## Multiple homoclinic orbits for autonomous, singular potentials.(English)Zbl 0812.58088

This work shows that the boundary value problem \left\{ \begin{aligned} \ddot u &= -V'(u)\\ u& (-\infty) = u(+\infty) = 0 \end{aligned} \right. where $$u \in \mathbb{R}$$, $$n \geq 2$$ and $$V \in C^ 2(\mathbb{R}^ n \setminus \text{e}, \mathbb{R})$$ is a potential with an absolute maximum at 0 and is such that $$V(x) \rightarrow -\infty$$ as $$x \rightarrow \text{e}$$, has at least $$n - 1$$ geometrically distinct solutions under a rather complicated set of somewhat contrived conditions. The result is supposed to have application in finding homoclinic orbits for the potential in a Hamiltonian system.

### MSC:

 58J32 Boundary value problems on manifolds 49Q20 Variational problems in a geometric measure-theoretic setting 70H05 Hamilton’s equations
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### References:

 [1] Bessi, Rend. Sem. Mat. Univ. Padova 85 pp 201– (1991) [2] Al’ber, Soviet Math. Dokl. 5 pp 312– (1964) [3] DOI: 10.1070/RM1970v025n04ABEH001261 · Zbl 0222.58002 [4] Tanaka, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 pp 427– (1990) · Zbl 0712.58026 [5] DOI: 10.1007/BF01444526 · Zbl 0731.34050 [6] Rabinowitz, Proc. Roy. Soc. Edinburgh Sect. A 114 pp 33– (1990) · Zbl 0705.34054 [7] Mel’nikov, Trans. Moscow Math. Soc. 12 pp 1– (1963) [8] Giannoni, Ann. Scuola Norm. Sup. Pisa 15 pp 467– (1988) [9] DOI: 10.1090/S0894-0347-1991-1119200-3 [10] DOI: 10.1016/0040-9383(66)90002-4 · Zbl 0138.18302
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