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Multibump solutions and critical groups
Dipartimento di Matematica, Politecnico di Milano, Milano, Italy.
Stockholm University, Faculty of Science, Department of Mathematics. matematik.
Department of Mathematical Sciences, Tsinghua University, Beijing, China.
2009 (English)In: Transactions of the American Mathematical Society, ISSN 0002-9947, Vol. 361, no 6, 33 p.3159-3187 p.Article in journal (Refereed) Published
Abstract [en]

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 $2\le k\le\iy$ 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 Schr\"odinger equation $-\Delta u+V(x)u = g(x,u)$ in $\RN$, where $V$, $g$ are periodic in $x_1,\ldots,x_N$, $\sigma(-\Delta+V)\subset (0,\iy)$, 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.

Place, publisher, year, edition, pages
Providence, R.I.: American Mathematical Society , 2009. Vol. 361, no 6, 33 p.3159-3187 p.
Keyword [en]
Multibump solution, critical group, Bernoulli shift, Newtonian system, Schrödinger equation, critical exponent
National Category
Mathematical Analysis
Research subject
URN: urn:nbn:se:su:diva-20661DOI: S 0002-9947(09)04669-8ISI: 000264881500014OAI: diva2:187187
Available from: 2007-11-28 Created: 2007-11-28 Last updated: 2010-01-04Bibliographically approved

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Szulkin, Andrzej
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