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Multibump solutions of differential equations: An overview. (English) Zbl 0968.37019

From the text: This paper describes and surveys recent advances in the existence of so-called multibump solutions of both ordinary and partial differential equations using methods from the calculus of variations.
Our goal here is to describe some of the results, ideas, and methods involved. In §1, we explain what we mean by one-bump solutions of differential equations and present examples of and results for problems having such solutions. The existence of multibump solutions will then be discussed for the settings of these examples in §2. In §3-5, our simplest model case: a singular Hamiltonian system in \(\mathbb{R}^2\) will be studied in more detail. In particular a detailed sketch of the existence of one-bump solutions will be given in §3. Then the existence of one-bump solutions for some variants of the model case of §3 will be carried out in §4. Lastly §5 shows how to obtain multibump solutions for the setting of §3.
This paper is an expanded version of lectures given at National Chung Cheng University and in an abbreviated form at Academia Sinica, National Changhua University and National Chiao-Tung University.

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35J60 Nonlinear elliptic equations
70H05 Hamilton’s equations
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