American Institute of Mathematical Sciences

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

Infinitely many solutions for a perturbed Schrödinger equation

Rossella Bartolo 1, , Anna Maria Candela 2, and  Addolorata Salvatore 3,

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.
Keywords: Nonlinear Schrödinger equation, broken symmetry, variational approach, unbounded domain., perturbative method.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35J60, 35J20, 58E05, 35B3.
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

show all references

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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