A coupled system of k-Hessian equations

In this paper, we consider the following system coupled by multiparameter k-Hessian equations Sk(D2u1)=lambda 1f1(-u2)in omega,Sk(D2u2)=lambda 2f2(-u1)in omega,u1=u2=0on partial differential omega. Here, lambda(1) and lambda(2) are positive parameters, omega is the unit ball in R-n, S-k(D(2)u) is the k-Hessian operator of u, n2<k <= n. Applying the eigenvalue theory in cones, several new results are obtained for the existence and multiplicity of nontrivial radial solutions for the above k-Hessian system. In particular, we study the dependence of the nontrivial radial solution u=(u lambda 1,u lambda 2) on the parameter lambda(1) and lambda(2). This is probably the first time that a system of equations, especially with fully nonlinear equations, has been studied by applying this technique. Finally, as an application, we obtain sufficient conditions for the existence of nontrivial radial solutions of the power-type coupled system of k-Hessian equations, which is new even for the special case k=1 and extends a previous result for the case k=n.