Existence And Regularity Of Entropy Solutions For A Class Of Elliptic Equations

In physics,geometry and other related problems,for the phenomenon with multiple variables,such as spatial variables and time variables,it is often necessary to establish a partial differential equations（PDEs）model to solve the problem.With the development of natural science and engineering technology,related nonlinear problems are proposed,and nonlinear PDEs have entered people’s field of vision.However,the research on nonlinear problems with p（x）growth condition in the space of constant exponential function is relatively limited.Hence,many researchers extend it to the space of variable exponential function to provide a theoretical basis for such problems.Therefore,the study of PDEs in the variable exponential function space has important theoretical and actual meaning.In this paper,the properties of entropy solutions are considered for the elliptic system with Dirichlet boundary data in the space of variable exponential functions.Firstly,in the case of weakening the operator conditions,the existence of entropy solutions for elliptic equations is discussed.The approximation problem of elliptic system is established,and the approximation solution sequence is estimated a priori by using Sobolev embedding theorem,Riesz’s theorem and Poincaré’s inequality.According to Vitali’s theorem and H?lder’s inequality,the gradient convergence of the truncated function sequence and the strong convergence of low-order terms are obtained.The existence of the entropy solution for a class of A-harmonic equations is proved.Based on the existence of entropy solutions for this system,the regularity of entropy solutions is further studied.Since the elliptic equation studied in this paper has low-order terms,after establishing the approximation problem,it is necessary to reselect the appropriate test function and combine the operator assumptions to process the low-order terms.Using Young’s inequality,Minkowski’s inequality and other tools,the high-order integrability of the weak solution for the approximation problem is proved.Finally,by establishing the approximation problem,we get its a priori estimate,so the regularity of the entropy solution is obtained.