Abstract
We study the nonexistence of nontrivial solutions for the nonlinear elliptic system
{ (-Delta(x))(alpha/2)u + vertical bar x vertical bar(2 delta) (-Delta(y))(beta/2)u + vertical bar x vertical bar(2 eta)vertical bar y vertical bar(2 theta) (-Delta(z))(gamma/2)u v(p), (-Delta(x))(mu/2)v + vertical bar x vertical bar(2 delta) (-Delta(y))(v/2)v + vertical bar x vertical bar(2 eta)vertical bar y vertical bar(2 theta) (-Delta(z))(sigma/2)v = u(q),
where ( x, y, z) is an element of R-N1 x R-N2 x R-N3, 0 < alpha, beta, gamma, mu, v, sigma <= 2, delta, eta, theta >= 0, and p, q > 1. Here, (-Delta(x))(alpha/2), 0 < alpha < 2, is the fractional Laplacian operator of order alpha/2 with respect to the variable x is an element of R-N1, (-Delta(y))(beta/2), 0 < beta < 2, is the fractional Laplacian operator of order beta/2 with respect to the variable y is an element of R-N2, and (-Delta(z))(gamma/2), 0 < gamma < 2, is the fractional Laplacian operator of order gamma/2 with respect to the variable z is an element of R-N3. Using a weak formulation approach, sufficient conditions are provided in terms of space dimension and system parameters.