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.

[2]

A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc., 267 (1981), 1-32.

[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.

[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.

[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.

[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.

[7]

V. Benci and D. Fortunato, Discreteness conditions of the spectrum of Schrödinger operators, J. Math. Anal. Appl., 64 (1978), 695-700.

[8]

F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Mathematics and its Applications, (Soviet Series) 66, Kluwer Academic Publishers, Dordrecht, 1991.

[9]

P. Bolle, On the Bolza problem, J. Differential Equations, 152 (1999), 274-288.

[10]

P. Bolle, N. Ghoussoub and H. Tehrani, The multiplicity of solutions in non-homogeneous boundary value problems, Manuscripta Math., 101 (2000), 325-350.

[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.

[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.

[13]

D. E. Edmunds and W.D. Evans, Spectral Theory and Differential Operators, Oxford Mathematical Monographs, New York, 1987.

[14]

P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309-318.

[15]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.

[16]

P. H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc., 272 (1982), 753-769.

[17]

A. Salvatore, Multiple solutions for perturbed elliptic equations in unbounded domains, Adv. Nonlinear Stud., 3 (2003), 1-23.

[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.

[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.

[20]

M. Struwe, Infinitely many solutions of superlinear boundary value problems with rotational symmetry, Arch. Math., 36 (1981), 360-369.

[21]

M. Struwe, Superlinear elliptic boundary value problems with rotational symmetry, Arch. Math., 39 (1982), 233-240.

[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.

[23]

W. Zou and M. Schechter, Critical Point Theory and Its Applications, Springer, New York, 2006.

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.

[2]

A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc., 267 (1981), 1-32.

[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.

[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.

[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.

[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.

[7]

V. Benci and D. Fortunato, Discreteness conditions of the spectrum of Schrödinger operators, J. Math. Anal. Appl., 64 (1978), 695-700.

[8]

F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Mathematics and its Applications, (Soviet Series) 66, Kluwer Academic Publishers, Dordrecht, 1991.

[9]

P. Bolle, On the Bolza problem, J. Differential Equations, 152 (1999), 274-288.

[10]

P. Bolle, N. Ghoussoub and H. Tehrani, The multiplicity of solutions in non-homogeneous boundary value problems, Manuscripta Math., 101 (2000), 325-350.

[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.

[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.

[13]

D. E. Edmunds and W.D. Evans, Spectral Theory and Differential Operators, Oxford Mathematical Monographs, New York, 1987.

[14]

P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309-318.

[15]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.

[16]

P. H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc., 272 (1982), 753-769.

[17]

A. Salvatore, Multiple solutions for perturbed elliptic equations in unbounded domains, Adv. Nonlinear Stud., 3 (2003), 1-23.

[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.

[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.

[20]

M. Struwe, Infinitely many solutions of superlinear boundary value problems with rotational symmetry, Arch. Math., 36 (1981), 360-369.

[21]

M. Struwe, Superlinear elliptic boundary value problems with rotational symmetry, Arch. Math., 39 (1982), 233-240.

[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.

[23]

W. Zou and M. Schechter, Critical Point Theory and Its Applications, Springer, New York, 2006.

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