Symmetry results for positive solutions of mixed integro-differential equations

Abstract

In this paper, we study symmetry property for positive solutions of mixed integro-differential equations

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where N, M >= 1, x is an element of B-1(N)(0) = {x is an element of R-N : vertical bar x vertical bar < 1}, y is an element of B-1(M) (0) = {y is an element of R-M : vertical bar y vertical bar < 1}, the operator (-Delta)(x)(alpha 1) denotes the fractional Laplacian of exponent alpha(1) is an element of (0,1) with respect to x, (-Delta)(y)(alpha 2) denotes the fractional Laplacian of exponent alpha(2) is an element of (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