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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Multibump solutions and critical groups
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by Gianni Arioli, Andrzej Szulkin and Wenming Zou PDF
Trans. Amer. Math. Soc. 361 (2009), 3159-3187 Request permission


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 $g(x,u)=|u|^{2^*-2}u$, $N\ge 4$ 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.
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Additional Information
  • Gianni Arioli
  • Affiliation: Dipartimento di Matematica, Politecnico di Milano, Piazza L. da Vinci 32, 20133 Milano, Italy
  • Email:
  • Andrzej Szulkin
  • Affiliation: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden
  • MR Author ID: 210814
  • Email:
  • Wenming Zou
  • Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
  • MR Author ID: 366305
  • Email:
  • 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
  • © Copyright 2009 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 3159-3187
  • MSC (2000): Primary 37J45; Secondary 34C28, 35J20, 35Q55
  • DOI:
  • MathSciNet review: 2485422