Two Types Of P (x) - Laplace Elliptic System Existence Of Solutions

In this paper, we investigate some solutions top(x)-Laplacian system of elliptic equations, including existence and multiplicity.In chapter1, some definitions and basic theories are introduced.In chapter2, we study the existence of solutions of the p(x)-Laplacian system problem (P) is discussed where Ω is a bounded domain in RN with smooth boundary,F∈C1(Ω×R2,R),h1,h,2are nonnegative measurable functions.This chapter introduces some basic properties of W1,p(x) firstly,then we prove that the problem has a weak solution by the meanings of weak solutions.In the last section,we also make use of the Fountain theorem to catch a conclusion which the problem has infinitely many pairs of weak solutions. The difficulty of this chapter is to prove the functional J satisfies the items included in Fountain theorem. At last,we give another situations about (P) has infinitely many pairs of weak solutions,too.In chapter3, we study the existence of solutions of the generalized p(x)-Laplacian system problem (F)Here F satisfies the conditions in Chapter2.g1, g2are Caratheodory functions.In this chapter,we get some similar conclusions to Chapter2by set the proper conditions to the functions g1and g2.