A Class Of Neumann Problems With Variable Exponential Growth

In this paper,we consider the following semilinear Neumann problem with variable exponential growth by means of spatial decomposition technique,variational method,concentration-compactness principle and so on.where ?(?)RN(N?3)is a bounded domin with smooth boundary,v is an external unit normal vector,??0,Q(x)?C(?),f(x,u)?(? x R,R).Firstly,we consider the case of sublinear exponential growth,that is,1<P?p(x)?p+<2.The existence of solutions to problem(0.1)is obtained by using the saddle point reduction method when Q(x)?1,?=0.Secondly,we consider the case of superlinear subcritical exponential growth,that is,2<p-?p(x)?p+<2*,Q(x)is a sign-changing function and f(x,u)satisfies the sublinear growth condition.The existence of two nontrivial solutions to problem(0.1)is obtained by using space decomposition and variational methods.Finally,we consider the case of critical exponential growth,that is 2<p-?p(x)?p+?2*,Q(x)is a sign-changing function and f(x,u)satisfies the sublinear growth condition.we use the concentration-compactness principle to overcome the lack of compactness,the existence of two nontrivial solutions to problem(0.1)is ob-tained by using the variational method.