Multiplicity Of Solutions For A Class Of Double Phase Elliptic Equations With Critical Growth

With the development of the objects studied in mathematical physics,the application of PDEs is more extensive.The study of nonlinear partial differential equations with variable exponential growth is a hot research topic.There are a lot of researches on the existence,uniqueness and regularity of solutions to these problems.In the framework of variable exponent Orlicz-Sobolev space,we mainly discuss the existence of multiple solutions for a class of double phase elliptic equations with critical growth:#12 where ??C(RN×R+,R+)and satisfies ?(x,t)=?t'(x,t).When t is very small or large,?(x,t)has different growth.In addition,the right-hand term function f(x,t)has sub critical growth,p*(x)=Np(x)/(N-p(x))is a critical exponent and 1??(x)?p(x)?q(x)?min {N,p*(x)}.In this paper,we mainly discuss the multiplicity of weak solution of the above equation by using the critical point theory.First of all,the concentration compactness theorem is established in the variable exponent Orlicz-Sobolev space.Then,based on the results of this kind of concentration compactness and the symmetric mountain pass theorem,we obtain a sequence of critical points for the energy functional related to the equation.These critical points are radial symmetry and their energy value tends to infinity.Thus we obtain the multiplicity of weak solutions for the equation.