July  2015, 14(4): 1275-1326. doi: 10.3934/cpaa.2015.14.1275

Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations

1. 

LAGA (UMR 7539), Institut Galilée, Université Paris 13, 99, avenue Jean-Baptiste Clément, 93430 Villetaneuse, France

2. 

Department of Mathematics, University of Chicago, Chicago, Illinois, 60637–1514, United States

3. 

Université de Cergy-Pontoise and IHES, Laboratoire de mathématiques, UMR CNRS 8088, 2, av. Adolphe Chauvin, 95302 Cergy-Pontoise cedex

Received  October 2013 Revised  March 2014 Published  April 2015

Consider a bounded solution of the focusing, energy-critical wave equation that does not scatter to a linear solution. We prove that this solution converges in some weak sense, along a sequence of times and up to scaling and space translation, to a sum of solitary waves. This result is a consequence of a new general compactness/rigidity argument based on profile decomposition. We also give an application of this method to the energy-critical Schrödinger equation.
Citation: Thomas Duyckaerts, Carlos E. Kenig, Frank Merle. Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1275-1326. doi: 10.3934/cpaa.2015.14.1275
References:
[1]

Takafumi Akahori and Hayato Nawa, Blowup and scattering problems for the nonlinear Schrödinger equations, Kyoto J. Math., 53 (2013), 629-672. doi: 10.1215/21562261-2265914.

[2]

Bulut Aynur, Maximizers for the Strichartz inequalities for the wave equation, Differential Integral Equations, 23 (2010), 1035-1072.

[3]

Hajer Bahouri and Patrick Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.

[4]

Aynur Bulut, Magdalena Czubak, Dong Li, Nataša Pavlović and Xiaoyi Zhang, Stability and unconditional uniqueness of solutions for energy critical wave equations in high dimensions, Comm. Partial Differential Equations, 38 (2013), 575-607. doi: 10.1080/03605302.2012.756520.

[5]

Thierry Cazenave and Fred B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.

[6]

Demetrios Christodoulou and A. Shadi Tahvildar-Zadeh, On the asymptotic behavior of spherically symmetric wave maps, Duke Math. J., 71 (1993), 31-69. doi: 10.1215/S0012-7094-93-07103-7.

[7]

Peter Constantin and Jean-Claude Saut, Local smoothing properties of Schrödinger equations, Indiana Univ. Math. J., 38 (1989), 791-810. doi: 10.1512/iumj.1989.38.38037.

[8]

Raphaël Côte, Soliton resolution for equivariant wave maps to the sphere,, Preprint, (). 

[9]

Manuel del Pino, Personal communication., \quad, (). 

[10]

Manuel del Pino, Monica Musso, Frank Pacard and Angela Pistoia, Large energy entire solutions for the Yamabe equation, J. Differential Equations, 251 (2011), 2568-2597. doi: 10.1016/j.jde.2011.03.008.

[11]

Manuel del Pino, Monica Musso, Frank Pacard and Angela Pistoia, Torus action on $S^n$ and sign-changing solutions for conformally invariant equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 209-237.

[12]

Weiyue Ding, On a conformally invariant elliptic equation on $R^n$, Comm. Math. Phys., 107 (1986), 331-335.

[13]

Benjamin Dodson, Global well-posedness and scattering for the defocusing, $L^{2}$-critical nonlinear Schrödinger equation when $d\geq3$, J. Amer. Math. Soc., 25 (2012), 429-463. doi: 10.1090/S0894-0347-2011-00727-3.

[14]

Roland Donninger and Joachim Krieger, Nonscattering solutions and blowup at infinity for the critical wave equation, Math. Ann., 357 (2013), 89-163. doi: 10.1007/s00208-013-0898-1.

[15]

Thomas Duyckaerts, Justin Holmer and Svetlana Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250. doi: 10.4310/MRL.2008.v15.n6.a13.

[16]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation, J. Eur. Math. Soc. (JEMS), 13 (2011), 533-599. doi: 10.4171/JEMS/261.

[17]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Profiles of bounded radial solutions of the focusing, energy-critical wave equation, Geom. Funct. Anal., 22 (2012), 639-698. doi: 10.1007/s00039-012-0174-7.

[18]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case, J. Eur. Math. Soc. (JEMS), 14 (2012), 1389-1454. doi: 10.4171/JEMS/336.

[19]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Classification of radial solutions of the focusing, energy-critical wave equation, Cambridge Journal of Mathematics, 1 (2013), 75-144. doi: 10.4310/CJM.2013.v1.n1.a3.

[20]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Scattering for radial, bounded solutions of focusing supercritical wave equations, International Mathematics Research Notices, (2014), 224-258.

[21]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case,, Corrected version, ().  doi: 10.4171/JEMS/336.

[22]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Solutions of the focusing, energy-critical wave equation with the compactness property,, Preprint, (). 

[23]

Daoyuan Fang, Jian Xie and Thierry Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062. doi: 10.1007/s11425-011-4283-9.

[24]

Leo Glangetas and Frank Merle, A geometrical approach of existence of blow up solutions in $H^1$ for nonlinear Schrödinger equation, Preprint, Laboratoire d'Analyse Numérique, Univ. Pierre and Marie Curie, 1995.

[25]

Cristi Guevara, Global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schr\"odinger equation, Appl. Math. Res. Express. AMRX, (2014), 177-243.

[26]

Justin Holmer and Svetlana Roudenko, On blow-up solutions to the 3D cubic nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX, (2007), Art. ID abm004, 31.

[27]

Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4.

[28]

Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212. doi: 10.1007/s11511-008-0031-6.

[29]

Carlos E. Kenig, Andrew Lawrie and Wilhelm Schlag, Relaxation of wave maps exterior to a ball to harmonic maps for all data, Geom. Funct. Anal., 24 (2014), 610-647. doi: 10.1007/s00039-014-0262-y.

[30]

Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212. doi: 10.1007/s11511-008-0031-6.

[31]

Sahbi Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392. doi: 10.1006/jdeq.2000.3951.

[32]

Sahbi Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal., 235 (2006), 171-192. doi: 10.1016/j.jfa.2005.10.005.

[33]

Rowan Killip and Monica Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424. doi: 10.1353/ajm.0.0107.

[34]

Joachim Krieger, Wilhelm Schlag and Daniel Tataru, Slow blow-up solutions for the $H^1(R^3)$ critical focusing semilinear wave equation, Duke Math. J., 147 (2009), 1-53. doi: 10.1215/00127094-2009-005.

[35]

Yvan Martel and Frank Merle, A Liouville theorem for the critical generalized Korteweg-de Vries equation, J. Math. Pures Appl., 79 (2000), 339-425. doi: 10.1016/S0021-7824(00)00159-8.

[36]

Frank Merle and Pierre Raphael, On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation, Invent. Math., 156 (2004), 565-672. doi: 10.1007/s00222-003-0346-z.

[37]

Ruipeng Shen, On the energy subcritical, nonlinear wave equation in $\mathbbR^3$ with radial data, Anal. PDE, 6 (2013), 1929-1987. doi: 10.2140/apde.2013.6.1929.

[38]

Per Sjölin, Convergence properties for the Schrödinger equation, Rend. Sem. Mat. Fis. Milano, 57 (1987), 293-297. doi: 10.1007/BF02925057.

[39]

Jacob Sterbenz and Daniel Tataru, Energy dispersed large data wave maps in $2+1$ dimensions, Comm. Math. Phys., 298 (2010), 139-230. doi: 10.1007/s00220-010-1061-4.

[40]

Jacob Sterbenz and Daniel Tataru, Regularity of wave-maps in dimension $2+1$, Comm. Math. Phys., 298 (2010), 231-264. doi: 10.1007/s00220-010-1062-3.

[41]

Michael Struwe, Equivariant wave maps in two space dimensions, Comm. Pure Appl. Math., 56 (2003), 815-823. doi: 10.1002/cpa.10074.

[42]

Terence Tao, Monica Visan and Xiaoyi Zhang, Minimal-mass blowup solutions of the mass-critical NLS, Forum Math., 20 (2008), 881-919. doi: 10.1515/FORUM.2008.042.

[43]

Luis Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc., 102 (1988), 874-878. doi: 10.2307/2047326.

show all references

References:
[1]

Takafumi Akahori and Hayato Nawa, Blowup and scattering problems for the nonlinear Schrödinger equations, Kyoto J. Math., 53 (2013), 629-672. doi: 10.1215/21562261-2265914.

[2]

Bulut Aynur, Maximizers for the Strichartz inequalities for the wave equation, Differential Integral Equations, 23 (2010), 1035-1072.

[3]

Hajer Bahouri and Patrick Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.

[4]

Aynur Bulut, Magdalena Czubak, Dong Li, Nataša Pavlović and Xiaoyi Zhang, Stability and unconditional uniqueness of solutions for energy critical wave equations in high dimensions, Comm. Partial Differential Equations, 38 (2013), 575-607. doi: 10.1080/03605302.2012.756520.

[5]

Thierry Cazenave and Fred B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.

[6]

Demetrios Christodoulou and A. Shadi Tahvildar-Zadeh, On the asymptotic behavior of spherically symmetric wave maps, Duke Math. J., 71 (1993), 31-69. doi: 10.1215/S0012-7094-93-07103-7.

[7]

Peter Constantin and Jean-Claude Saut, Local smoothing properties of Schrödinger equations, Indiana Univ. Math. J., 38 (1989), 791-810. doi: 10.1512/iumj.1989.38.38037.

[8]

Raphaël Côte, Soliton resolution for equivariant wave maps to the sphere,, Preprint, (). 

[9]

Manuel del Pino, Personal communication., \quad, (). 

[10]

Manuel del Pino, Monica Musso, Frank Pacard and Angela Pistoia, Large energy entire solutions for the Yamabe equation, J. Differential Equations, 251 (2011), 2568-2597. doi: 10.1016/j.jde.2011.03.008.

[11]

Manuel del Pino, Monica Musso, Frank Pacard and Angela Pistoia, Torus action on $S^n$ and sign-changing solutions for conformally invariant equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 209-237.

[12]

Weiyue Ding, On a conformally invariant elliptic equation on $R^n$, Comm. Math. Phys., 107 (1986), 331-335.

[13]

Benjamin Dodson, Global well-posedness and scattering for the defocusing, $L^{2}$-critical nonlinear Schrödinger equation when $d\geq3$, J. Amer. Math. Soc., 25 (2012), 429-463. doi: 10.1090/S0894-0347-2011-00727-3.

[14]

Roland Donninger and Joachim Krieger, Nonscattering solutions and blowup at infinity for the critical wave equation, Math. Ann., 357 (2013), 89-163. doi: 10.1007/s00208-013-0898-1.

[15]

Thomas Duyckaerts, Justin Holmer and Svetlana Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250. doi: 10.4310/MRL.2008.v15.n6.a13.

[16]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation, J. Eur. Math. Soc. (JEMS), 13 (2011), 533-599. doi: 10.4171/JEMS/261.

[17]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Profiles of bounded radial solutions of the focusing, energy-critical wave equation, Geom. Funct. Anal., 22 (2012), 639-698. doi: 10.1007/s00039-012-0174-7.

[18]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case, J. Eur. Math. Soc. (JEMS), 14 (2012), 1389-1454. doi: 10.4171/JEMS/336.

[19]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Classification of radial solutions of the focusing, energy-critical wave equation, Cambridge Journal of Mathematics, 1 (2013), 75-144. doi: 10.4310/CJM.2013.v1.n1.a3.

[20]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Scattering for radial, bounded solutions of focusing supercritical wave equations, International Mathematics Research Notices, (2014), 224-258.

[21]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case,, Corrected version, ().  doi: 10.4171/JEMS/336.

[22]

Thomas Duyckaerts, Carlos Kenig and Frank Merle, Solutions of the focusing, energy-critical wave equation with the compactness property,, Preprint, (). 

[23]

Daoyuan Fang, Jian Xie and Thierry Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062. doi: 10.1007/s11425-011-4283-9.

[24]

Leo Glangetas and Frank Merle, A geometrical approach of existence of blow up solutions in $H^1$ for nonlinear Schrödinger equation, Preprint, Laboratoire d'Analyse Numérique, Univ. Pierre and Marie Curie, 1995.

[25]

Cristi Guevara, Global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schr\"odinger equation, Appl. Math. Res. Express. AMRX, (2014), 177-243.

[26]

Justin Holmer and Svetlana Roudenko, On blow-up solutions to the 3D cubic nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX, (2007), Art. ID abm004, 31.

[27]

Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4.

[28]

Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212. doi: 10.1007/s11511-008-0031-6.

[29]

Carlos E. Kenig, Andrew Lawrie and Wilhelm Schlag, Relaxation of wave maps exterior to a ball to harmonic maps for all data, Geom. Funct. Anal., 24 (2014), 610-647. doi: 10.1007/s00039-014-0262-y.

[30]

Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212. doi: 10.1007/s11511-008-0031-6.

[31]

Sahbi Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392. doi: 10.1006/jdeq.2000.3951.

[32]

Sahbi Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal., 235 (2006), 171-192. doi: 10.1016/j.jfa.2005.10.005.

[33]

Rowan Killip and Monica Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424. doi: 10.1353/ajm.0.0107.

[34]

Joachim Krieger, Wilhelm Schlag and Daniel Tataru, Slow blow-up solutions for the $H^1(R^3)$ critical focusing semilinear wave equation, Duke Math. J., 147 (2009), 1-53. doi: 10.1215/00127094-2009-005.

[35]

Yvan Martel and Frank Merle, A Liouville theorem for the critical generalized Korteweg-de Vries equation, J. Math. Pures Appl., 79 (2000), 339-425. doi: 10.1016/S0021-7824(00)00159-8.

[36]

Frank Merle and Pierre Raphael, On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation, Invent. Math., 156 (2004), 565-672. doi: 10.1007/s00222-003-0346-z.

[37]

Ruipeng Shen, On the energy subcritical, nonlinear wave equation in $\mathbbR^3$ with radial data, Anal. PDE, 6 (2013), 1929-1987. doi: 10.2140/apde.2013.6.1929.

[38]

Per Sjölin, Convergence properties for the Schrödinger equation, Rend. Sem. Mat. Fis. Milano, 57 (1987), 293-297. doi: 10.1007/BF02925057.

[39]

Jacob Sterbenz and Daniel Tataru, Energy dispersed large data wave maps in $2+1$ dimensions, Comm. Math. Phys., 298 (2010), 139-230. doi: 10.1007/s00220-010-1061-4.

[40]

Jacob Sterbenz and Daniel Tataru, Regularity of wave-maps in dimension $2+1$, Comm. Math. Phys., 298 (2010), 231-264. doi: 10.1007/s00220-010-1062-3.

[41]

Michael Struwe, Equivariant wave maps in two space dimensions, Comm. Pure Appl. Math., 56 (2003), 815-823. doi: 10.1002/cpa.10074.

[42]

Terence Tao, Monica Visan and Xiaoyi Zhang, Minimal-mass blowup solutions of the mass-critical NLS, Forum Math., 20 (2008), 881-919. doi: 10.1515/FORUM.2008.042.

[43]

Luis Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc., 102 (1988), 874-878. doi: 10.2307/2047326.

[1]

Hongqiu Chen, Jerry L. Bona. Periodic traveling--wave solutions of nonlinear dispersive evolution equations. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4841-4873. doi: 10.3934/dcds.2013.33.4841

[2]

Fabrício Cristófani, Ademir Pastor. Nonlinear stability of periodic-wave solutions for systems of dispersive equations. Communications on Pure and Applied Analysis, 2020, 19 (10) : 5015-5032. doi: 10.3934/cpaa.2020225

[3]

Jerry L. Bona, Laihan Luo. More results on the decay of solutions to nonlinear, dispersive wave equations. Discrete and Continuous Dynamical Systems, 1995, 1 (2) : 151-193. doi: 10.3934/dcds.1995.1.151

[4]

Nakao Hayashi, Seishirou Kobayashi, Pavel I. Naumkin. Nonlinear dispersive wave equations in two space dimensions. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1377-1393. doi: 10.3934/cpaa.2015.14.1377

[5]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450

[6]

Xiaoyu Zeng. Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1749-1762. doi: 10.3934/dcds.2017073

[7]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316

[8]

Jaeyoung Byeon, Ohsang Kwon, Yoshihito Oshita. Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2015, 14 (3) : 825-842. doi: 10.3934/cpaa.2015.14.825

[9]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, 2021, 29 (4) : 2561-2597. doi: 10.3934/era.2021002

[10]

Jerry Bona, Hongqiu Chen. Well-posedness for regularized nonlinear dispersive wave equations. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1253-1275. doi: 10.3934/dcds.2009.23.1253

[11]

Shuangjie Peng, Huirong Pi. Spike vector solutions for some coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2205-2227. doi: 10.3934/dcds.2016.36.2205

[12]

Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071

[13]

Juncheng Wei, Wei Yao. Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1003-1011. doi: 10.3934/cpaa.2012.11.1003

[14]

Chuangye Liu, Rushun Tian. Normalized solutions for 3-coupled nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5115-5130. doi: 10.3934/cpaa.2020229

[15]

Seunghyeok Kim. On vector solutions for coupled nonlinear Schrödinger equations with critical exponents. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1259-1277. doi: 10.3934/cpaa.2013.12.1259

[16]

Xing Cheng, Ze Li, Lifeng Zhao. Scattering of solutions to the nonlinear Schrödinger equations with regular potentials. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2999-3023. doi: 10.3934/dcds.2017129

[17]

Laurent Di Menza, Olivier Goubet. Stabilizing blow up solutions to nonlinear schrÖdinger equations. Communications on Pure and Applied Analysis, 2017, 16 (3) : 1059-1082. doi: 10.3934/cpaa.2017051

[18]

Zuji Guo. Nodal solutions for nonlinear Schrödinger equations with decaying potential. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1125-1138. doi: 10.3934/cpaa.2016.15.1125

[19]

Tai-Chia Lin, Tsung-Fang Wu. Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2911-2938. doi: 10.3934/dcds.2013.33.2911

[20]

Miao Du, Lixin Tian. Infinitely many solutions of the nonlinear fractional Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3407-3428. doi: 10.3934/dcdsb.2016104

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (97)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]