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Multiplicity of positive solutions of nonlinear Schrödinger equations concentrating at a potential well

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Abstract

We consider singularly perturbed nonlinear Schrödinger equations

$$\begin{aligned} - \varepsilon ^2 \Delta u + V(x)u = f(u), \, \, u > 0, \, \, v \in H^1( \mathbb {R}^N) \end{aligned}$$
(0.1)

where \(V \in C(\mathbb {R}^N, \mathbb {R})\) and \(f\) is a nonlinear term which satisfies the so-called Berestycki–Lions conditions. We assume that there exists a bounded domain \(\Omega \subset \mathbb {R}^N\) such that

$$\begin{aligned} m_0 \equiv \inf _{x \in \Omega } V(x) < \inf _{x \in \partial \Omega } V(x) \end{aligned}$$

and we set \(K = \{ x \in \Omega \ | \ V(x) = m_0\}\). For \(\varepsilon >0\) small we prove the existence of at least \({\mathrm{cupl}}(K) + 1\) solutions to (0.1) concentrating, as \(\varepsilon \rightarrow 0\) around \(K\). We remark that, under our assumptions of \(f\), the search of solutions to (0.1) cannot be reduced to the study of the critical points of a functional restricted to a Nehari manifold.

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References

  1. Ambrosetti, A., Badiale, M., Cingolani, S.: Semiclassical states of nonlinear Schrödinger equations. Arch. Rational Mech. Anal. 140(3), 285–300 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ambrosetti, A., Malchiodi, A., Ni, W.M.: Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres I. Comm. Math. Phys. 235, 427–466 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ambrosetti, A., Malchiodi, A., Secchi, S.: Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Ration. Mech. Anal. 159, 253–271 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bartsch, T.: Note on category and cup-length, private communication (2012).

  5. Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82(4), 313–345 (1983)

    MATH  MathSciNet  Google Scholar 

  6. Bahri, A., Coron, J.M.: On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain. Comm. Pure Appl. Math. 41(3), 253–294 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bartsch, T., Weth, T.: The effect of the domain’s configuration space on the number of nodal solutions of singularly perturbed elliptic equations. Topol. Methods Nonlinear Anal. 26, 109–133 (2005)

    MATH  MathSciNet  Google Scholar 

  8. Benci, V., Cerami, G.: The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems. Arch. Rational Mech. Anal. 114(1), 79–93 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  9. Benci, V., Cerami, G.: Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology. Calc. Var. Partial Differ. Equ. 2(1), 29–48 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  10. Benci, V., Cerami, G., Passaseo, D.: On the number of the positive solutions of some nonlinear elliptic problems, Nonlinear Analysis, tribute in honor of G. Prodi, Quaderno Scuola Normale Sup. Pisa, pp. 93–107 (1991)

  11. Byeon, J., Jeanjean, L.: Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch. Ration. Mech. Anal. 185(2), 185–200 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Byeon, J., Jeanjean, L., Tanaka, K.: Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases. Comm. Partial Differ. Equ. 33(6), 1113–1136 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Byeon, J., Tanaka, K.: Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential, J. Euro. Math. Soc. (JEMS) 15(5), 1859–1899 (2013)

  14. Byeon, J., Tanaka, K.: Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations, Mem. Amer. Math. Soc. 229(1076) (2014)

  15. Byeon, J., Wang, Z.-Q.: Standing waves with a critical frequency for nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 165, 295–316 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Cerami, G., Passaseo, D.: Existence and multiplicity of positive solutions for nonlinear elliptic problems in exterior domains with “rich” topology. Nonlinear Anal. TMA 18, 109–119 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  17. Cingolani, S.: Semiclassical stationary states of nonlinear Schrödinger equations with an external magnetic field. J. Differ. Equ. 188(1), 52–79 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Cingolani, S., Jeanjean, L., Secchi, S.: Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions. ESAIM Control Optim. Calc. Var. 15(3), 653–675 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. Cingolani, S., Lazzo, M.: Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations. Topol. Methods Nonlinear Anal. 10(1), 1–13 (1997)

    MATH  MathSciNet  Google Scholar 

  20. Cingolani, S., Lazzo, M.: Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Differ. Equ. 160(1), 118–138 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  21. Cingolani, S., Nolasco, M.: Multi-peak periodic semiclassical states for a class of nonlinear Schrdinger equations, Proceedings of the Royal Society of Edinburgh, vol. 128 A, pp. 1249–1260 (1998)

  22. Cingolani, S., Secchi, S.: Semiclassical limit for nonlinear Schrödinger equations with magnetic fields. J. Math. Anal. Appl. 275, 108–130 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. Cortazar, C., Elgueta, M., Felmer, P.: Uniqueness of positive solutions of \(\Delta u + f(u) = 0\) in \({\mathbb{R}}^{N}\), \(N \ge 3\). Arch. Ration. Mech. Anal. 142, 127–141 (1998)

  24. Dancer, E.N.: Peak solutions without non-degeneracy conditions. J. Diff. Equ. 246, 3077–3088 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Dancer, E.N., Lam, K.Y., Yan, S.: The effect of the graph topology on the existence of multipeak solutions for nonlinear Schrödinger equations. Abstr. Appl. Anal. 3(3–4), 293–318 (1998)

    Article  MathSciNet  Google Scholar 

  26. Dancer, E.N., Wei, J.: On the effect of domain topology in a singular perturbation problem. Top. Methods Nonlinear Anal. 4, 347–368 (1999)

    MATH  Google Scholar 

  27. Dancer, E.N., Yan, S.: A singularly perturbed elliptic problem in bounded domains with nontrivial topology. Adv. Differ. Equ. 4(3), 347–368 (1999)

    MATH  MathSciNet  Google Scholar 

  28. Dancer, E.N., Yan, S.: On the existence of multipeak solutions for nonlinear field equations on \({\mathbb{R}}^{N}\). Discrete Contin. Dynam. Systems 6, 39–50 (2000)

  29. J. Davila, M. del Pino, I. Guerra, Non-uniqueness of positive ground states of non-linear Schrödinger equations, Proc. London Math. Soc. (3) 106 (2013) 318–344.

  30. D’Avenia, P., Pomponio, A., Ruiz, D.: Semiclassical states for the nonlinear Schrödinger equation on saddle points of the potential via variational methods. J. Funct. Anal. 262, 4600–4633 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  31. del Pino, M., Felmer, P.: Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4(2), 121–137 (1996)

    Article  MATH  Google Scholar 

  32. del Pino, M., Felmer, P.: Semi-classical states for nonlinear Schrödinger equations. J. Funct. Anal. 149, 245–265 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  33. del Pino, M., Felmer, P.L.: Multi-peak bound states for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré 15, 127–149 (1998)

    Article  MATH  Google Scholar 

  34. del Pino, M., Felmer, P.L.: Semi-classical states for nonlinear Schrödinger equations: a variational reduction method. Math. Ann. 324(1), 1–32 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  35. del Pino, M., Kowalczyk, M., Wei, J.: Concentration on curves for nonlinear Schrödinger equations. Comm. Pure Appl. Math 60, 113–146 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  36. Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69(3), 397–408 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  37. G. Fournier, M. Willem, Multiple solutions of the forced double pendulum equation, Ann. Inst. H. Poincaré Anal. Non Lineaire 6 (suppl.), 259–281 (1989)

  38. Fournier, G., Willem, M.: Relative category and the calculus of variations, variational methods. Progr. Nonlinear Differ. Equ. Appl. 4, 95–104 (1990)

    MathSciNet  Google Scholar 

  39. Gui, C.: Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method. Comm. in P.D.E., 21, 787–820 (1996).

  40. Kang, X., Wei, J.: On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Adv. Differ. Equ. 5, 899–928 (2000)

    MATH  MathSciNet  Google Scholar 

  41. Jeanjean, L., Tanaka, K.: A remark on least energy solutions in \({\mathbb{R}}^{N}\). Proc. Amer. Math. Soc. 131(8), 2399–2408 (2003)

  42. Jeanjean, L., Tanaka, K.: Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc. Var. Partial Diff. Eq. 21(3), 287–318 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  43. Li, Y.Y.: On a singularly perturbed elliptic equation. Adv. Diff. Equ. 2, 955–980 (1997)

    MATH  Google Scholar 

  44. Oh, Y.G.: Existence of semiclassical bound states of nonlinear Schrödinger equations. Comm. Partial Diff. Eq. 13, 1499–1519 (1988)

    Article  MATH  Google Scholar 

  45. Oh, Y.G.: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Comm. Math. Phys. 131, 223–253 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  46. Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43(2), 270–291 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  47. Spanier, E.H.: Algebraic topology. McGraw Hill, New York (1966)

    MATH  Google Scholar 

  48. Wang, X.: On concentration of positive bound states of nonlinear Schrödinger equations. Comm. Math. Phys. 153, 229–244 (1993)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors would like to thank Professor Thomas Bartsch for providing them with a proof of Lemma 5.5 and for very helpful discussions on relative category and cup-length. The first author would like to thank Professor Marco Degiovanni for very helpful discussions on relative category and for suggesting the counter-example in Remark 4.3.

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Authors and Affiliations

  1. Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via E. Orabona 4, 70125 , Bari, Italy

    Silvia Cingolani

  2. Laboratoire de Mathématiques (UMR CNRS 6623), Université de Franche-Comté, 16, Route de Gray, 25030 , Besançon Cedex, France

    Louis Jeanjean

  3. Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shijuku-ku, Tokyo , 169-8555, Japan

    Kazunaga Tanaka

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  1. Silvia Cingolani
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Correspondence to Silvia Cingolani.

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Communicated by P. Rabinowitz.

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Cingolani, S., Jeanjean, L. & Tanaka, K. Multiplicity of positive solutions of nonlinear Schrödinger equations concentrating at a potential well. Calc. Var. 53, 413–439 (2015). https://doi.org/10.1007/s00526-014-0754-5

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