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Homoclinic solutions of nonlinear second-order Hamiltonian systems. (English) Zbl 1378.37108

The article addresses the existence of homoclinic solutions to a nonautonomous differential equation of the form \[ -\ddot{x}(t) = B(t)x(t)+\nabla_x V(t,x(t)). \] The condition \[ \lim_{|t|\to\infty} \int_t^{t+1} |b_{jk}(s)|\,\mathrm{d}s =0 \] on the components \(b_{jk}(t)\) of the symmetric matrix \(B(t)\) implies that the essential spectrum \([0,\infty)\) of the linear part is the same as for the case \(B(t)= 0\).
Using linking methods and Sandwich Pair theorems the author shows the existence of a homoclinic solution under different assumptions including sublinear and superlinear growth. In particular, conditions are given under which the system possesses ground state solutions, i.e., solutions minimizing the energy functional.

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37C60 Nonautonomous smooth dynamical systems
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
49J35 Existence of solutions for minimax problems
49K15 Optimality conditions for problems involving ordinary differential equations
34L30 Nonlinear ordinary differential operators
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References:

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