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Multibump solutions and critical groups. (English) Zbl 1175.37066

The topic of the paper is the existence of multibump solutions for Newtonian systems of ordinary differential equations and for semilinear partial differential equations of Schrödinger type. At first a Newtonian system \(-\ddot{q}+B(t) q=W_q(q,t)\) is considered where \(B,W\) are periodic in \(t\) and \(B\) is positive definite. Then for an isolated homoclinic solution \(q_0\) with non-trivial critical group multibump solutions are constructed by gluing translates of \(q_0.\) An analogous result is shown for a Schrödinger equation \(-\Delta u+v(x)u=g(x,u)\) in \(\mathbb{R}^N.\) Here \(V,g\) are periodic in \(x_1,\ldots, x_N\) and the spectrum satisfies: \(\sigma\left(-\Delta+v\right)\subset (0,\infty).\)

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34C28 Complex behavior and chaotic systems of ordinary differential equations
35J20 Variational methods for second-order elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
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