Solutions to the nonlinear Schrödinger systems involving the fractional Laplacian

摘要

In this paper, we consider the following nonlinear Schrödinger system involving the fractional Laplacian operator: {(−Δ)α2u+au=f(v),(−Δ)β2v+bv=g(u),on Ω⫅Rn, where a,b≥0. When Ω is the unit ball or Rn, we prove that the solutions (u,v) are radially symmetric and decreasing. When Ω is the parabolic domain on Rn, we prove that the solutions (u,v) are increasing. Furthermore, if Ω is the R+n, then we also derive the nonexistence of positive solutions to the system on the half-space. We assume that the nonlinear terms f, g and the solutions u, v satisfy some amenable conditions in different cases.