Existence Of Solutions For Several Kinds Of Nonlinear Elliptic Equations With P(x)-growth

Sobolev space is an important mathematical concept, but with the emergement of many nonlinear problems in natural science and engineering technology, Sobolev is limited in its field of application, such as the research on exponential growth. Therefore, under the condition of exponential growth nonlinear problem has become a new research topic. On such nonlinear problems, the variable exponent function space has given theoretical support.This paper, mainly set in the variable exponent Lebesgue space Lp(x)(Ω) and Sobolev space Wk,p(x)(Ω). Further studies whether it exists in Nonlinear Elliptic Equations with variable index. Because p(x) is function, and compared with p-Laplace, the operator p(x)-Laplace is more complicatedly nonlinear. It was originally in ordinary index case method for variable index case is no longer in use. In this paper, by means of variable exponent spaces, the known constant exponential equation and get the variable exponent Nonlinear Elliptic Equations, for the nonlinear elliptic equation with singular term existence of Solutions. In order to solve this problem, firstly the definition of Sobolev spaces and its properties, and on this basis, the establishment of the basic framework of Sobolev space. Then by using the critical point theory to obtain the sufficient and necessary conditions for existence of solution of equation. And then from the superlinear and sublinear condition, that is to change the existence of solutions to equations into that of critical point. At the same time, combined with the mountain pass theorem and embedding theorem proving the existence of solutions, the final conclusion, the existence of nontrivial weak solution.