The Study Of Standing Wave Solutions For Logarithmic Klein-Gordon Equation

The nonlinear Klein-Gordon equation is an important class of equations in many scientific fields such as quantum mechanics and nonlinear optics.It is used to describe the motion of charged particles in relativistic quantum mechanics and quantum field theory,and has important practical background and physical significance.When describing the motion of spin particles in a charged magnetic field,logarithmic nonlinear terms are introduced.Klein-Gordon equation with logarithmic nonlinear terms has extensive applications in physics,such as nuclear physics,geophysics,etc.In addition,the Klein-Gordon equation with logarithmic nonlinear terms has also been introduced into the quantum field and has important applications in expanding cosmology and field theory.This article mainly studies the existence of standing wave solutions for the Klein-Gordon equation with logarithmic nonlinearity.We prove the existence of standing wave solutions by using variational methods,Gagliardo-Nirenberg inequality,compactness theorem,Sobolev embedding theorem,and lower semicontinuity of limits.Then,we proved two properties of the solution by using the contradiction method,Taylor formula.