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A Sign-Changing Solution for an Asymptotically Linear Schrödinger Equation

Published online by Cambridge University Press:  05 January 2015

Liliane A. Maia ,
Olimpio H. Miyagaki  and
Sergio H. M. Soares
Liliane A. Maia
Affiliation:
Universidade de Brasília, Departamento de Matemática, 70910-900 Brasília, Distrito Federal, Brazil ([email protected])
Olimpio H. Miyagaki
Affiliation:
Universidade Federal de Juiz de Fora, Departamento de Matemática, 36036-330 Juiz de Fora, Minas Gerais, Brazil ([email protected])
Sergio H. M. Soares
Affiliation:
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Campus de São Carlos, Caixa Postal 668, 13560-970 Sa˜o Carlos, São Paulo, Brazil ([email protected])
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Abstract

The aim of this paper is to present a sign-changing solution for a class of radially symmetric asymptotically linear Schrödinger equations. The proof is variational and the Ekeland variational principle is employed as well as a deformation lemma combined with Miranda’s theorem.

Keywords

Schrödinger equationasymptotically linear elliptic equationvariational methodPrimary 35J6135Q5535Q60
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 3 , October 2015 , pp. 697 - 716
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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References

1.Agrawal, G. P. and Kivshar, Y. S., Optical solitons: from fibers to photonic crystals (Academic Press, 2003).Google Scholar
2.Ambrosetti, A. and Rabinowitz, R., Dual variational methods in critical points theory and applications, J. Funct. Analysis 14 (1973), 349–381.CrossRefGoogle Scholar
3.Bartolo, P., Benci, V. and Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlin. Analysis 7 (1983), 981–1012.Google Scholar
4.Bartsch, T. and Weth, T., A note on additional properties of sign changing solutions to superlinear Schrödinger equations, Topolog. Meth. Nonlin. Analysis 22 (2003), 1–14.Google Scholar
5.Bartsch, T. and Willem, M., Infinitely many radial solutions of a semilinear elliptic problem on ℝN, Arch. Ration. Mech. Analysis 124(3 (1993), 261–276.Google Scholar
6.Bartsch, T., Weth, T. and Willem, M., Partial symmetry of least energy nodal solutions to some variational problems, J. Analysis Math. 96 (2005), 1–18.Google Scholar
7.Berestycki, H. and Lions, P. L., Nonlinear scalar field equations, I, Existence of a ground state, Arch. Ration. Mech. Analysis 82 (1983), 313–346.Google Scholar
8.Castro, A., Cossio, J. and Neuberger, J. M., A sign-changing solution for a superlinear Dirichlet problem, Rocky Mt. J. Math. 27(4) (1997), 1041–1053.Google Scholar
9.Chabrowski, J., On nodal radial solutions of an elliptic problem involving critical Sobolev exponent, Commentat. Math. Univ. Carolinae 37 (1996), 1–16.Google Scholar
10.Conti, M., Merizzi, L. and Terracini, S., Radial solutions of superlinear equations on ℝN, Part I, A global variational approach, Arch. Ration. Mech. Analysis 153 (2000), 291–316.Google Scholar
11.Conti, M., Terracini, S. and Verzini, G., Nehari’s problem and competing species systems, Annales Inst. H. Poincaré Analyse Non Linéaire 19(6 (2002), 871–888.Google Scholar
12.Costa, D. G. and Magalhães, C. A., Variational elliptic problems which are non-quadratic at infinity, Nonlin. Analysis 23 (1994), 1401–1412.Google Scholar
13.Costa, D. G. and Tehrani, H., On a class of asymptotically linear elliptic problems in ℝN, J. Diff. Eqns 173 (2001), 470–494.CrossRefGoogle Scholar
14.Gazzola, F., Serrin, J. and Tang, M., Existence of ground states and free boundary problems for quasilinear elliptic operators, Adv. Diff. Eqns 5(1–3) (2000), 1–30.Google Scholar
15.Gladiali, F., Pacella, F. and Weth, T., Symmetry and nonexistence of low Morse index solutions in unbounded domains, J. Math. Pures Appl. 93 (2010), 536–558.CrossRefGoogle Scholar
16.Jeanjean, L., On the existence of bounded Palais-Smale sequences and application to a Landesmann-Lazer-type problem set on ℝN, Proc. R. Soc. Edinb. A 129 (1999), 787–809.Google Scholar
17.Jeanjean, L. and Tanaka, K., A positive solution for an asymptotically linear elliptic problem on ℝN autonomous at infinity, ESAIM: Control Optim. Calc. Variations 7 (2002), 597–614.Google Scholar
18.Li, G. and Zheng, G. F., The existence of positive solution to some asymptotically linear elliptic equations in exterior domains, Rev. Mat. Ibero. 22(2) (2006), 559–590Google Scholar
19.Lucia, M., Magrone, P. and Zhou, H. S., A Dirichlet problem with asymptotically linear and changing sign nonlinearity, Rev. Mat. Complut. 16 (2003), 465–481.Google Scholar
20.Miranda, C., Un’osservazione su un teorema di Brouwer, Boll. UMI 3 (1940), 5–7.Google Scholar
21.Nehari, Z., Characteristic values associated with a class of nonlinear second order differential equations, Acta Math. 105 (1961), 141–175.Google Scholar
22.Pohozaev, S., Eigenfunctions of the equations Δ u+λf(u) = 0, Sov. Math. Dokl. 6 (1965), 1408–1411.Google Scholar
23.Serrin, J. and Tang, M., Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J. 49(3 (2000), 897–923.CrossRefGoogle Scholar
24.Struwe, M., Superlinear elliptic boundary value problems with rotational symmetry, Arch. Math. 39 (1982), 233–240.Google Scholar
25.Stuart, C. A., Guidance properties of nonlinear planar waveguides, Arch. Ration. Mech. Analysis 125(2) (1993), 145–200.Google Scholar
26.Stuart, C. A. and Zhou, H. S., Applying the mountain pass theorem to an asymptotically linear elliptic equation on ℝN, Commun. PDEs 24(9–10) (1999), 1731–1758.Google Scholar
27.Terracini, S. and Verzini, G., Solutions of prescribed number of zeroes to a class of superlinear ODE’s systems, NoDEA. Nonlin. Diff. Eqns Applic. 8 (2001), 323–341.Google Scholar
28.Willem, M., Minimax theorems, Volume 24 (Birkhäuser, 1996).Google Scholar