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Transactions of the American Mathematical Society
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
Published electronically: January 22, 2009
MathSciNet review: 2485422
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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.


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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: http://dx.doi.org/10.1090/S0002-9947-09-04669-8
PII: S 0002-9947(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