REGULARITY AND NONEXISTENCE OF SOLUTIONS FOR A SYSTEM INVOLVING THE FRACTIONAL LAPLACIAN

We consider a system involving the fractional Laplacian { (-Delta)(alpha 1/2) u = u(p1) v(q1) in R-+(N), (-Delta)(alpha 2/2) v = u(p2) v(q2) in R-+(N), (1) u = v = 0, in R-N\R-+(N), where alpha(i) is an element of (0, 2), p(i), q(i) > 0, i = 1, 2. Based on the uniqueness of alpha-harmonic function ([9]) on half space, the equivalence between (1) and integral equations { u(x) = C(1)x(N) (alpha 1/2) + integral(R+N) G(infinity)(1)(x, y)u(p1) (y)v(q1) (y)dy, (2) v(x) = C(2)x(N) (alpha 2/2) + integral(R+N) G(infinity)(2)(x, y)u(p2) (y)v(q2) (y)dy. are derived Based on this result we deal with integral equations (2) instead of (1) and obtain the regularity. Especially, by the method of moving planes in integral forms which is established by Chen-Li-Ou [12] we obtain the nonexistence of positive solutions of integral equations (2) under only local integrability assumptions.