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Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potentials

Published online by Cambridge University Press:  03 June 2015

Anouar Bahrouni ,
Hichem Ounaies  and
Vicenţiu D. Rădulescu
Anouar Bahrouni
Affiliation:
Mathematics, Faculty of Sciences, University of Monastir, 5019 Monastir, Tunisia, (bahrounianouar@yahoo.fr; hichem.ounaies@fsm.rnu.tn)
Hichem Ounaies
Affiliation:
Mathematics, Faculty of Sciences, University of Monastir, 5019 Monastir, Tunisia, (bahrounianouar@yahoo.fr; hichem.ounaies@fsm.rnu.tn)
Vicenţiu D. Rădulescu
Affiliation:
Department of Mathematics, Faculty of Science, King Abdulaziz University, PO Box 80203, Jeddah 21589, Saudi Arabia, (vicentiu.radulescu@math.cnrs.fr) and ‘Simion Stoilow’ Institute of Mathematics of the Romanian Academy, PO Box 1–764, 014700 Bucharest, Romania
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Abstract

In this paper we are concerned with qualitative properties of entire solutions to a Schrödinger equation with sublinear nonlinearity and sign-changing potentials. Our analysis considers three distinct cases and we establish sufficient conditions for the existence of infinitely many solutions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 3 , June 2015 , pp. 445 - 465
Copyright
Copyright © Royal Society of Edinburgh 2015 

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References

1 Ablowitz, M. J., Prinari, B. and Trubatch, A. D.. Discrete and continuous nonlinear Schrödinger systems (Cambridge University Press, 2004).Google Scholar
2 Adachi, S. and Tanaka, K.. Four positive solutions for the equation −Δu + u = a(x)u p + f(x) in ℝ N . Calc. Var. PDEs 11 (2000), 63–95.Google Scholar
3 Ambrosetti, A. and Badiale, M.. The dual variational principle and elliptic problems with discontinuous nonlinearities. J. Math. Analysis Applic. 140 (1989), 363–373.Google Scholar
4 Ambrosetti, A. and Rabinowitz, P. H.. Dual variational methods in critical point theory and applications. J. Funct. Analysis 14 (1973), 349–381.Google Scholar
5 Avron, J., Herbst, I. and Simon, B.. Schrödinger operators with electromagnetic fields. III. Atoms in homogeneous magnetic field. Commun. Math. Phys. 79 (1981), 529–572.Google Scholar
6 Bahri, A. and Lions, P.-L.. On the existence of a positive solution of semilinear elliptic equations in unbounded domains. Annales Inst. H. Poincaré Analyse Non Linéaire 14 (1997), 365–413.Google Scholar
7 Balabane, M., Dolbeault, J. and Ounaies, H.. Nodal solutions for a sublinear elliptic equation. Nonlin. Analysis 52 (2003), 219–237.Google Scholar
8 Bartsch, T., Liu, Z. and Weth, T.. Sign changing solutions for superlinear Schrödinger equations. Commun. PDEs 29 (2004), 25–42.Google Scholar
9 Benrhouma, M.. Study of multiplicity and uniqueness of solutions for a class of nonhomogeneous sublinear elliptic equations. Nonlin. Analysis 74 (2011), 2682–2694.Google Scholar
10 Benrhouma, M. and Ounaies, H.. Existence and uniqueness of positive solution for nonhomogeneous sublinear elliptic equation. J. Math. Analysis Applic. 358 (2009), 307–319.Google Scholar
11 Benrhouma, M. and Ounaies, H.. Existence of solutions for a perturbation sublinear elliptic equation in ℝ N . Nonlin. Diff. Eqns Applic. 5 (2010), 647–662.Google Scholar
12 Brezis, H. and Kamin, S.. Sublinear elliptic equations in R n . Manuscr. Math. 74 (1992), 87–106.Google Scholar
13 Brezis, H. and Lieb, E. H.. A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88 (1983), 486–490.Google Scholar
14 Brezis, H. and Oswald, L.. Remarks on sublinear elliptic equations. Nonlin. Analysis 10 (1986), 55–64.Google Scholar
15 Byeon, J. and Wang, Z. Q.. Standing waves with a critical frequency for nonlinear Schrödinger equations. Arch. Ration. Mech. Analysis 165 (2002), 295–316.Google Scholar
16 Chabrowski, J. and Costa, D. G.. On a class of Schrödinger-type equations with indefinite weight functions. Commun. PDEs 33 (2008), 1368–1394.Google Scholar
17 Costa, D. G. and Tehrani, H.. Existence of positive solutions for a class of indefinite elliptic problems in R N . Calc. Var. PDEs 13 (2001), 159–189.Google Scholar
18 Costa, D. G. and Tehrani, H.. Unbounded perturbations of resonant Schrödinger equations. In Variational methods: open problems, recent progress, and numerical algorithms. Contemporary Mathematics, vol. 357, pp. 101–110 (Providence, RI: American Mathematical Society, 2004).Google Scholar
19 Costa, D. G., Tehrani, H. and Ramos, M.. Non-zero solutions for a Schrödinger equation with indefinite linear and nonlinear terms. Proc. R. Soc. Edinb. A 134 (2004), 249–258.Google Scholar
20 Dellacherie, C. and Meyer, P. A.. Probabilités et potentiel (Paris: Hermann, 1983).Google Scholar
21 Dong, W. and Mei, L.. Multiple solutions for an indefinite superlinear elliptic problem on R N . Nonlin. Analysis 73 (2010), 2056–2070.Google Scholar
22 Du, Y.. Multiplicity of positive solutions for an indefinite superlinear elliptic problem on ∝ N . Annales Inst. H. Poincaré Analyse Non Linéaire 21 (2004), 657–672.Google Scholar
23 Einstein, A.. Ideas and opinions (New York: Crown Trade Paperbacks, 1954).Google Scholar
24 Floer, A. and Weinstein, A.. Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Analysis 69 (1986), 397–408.Google Scholar
25 Furtado, M. F., Maia, L. A. and Medeiros, E. S.. Positive and nodal solutions for a nonlinear Schrödinger equation with indefinite potential. Adv. Nonlin. Studies 8 (2008), 353–373.Google Scholar
26 Hasegawa, A. and Kodama, Y.. Solitons in optical communications (Academic Press, 1995).Google Scholar
27 Kajikiya, R.. A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. J. Funct. Analysis 225 (2005), 352–370.Google Scholar
28 Kajikiya, R.. Multiple solutions of sublinear Lane–Emden elliptic equations. Calc. Var. PDEs 26 (2006), 29–48.Google Scholar
29 Kato, T.. Remarks on holomorphic families of Schrödinger and Dirac operators. In Differential equations (ed. Knowles, I. and Lewis, R.). North-Holland Mathematics Studies, vol. 92, pp. 341–352 (Amsterdam: North-Holland, 1984).Google Scholar
30 Kristaly, A.. Multiple solutions of a sublinear Schrödinger equation. Nonlin. Diff. Eqns Applic. 14 (2007), 291–302.Google Scholar
31 Lions, P.-L.. The concentration–compactness principle in the calculus of variations. The locally compact case. Part 2. Annales Inst. H. Poincaré Analyse Non Linéaire 2 (1984), 223–283.Google Scholar
32 Malomed, B. A.. Variational methods in nonlinear fiber optics and related fields. Prog. Opt. 43 (2002), 69–191.Google Scholar
33 Oh, Y. G.. Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class (Va ). Commun. PDEs 13 (1988), 1499–1519.Google Scholar
34 Onorato, M., Osborne, A. R., Serio, M. and Bertone, S.. Freak waves in random oceanic sea states. Phys. Rev. Lett. 86 (2001), 5831–5834.Google Scholar
35 Rabinowitz, P. H.. On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43 (1992), 270–291.Google Scholar
36 Sulem, C. and Sulem, P.-L.. The nonlinear Schrödinger equation. Self-focusing and wave collapse. Applied Mathematical Sciences, vol. 139 (Springer, 1999).Google Scholar
37 Taira, K. and Umezu, K.. Positive solutions of sublinear elliptic boundary value problems. Nonlin. Analysis 29 (1997), 711–761.Google Scholar
38 Tehrani, H.. Infinitely many solutions for indefinite semilinear elliptic equations without symmetry. Commun. PDEs 21 (1996), 541–557.Google Scholar
39 Tehrani, H.. Existence results for an indefinite unbounded perturbation of a resonant Schrödinger equation. J. Diff. Eqns 236 (2007), 1–28.Google Scholar
40 Willem, M.. Minimax theorems. Progress in Nonlinear Differential Equations and Their Applications, vol. 24 (Birkhäuser, 1996).Google Scholar
41 Wu, T. F.. On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J. Math. Analysis Applic. 318 (2006), 253–276.Google Scholar
42 Wu, T. F.. Multiple positive solutions for a class of concave-convex elliptic problems in ∝ N involving sign-changing weight. J. Funct. Analysis 258 (2010), 99–131.Google Scholar
43 Zakharov, V. E.. Collapse and self-focusing of Langmuir waves. In Basic plasma physics II (ed. Galeev, A. A. and Sudan, R. N.). Handbook of Plasma Physics, vol. 2, pp. 81–121 (Elsevier, 1984).Google Scholar
44 Zhang, Q. and Wang, Q.. Multiple solutions for a class of sublinear Schrödinger equations. J. Math. Analysis Applic. 389 (2012), 511–518.Google Scholar