Multiple Solutions for a Class of p(x)-Laplacian Systems

Since the space L x and W1,p x were thoroughly studied by Kováčik and Rákosnı́k 1 , variable exponent Sobolev spaces have been used in the last decades to model various phenomena. In 2 , Růžička presented the mathematical theory for the application of variable exponent spaces in electro-rheological fluids. In recent years, the differential equations and variational problems with p x -growth conditions have been studied extensively; see for example 3–6 . In 7 , De Figueiredo and Ding discussed the multiple solutions for a kind of elliptic systems on a smooth bounded domain. Motivated by their work, we will consider the following sort of p x -Laplacian systems with “concave and convex nonlinearity”: