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
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 $g(x,u)=|u|^{2^*-2}u$, $N\ge 4$ 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.References
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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
- MR Author ID: 210814
- Email: andrzejs@math.su.se
- Wenming Zou
- Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
- MR Author ID: 366305
- Email: wzou@math.tsinghua.edu.cn
- 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: https://doi.org/10.1090/S0002-9947-09-04669-8
- MathSciNet review: 2485422