Classification and existence of positive entire k-convex radial solutions for generalized nonlinear k-Hessian system

In this paper, we consider the following generalized nonlinear k-Hessian system {G(S-k(1/k)(lambda(D(2)z(1)))) S-1/kk(lambda(D(2)z(1))) = phi(vertical bar x vertical bar, z(1), z(2)), x epsilon R-N, G(S-k(1/k)(lambda(D(2)z(2)))) S-1/kk(lambda(D(2)z(2))) = psi(vertical bar x vertical bar, z(1), z(2)), x epsilon R-N,R- where G is a nonlinear operator and S-k (lambda(D(2)z)) stands for the k-Hessian operator. We first are interested in the classification of positive entire k-convex radial solutions for the k-Hessian system if phi(vertical bar x vertical bar, z(1), z(2)) = b(vertical bar x vertical bar)phi(z(1), z(2)) and psi(vertical bar x vertical bar, z(1), z(2)) = h(vertical bar x vertical bar)psi(z(1)). Moreover, with the help of the monotone iterative method, some new existence results on the positive entire k-convex radial solutions of the k-Hessian system with the special non-linearities psi,phi are given, which improve and extend many previous works.