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A local mountain pass type result for a system of nonlinear Schrödinger equations

  • Norihisa IkomaEmail author
  • Kazunaga Tanaka
Article

Abstract

We consider a singular perturbation problem for a system of nonlinear Schrödinger equations:
$$ \begin{array}{l} -\varepsilon^2\Delta v_1 +V_1(x)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\varepsilon^2\Delta v_2 +V_2(x)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N), \end{array} \quad\quad\quad\quad\quad (*) $$
where N = 2, 3, μ 1, μ 2, β > 0 and V 1(x), V 2(x): R N → (0, ∞) are positive continuous functions. We consider the case where the interaction β > 0 is relatively small and we define for \({P\in{\bf R}^N}\) the least energy level m(P) for non-trivial vector solutions of the rescaled “limit” problem:
$$ \begin{array}{l} -\Delta v_1 +V_1(P)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\Delta v_2 +V_2(P)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N). \end{array} \quad\quad\quad\quad\quad\quad (**) $$
We assume that there exists an open bounded set \({\Lambda\subset{\bf R}^N}\) satisfying
$$ {\mathop {\rm inf} _{P\in\Lambda} m(P)} < {\mathop {\rm inf}_{P\in\partial\Lambda} m(P)}. $$
We show that (*) possesses a family of non-trivial vector positive solutions \({\{(v_{1\varepsilon}(x), v_{2\varepsilon} (x))\}_{\varepsilon\in (0,\varepsilon_0]}}\) which concentrates—after extracting a subsequence ε n → 0—to a point \({P_0\in\Lambda}\) with \({m(P_0)={\rm inf}_{P\in\Lambda}m(P)}\). Moreover (v 1ε (x), v 2ε (x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling.

Mathematics Subject Classification (2000)

35B25 35J65 58E05 

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References

  1. 1.
    Ambrosetti A., Badiale M., Cingolani S.: Semiclassical states of nonlinear Schrödinger equations. Arch. Rational Mech. Anal. 140(3), 285–300 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ambrosetti A., Colorado E.: Standing waves of some coupled nonlinear Schrödinger equations. J. Lond. Math. Soc. (2) 75(1), 67–82 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bartsch T., Wang Z.-Q.: Note on ground states of nonlinear Schrödinger systems. J. Partial Differ. Eq. 19(3), 200–207 (2006)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bartsch T., Wang Z.-Q., Wei J.: Bound states for a coupled Schrödinger system. J. Fixed Point Theory Appl. 2(2), 353–367 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bartsch T., Dancer N., Wang Z.-Q.: A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system. Calc. Var. Partial Diff. Eq. 37 3–4, 345–361 (2010)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Berestycki H., Lions P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82(4), 313–345 (1983)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Busca J., Sirakov B.: Symmetry results for semilinear elliptic systems in the whole space. J. Differ. Equ. 163(1), 41–56 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Byeon J., jeanjean L.: Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch. Ration. Mech. Anal. 185(2), 185–200 (2008) 190(3), 549–551 (2008)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Byeon J., Jeanjean L.: Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity. Discr. Contin. Dyn. Syst. 19(2), 255–269 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Byeon J., Jeanjean L., Tanaka K.: Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases. Commun. Partial Differ. Equ. 33(6), 1113–1136 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dancer N., Wei J.: Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction. Trans. Am. Math. Soc. 361(3), 1189–1208 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    de Figueiredo D.G., Lopes O.: Solitary waves for some nonlinear Schrödinger systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(1), 149–161 (2008)zbMATHCrossRefGoogle Scholar
  13. 13.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    del Pino M., Felmer P.: Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math. Ann. 324(1), 1–32 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    del Pino M., Kowalczyk M., Wei J.: Concentration on curves for nonlinear Schrödinger equations. Commun. Pure Appl. Math. 60(1), 113–146 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Floer A., Weinstein A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69(3), 397–408 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gui C.: Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method. Commun. Partial Differ. Equ. 21(5–6), 787–820 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Hirano N., Shioji N.: Multiple existence of solutions for coupled nonlinear Schrödinger equations. Nonlinear Anal. 68(12), 3845–3859 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ikoma N.: Uniqueness of positive solutions for a nonlinear elliptic system. Nonlinear Differ. Equ. Appl. 16(5), 555–567 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Ikoma, N.: Existence of standing waves for coupled nonlinear Schrödinger equations. Tokyo J. Math. (to appear)Google Scholar
  21. 21.
    Jeanjean L., Tanaka K.: Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc. Var. Partial Differ. Equ. 21(3), 287–318 (2004)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Li Y.Y.: On a singularly perturbed elliptic equation. Adv. Differ. Equ. 2(6), 955–980 (1997)zbMATHGoogle Scholar
  23. 23.
    Lin T.-C., Wei J.: Spikes in two coupled nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(4), 403–439 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Lin T.-C., Wei J.: Ground state of N coupled nonlinear Schrödinger equations in R n , n ≤ 3. Commun. Math. Phys. 255(3), 629–653 (2005) and Commun. Math. Phys. 277(2), 573–576 (2008); 277(2), 573–576 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Lin T.-C., Wei J.: Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials. J. Differ. Equ. 229(2), 538–569 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Maia L.A., Montefusco E., Pellacci B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differ. Equ. 229(2), 743–767 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Montefusco E., Pellacci B., Squassina M.: Semiclassical states for weakly coupled nonlinear Schrödinger systems. J. Eur. Math. Soc. 10(1), 47–71 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Pomponio A.: Coupled nonlinear Schrödinger systems with potentials. J. Differ. Equ. 227(1), 258–281 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Rabinowitz P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43(2), 270–291 (1982)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Sirakov B.: Least energy solitary waves for a system of nonlinear Schrödinger equations in R n. Commun. Math. Phys. 271(1), 199–221 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Wei G.-M.: Existence and concentration of ground states of coupled nonlinear Schrödinger equations. J. Math. Anal. Appl. 332(2), 846–862 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Wei G.-M.: Existence and concentration of ground states of coupled nonlinear Schrödinger equations with bounded potentials. Chin. Ann. Math. Ser. B 29(3), 247–264 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Wei J., Weth T.: Nonradial symmetric bound states for a system of coupled Schrödinger equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18(3), 279–293 (2007)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Wei, J., Yao, W.: Note on uniqueness of positive solutions for some coupled nonlinear Schrödinger equations. Methods Anal. Appl. (to appear)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsSchool of Science and Engineering, Waseda UniversityTokyoJapan

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