NONEXISTENCE AND SYMMETRY OF SOLUTIONS FOR SCHRODINGER SYSTEMS INVOLVING FRACTIONAL LAPLACIAN

In this paper, we consider the following Schrodinger systems involving pseudo-differential operator in R-n { (-Delta)<SUP>alpha/2 u(x) = ubeta1 (x)vtau 1 (x), in </SUP> Rn, (1) (-Delta)<SUP>gamma/2 v(x) = ubeta2 (x)vtau 2 (x), in </SUP> Rn, where alpha and gamma are any number between 0 and 2, alpha does not identically equal to gamma. We employ a direct method of moving planes to partial differential equations (PDEs) (1). Instead of using the Caffarelli-Silvestre's extension method and the method of moving planes in integral forms, we directly apply the method of moving planes to the nonlocal fractional order pseudo-differential system. We obtained radial symmetry in the critical case and non-existence in the subcritical case for positive solutions. In the proof, combining a new approach and the integral definition of the fractional Laplacian, we derive the key tools, which are needed in the method of moving planes, such as, narrow region principle, decay at infinity. The new idea may hopefully be applied to many other problems.