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

1.   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 $ \alpha, \beta \in \left(0, \infty \right) $, $ k_{1}, k_{2}, k_{3}, k_{4} \in \{1, 2, ..., N\} $ with $ k_{1} > k_{2} $ and $ k_{3} > k_{4} $, $ S_{k_{i}}\left(\lambda \left(D^{2}\left(\circ \right) \right) \right) $ ($ i = 1, 2, 3, 4 $) stands for the $ k_{i} $-Hessian operator defined as the sum of all $ k_{i}\times k_{i} $ principal minors of the Hessian matrix $ D^{2}\left(\circ \right) $ and the functions $ p $, $ q $, $ f $ and $ g $ satisfy some suitable conditions.

In the case $ \alpha = \beta = 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

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\left(v\right) = v^{a} $ and $ g\left(u\right) = u^{b} $ with $ 0 < ab\leq 1 $ and noticed that (1.2) has a positive entire large solution if and only if the nonnegative spherically symmetric continuous functions $ p $ and $ q $ satisfy

Moreover, if $ a\cdot 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 $ \alpha, \beta \in \left[0, \infty \right) $, $ 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) under hypotheses that $ p $, $ q:\left[0, \infty \right) \rightarrow \left[0, \infty \right) $ are spherically symmetric continuous functions and $ f $, $ g:\left[0, \infty \right) \rightarrow \left[0, \infty \right) $ are continuous, monotone non-decreasing nonlinearities such that

and

Here, the results of ^[5] are included for $ a, b\in \left(0, 1\right] $, i.e. $ f $ and $ g $ are sublinear. Hence, it remains unknown the case $ 0 < a\cdot b\leq 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.

2.   The main result and auxiliary lemmas

For the purpose of the paper, the following basic class of functions are considered

(P1) $ p $, $ q:\left[0, \infty \right) \rightarrow \left[0, \infty \right) $ continuous functions;

(C1) $ f $, $ g:\left[0, \infty \right) \rightarrow \left[0, \infty \right) $ are continuous and monotone non-decreasing such that

and

(C2) $ \int_{1}^{\infty }\frac{dt}{\left(1+f\left(\left(1+g\left(t\right) \right) ^{1/k_{3}}\right) \right) ^{1/k_{1}}} = \infty $ and $ \int_{1}^{\infty }\frac{dt}{\left(1+g\left(\left(1+f\left(t\right) \right) ^{1/k_{1}}\right) \right) ^{1/k_{3}}} = \infty $.

Our main interest is to prove the following theorem.

Theorem 1. If $ p, $ $ q $ satisfy (P1) and $ f, $ $ g $ satisfy (C1), (C2), then the system (1.1) has one positive entire radial solution $ \left(u, v\right) \in C^{2}\left(\mathbb{R}^{N}\right) \times C^{2}\left(\mathbb{R }^{N}\right) $ such that

where

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].

Lemma 1. Let $ k\in \{1, 2, ..., N\} $. Assume $ y\in C^{2}\left[0, R\right) $ is radially symmetric with $ y^{\prime }\left(0\right) = 0 $. Then, the function $ u $ defined by $ u\left(x\right) = y\left(r\right) $ where $ r = \left\vert x\right\vert < R $ is $ C^{2}\left(B_{R}\right) $, and

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].

Lemma 2. Setting

the following hold:

1. $ \varphi _{i}\left(t_{0}^{i}\right) = \frac{k_{i+1}-k_{i}}{k_{i}} \left(\frac{k_{i+1}}{k_{i}}\right) ^{k_{i+1}/\left(k_{i}-k_{i+1}\right) } < 0 $, $ \varphi _{i}\left(1\right) = 0 $ and $ \varphi _{i}\left(\infty \right) : = \lim_{t\rightarrow \infty }\varphi _{i}\left(t\right) = \infty; $

2. $ \varphi _{i}:\left[t_{0}^{i}, \infty \right) \rightarrow \left[\varphi _{i}\left(t_{0}^{i}\right), \infty \right) $ is strictly increasing for $ t > t_{0}^{i} $ and in fact has a uniquely defined inverse function $ \phi _{i}:\left[\varphi _{i}\left(t_{0}^{i}\right), \infty \right) \rightarrow \left[t_{0}^{i}, \infty \right) $ with $ \phi _{i}\left(0\right) = 1 $;

3. $ \phi _{i}:\left[\varphi _{i}\left(t_{0}^{i}\right), \infty \right) \rightarrow \left[t_{0}^{i}, \infty \right) $ is analytic, strictly increasing for $ t > \varphi _{i}\left(t_{0}^{i}\right) $ and concave. In particular, $ \phi _{i}\left(t\right) \geq 1 $ for all $ t\geq 0 $, $ \phi _{i}\left(\infty \right) : = \lim_{t\rightarrow \infty }\phi _{i}\left(t\right) = \infty $ and for $ t > \varphi _{i}\left(t_{0}^{i}\right) $ it hold

4. $ \phi _{i}\left(s\xi \right) \leq \xi ^{1/k_{i}}\phi _{i}\left(s\right) $ for all $ s\geq 0 $ and $ \xi \geq 1 $.

3.   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

of a vector

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 $ \left(u, v\right) $ of (3.1) with the initial conditions

Integrating from $ 0 $ to $ r > 0 $ in (3.1) we obtain

or, equivalently

Using, the definition of $ \phi _{i} $ given in Lemma 2, we rewrite (3.3) in an equivalent form

which yields

Since

via L'Hôpital's rule and (3.2), the equations in (3.4) can be extended by continuity at $ r = 0 $. Then, the system (3.1) with the initial conditions (3.2) can be equivalently written as an integral system of equations

Let us now construct a sequence

in such a way

By construction, for all $ r\geq 0 $ and $ n\in \mathbb{N} $ $ \ $we have

Moreover, proceeding by induction we conclude

is a non-decreasing sequence on

We note that, for all $ r > 0 $ the sequence

satisfies

Integrating (3.6) from $ 0 $ to $ r > 0 $ we get (3.5). We now briefly, (3.5) imply

and

Setting

we have

By the monotonicity of the sequence $ \left\{ \left(u_{n}, v_{n}\right) \right\} _{n\geq 0} $ respectively of $ f $ and $ g $, the inequalities in (3.7) and with the use of Lemma 2 for

we have

and, similarly

Finally

Integrating (3.8) from $ 0 $ to $ r > 0 $ we get

where

Choose $ R > 0 $. We are now ready to show that

both of which are non-negative, are bounded above independent of $ n $. To solve this problem, we observe that

On the other hand, since

is a bijection map for all $ s > c_{i} $ with the inverse denoted by $ H_{i, c_{i}}^{-1}\left(s\right) $ on $ \left[0, \infty \right) $ such that

we see that

which proved that

is an uniformly bounded independent of $ n $ sequence on

for arbitrary $ R > 0 $. On the other hand, using this result in (3.8) the same is true for

We finished the proof that the sequences

are bounded above independent of $ n $ which coupled with the fact that

is non-decreasing on $ \left[0, \infty \right) \times \left[0, \infty \right) $ we see that

itself converges to a function

and the limit $ \left(u\left(r\right), v\left(r\right) \right) $ 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

is in the space

and moreover is convex for any $ x\in \mathbb{R}^{N} $. This is the end of the proof of the theorem.

4.   Conclusions

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

Acknowledgments

The author is grateful to the anonymous referees for their careful reading and valuable comments.

Conflict of interest

All author declare no conflicts of interest in this paper.