A coupled system of k‐Hessian equations

Abstract

In this paper, we consider the following system coupled by multiparameter k‐Hessian equations Skfalse(D2u1false)=λ1f1false(−u2false)1emin1emnormalΩ,Skfalse(D2u2false)=λ2f2false(−u1false)1emin1emnormalΩ,u1=u2=01emon1em∂normalΩ. Here, λ1 and λ2 are positive parameters, Ω is the unit ball in Rn, Sk(D2u) 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=false(uλ1,uλ2false) 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.

Convex solutions of Monge-Ampère equations and systems: Existence, uniqueness and asymptotic behavior

“…which have the same meaning as T . Next, we consider the existence results of system (S λ ,••• ,n ) by using a completely di erent method from that of [2,8,10,11,[14][15][16]20,21,29,30,32,35,36,40,41,46,49,50,[53][54][55]61], namely the eigenvalue theory:…”

“…At the same time, we notice that many authors have paid more attention to various of system problems, for example, see [2,10,[14][15][16]20,21,29,30,32,35,40,46,49,50]. Specially, Lair and Wood [36] analyzed the existence of entire positive solutions of system ∆u = p(|x|)v α , ∆v = q(|x|)u β ,…”

On the solutions to weakly coupled system of $\boldsymbol{k_i}$-Hessian equations

“…For example, by using fixed point theorem, Wang [24] established the existence, multiplicity and nonexistence of convex radial solutions to a coupled system of Monge-Ampère equations in superlinear and sublinear cases. In [7], the authors studied the existence and multiplicity of nontrivial radial solutions for system coupled by multiparameter k-Hessian equations and obtained sufficient conditions for the existence of nontrivial radial solutions to power-type coupled k-Hessian system based on a eigenvalue theory in cones. In particular, Cui considered a Hessian type system coupled by different k-Hessian equations and obtained the existence of entire k-convex radial solutions, see [4].…”

“…In this paper, we shall establish the existence and multiplicity of nontrivial radial k-admissible solutions of the weakly coupled degenerated system (1.1). It is worth to notice that the system (1.1) contains a variety of different k-Hessian equations which is significantly different from that in [5,7,23,29] such that the problem we considered can contain Laplace equations and Monge-Ampère equations at the same time. This kind of system can represent the coupling of different types of elliptic equations, which makes our problem more comprehensive and more applicable.…”

“…Here, we call them α i or β i -asymptotic growth condition, super-α i or β i -asymptotic growth condition and sub-α i or β i -asymptotic growth condition. Compared with some N -asymptotic growth (see, for instance [6,8,24] where the constants α i = β i = N ) in studying Monge-Ampère equations and some k-asymptotic growth (see, for instance [7,26,27] where the constants α i = β i = k) in studying k-Hessian equations, our conditions are more flexible. By imposing suitable conditions on…”