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# memo_has_moved_text();Semiclassical standing waves with clustering peaks, for nonlinear Schrödinger equations

Jaeyoung Byeon and Kazunaga Tanaka

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 229, Number 1076
ISBNs: 978-0-8218-9163-6 (print); 978-1-4704-1530-3 (online)
DOI: http://dx.doi.org/10.1090/memo/1076
Published electronically: October 10, 2013
Keywords:Nonlinear Schödinger equations, Singular perturbation, semi-classical standing waves, local variational method, interaction estimate

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Chapters

• Chapter 1. Introduction and results
• Chapter 2. Preliminaries
• Chapter 3. Local centers of mass
• Chapter 4. Neighborhood $\Omega _\varepsilon (\rho ,R,\beta )$ and minimization for a tail of $u$ in $\Omega _\varepsilon$
• Chapter 5. A gradient estimate for the energy functional
• Chapter 6. Translation flow associated to a gradient flow of $V(x)$ on ${\bf R}^N$
• Chapter 7. Iteration procedure for the gradient flow and the translation flow
• Chapter 8. An $(N+1)\ell _0$-dimensional initial path and an intersection result
• Chapter 9. Completion of the proof of Theorem 1.3
• Chapter 10. Proof of Proposition 8.3
• Chapter 11. Proof of Lemma 6.1
• Chapter 12. Generalization to a saddle point setting

### Abstract

We study the following singularly perturbed problem

Our main result is the existence of a family of solutions with peaks that cluster near a local maximum of $V(x)$. A local variational and deformation argument in an infinite dimensional space is developed to establish the existence of such a family for a general class of nonlinearities $f$. Earlier works in this direction can be found in [KW, DLY, DY, NY] for $f(\xi )=\xi ^p$ ($1 < p < \frac { N+2}{N-2}$ when $N \geq 3$, $1< p < \infty$ when $N=1,2$). These papers use the Lyapunov-Schmidt reduction method, where it is essential to have information about the null space of the linearization of a solution of the limit equation $-\Delta u+u =u^p$. Such spectral information is difficult to get and can only be obtained for very special $f's$. Our new approach in this memoir does not require such a detailed knowledge of the spectrum and works for a much more general class of nonlinearities $f$.