Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments

Transactions of the American Mathematical Society

Volumn 361, Issue 10, 2009, Pages 5401-5416

  Jeanjean, Louis ^a   Le Coz, Stefan ^a,b  

^a UNIVERSITÉ DE FRANCHE COMTÉ   (France)

^b SISSA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77950638323     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-09-04790-4     Document Type: Article

References (26)

1. R. Adams, Sobolev spaces, Pure and Applied Mathematics, 65, Academic Press, (1975). MR0450957 (56:9247)
2. H. Berestycki and T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris 293, (1981), 489-492 and Publications du Laboratoire d'Analyse Numérique, Université de Paris VI. MR646873 (84f:35120)
3. H. Berestycki, T. Gallouet and O. Kavian, Équations de champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris 297, (1983), 307-310 and Publications du Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie. MR734575 (85e:35041)
4. H. Berestycki and P.L. Lions, Nonlinear scalar field equations I, Arch. Ration. Mech. Anal., 82, (1983), 313-346. MR695535 (84h:35054a)
5. J. Byeon, L. Jeanjean and K. Tanaka, Standing waves for nonlinear Schrödinger equations with a general nonlinearity: One and two dimensional cases, Comm. Partial Differential Equations 33, 4-6, (2008), 1113-1136. MR2424391
6. T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10, (2003). MR2002047 (2004j:35266)
7. T. Cazenave and A. Haraux, An introduction to semilinear evolution equations, Oxford Lecture Series in Mathematics and its Applications, 13, Oxford University Press, (1998). MR1691574 (2000e:35003)
8. T. Cazenave and P.L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85, 4, (1982), 549-561. MR677997 (84i:81015)
9. J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation, Phys. D 227, (2007), 142-148. MR2332502 (2008c:35301)
10. S. Cingolani, L. Jeanjean and S. Secchi, Multi-peak solutions for magnetic NLS equations without non-degeneracy condition, to appear in ESAIM Contrôle Optim. Calc. Var. DOI: 10.1051/cocv:2008055.
11. S. Coleman, V. Glaser and A. Martin, Action minima among solutions to a class of Euclidean scalar field equations, Comm. Math. Phys. 58, 2, (1978), 211-221. MR0468913 (57:8716)
12. S. Ibrahim, M. Majdoub and N. Masmoudi, Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type nonlinearity, Comm. Pure Appl. Math. 59, 11, (2006), 1639-1658. MR2254447 (2007h:35229)
13. N, Proc. Amer. Math. Soc. 131, 8, (2003), 2399-2408. MR1974637 (2004c:35127)
14. L. Jeanjean and K. Tanaka, A note on a mountain pass characterization of least energy solutions, Adv. Nonlinear Stud. 3, 4, (2003), 445-455. MR2017241 (2004i:34228)
15. S. Le Coz, A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations, Adv. Nonlinear Stud. 8, 3, (2008), 455-463. MR2426909
16. Y. Liu, Strong instability of solitary-wave solutions to a Kadomtsev-Petviashvili equation in three dimensions, J. Differential Equations 180, 1, (2002), 153-170. MR1890602 (2003j:35278)
17. Y. Liu, M. Ohta and G. Todorova, Strong instability of solitary waves for nonlinear Klein-Gordon equations and generalized Boussinesq equations, Ann. Henri Poincaré 24, 4, (2007), 539-548. MR2334991 (Pubitemid 46907421)
18. Y. Liu, X.-P. Wang and K. Wang, Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity, Trans. Amer. Math. Soc. 358, (2006), 2105-2122. MR2197450 (2006k:35275)
19. H. Nawa, Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity, J. Math. Soc. Japan 46, 4, (1994), 557-586. MR1291107 (95g:35195)
20. O. A. Nielsen, An introduction to integration and measure theory, Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley-Interscience, (1997). MR1468232 (98j:28002)
21. M. Nakamura and T. Ozawa, The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces, Publ. Res. Inst. Math. Sci. 37, 3, (2001), 255-293. MR1855424 (2002k:35213)
22. M. Ohta and G. Todorova, Strong instability of standing waves for nonlinear Klein-Gordon equations, Discrete Contin. Dyn. Syst. 12, 2, (2005), 315-322. MR2122169 (2005k:35289)
23. M. Ohta and G. Todorova, Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system, SIAM J. Math. Anal. 38, 6, (2007), 1912-1931. MR2299435 (2008a:35198) (Pubitemid 350121922)
24. J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc. 290, 2, (1985), 701-710. MR792821 (86k:35088)
25. J. Shatah and W. Strauss, Instability of nonlinear bound states, Comm. Math. Phys. 100, 2, (1985), 173-190. MR804458 (87b:35159)
26. W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55, 2, (1977), 149-162. MR0454365 (56:12616)