Symmetry Of Solutions To The Fractional Double-Phase Elliptic Equations

The fractional double-phase elliptic equations come from anisotropic materials,elastic mechanics and homogenization.In recent decades,they have been widely applied in many fields,such as financial markets,quantum mechanics,anomalous diffusion,image processing,and they attracted extensive attention from reseachers at home and overseas.The symmetry of solutions in partial differential equations has always been a subject of deep research among scholars,and the method of moving planes is an important tool to study this subject.In this paper,we mainly study the symmetry of positive solutions of the fractional double-phase elliptic equation with and s,t?(0,1),p,q?(1,?),q?p?1,P.V.stands for the Cauchy's principal value.The specific content is as follows:Firstly,for this equation,we establish maximum principle,maximum principle for anti-symmetric functions,narrow region principle and decay at infinity,which are the key elements to prove the symmetry of solutions.Secondly,we employ the method of moving planes to study symmetry of positive solutions of this equation in unit sphere and the whole space respectively,and under given conditions we obtain the conclusion that the radial symmetry of positive solution u in the region.Finally,the symmetry of positive solutions of the equation with more general nonlinear terms in the bounded region and the whole space are studied respectively.