(S)_+-property Of Quasilinear Elliptic Operators And Equations Of P(x)-Laplacian Type

In this paper,under some weaker conditions we have improved some theories of the vector valued Musielak-Orlicz spaces and give some useful variants of Radon-Riesz theorem in the uniformly convex spaces.We studies the property of the convex modular which is generated by a Musielak-Orlicz function,using the uniform convexity of the MusielakOrlicz function,we give a sufficient condition for the(S)_+-property of the subdifferential mapping of the convex modular generated by it;And based on this we obtain the(S)_+-property of a wide class of the quasilinear elliptic operators with the variational structure. We introduce the new notion of a special subclass of the Musielak-Orlicz function,which is called of type with variable exponent,and the p(x)-Laplacian type operators which include the p(x)-Laplacian operator as a special case.We have showed the(S)_+-property of the p(x)-Laplacian type operators.As an application of this result,we obtain the existence and multiplicity of solutions for the class of equations of the p(x)-Laplacian type. The notion of p(x)-Laplacian type operators and the results obtained in this paper are the essence of the improvement and development of p-Laplacian type operators and the corresponding results in[10]by De Nápoli and Mariani.