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Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities

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Abstract

We study the following nonlinear Schrödinger equations

$$\begin{array}{lll}(-i\varepsilon\nabla+A(x))^2 w + V(x)w = W(x)g(|w|)w; \quad \quad \quad \quad \quad \quad \quad \quad \quad (0.1)\\(-i\varepsilon\nabla+A(x))^2 w + V(x)w = W(x)\left(g(|w|)+|w|^{2^*-2}\right)w,\quad \quad \quad\,\,(0.2)\end{array}$$

for \({w \in H^1\left( \mathbb{R}^N, \mathbb{C} \right)}\) , where g(|w|)w is super linear and subcritical, 2* = 2N/(N − 2) if N > 2 and = ∞ if N = 2, min V > 0 and inf W > 0. Under proper assumptions we explore the existence and concentration phenomena of semiclassical solutions of (0.1). The most interesting result obtained here refers to the critical case. We establish the existence and describe the concentration of semiclassical ground states of (0.2) provided either min V <  τ 0 for some τ0 > 0, or \({\max W > \kappa_{0}}\) for some \({\kappa_0 > 0}\) .

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References

  1. Ackermann N.: On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z. 248, 423–443 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Ambrosetti A., Malchiodi A.: Perturbation Methods and Semilinear Elliptic Problems on \({\mathbb{R}^n}\) . Birkhäuser Verlag, Basel (2006)

    MATH  Google Scholar 

  4. Arioli G., Szulkin A.: A semilinear Schrödinger equation in the presence of a magnetic field. Arch. Rational Mech. Anal. 170, 277–295 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barile S.: Multiple semiclassical states for singular magnetic nonlinear Schrödinger equations. Electron. J. Differ. Equ 37, 1–18 (2008)

    MathSciNet  Google Scholar 

  6. Barile S.: A multiplicity result for singular NLS equations with magnetic potentials. Nonlinear Anal. 68, 3525–3540 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Barile S., Cingolani S., Secchi S.: Single-peaks for a magnetic Schrödinger equation with critical growth. Adv. Differ. Equ. 11, 1135–1166 (2006)

    MathSciNet  MATH  Google Scholar 

  8. Bartsch T., Pankov A., Wang Z.-Q.: Nonlinear Schrödinger equations with steep potential well. Commun. Contemp. Math. 3, 549–569 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Byeon J., Wang Z.-Q.: Standing waves with a critical frequency for nonlinear Schrödinger equations, II. Calc. Var. Partial Differ. Equ. 18, 207–219 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chabrowski J., Szulkin A.: On the Schrödinger equation involving a critical Sobolev exponent and magnetic field. Topol. Methods Nonlinear Anal. 25, 3–21 (2005)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  14. Cingolani S., Simone S.: Semiclassical states for NLS equations with magnetic potentials having polynomial growths. J. Math. Phys. 46, 19 (2005)

    Article  Google Scholar 

  15. Coti-Zelati V., Rabinowitz P.: Homoclinic type solutions for a semilinear elliptic PDE on \({\mathbb{R}^n}\) . Commun. Pure Appl. Math. 46, 1217–1269 (1992)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Ding, Y.H.: Variational Methods for Strongly Indefinite Problems. Interdisciplinary Mathematical Science, vol. 7. World Scientific Publishing, Singapore (2007)

  18. Ding Y.H., Szulkin A.: Bound states for semilinear Schrödinger equations with sign-changing potential. Calc. Var. Partial Differ. Equ. 29, 397–419 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ding Y.H., Tanaka K.: Multiplicity of positive Solutions of a nonlinear Schrödinger equation. Manuscripta Math. 112, 109–135 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ding Y.H., Wang Z.-Q.: Bound states of nonlinear Schrödinger equations with magnetic fields. Annali di Matematica 190, 427–451 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ding Y.H., Wei J.C.: Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials. J. Funct. Anal. 251, 546–572 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Esteban M.J., Lions P.L.: Stationary solutions of nonlinear Schrödinger equations with an external magnetic field. In: Colombini, F., Marino, A., Modica, L., Spagnolo, S. (eds) Partial Differential Equations and the Calculus of Variations, vol 1., pp. 401–449. Birkhäuser, Boston (1989)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Gui C.: Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method. Commun. Partial Differ. Equ. 21, 787–820 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. Han P.: Solutions for singular critical growth Schrödinger equations with magnetic field. Port. Math. (N.S.) 63, 37–45 (2006)

    MathSciNet  MATH  Google Scholar 

  26. Kurata K.: Existence and semiclassical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic field. Nonlinear Anal. 41, 763–778 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liang S., Zhang J.: Solutions of perturbed Schrödinger equations with electromagnetic fields and critical nonlinearity. Proc. Edinb. Math. Soc. (2) 54, 131–147 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ni W.M., Wei J.C.: On positive solutions concentration on spheres for the Gierer-Meinhardt system. J. Differ. Equ. 221, 158–189 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pankov A.: On nontrivial solutions of nonlinear Schrödinger equation with external magnetic field (Russian). Funktsioal Anal. i Prilozhen 37, 88–91 (2003)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Sirakov B.: Standing wave solutions of the nonlinear Schrödinger equations in \({\mathbb{R}^n}\) . Annali di Matematica 183, 73–83 (2002)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  33. Wang F.: On an electromagnetic Schrödinger equation with critical growth. Nonlinear Anal. 69, 4088–4098 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Willem M.: Minimax Theorems. Birkhäuser, Boston (1996)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Yanheng Ding

  2. Department of Mathematics, Xuzhou Normal University, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Xiaoying Liu

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  1. Yanheng Ding
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  2. Xiaoying Liu
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Correspondence to Yanheng Ding.

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Ding, Y., Liu, X. Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities. manuscripta math. 140, 51–82 (2013). https://doi.org/10.1007/s00229-011-0530-1

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