LIOUVILLE-TYPE THEOREMS FOR AN ELLIPTIC SYSTEM INVOLVING FRACTIONAL LAPLACIAN OPERATORS WITH MIXED ORDER

We study the nonexistence of nontrivial solutions for the nonlinear elliptic system G(alpha,beta,theta)(u(p),u(q)) = v(r) G(lambda,mu,theta)(v(s),v(t)) = u(m) u, v >= 0, where 0 < alpha,beta,lambda,mu <= 2, theta > 0, m > q >= p >= 1, r > t >= s >= 1, and G(alpha,beta,theta) is the fractional operator of mixed orders alpha,beta, defined by G(alpha,beta,theta)(u,v) (-Delta(x))(alpha/2)u+ broken vertical bar x broken vertical bar(2 theta)(-Delta(y))(beta/2) v, in R-1(N) x R-2(N). 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-1(N), and (-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-2(N). Via a weak formulation approach, sufficient conditions are provided in terms of space dimension and system parameters.