A coupled system of k‐Hessian equations

Abstract

In this paper, we consider the following system coupled by multiparameter k‐Hessian equations Here, λ1 and λ2 are positive parameters, Ω is the unit ball in Rn, Sk(D2u) is the k‐Hessian operator of u, . 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 on the parameter λ1 and λ2. 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.