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In this paper, we study a nonlinear elliptic system involving a fractional Laplacion:  where 0  &lt; α &lt;  2 , p, q &gt;  0  and  max{ p, q }  ≥  1 , τ ≥  0 , n ≥  2. There are two cases to be considered. The first one is where the domain is bounded, and the second one is where the domain is the whole space. First of all, we consider the above system in the star-shaped and bounded domain Ω, by using the  Pohozaev identity.  We prove the nonexistence of a positive solution in the critical and supercritical case  p  +  q  + 1 ≥   . In addition, we show that the positive solutions of the above system are radially symmetric and decreasing about the origin by using the method of  moving planes  in ℝn .  Moreover, while in the subcritical case   , we prove the nonexistence of a positive solution for the above system in ℝn. Then, through the  doubling lemma  we obtain the singularity estimates of the positive solutions on a bounded domain   .

Symmetry and nonexistence of positive solutions for an elliptic system involving the Fractional Laplacian