Study On The Solutions Of Two Kinds Of Generalized Born-infeld Models

Born-Infeld nonlinear electrodynamics has a widely used in both mathematics and physics.In this paper,we select two kinds of generalized Born-Infeld models to study,namely,rationalfunction model on bounded interval of one-dimensional space and logarithm model in the whole R3 space.For the one-dimensional problem derived from the rational-function model,we firstly use the time-map method to transform the differential problem into the integral problem.Next,constructing the auxiliary function h(λ)and l(λ)to determine the exact number of positive solutions through analyzing their properties.And then we consider the influence of parameterλ on the number of positive solutions.For the three-dimensional problems derived from the logarithmic function model,we use the variational method.Firstly,the existence and uniqueness of minimizer is obtained for the weak lower semi-continuity and convexity of the functional.Then we get the existence and uniqueness of the weak solution in the radial case of problem under the assumption that the charge density p is radially distributed.Finally,the related properties of the minimizer of the functional and weak solution are analyzed.