# American Institute of Mathematical Sciences

2015, 2015(special): 94-102. doi: 10.3934/proc.2015.0094

## Infinitely many solutions for a perturbed Schrödinger equation

 1 Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via E. Orabona 4, 70125 Bari, Italy 2 Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Campus-via E. Orabona 4, 70125 BARI 3 Dipartimento di Matematica, Università degli Studi di Bari "Aldo Moro", Via E. Orabona 4, 70125 Bari

Received  September 2014 Revised  August 2015 Published  November 2015

We find multiple solutions for a nonlinear perturbed Schrödinger equation by means of the so--called Bolle's method.
Citation: Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many solutions for a perturbed Schrödinger equation. Conference Publications, 2015, 2015 (special) : 94-102. doi: 10.3934/proc.2015.0094
##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. Google Scholar [2] A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc., 267 (1981), 1-32. Google Scholar [3] A. Bahri and P. L. Lions, Morse index of some min-max critical points. I. Applications to multiplicity results, Comm. Pure Appl. Math., 41 (1988), 1027-1037. Google Scholar [4] S. Barile and A. Salvatore, Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains, Discrete Contin. Dyn. Syst. Supplement 2013, (2013), 41-49. Google Scholar [5] S. Barile and A. Salvatore, Multiplicity results for some perturbed elliptic problems in unbounded domains with non-homogeneous boundary conditions, Nonlinear Analysis, 110 (2014), 47-60. Google Scholar [6] T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbbR^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. Google Scholar [7] V. Benci and D. Fortunato, Discreteness conditions of the spectrum of Schrödinger operators, J. Math. Anal. Appl., 64 (1978), 695-700. Google Scholar [8] F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Mathematics and its Applications, (Soviet Series) 66, Kluwer Academic Publishers, Dordrecht, 1991. Google Scholar [9] P. Bolle, On the Bolza problem, J. Differential Equations, 152 (1999), 274-288. Google Scholar [10] P. Bolle, N. Ghoussoub and H. Tehrani, The multiplicity of solutions in non-homogeneous boundary value problems, Manuscripta Math., 101 (2000), 325-350. Google Scholar [11] A. Candela, G. Palmieri and A. Salvatore, Radial solutions of semilinear elliptic equations with broken symmetry, Topol. Methods Nonlinear Anal., 27 (2006), 117-132. Google Scholar [12] M. Clapp, Y. Ding and S. Hernández-Linares, Strongly indefinite functionals with perturbed symmetries and multiple solutions of nonsymmetric elliptic systems, Electron. J. Differential Equations, 100 (2004), 18 pp. Google Scholar [13] D. E. Edmunds and W.D. Evans, Spectral Theory and Differential Operators, Oxford Mathematical Monographs, New York, 1987. Google Scholar [14] P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309-318. Google Scholar [15] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. Google Scholar [16] P. H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc., 272 (1982), 753-769. Google Scholar [17] A. Salvatore, Multiple solutions for perturbed elliptic equations in unbounded domains, Adv. Nonlinear Stud., 3 (2003), 1-23. Google Scholar [18] A. Salvatore, M. Squassina, Deformation from symmetry for nonhomogeneous Schrödinger equations of higher order on unbounded domains, Electron. J. Differential Equations, 65 (2003), 1-15. Google Scholar [19] M. Struwe, Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscripta Math., 32 (1980), 335-364. Google Scholar [20] M. Struwe, Infinitely many solutions of superlinear boundary value problems with rotational symmetry, Arch. Math., 36 (1981), 360-369. Google Scholar [21] M. Struwe, Superlinear elliptic boundary value problems with rotational symmetry, Arch. Math., 39 (1982), 233-240. Google Scholar [22] K. Tanaka, Morse indices at critical points related to the symmetric mountain pass theorem and applications, Comm. Partial Differential Equations, 14 (1989), 99-128. Google Scholar [23] W. Zou and M. Schechter, Critical Point Theory and Its Applications, Springer, New York, 2006. Google Scholar

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##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. Google Scholar [2] A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc., 267 (1981), 1-32. Google Scholar [3] A. Bahri and P. L. Lions, Morse index of some min-max critical points. I. Applications to multiplicity results, Comm. Pure Appl. Math., 41 (1988), 1027-1037. Google Scholar [4] S. Barile and A. Salvatore, Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains, Discrete Contin. Dyn. Syst. Supplement 2013, (2013), 41-49. Google Scholar [5] S. Barile and A. Salvatore, Multiplicity results for some perturbed elliptic problems in unbounded domains with non-homogeneous boundary conditions, Nonlinear Analysis, 110 (2014), 47-60. Google Scholar [6] T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbbR^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. Google Scholar [7] V. Benci and D. Fortunato, Discreteness conditions of the spectrum of Schrödinger operators, J. Math. Anal. Appl., 64 (1978), 695-700. Google Scholar [8] F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Mathematics and its Applications, (Soviet Series) 66, Kluwer Academic Publishers, Dordrecht, 1991. Google Scholar [9] P. Bolle, On the Bolza problem, J. Differential Equations, 152 (1999), 274-288. Google Scholar [10] P. Bolle, N. Ghoussoub and H. Tehrani, The multiplicity of solutions in non-homogeneous boundary value problems, Manuscripta Math., 101 (2000), 325-350. Google Scholar [11] A. Candela, G. Palmieri and A. Salvatore, Radial solutions of semilinear elliptic equations with broken symmetry, Topol. Methods Nonlinear Anal., 27 (2006), 117-132. Google Scholar [12] M. Clapp, Y. Ding and S. Hernández-Linares, Strongly indefinite functionals with perturbed symmetries and multiple solutions of nonsymmetric elliptic systems, Electron. J. Differential Equations, 100 (2004), 18 pp. Google Scholar [13] D. E. Edmunds and W.D. Evans, Spectral Theory and Differential Operators, Oxford Mathematical Monographs, New York, 1987. Google Scholar [14] P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309-318. Google Scholar [15] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. Google Scholar [16] P. H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc., 272 (1982), 753-769. Google Scholar [17] A. Salvatore, Multiple solutions for perturbed elliptic equations in unbounded domains, Adv. Nonlinear Stud., 3 (2003), 1-23. Google Scholar [18] A. Salvatore, M. Squassina, Deformation from symmetry for nonhomogeneous Schrödinger equations of higher order on unbounded domains, Electron. J. Differential Equations, 65 (2003), 1-15. Google Scholar [19] M. Struwe, Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscripta Math., 32 (1980), 335-364. Google Scholar [20] M. Struwe, Infinitely many solutions of superlinear boundary value problems with rotational symmetry, Arch. Math., 36 (1981), 360-369. Google Scholar [21] M. Struwe, Superlinear elliptic boundary value problems with rotational symmetry, Arch. Math., 39 (1982), 233-240. Google Scholar [22] K. Tanaka, Morse indices at critical points related to the symmetric mountain pass theorem and applications, Comm. Partial Differential Equations, 14 (1989), 99-128. Google Scholar [23] W. Zou and M. Schechter, Critical Point Theory and Its Applications, Springer, New York, 2006. Google Scholar
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