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Standing waves for nonlinear Klein-Gordon equations with nonnegative potentials. (English) Zbl 1070.35085

Summary: This paper is concerned with the standing waves for nonlinear Klein-Gordon equations with nonnegative potentials. First, the existence of standing waves associated with the ground states is obtained by using variational calculus as well as a compactness lemma. Next, a series of sharp conditions for global existence of nonlinear Klein-Gordon equations with nonnegative potentials are established in terms of the characteristics of the ground state and the local theory. Then, how small the initial data are, the existence of global solutions is obtained. Finally, the instability of standing waves is shown by combining those results.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
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