A remark on the existence of positive radial solutions to a Hessian system

: We give new conditions for the study of existence of positive radial solutions for a system involving the Hessian operator. The solutions to be obtained are given by successive-approximation. Our interest is to improve the works that deal with such systems at the present and to give future directions of research related to this work for researchers.


Introduction
This paper is devoted to develop the mathematical theory for the study of existence of positive radial solutions of a system of partial differential equations (PDE) of the form where α, β ∈ (0, ∞), k 1 , k 2 , k 3 , k 4 ∈ {1, 2, ..., N} with k 1 > k 2 and k 3 > k 4 , S k i λ D 2 (•) (i = 1, 2, 3, 4) stands for the k i -Hessian operator defined as the sum of all k i ×k i principal minors of the Hessian matrix D 2 (•) and the functions p, q, f and g satisfy some suitable conditions. In the case α = β = 0 and k 1 = k 3 = 1, there are several works that deals with the existence of radially symmetric solution for (1.1), in which situation the system become ∆u = p (|x|) f (v) , x ∈ R N , (N ≥ 3), ∆v = q (|x|) g (u) , x ∈ R N , (N ≥ 3). (1.2) Some of these are analyzed in the following. For example, [5] considered the existence of entire large solutions for the system (1.2) in the case f (v) = v a and g (u) = u b with 0 < ab ≤ 1 and noticed that Moreover, if a · b > 1 he showed that the system (1.2) has a positive entire large solution if the radial functions p and q satisfy one of the two inequalities Recently, for the particular case α, β ∈ [0, ∞), k 1 = k 3 = N and k 2 = k 4 = 1, the authors [7] obtained the existence of entire radial large solutions for the system (1. (1.7) Here, the results of [5] are included for a, b ∈ (0, 1], i.e. f and g are sublinear. Hence, it remains unknown the case 0 < a · b ≤ 1, i.e. if an analogous result obtained by [5] holds for the more general system (1.1). In our paper, we give a new methodology for proving existence results under a class of general nonlinearities considered in other frameworks (see e.g. Orlicz Spaces Theory) including such the sublinear and superlinear class of functions discussed in [5]. This may be used in tackling other related problems. The reminder of this paper is organized as follows. Section 2 contains our main result and some lemmas. In Section 3 we give the proof of our main result. f (t · s) ≤ f (t) · f (s) and g (t · s) ≤ g (t) · g (s) for all s, t ≥ 0;
Moreover, when p and q are non-decreasing, u and v are convex.
As we see from the paper of Zhang-Liu [7], our Theorem 1 represent a consistent generalization from the mathematical point of view. This is due to the fact that we deal with more general nonlinearities f and g that was considered by [7] and with a mixed nonlinear k i −Hessian system of equations.
Next, let us recall the radial form of the k-Hessian operator, see for example [6] and [3].
where the prime denotes differentiation with respect to r.
Before to consider the proof of our main result, we give an useful lemma that can be easy proved as in the papers of Zhang-Liu [7] and Kusano-Swanson [4].

The proof of Theorem 1
The main references for proving Theorem 1 are the works of [7] and [2]. In the next, r is referred for the Euclidean norm We are ready to prove the existence of a radial solution to the problem (1.1). For beginning, we observe that we can rewrite (1.1) as follows and that, the radial solution of (3.1) is a solution (u, v) of (3.1) with the initial conditions Integrating from 0 to r > 0 in (3.1) we obtain Using, the definition of φ i given in Lemma 2, we rewrite (3.3) in an equivalent form Let us now construct a sequence Moreover, proceeding by induction we conclude We note that, for all r > 0 the sequence Integrating (3.6) from 0 to r > 0 we get (3.5). We now briefly, (3.5) imply u n (r) = c 1 + α N;k 1 ,k 2 r 0 tφ 1 ( By the monotonicity of the sequence {(u n , v n )} n≥0 respectively of f and g, the inequalities in (3.7) and with the use of Lemma 2 for ≤ α N;k 1 ,k 2 r 1 + f 1 + g (u n (r)) 1/k 3 and, similarly v n (r) = β N;k 3 ,k 4 rφ 3 ( (3.8) Integrating (3.8) from 0 to r > 0 we get H 1,c 1 (u n (r)) ≤ Λ p,α (r) and H 2,c 2 (v n (r)) ≤ Λ q,β (r) , Choose R > 0. We are now ready to show that {(u n (r) , v n (r))} n≥0 and { u n (r) , v n (r) } n≥0 , for r ∈ [0, R] , both of which are non-negative, are bounded above independent of n. To solve this problem, we observe that We finished the proof that the sequences {(u n (r) , v n (r))} n≥0 and { u n (r) , v n (r) } n≥0 , are bounded above independent of n which coupled with the fact that (u n (r) , v n (r)) , is non-decreasing on [0, ∞) × [0, ∞) we see that {u n (r) , v n (r)} n≥0 itself converges to a function (u (r) , v (r)) as n → ∞, and the limit (u (r) , v (r)) is a positive entire radial solution of equation (1.1). Clearly, the arguments in Zhang and Liu [7] (see also [2]) guarantees that the solution (u (x) , v (x)) := (u (|x|) , v (|x|)) is in the space C 2 R N × C 2 R N and moreover is convex for any x ∈ R N . This is the end of the proof of the theorem.

Conclusions
We have obtained new conditions for the study of existence of positive radial solutions for a system involving the Hessian operator.