Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential

Abstract

In this paper, we apply a cross-constrained variational approach forthe nonlinear Klein-Gordon equations with an inverse squarepotential in three space dimensions (which is a representative ofthe class of equations of interest) based on the relationshipbetween a type of cross-constrained variational problem and energy.By constructing a type of cross-constrained variational problem andestablishing so-called cross-invariant manifolds of the evolutionflow, we first derive a sharp threshold for global existence andblow-up of solutions to the Cauchy problem for the equations understudy. On the other hand, we get an answer of the question: howsmall are the initial data, the global solutions exist?