May  2017, 16(3): 953-972. doi: 10.3934/cpaa.2017046

Multiple positive standing wave solutions for schrödinger equations with oscillating state-dependent potentials

1. 

Institut Elie Cartan de Lorraine and CNRS, UMR 7502, Université de Lorraine, Metz, 57045, France

2. 

Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj, 400084, Romania

3. 

Department of Mathematics, University of Pécs, Pécs, 7624, Hungary

* Corresponding author

Received  August 2016 Revised  December 2016 Published  February 2017

Motivated by relevant physical applications, we study Schrödinger equations with state-dependent potentials. Existence, localization and multiplicity results are established for positive standing wave solutions in the case of oscillating potentials. To this aim, a localized Pucci-Serrin type critical point theorem is first obtained. Two examples are then given to illustrate the new theory.

Citation: Renata Bunoiu, Radu Precup, Csaba Varga. Multiple positive standing wave solutions for schrödinger equations with oscillating state-dependent potentials. Communications on Pure & Applied Analysis, 2017, 16 (3) : 953-972. doi: 10.3934/cpaa.2017046
References:
[1]

N. N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams, Chapman and Hall, London, 1997. Google Scholar

[2]

A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511618260.  Google Scholar

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A. AmbrosettiA. Malchiodi and S. Secchi, Multiplicity results for some nonlinear singularly perturbed elliptic problems on Rn, Arch. Rat. Mech. Anal., 159 (2001), 253-271.  doi: 10.1007/s002050100152.  Google Scholar

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T. BartschN. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds. and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

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V. Benci and D. Fortunato, Variational Methods in Nonlinear Field Equations, Springer, Cham, 2014. doi: 10.1007/978-3-319-06914-2.  Google Scholar

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P. Böhi, Coherent manipulation of Bose-Einstein condensates with state-dependent microwave potentials on an atom chip, Nature Physics, 5 (2009), 592-597.   Google Scholar

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G. Bonanno and G. Molica Bisci, Three weak solutions for elliptic Dirichlet problems, J. Math. Anal. Appl, 382 (2011), 1-8.  doi: 10.1016/j.jmaa.2011.04.026.  Google Scholar

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G. Bonanno and G. Molica Bisci, O'Regan, Infinitely many weak solutions for a class of quasilinear elliptic systems, Math. Comput. Model, 52 (2010), 152-160.  doi: 10.1016/j.mcm.2010.02.004.  Google Scholar

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J. Bourgain, Global Solutions of Nonlinear Schrodinger Equations, AMS, 1999. doi: 10.1090/coll/046.  Google Scholar

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J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200.  doi: 10.1007/s00205-006-0019-3.  Google Scholar

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J. Byeon and Z. Q. Wang, Standing waves for nonlinear Schrödinger equations with singular potentials, Ann. I. H. Poincar′e -AN, 26 (2009), 943-958.  doi: 10.1016/j.anihpc.2008.03.009.  Google Scholar

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R. Carles, Semi-classical Analysis for Nonlinear Schrödinger Equations, World Scientific, Singapore, 2008. doi: 10.1142/9789812793133.  Google Scholar

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T. Cazenave, Semilinear Schrödinger Equations, Amer. Math. Soc., CIMS 10, 2003. doi: 10.1090/cln/010.  Google Scholar

[14]

M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32.  doi: 10.1007/s002080200327.  Google Scholar

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I. Ekeland, On the variational principle, J. Math. Anal. Appl, 47 (1974), 443-474.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

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M. Garcia-HuidobroR. Manasevich and K. Schmitt, Positive radial solutions of quasilinear elliptic partial differential equations on a ball, Nonlinear Anal., 35 (1999), 175-190.  doi: 10.1016/S0362-546X(97)00613-5.  Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

[18]

D. D. Hai and K. Schmitt, On radial solutions of quasilinear boundary value problems, in Progress in Nonlinear Differential equations and Their Applications (J. Escher and G. Simonett eds. ), Birkhäuser, (1999), 349-361.  Google Scholar

[19]

M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.  Google Scholar

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A. Kristály, V. Rădulescu and C. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511760631.  Google Scholar

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Y.-G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potentials, Commun. Math. Phys., 131 (1990), 223-253.   Google Scholar

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P. Omari and F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential, Comm. Partial Differential Equations, 21 (1996), 721-733.  doi: 10.1080/03605309608821205.  Google Scholar

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L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Clarendon Press, Oxford, 2003.  Google Scholar

[24]

R. Precup, On a bounded critical point theorem of Schechter, Studia Univ. Babeş-Bolyai Math., 58 (2013), 87-95.   Google Scholar

[25]

R. Precup, Critical point theorems in cones and multiple positive solutions of elliptic problems, Nonlinear Anal., 75 (2012), 834-851.  doi: 10.1016/j.na.2011.09.016.  Google Scholar

[26]

R. Precup, Critical point localization theorems via Ekeland's variational principle, Dynam. Systems Appl., 22 (2013), 355-370.   Google Scholar

[27]

R. Precup, P. Pucci and C. Varga, A three critical points result in a bounded domain of a Banach space and applications, Differential Integral Equations, to appear. Google Scholar

[28]

P. Pucci and J. Serrin, Extensions of mountain pass theorem, J. Funct. Anal., 59 (1984), 185-210.  doi: 10.1016/0022-1236(84)90072-7.  Google Scholar

[29]

P. Pucci and J. Serrin, A mountain pass theorem, J. Differential Equations, 60 (1985), 142-149.  doi: 10.1016/0022-0396(85)90125-1.  Google Scholar

[30]

P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007.  Google Scholar

[31]

B. Ricceri, On a three critical points theorem, Arch. Math. (Basel), 75 (2000), 220-226.  doi: 10.1007/s000130050496.  Google Scholar

[32]

B. Ricceri, A further three critical points theorem, Nonlinear Anal., 71 (2009), 4151-4157.  doi: 10.1016/j.na.2009.02.074.  Google Scholar

[33]

M. Schechter, A bounded mountain pass lemma without the (PS) condition and applications, Trans. Amer. Math. Soc., 331 (1992), 681-703.  doi: 10.2307/2154135.  Google Scholar

[34]

M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Basel, 1999. doi: 10.1007/978-1-4612-1596-7.  Google Scholar

[35]

M. Struwe, Variational Methods, Springer, Berlin, 1990. doi: 10.1007/978-3-662-02624-3.  Google Scholar

[36]

C. Sulem and P. -L. Sulem, The Nonlinear Schrödinger Equation, Springer, New York, 1999.  Google Scholar

[37] M. E. Taylor, Partial Differential Equations, Vols. Ⅰ-Ⅲ, Springer, New York, 1996.  doi: 10.1007/978-1-4684-9320-7.  Google Scholar
[38]

R. Tian and Z.-Q Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.   Google Scholar

show all references

References:
[1]

N. N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams, Chapman and Hall, London, 1997. Google Scholar

[2]

A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511618260.  Google Scholar

[3]

A. AmbrosettiA. Malchiodi and S. Secchi, Multiplicity results for some nonlinear singularly perturbed elliptic problems on Rn, Arch. Rat. Mech. Anal., 159 (2001), 253-271.  doi: 10.1007/s002050100152.  Google Scholar

[4]

T. BartschN. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds. and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

[5]

V. Benci and D. Fortunato, Variational Methods in Nonlinear Field Equations, Springer, Cham, 2014. doi: 10.1007/978-3-319-06914-2.  Google Scholar

[6]

P. Böhi, Coherent manipulation of Bose-Einstein condensates with state-dependent microwave potentials on an atom chip, Nature Physics, 5 (2009), 592-597.   Google Scholar

[7]

G. Bonanno and G. Molica Bisci, Three weak solutions for elliptic Dirichlet problems, J. Math. Anal. Appl, 382 (2011), 1-8.  doi: 10.1016/j.jmaa.2011.04.026.  Google Scholar

[8]

G. Bonanno and G. Molica Bisci, O'Regan, Infinitely many weak solutions for a class of quasilinear elliptic systems, Math. Comput. Model, 52 (2010), 152-160.  doi: 10.1016/j.mcm.2010.02.004.  Google Scholar

[9]

J. Bourgain, Global Solutions of Nonlinear Schrodinger Equations, AMS, 1999. doi: 10.1090/coll/046.  Google Scholar

[10]

J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200.  doi: 10.1007/s00205-006-0019-3.  Google Scholar

[11]

J. Byeon and Z. Q. Wang, Standing waves for nonlinear Schrödinger equations with singular potentials, Ann. I. H. Poincar′e -AN, 26 (2009), 943-958.  doi: 10.1016/j.anihpc.2008.03.009.  Google Scholar

[12]

R. Carles, Semi-classical Analysis for Nonlinear Schrödinger Equations, World Scientific, Singapore, 2008. doi: 10.1142/9789812793133.  Google Scholar

[13]

T. Cazenave, Semilinear Schrödinger Equations, Amer. Math. Soc., CIMS 10, 2003. doi: 10.1090/cln/010.  Google Scholar

[14]

M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32.  doi: 10.1007/s002080200327.  Google Scholar

[15]

I. Ekeland, On the variational principle, J. Math. Anal. Appl, 47 (1974), 443-474.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[16]

M. Garcia-HuidobroR. Manasevich and K. Schmitt, Positive radial solutions of quasilinear elliptic partial differential equations on a ball, Nonlinear Anal., 35 (1999), 175-190.  doi: 10.1016/S0362-546X(97)00613-5.  Google Scholar

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

[18]

D. D. Hai and K. Schmitt, On radial solutions of quasilinear boundary value problems, in Progress in Nonlinear Differential equations and Their Applications (J. Escher and G. Simonett eds. ), Birkhäuser, (1999), 349-361.  Google Scholar

[19]

M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.  Google Scholar

[20]

A. Kristály, V. Rădulescu and C. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511760631.  Google Scholar

[21]

Y.-G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potentials, Commun. Math. Phys., 131 (1990), 223-253.   Google Scholar

[22]

P. Omari and F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential, Comm. Partial Differential Equations, 21 (1996), 721-733.  doi: 10.1080/03605309608821205.  Google Scholar

[23]

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Clarendon Press, Oxford, 2003.  Google Scholar

[24]

R. Precup, On a bounded critical point theorem of Schechter, Studia Univ. Babeş-Bolyai Math., 58 (2013), 87-95.   Google Scholar

[25]

R. Precup, Critical point theorems in cones and multiple positive solutions of elliptic problems, Nonlinear Anal., 75 (2012), 834-851.  doi: 10.1016/j.na.2011.09.016.  Google Scholar

[26]

R. Precup, Critical point localization theorems via Ekeland's variational principle, Dynam. Systems Appl., 22 (2013), 355-370.   Google Scholar

[27]

R. Precup, P. Pucci and C. Varga, A three critical points result in a bounded domain of a Banach space and applications, Differential Integral Equations, to appear. Google Scholar

[28]

P. Pucci and J. Serrin, Extensions of mountain pass theorem, J. Funct. Anal., 59 (1984), 185-210.  doi: 10.1016/0022-1236(84)90072-7.  Google Scholar

[29]

P. Pucci and J. Serrin, A mountain pass theorem, J. Differential Equations, 60 (1985), 142-149.  doi: 10.1016/0022-0396(85)90125-1.  Google Scholar

[30]

P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007.  Google Scholar

[31]

B. Ricceri, On a three critical points theorem, Arch. Math. (Basel), 75 (2000), 220-226.  doi: 10.1007/s000130050496.  Google Scholar

[32]

B. Ricceri, A further three critical points theorem, Nonlinear Anal., 71 (2009), 4151-4157.  doi: 10.1016/j.na.2009.02.074.  Google Scholar

[33]

M. Schechter, A bounded mountain pass lemma without the (PS) condition and applications, Trans. Amer. Math. Soc., 331 (1992), 681-703.  doi: 10.2307/2154135.  Google Scholar

[34]

M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Basel, 1999. doi: 10.1007/978-1-4612-1596-7.  Google Scholar

[35]

M. Struwe, Variational Methods, Springer, Berlin, 1990. doi: 10.1007/978-3-662-02624-3.  Google Scholar

[36]

C. Sulem and P. -L. Sulem, The Nonlinear Schrödinger Equation, Springer, New York, 1999.  Google Scholar

[37] M. E. Taylor, Partial Differential Equations, Vols. Ⅰ-Ⅲ, Springer, New York, 1996.  doi: 10.1007/978-1-4684-9320-7.  Google Scholar
[38]

R. Tian and Z.-Q Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.   Google Scholar

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