg spin chain,, \emph{Phys. A}, 110 (1982), 41.  doi: 10.1016/0378-4371(82)90104-2.  Google Scholar

[34]

B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions,, \emph{Phys. Rev. E}, 50 (1994), 687.  doi: 10.1103/PhysRevE.50.R687.  Google Scholar

[35]

Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations,, \emph{Nonlinear Anal. TMA.}, 80 (2013), 194.  doi: 10.1016/j.na.2012.10.005.  Google Scholar

[36]

E.A.B. Silva and G.F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth,, \emph{Calc. Var. Partial Differential Equations}, 39 (2010), 1.  doi: 10.1007/s00526-009-0299-1.  Google Scholar

[37]

J. Yang, Y. Wang and A.A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations,, \emph{J. Math. Phys.}, 54 (2013).  doi: 10.1063/1.4811394.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Func. Anal.}, 14 (1973), 349.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 437.  doi: 10.1002/cpa.3160360405.  Google Scholar

[3]

João M. Bezerra do Ó, Olímpio H. Miyagaki and Sérgio H.M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth,, \emph{J. Differential Equations}, 248 (2010), 722.  doi: 10.1016/j.jde.2009.11.030.  Google Scholar

[4]

H. Brandi, C. Manus, G. Mainfray, T. Lehner and G. Bonnaud, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma,, \emph{Phys. Fluids B}, 5 (1993), 3539.  doi: 10.1063/1.860828.  Google Scholar

[5]

F.G. Bass and N.N. Nasanov, Nonlinear electromagnetic-spin waves,, \emph{Phys. Rep.}, 189 (1990), 165.  doi: 10.1016/0370-1573(90)90093-H.  Google Scholar

[6]

L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations,, \emph{Exposition. Math.}, 4 (1986), 279.   Google Scholar

[7]

X.L. Chen and R.N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma,, \emph{Phys. Rev. Lett.}, 70 (1993), 2082.  doi: 10.1103/PhysRevLett.70.2082.  Google Scholar

[8]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach,, \emph{Nonlinear Anal. TMA.}, 56 (2004), 213.  doi: 10.1016/j.na.2003.09.008.  Google Scholar

[9]

S. Cuccagna, On instability of excited states of the nonlinear Schrödinger equation,, \emph{Physica D}, 238 (2009), 38.  doi: 10.1016/j.physd.2008.08.010.  Google Scholar

[10]

A. De Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation,, \emph{Commun. Math. Phys.}, 189 (1997), 73.  doi: 10.1007/s002200050191.  Google Scholar

[11]

Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent,, \emph{J. Math. Phys.}, 54 (2013).  doi: 10.1063/1.4774153.  Google Scholar

[12]

Y. Deng, S. Peng and J. Wang, Infinitely many sign-changing solutions for quasilinear Schrödinger equations in $R^N$,, \emph{Commun. Math. Sci.}, 9 (2011), 859.  doi: 10.4310/CMS.2011.v9.n3.a9.  Google Scholar

[13]

Y. Deng, S. Peng and S. Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth,, \emph{J. Differential Equations}, 258 (2015), 115.  doi: 10.1016/j.jde.2014.09.006.  Google Scholar

[14]

Y. Deng, S. Peng and S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations,, \emph{J. Differential Equations}, 260 (2016), 1228.  doi: 10.1016/j.jde.2015.09.021.  Google Scholar

[15]

Q. Han and F. Lin, Elliptic Partial Differential Equations,, Courant Lecture Notes in Mathematics, (1997).   Google Scholar

[16]

R.W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations,, \emph{Z. Phys. B}, 37 (1980), 83.   Google Scholar

[17]

A.M. Kosevich, B.A. Ivanov and A.S. Kovalev, Magnetic solitons,, \emph{Phys. Rep.}, 194 (1990), 117.  doi: doi:10.1016/0370-1573(90)90130-T.  Google Scholar

[18]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films,, \emph{J. Phys. Soc. Japan}, 50 (1981), 3262.  doi: 10.1143/JPSJ.50.3262.  Google Scholar

[19]

E. Laedke, K. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions,, \emph{J. Math. Phys.}, 24 (1983), 2764.  doi: 10.1063/1.525675.  Google Scholar

[20]

P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part I},, \emph{Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire}, 1 (1984), 109.   Google Scholar

[21]

P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part II,, \emph{Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire}, 1 (1984), 223.   Google Scholar

[22]

H. Lange, M. Poppenberg and H. Teismann, Nash-Moser methods for the solution of quasilinear Schrödinger equations,, \emph{Comm. Partial Differential Equations}, 24 (1999), 1399.  doi: 10.1080/03605309908821469.  Google Scholar

[23]

J. Liu, Y. Wang and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. II,, \emph{J. Differential Equations}, 187 (2003), 473.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[24]

J. Liu, Y. Wang and Z. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method,, \emph{Comm. Partial Differential Equations}, 29 (2004), 879.  doi: 10.1081/PDE-120037335.  Google Scholar

[25]

J. Liu and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. I,, \emph{Proc. Amer. Math. Soc.}, 131 (2003), 441.  doi: 10.1090/S0002-9939-02-06783-7 .  Google Scholar

[26]

X. Liu, J. Liu and Z. Wang, Quasilinear elliptic equations via perturbation method,, \emph{Proc. Amer. Math. Soc.}, 141 (2013), 253.  doi: 10.1090/S0002-9939-2012-11293-6 .  Google Scholar

[27]

X. Liu, J. Liu and Z. Wang, Quasilinear elliptic equations with critical growth via perturbation method,, \emph{J. Differential Equations}, 254 (2013), 102.  doi: 10.1016/j.jde.2012.09.006.  Google Scholar

[28]

X. Liu, J. Liu and Z. Wang, Ground states for quasilinear Schrödinger equations with critical growth,, \emph{Calc. Var. Partial Differential Equations}, 46 (2013), 641.  doi: 10.1007/s00526-012-0497-0.  Google Scholar

[29]

A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $R^N$,, \emph{J. Differential Equations}, 229 (2006), 570.  doi: 10.1016/j.jde.2006.07.001.  Google Scholar

[30]

V.G. Makhankov and V.K. Fedyanin, Nonlinear effects in quasi-one-dimensional models and condensed matter theory,, \emph{Phys. Rep.}, 104 (1984), 1.  doi: 10.1016/0370-1573(84)90106-6.  Google Scholar

[31]

P. Pucci and J. Serrin, A general variational idnetity,, \emph{Indiana Univ. Math. J.}, 35 (1986), 681.  doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[32]

M. Poppenberg, K. Schmitt and Z. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations,, \emph{Calc. Var. Partial Differential Equations}, 14 (2002), 329.  doi: 10.1007/s005260100105.  Google Scholar

[33]

G.R.W. Quispel and H.W. Capel, Equation of motion for the Heisenberg spin chain,, \emph{Phys. A}, 110 (1982), 41.  doi: 10.1016/0378-4371(82)90104-2.  Google Scholar

[34]

B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions,, \emph{Phys. Rev. E}, 50 (1994), 687.  doi: 10.1103/PhysRevE.50.R687.  Google Scholar

[35]

Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations,, \emph{Nonlinear Anal. TMA.}, 80 (2013), 194.  doi: 10.1016/j.na.2012.10.005.  Google Scholar

[36]

E.A.B. Silva and G.F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth,, \emph{Calc. Var. Partial Differential Equations}, 39 (2010), 1.  doi: 10.1007/s00526-009-0299-1.  Google Scholar

[37]

J. Yang, Y. Wang and A.A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations,, \emph{J. Math. Phys.}, 54 (2013).  doi: 10.1063/1.4811394.  Google Scholar

[1]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179
[2]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
[3]

Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99
[4]

Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120
[5]

Jianhua Chen, Xianhua Tang, Bitao Cheng. Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 493-517. doi: 10.3934/cpaa.2019025
[6]

Yanfang Xue, Chunlei Tang. Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1121-1145. doi: 10.3934/cpaa.2018054
[7]

Xiang-Dong Fang. A positive solution for an asymptotically cubic quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (1) : 51-64. doi: 10.3934/cpaa.2019004
[8]

GUANGBING LI. Positive solution for quasilinear Schrödinger equations with a parameter. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1803-1816. doi: 10.3934/cpaa.2015.14.1803
[9]

Gui-Dong Li, Chun-Lei Tang. Existence of positive ground state solutions for Choquard equation with variable exponent growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2035-2050. doi: 10.3934/dcdss.2019131
[10]

Kenji Nakanishi, Tristan Roy. Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2023-2058. doi: 10.3934/cpaa.2016026
[11]

Maoding Zhen, Jinchun He, Haoyuan Xu, Meihua Yang. Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6523-6539. doi: 10.3934/dcds.2019283
[12]

Zhanping Liang, Yuanmin Song, Fuyi Li. Positive ground state solutions of a quadratically coupled schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (3) : 999-1012. doi: 10.3934/cpaa.2017048
[13]

Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025
[14]

Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang. Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104
[15]

Qian Shen, Na Wei. Stability of ground state for the Schrödinger-Poisson equation. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020095
[16]

Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004
[17]

Xia Sun, Kaimin Teng. Positive bound states for fractional Schrödinger-Poisson system with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3735-3768. doi: 10.3934/cpaa.2020165
[18]

Yinbin Deng, Wei Shuai. Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2273-2287. doi: 10.3934/cpaa.2014.13.2273
[19]

Daniele Garrisi, Vladimir Georgiev. Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4309-4328. doi: 10.3934/dcds.2017184
[20]

Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences