Standing waves for nonlinear Schrödinger equations with a general nonlinearity: One and two dimensional cases

Communications in Partial Differential Equations

Volumn 33, Issue 6, 2008, Pages 1113-1136

  Byeon, Jaeyoung ^a   Jeanjean, Louis ^b   Tanaka, Kazunaga ^c  

^a Pohang University of Science and Technology (POSTECH)   (South Korea)

^b UNIVERSITÉ DE FRANCHE COMTÉ   (France)

^c WASEDA UNIVERSITY   (Japan)

Author keywords

Berestycki Lions conditions; Nonlinear Schr dinger equations; Standing waves; Variational methods

Indexed keywords

EID: 45849139773     PISSN: 03605302     EISSN: 15324133     Source Type: Journal    
DOI: 10.1080/03605300701518174     Document Type: Article

References (33)

1. N and their best exponents. Proc. Amer. Math. Soc. 128:2051-2057.
2. Ambrosetti, A., Badiale, M., Cingolani, S. (1997). Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 140:285-300.
3. Ambrosetti, A., Malchiodi, A., Secchi, S. (2001). Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Ration. Mech. Anal. 159:253-271.
4. Berestycki, H., Lions, P. L. (1983). Nonlinear scalar field equations I. Arch. Ration. Mech. Anal. 82:313-346.
5. Berestycki, H., Gallouët, T., Kavian, O. (1984). Equations de Champs scalaires euclidens non linéaires dans le plan. C.R. Acad. Sci; Paris Ser. I Math. 297:307-310 and Publications du Laboratoire d'Analyse Numérique, Université de Paris VI.
6. Byeon, J., Jeanjean, L. (2007). Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch. Ration. Mech. Anal. 185:185-200.
7. Byeon, J., Wang, Z.-Q. (2002). Standing waves with a critical frequency for nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 165:295-316.
8. Byeon, J., Wang, Z.-Q. (2003). Standing waves with a critical frequency for nonlinear Schrödinger equations II. Calculus of Variations and PDE 18:207-219.
9. Byeon, J., Oshita, Y. (2004). Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations. Comm. PDE 29:1877-1904.
10. Cingolani S., Nolasco, M. (1998). Multi-peak periodic semiclassical states for a class of nonlinear Schrödinger equations. Proc. Roy. Soc. Edinburgh Sect. A 128:1249-1260.
11. Cingolani, S., Lazzo, M. (2000). Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Differential Equations 160:118-138.
12. N. Comm. Pure Appl. Math. 45:1217-1269.
13. N. Discrete Contin. Dynam. Systems 6:39-50.
14. Dancer, E. N., Lam, K. Y., Yan, S. (1998). The effect of the graph topology on the existence of multipeak solutions for nonlinear Schrödinger equations. Abstr. Appl. Anal. 3:293-318.
15. Del Pino, M., Felmer, P. L. (1996). Local mountain passes for semilinear elliptic problems in unbounded domains. Calculus of Variations and PDE 4:121-137.
16. Del Pino, M., Felmer, P. L. (1997). Semi-classical states for nonlinear Schrödinger equations. J. Functional Analysis 149:245-265.
17. Del Pino, M., Felmer, P. L. (1998). Multi-peak bound states for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré 15:127-149.
18. Del Pino, M., Felmer, P. L. (2002). Semi-classical states for nonlinear Schrödinger equations: a variational reduction method. Math. Ann. 324(1):1-32.
19. Floer, A., Weinstein, A. (1986). Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential. J. Functional Analysis 69:397-408.
20. Gilbarg, D., Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order. 2nd ed. Grundlehren 224, Berlin, Heidelberg, New York and Tokyo: Springer.
21. Gui, C. (1996). Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method. Comm. PDE 21:787-820.
22. N. Proc. Amer. Math. Soc. 131:2399-2408.
23. Jeanjean, L., Tanaka, K. (2003b). A note on a mountain pass characterization of least energy solutions. Advanced Nonlinear Studies 3:461-471.
24. Jeanjean, L., Tanaka, K. (2004). Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calculus of Variations and PDE 21:287-318.
25. Kang, X., Wei, J. (2000). On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Adv. Differential Equations 5:899-928.
26. Li, Y. Y. (1997). On a singularly perturbed elliptic equation. Adv. Differential Equations 2:955-980.
27. Lions, P. L. (1984). The concentration-compactness principle in the calculus of variations. The locally compact case, part II. Ann. Inst. Henri Poincaré 1:223-283.
28. a. Comm. PDE 13:1499-1519.
29. Oh, Y. G. (1990). On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Comm. Math. Phys. 131:223-253.
30. Ogawa, T. (1990). A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations. Nonlinear Anal. 14:765-769.
31. Protter, M. H, Weinberger, H. F. (1984). Maximum Principles in Differential Equations. New York, Berlin, Heidelberg and Tokyo: Springer-Verlag.
32. Rabinowitz, P. H. (1992). On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43:270-291.
33. Struwe, M. (1990). Variational Methods; Application to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer-Verlag.