A coupled system of k‐Hessian equations. (26th November 2019)

• Record Type: Journal Article
• Authors: Feng, Meiqiang; Zhang, Xuemei
• Other Names: Tosun Murat guestEditor.; Kara Emrah Evren guestEditor.; Usta Fuat guestEditor.
• Is Part Of: Mathematical methods in the applied sciences. Volume 44:Number 9(2021)
• Page Start: 7377
• Page End: 7394
• Publication Date: 2019-11-26
• Subjects: eigenvalue theory in cones -- nontrivial radial solutions -- parameter dependence -- system of k‐Hessian equations; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.6053
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : In this paper, we consider the following system coupled by multiparameter k ‐Hessian equations S k ( D 2 u 1 ) = λ 1 f 1 ( − u 2 ) in Ω, S k ( D 2 u 2 ) = λ 2 f 2 ( − u 1 ) in Ω, u 1 = u 2 = 0 on ∂ Ω . Here, λ 1 and λ 2 are positive parameters, Ω is the unit ball in R n, S k ( D 2 u ) is the k ‐Hessian operator of u, n 2 < 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 λ 1, u λ 2 ) on the parameter λ 1 and λ 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 .