Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities

  • Louis JeanjeanEmail author
  • Kazunaga Tanaka


We consider a class of equations of the form \(-\varepsilon^2\Delta u + V(x)u = f(u), \quad u\in H^1({\bf R}^N).\) By variational methods, we show the existence of families of positive solutions concentrating around local minima of the potential V(x), as \(\varepsilon\to 0\). We do not require uniqueness of the ground state solutions of the associated autonomous problems nor the monotonicity of the function \(\xi\mapsto \frac{f(\xi)}{\xi}\). We deal with asymptotically linear as well as superlinear nonlinearities.


Local Minimum State Solution Variational Method Potential Versus Elliptic Problem 
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  1. 1.
    Ambrosetti, A., Badiale, M., Cingolani, S.: Semiclassical states of nonlinear Schrödinger equations. Arch. Rational Mech. Anal. 140, 285-300 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Berestycki, H., Lions, P.L.: Nonlinear scalar field equations I. Arch. Rat. Mech. Anal. 82, 313-346 (1983)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Berestycki, H., Gallouët, T., Kavian, O.: Equations de Champs scalaires euclidiens non lin\’ eaires dans le plan. C.R. Acad. Sci; Paris Ser. I Math. 297, 307-310 (1983) and Publications du Laboratoire d’Analyse Numérique, Université de Paris VI (1984)Google Scholar
  4. 4.
    Brezis, H.: Analyse fonctionnelle. Masson 1983Google Scholar
  5. 5.
    del Pino, M., Felmer, P.: Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var. PDE 4, 121-137 (1996)CrossRefzbMATHGoogle Scholar
  6. 6.
    del Pino, M., Felmer, P.: Multi-peak bound states of nonlinear Schrödinger equations. Ann. IHP, Analyse Nonlineaire 15, 127-149 (1998)Google Scholar
  7. 7.
    del Pino, M., Felmer, P.: Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math. Ann. 324 (1), 1-32 (2002)CrossRefzbMATHGoogle Scholar
  8. 8.
    del Pino, M., Felmer, P., Tanaka, K.: An elementary construction of complex patterns in nonlinear Schrödinger equations. Nonlinearity 15 (5), 1653-1671 (2002)CrossRefzbMATHGoogle Scholar
  9. 9.
    Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69 (3), 397-408 (1986)CrossRefzbMATHGoogle Scholar
  10. 10.
    Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry of positive solutions of nonlinear equations in \({\bf R}^N\). In: Nachbin, L. (ed.) Math. Anal. and Applications, Part A, Advances in Math. Suppl. Studies 7A, pp. 369-402. Academic Press 1981Google Scholar
  11. 11.
    Grossi, M.: Some results on a class of nonlinear Schrödinger equations. Math. Zeit. 235, 687-705 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Gui, C.: Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method. Comm. Partial Differential Equations 21, 787-820 (1996)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Jeanjean, L.: On the existence of bounded Palais-Smale sequences and applications to a Landesman-Lazer-type problem set on \({\bf R}^N\). Proc. Roy. Soc. Edinburgh 129A, 787-809 (1999)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Jeanjean, L., Tanaka, K.: A positive solution for an asymptotically linear elliptic problem on \({\bf R}^N\) autonomous at infinity. ESAIM Control Optim. Calc. Var. 7, 597-614 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Jeanjean, L., Tanaka, K.: A remark on least energy solutions in \({\bf R}^N\). Proc. Amer. Math. Soc. 131, 2399-2408 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Kang, X., Wei, J.: On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Advances Diff. Eq. 5, 899-928 (2000)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Li, Y.-Y.: On asingularly perturbed elliptic equation. Adv. Differential Equations 2, 955-980 (1997)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 109-145, (1984) and II. l Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 223-283 (1984)zbMATHGoogle Scholar
  19. 19.
    Nakashima, K., Tanaka, K.: Clustering layers and boundary layers in spatially inhomogeneous phase transition problems. Ann. I. H. Poincaré Anal. Non Linéaire 20, 107-143 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Oh, Y.-G.: Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (V)a. Comm. Partial Differential Equations 13 (12), 1499-1519 (1988)zbMATHGoogle Scholar
  21. 21.
    Oh, Y.-G.: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Comm. Math. Phys. 131 (2), 223-253 (1990)zbMATHGoogle Scholar
  22. 22.
    Pistoia, A.: Multi-peak solutions for a class of nonlinear Schrödinger equations. NoDEA Nonlinear Diff. Eq. Appl. 9, 69-91 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Rabinowitz, P.: On a class of nonlinear Schrödinger equations. Z. Angew Math Phys 43, 270-291 (1992)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Struwe, M.: A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187, 511-517 (1984)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Stuart, C.A.: Personal communication. Summer 2000Google Scholar
  26. 26.
    Wang, X.: On concentration of positive bound states of nonlinear Schrödinger equations. Comm. Math. Phys. 153, 229-244 (1993)MathSciNetzbMATHGoogle Scholar

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© Springer-Verlag Berlin/Heidelberg 2004

Authors and Affiliations

  1. 1.Equipe de Mathématiques (UMR CNRS 6623)Université de Franche-ComtéBesançonFrance

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