Symmetry results for positive solutions of mixed integro-differential equations

In this paper, we study symmetry property for positive solutions of mixed integro-differential equationsequation(0.1){(−Δ)xα1u+(−Δ)yα2u=f(u)inB1N(0)×B1M(0),u=0in(RN×RM)∖(B1N(0)×B1M(0)), where N  , M≥1M≥1, x∈B1N(0)={x∈RN:|x|<1}, y∈B1M(0)={y∈RM:|y|<1}, the operator (−Δ)xα1 denotes the fractional Laplacian of exponent α1∈(0,1)α1∈(0,1) with respect to x  , (−Δ)yα2 denotes the fractional Laplacian of exponent α2∈(0,1)α2∈(0,1) with respect to y. We make use of the Maximum Principle for small domain to start the moving planes to obtain the symmetry results for positive solutions.