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

Renata Bunoiu; Radu Precup; Csaba Varga

doi:10.3934/cpaa.2017046

Article Contents

2017, Volume 16, Issue 3: 953-972. Doi: 10.3934/cpaa.2017046

This issue Previous Article Non-collapsing for a fully nonlinear inverse curvature flow Next Article Equivalence of sharp Trudinger-Moser-Adams Inequalities

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
^* Corresponding author

Received: July 31, 2016

Revised: November 30, 2016

Published: January 2018

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35J20, 47J30, 58E05.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	N. N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams, Chapman and Hall, London, 1997.
[2]	A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511618260.
[3]	A. Ambrosetti, A. 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.
[4]	T. Bartsch, N. 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.
[5]	V. Benci and D. Fortunato, Variational Methods in Nonlinear Field Equations, Springer, Cham, 2014. doi: 10.1007/978-3-319-06914-2.
[6]	P. Böhi, et al., Coherent manipulation of Bose-Einstein condensates with state-dependent microwave potentials on an atom chip, Nature Physics, 5 (2009), 592-597.
[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.
[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.
[9]	J. Bourgain, Global Solutions of Nonlinear Schrodinger Equations, AMS, 1999. doi: 10.1090/coll/046.
[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.
[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.
[12]	R. Carles, Semi-classical Analysis for Nonlinear Schrödinger Equations, World Scientific, Singapore, 2008. doi: 10.1142/9789812793133.
[13]	T. Cazenave, Semilinear Schrödinger Equations, Amer. Math. Soc., CIMS 10, 2003. doi: 10.1090/cln/010.
[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.
[15]	I. Ekeland, On the variational principle, J. Math. Anal. Appl, 47 (1974), 443-474. doi: 10.1016/0022-247X(74)90025-0.
[16]	M. Garcia-Huidobro, R. 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.
[17]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, (1983). doi: 10.1007/978-3-642-61798-0.
[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.
[19]	M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.
[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.
[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.
[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.
[23]	L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Clarendon Press, Oxford, 2003.
[24]	R. Precup, On a bounded critical point theorem of Schechter, Studia Univ. Babeş-Bolyai Math., 58 (2013), 87-95.
[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.
[26]	R. Precup, Critical point localization theorems via Ekeland's variational principle, Dynam. Systems Appl., 22 (2013), 355-370.
[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.
[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.
[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.
[30]	P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007.
[31]	B. Ricceri, On a three critical points theorem, Arch. Math. (Basel), 75 (2000), 220-226. doi: 10.1007/s000130050496.
[32]	B. Ricceri, A further three critical points theorem, Nonlinear Anal., 71 (2009), 4151-4157. doi: 10.1016/j.na.2009.02.074.
[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.
[34]	M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Basel, 1999. doi: 10.1007/978-1-4612-1596-7.
[35]	M. Struwe, Variational Methods, Springer, Berlin, 1990. doi: 10.1007/978-3-662-02624-3.
[36]	C. Sulem and P. -L. Sulem, The Nonlinear Schrödinger Equation, Springer, New York, 1999.
[37]	M. E. Taylor, Partial Differential Equations, Vols. Ⅰ-Ⅲ, Springer, New York, 1996. doi: 10.1007/978-1-4684-9320-7.
[38]	R. Tian and Z.-Q Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.