Multiple Solutions To A Class Of P(x)-curl Systems

The multiple solutions problem plays an important role in the field of partial differential equations.In recent years,the subj ect of variable p(x)-curl system has gained more and more scholars’ attention,mainly due to the appearence of more and more nonlinear problems.Moreover,the existence and multiplicity of solutions becomes a main research emphasis for many scholars.In this paper,we will discuss the multiple solutions of a class of p(x)-curl systems,which is based on the variable exponential Sobolev space and Wp(x)(Ω)space by using variational method and critical point theory.where Ω∈R3,λ>0,a∈L∞(Ω),p(x),g(x)∈C(Ω(and p(x)∈(5/6,3),f(x,u)is a Caratheodory function,n is the unit vector of the outer normal vector on(?)Ω.In this paper,it is concluded that when a(x)=0,g(x)=0,the p(x)-curl systems can obtain at least two nontrivial solutions in the case when the nonlinear term f(x,u)is sublinear growth or superlinear growth.When A=1,if f(x,u)satisfies Ambrosetti-Rabinowitz condition,we will obtain at least two nontrivial solutions to p(x)-curl systems.