Existence Of Solutions For P(x)-Laplacian Problems On A Bounded Domain

The main content of this paper is to study the existence of multiple solutions of a p (x)-Laplacian problem, our study is on the base of the basic theory of the spaces L^p(x)and W^k,p(x).With the development of elastic mechanics, the study of p (x)-Laplacian problem with nonstandard growth conditions is a new topic developed in recent years. The p (x)-Laplacian equation arises naturally in various contexts of physics, for instance, in the study of non-Newtonian fluids (the case of Newtonian fluid corresponding to p =2), and in the study of nonlinear elasticity. So studying such problems has wide meaning of theory and practice. To study p (x)-Laplacian problem, we have various means. Recently, Critical Point Theory seems to be a very useful tool to solve partial differential equation. We can prove the existence of solutions of much partial differential equations successfully with this tool, especially for the Laplacian problem with non-standard growth conditions.In this paper, by means of the basic theory of generalized Lebesuge space L^p(x)and generalized Sobolev space W^k,p(x), especially the Embedding Theorems, we study the existence of multiple solutions of the p (x)-Laplacian problemWhereΩ(?)RN is a bounded domain with smooth boundary (?)Ω. p (x)is Lip-schitz continuous onΩand satisfies 2 < p ^-≤p( x)≤p⁺