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This paper studies second order elliptic equations in both divergence and non-divergence forms with measurable complex valued principle coefficients and measurable complex valued potentials. The PDE operators can be considered as generalized Schrödinger operators. Under some sufficient conditions, we prove existence, uniqueness, and regularity estimates in Sobolev spaces for solutions to the equations. We particularly show that the non-zero imaginary parts of the potentials are the main mechanisms that control the solutions. Our results can be considered as limiting absorption principle for Schrödinger operators with measurable coefficients and they could be useful in applications. The approach is based on the perturbation technique that freezes the potentials. The results of the paper not only generalize known results but also provide a key ingredient for the study of Lp-diffusion phenomena for dissipative wave equations. [ABSTRACT FROM AUTHOR]

The orbital instability of ground state standing waves eiωtφω(x) for the nonlinear Klein-Gordon equation has been known in the domain of all frequencies ω for the supercritical case and for frequencies strictly less than a critical frequency ωc in the subcritical case. We prove the strong instability of ground state standing waves for the entire domain above. For the case when the frequency is equal to the critical frequency ωc we prove strong instability for all radially symmetric standing waves eiωctφ(x). We prove similar strong instability results for the Klein-Gordon-Zakharov system. [ABSTRACT FROM AUTHOR]