Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Multibump solutions and critical groups


Authors: Gianni Arioli, Andrzej Szulkin and Wenming Zou
Journal: Trans. Amer. Math. Soc. 361 (2009), 3159-3187
MSC (2000): Primary 37J45; Secondary 34C28, 35J20, 35Q55
DOI: https://doi.org/10.1090/S0002-9947-09-04669-8
Published electronically: January 22, 2009
MathSciNet review: 2485422
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the Newtonian system $ -\ddot q+B(t)q = W_q(q,t)$ with $ B$, $ W$ periodic in $ t$, $ B$ positive definite, and show that for each isolated homoclinic solution $ q_0$ having a nontrivial critical group (in the sense of Morse theory), multibump solutions (with $ 2\le k\le\infty$ bumps) can be constructed by gluing translates of $ q_0$. Further we show that the collection of multibumps is semiconjugate to the Bernoulli shift. Next we consider the Schrödinger equation $ -\Delta u+V(x)u = g(x,u)$ in $ \mathbb{R}^N$, where $ V$, $ g$ are periodic in $ x_1,\ldots,x_N$, $ \sigma(-\Delta+V)\subset (0,\infty)$, and we show that similar results hold in this case as well. In particular, if $ g(x,u)=\vert u\vert^{2^*-2}u$, $ N\ge 4$ and $ V$ changes sign, then there exists a solution minimizing the associated functional on the Nehari manifold. This solution gives rise to multibumps if it is isolated.


References [Enhancements On Off] (What's this?)

  • 1. N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal. 234 (2006), 277-320. MR 2216902 (2007b:47182)
  • 2. N. Ackermann and T. Weith, Multibump solutions of nonlinear periodic Schrödinger equations in a degenerate setting, Comm. Contemp. Math. 7 (2005), 269-298. MR 2151860 (2006f:35069)
  • 3. S. Alama and Y.Y. Li, On ``multibump'' bound states for certain semilinear elliptic equations, Indiana Univ. Math. J. 41 (1992), 983-1026. MR 1206338 (94d:35044)
  • 4. S. Angenent, The shadowing lemma for elliptic PDE. In: Dynamics of infinite-dimensional systems, S.N. Chow and J.K. Hale eds., Springer-Verlag, Berlin, 1987, pp. 7-22. MR 921893 (89b:58165)
  • 5. G. Arioli, F. Gazzola and S. Terracini, Multibump periodic motions of an infinite lattice of particles, Math. Z. 223 (1996), 627-642. MR 1421960 (98d:58150)
  • 6. V. Benci and G. Cerami, Existence of positive solutions of the equation $ -\Delta u+a(x)u = u^{(N+2)/(N-2)}$ in $ \mathbb{R}^N$, J. Func. Anal. 88 (1990), 90-117. MR 1033915 (91f:35097)
  • 7. M. Berti and P. Bolle, Homoclinics and chaotic behaviour for perturbed second order systems, Ann. Mat. Pura Appl. (4) 176 (1999), 323-378. MR 1746547 (2001c:37065)
  • 8. J. Chabrowski and A. Szulkin, On a semilinear Schrödinger equation with critical Sobolev exponent, Proc. Amer. Math. Soc. 130 (2002), 85-93. MR 1855624 (2002i:35053)
  • 9. J. Chabrowski and A. Szulkin, On the Schrödinger equation involving a critical exponent and magnetic field, Topol. Meth. in Nonl. Anal. 25 (2005), 3-21. MR 2133390 (2005k:35110)
  • 10. K.C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problem, Birkhäuser, Boston, 1993. MR 1196690 (94e:58023)
  • 11. W.A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics 629, Springer-Verlag, Berlin, 1978. MR 0481196 (58:1332)
  • 12. V. Coti Zelati, I. Ekeland and E. Séré, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann. 288 (1990), 133-160. MR 1070929 (91g:58065)
  • 13. V. Coti Zelati and P.H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc. 4 (1991), 693-727. MR 1119200 (93e:58023)
  • 14. V. Coti Zelati and P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $ R\sp n$, Comm. Pure Appl. Math. 45 (1992), 1217-1269. MR 1181725 (93k:35087)
  • 15. V. Coti Zelati and P.H. Rabinowitz, Multibump periodic solutions of a family of Hamiltonian systems, Topol. Meth. Nonl. Anal. 4 (1994), 31-57. MR 1321808 (96a:58042)
  • 16. E.N. Dancer, Degenerate critical points, homotopy indices and Morse inequalities, J. Reine Angew. Math. 350 (1984), 1-22. MR 743531 (85i:58033)
  • 17. Y. Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms, Comm. Contemp. Math. 8 (2006), 453-480. MR 2258874 (2007h:37092)
  • 18. Y. Ding and M. Willem, Homoclinic orbits of a Hamiltonian system, Z. Angew. Math. Phys. 50 (1999), 759-778. MR 1721793 (2000k:37086)
  • 19. A. Dold, Lectures on Algebraic Topology, Springer-Verlag, Berlin, 1972. MR 0415602 (54:3685)
  • 20. L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $ \mathbf{R}^N$, Proc. Roy. Soc. Edinb. 129A (1999), 787-809. MR 1718530 (2001c:35034)
  • 21. W. Kryszewski and A. Szulkin, An infinite dimensional Morse theory with applications, Trans. Amer. Math. Soc. 349 (1987), 3181-3234. MR 1422612 (97m:58035)
  • 22. P. Kuchment, Floquet Theory for Partial Differential Equations, Birkhäuser, Basel, 1993. MR 1232660 (94h:35002)
  • 23. Y.Y. Li and Z.Q. Wang, Gluing approximate solutions of minimum type on the Nehari manifold, Electron. J. Diff. Eq., Conference 06, 2001, 215-223. MR 1804776 (2001m:58027)
  • 24. Z.L. Liu and Z.Q. Wang, Multi-bump type nodal solutions having a prescribed number of nodal domains, I and II, Ann. IHP, Analyse Non Linéaire 22 (2005), pp. 597-608 and 609-631. MR 2171994 (2006f:35094)
  • 25. R. Magnus, The implicit function theorem and multi-bump solutions of periodic partial differential equations, Proc. Roy. Soc. Edinburgh 136 (2006), 559-583. MR 2227808 (2007k:47116)
  • 26. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989. MR 982267 (90e:58016)
  • 27. A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math. 73 (2005), 259-287. MR 2175045 (2006h:35087)
  • 28. P.H. Rabinowitz, Variational methods for Hamiltonian systems. In: Handbook of Dynamical Systems, Vol. 1A, B. Hasselblatt and A. Katok eds., North-Holland, Amsterdam, 2002, pp. 1091-1127. MR 1928531 (2004a:37081)
  • 29. E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z. 209 (1992), 27-42. MR 1143210 (92k:58201)
  • 30. E. Séré, Looking for the Bernoulli shift, Ann. IHP, Analyse Non Linéaire 10 (1993), 561-590. MR 1249107 (95b:58031)
  • 31. B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447-526. MR 670130 (86b:81001a)
  • 32. C.A. Stuart, Bifurcation into spectral gaps, Bull. Belg. Math. Soc., Supplement, 1995. MR 1361485 (96m:47115)
  • 33. W. Szlenk, An Introduction to the Theory of Smooth Dynamical Systems, Polish Scientific Publishers, Warsaw and Wiley, New York, 1984. MR 791919 (86f:58042)
  • 34. F.A. van Heerden, Homoclinic solutions for a semilinear elliptic equation with an asymptotically linear nonlinearity, Calc. Var. PDE 20 (2004), 431-455. MR 2071929 (2005c:35098)
  • 35. S. Wiggins, Global Bifurcations and Chaos, Springer-Verlag, New York, 1988. MR 956468 (89m:58057)
  • 36. M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. MR 1400007 (97h:58037)
  • 37. H.Y. Xu, Critical exponent elliptic equations: gluing and the moving sphere method, Thesis, Rutgers, 2007.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37J45, 34C28, 35J20, 35Q55

Retrieve articles in all journals with MSC (2000): 37J45, 34C28, 35J20, 35Q55


Additional Information

Gianni Arioli
Affiliation: Dipartimento di Matematica, Politecnico di Milano, Piazza L. da Vinci 32, 20133 Milano, Italy
Email: gianni.arioli@polimi.it

Andrzej Szulkin
Affiliation: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden
Email: andrzejs@math.su.se

Wenming Zou
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
Email: wzou@math.tsinghua.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-09-04669-8
Keywords: Multibump solution, critical group, Bernoulli shift, Newtonian system, Schr\"odinger equation, critical exponent
Received by editor(s): July 12, 2007
Published electronically: January 22, 2009
Additional Notes: The first author was supported in part by the MIUR project “Equazioni alle derivate parziali e disuguaglianze funzionali: aspetti quantitativi, proprietà geometriche e qualitative, applicazioni”
The second author was supported in part by the Swedish Research Council
The third author was supported by NSFC (10571096 and 10871109), SRF-ROCS-SEM and the program of the Ministry of Education in China for New Century Excellent Talents in Universities of China
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society