Existence of radial solutions for $k$-Hessian system

1.   Introduction

In this paper, we consider the existence of radial solutions for the following $k$-Hessian system

where $\lambda, \mu$ are positive parameters, $\Omega = \{x\in \mathbb{R}^{N}:|x| < 1\}$, $k\leq N < 2k$, the nonlinear terms $f$ and $g$ are nonnegative continuous functions, $S_k(\sigma(D^2z))$ is the $k$-Hessian operator of $z$, $k = 1, 2, \cdots, N$. From a discrete perspective, the $k$-Hessian operator has the form

where $S_k^{ij} = \frac{\partial S_k(\sigma(D^2z))}{\partial z_{ij}}$, see details in ^[1,2].

In general, the $k$-Hessian operator is defined as

where $\Lambda = (\lambda_1, \lambda_2, \cdots, \lambda_N)$ are the eigenvalues of Hessian matrix $D^2z$. In particular, when $k = 1$, $k$-Hessian operator reduces to Laplace operator $S_{1}(\sigma(D^2z)) = \sum\limits_{i = 1}^{N}\lambda_{i} = \triangle z;$ when $k = N$, $k$-Hessian operator is Monge-Ampère operator $S_{N}(\sigma(D^2z)) = \prod\limits_{i = 1}^{N}\lambda_{i} =$det$(D^2z).$ About Laplace problem and Monge-Ampère problem, there are a lot of brilliant papers, see ^[3,4,5,6,7].

$k$-Hessian equations for $k\neq1$ are fully nonlinear partial differential equations ^[8,9], which have important applications in fluid mechanics, geometric analysis and other disciplines ^[10,11,12]. For single $k$-Hessian equation, in recent years, many interesting results have been obtained by different methods, such as the monotone iterative method^[13,14,15], upper and lower solution method^[16,17] and different kinds of fixed point theorems^[18,19,20]. However, there are few studies about $k$-Hessian system. In 2015, Zhang and Zhou ^[13] investigated the $k$-Hessian system

where $p, q: [0, \infty)\rightarrow (0, \infty)$ are continuous, $f, g: [0, \infty)\rightarrow [0, \infty)$ are continuous and increasing. The existence of entire positive radial solutions was obtained by using monotone iterative method. Recently, Yang and Bai^[21] considered a class of more general $k$-Hessian system with parameters

They were interested in the existence of at least one or two $k$-convex radial solutions by analysing the growth of $f$ and $g$ at the origin and infinity.

Some researchers focused on the blow-up radial solution of the $k$-Hessian equations, such as, Covei ^[14] established the necessary and sufficient conditions for the coupled $k$-Hessian system

For more results about blow-up solutions, we refer the reader to ^{[22,23,24,25]}.

Based on the above discussions, in this paper, we consider the existence of the radial solutions of the $k$-Hessian system (1.1). Three cases are considered for the growth of the nonlinear terms $f$ and $g$, that is, $k$-superlinear, $k$-sublinear and $k$-asymptotically linear at the origin and infinity, respectively.

The results obtained in this paper generalize and develop some known results from two aspects. The one is that the nonlinearities $f$ and $g$ have such more general form as $f(|x|, -z_{1}, -z_{2})$ and $g(|x|, -z_{1}, -z_{2})$ than the nonlinearity in ^[21], in which the nonlinear term is separable with the form $H(|x|)f(-z_1, -z_2)$. The other one is, to the best of our knowledge, there are few results about the existence of the entire radial solutions for $k$-Hessian equations, especially for $k$-Hessian system. In ^[26], the authors only discussed the existence of solutions for the single $k$-Hessian equation. From this perspective, parts of our main results generalize the results in ^[26].

The rest of the paper is organized as follows. In Section 2, we provide some preliminary results, which are useful in the following proof. In Section 3, under the conditions of $k$-superlinear or $k$-sublinear for nonlinear terms, the existence of solutions of (1.1) is obtained for any positive parameters $\lambda$ and $\mu$. For the case of $k$-asymptotically linear growth, the range of parameters is determined to ensure the existence of radial solutions of system (1.1). Four examples are given to verify some of our results in Section 4.

2.   Preliminaries

In this section we will provide basic lemmas which are necessary for the understanding of subsequent results.

Lemma 2.1. ^[27] Assume $y(r)\in C^{2}[0, 1)$, with $y'(0) = 0$. Then for $z(|x|) = y(r)$, $r = |x| < 1$, we have that $z(|x|)\in C^{2}(\Omega)$, and

and then

where $\Omega = \{x\in \mathbb{R}^{N}:|x| < 1\}$ and $\ C_{N}^{k} = \frac{N!}{k!(N-k)!}$.

Set space $X = C[0, 1]\times C[0, 1]$ with the norm $\|(y_1, y_2)\| = \|y_1\|_\infty+\|y_2\|_\infty$, where $\|\cdot\|_\infty$ denotes the max norm in $C[0, 1]$. It is well known that the space $(X, \|\cdot\|)$ is a Banach space. For any $\theta\in(0, \frac{1}{2})$, the cone $K$ is defined by

Throughout the paper, we assume that

(H) $f, g: [0, 1]\times[0, \infty)\times[0, \infty)\rightarrow [0, \infty)$ are continuous.

Define the operator $T:X\rightarrow X$ as follows

for any $(y_1, y_2)\in X$, and $T_i:X\rightarrow C[0, 1]$, $(i = 1, 2)$ be the operators

Now we can claim that the radial solutions of system (1.1) is equivalent to the fixed point of the operator $T$ in Banach space $X$. In fact, set $(z_{1}(|x|), z_{2}(|x|)) = (-y_{1}(r), -y_{2}(r))$ with $y'_{1}(0) = y'_{2}(0) = 0$. According to Lemma 2.1, system (1.1) can be transformed into the ordinary differential boundary value problems

Noticed that $C_{N-1}^{k} = \frac{N-k}{k}C_{N-1}^{k-1}$, the left side of the first equation of (2.1) can be reduced to

then the first equation of (2.1) can be rewritten as

Furthermore, integrating the above equation twice, we have

Similarly, from the second equation of (2.1) and boundary value conditions, we can get

Combining (2.2), (2.3) and the definition of operators $T_1$, $T_2$ and $T$, we can conclude that the radial solutions of system (1.1) is equivalent to the fixed point of the operator $T$ in Banach space $X$. In addition, standard arguments show that the operator $T:K\rightarrow K$ is completely continuous.

Lemma 2.2. ^[28] Let $X$ be a Banach space and $K\subset X$ be a cone in $X$. Assume $\Omega_{1}, \Omega_{2}$ are open subsets of $X$ with $0 \in \Omega_{1}, \ \overline{ \Omega}_{1} \subset \Omega_{2}$ and

is a completely continuous operator such that either

(1) $\|Ty\|\leq \|y\|$, $y\in K \cap\partial\Omega_1$ and $\|Ty\|\geq \|y\|$, $y\in K \cap\partial\Omega_2$

or

(2) $\|Ty\|\geq\|y\|$, $y\in K \cap\partial\Omega_1$ and $\|Ty\|\leq\|y\|$, $y\in K \cap\partial\Omega_2$.

Then $T$ has a fixed point in $K\cap(\overline{\Omega}_{2}\setminus\Omega_{1}).$

3.   Existence results

For the convenience, we firstly introduce some notation

Here we are in the position of our main results.

Theorem 3.1. Assume (H) holds, $f^{0} = g^{0} = 0$, $f_{\infty} = \infty$ or $g_{\infty} = \infty$. Then, for all $\lambda, \mu\in(0, \infty)$, the $k$-Hessian system (1.1) has at least one radial solution.

Proof. Let $L_{1} = \frac{k}{2k-N}\left(\frac{N}{C_{N}^{k}}\right)^\frac{1}{k}$. Since $f^{0} = g^{0} = 0$, we can choose $J_{1} > 0$ such that

for $r\in[0, 1], \ 0 < y_{1}+y_{2}\leq J_{1}$, where $\varepsilon > 0$ satisfies

Set

Then, for any $(y_{1}, y_{2})\in K\cap\partial\Omega_{1},$ we have

Therefore, we have

Similarly, we can get

Combining inequalities (3.1) and (3.2), for $(y_{1}, y_{2})\in K\cap\partial\Omega_{1}$, we have

For any $\theta\in (0, \frac12)$, denote

Next, considering $f_{\infty} = \infty$, an analogous estimate holds if $g_\infty = \infty$. There is $\hat{J} > 0$ such that

for any $r\in[\theta, 1-\theta], \ y_{1}+y_{2}\geq\hat{J}$, where $\delta > 0$ satisfies

Let $J_{2} = \max\{2J_{1}, \frac{1}{\theta}\hat{J}\}$ and

If $(y_{1}, y_{2})\in K\cap\partial\Omega_{2}$, then

Furthermore, from the above inequality and (3.3), we have

Therefore, if $(y_{1}, y_{2})\in K\cap\partial\Omega_{2}$, then

By part (1) of Lemma $2.2$, system (1.1) has at least one radial solution. □

Theorem 3.2. Assume (H) holds, $f^{\infty} = g^{\infty} = 0$, $f_{0} = \infty$ or $g_{0} = \infty$. Then, for all $\lambda, \mu\in(0, \infty)$, the $k$-Hessian system (1.1) has at least one radial solution.

Proof. Considering $f_{0} = \infty$, an analogous estimate holds if $g_{0} = \infty$. We can choose $J_{3} > 0$ such that

where $\eta > 0$ satisfies

here the positive constant $L_2$ is defined the same as in proof of Theorem 3.1.

Set

For any $(y_{1}, y_{2})\in K\cap \partial\Omega_{3}$, according to (3.4), we can get

furtheremore, we have

In order to construct $\Omega_{4}$, we define two new functions

it is easy to see $F(r, t)$ and $G(r, t)$ are nondecreasing about variable $t$. From the definitions of $f^{\infty}$ and $g^{\infty}$ and the expressions of $F$ and $G$, it can be seen that the following two limits are valid according to $f^{\infty} = g^{\infty} = 0$,

therefore, there is a constant $J_{4} > 2J_{3}$, such that

for $\rho > 0$ satisfying

here positive constant $L_1$ is defined the same as in proof of Theorem 3.1.

Set

For any $(y_{1}, y_{2})\subset K\cap\partial\Omega_{4}$, we have

Therefore, we have

Similarly, we can get

Combining inequalities (3.5) and (3.6), for $(y_{1}, y_{2})\in K\cap\partial\Omega_{4}$, one has

By part (2) of Lemma $2.2$, system (1.1) has at least one radial solution. □

Let

Theorem 3.3. Assume (H) holds, $f^0, \ g^0, \ f_\infty, \ g_\infty\in(0, \infty)$, $f_\infty > f^0$ and $g_\infty > g^0$. Then, for each $\lambda\in\left(\frac{1}{L_4^kf_\infty\theta^k}, \frac{1}{L_3^kf^02^k}\right)$ and $\mu\in\left(\frac{1}{L_4^kg_\infty\theta^k}, \frac{1}{L_3^kg^02^k}\right),$ the $k$-Hessian system (1.1) has at least one radial solution.

Proof. Let $\lambda$ and $\mu$ be given in (3.7). Now, let $\varepsilon > 0$ be chosen, such that

and

Since $f^0$, $g^0\in(0, \infty)$, for $\varepsilon > 0$, there exists $J_5 > 0$, such that

for $r\in[0, 1], \ 0 < y_1+y_2\leq J_5$. Therefore, choosing $(y_1, y_2)\in K$ with $\|(y_1, y_2)\| = J_5$, one gets

Therefore, we have

Similarly, we can get

Combining (3.11) and (3.12), we can get

Therefore, if we set

then, for $(y_1, y_2)\in K\cap\partial\Omega_5$,

In view of $f_\infty\in(0, \infty)$, for $\varepsilon > 0$, there is $\bar{J} > 0$, such that

Let $J_6 = \max\{2J_5, \frac{1}{\theta}\bar{J}\}$ and set

If $(y_1, y_2)\in K$ with $\|(y_1, y_2)\| = J_6$, then

From the above inequality and (3.10), we have

Furthermore, $\|T(y_1, y_2)\| = \|T_1(y_1, y_2)\|_\infty+\|T_2(y_1, y_2)\|_\infty\geq\|(y_1, y_2)\|.$ System (1.1) has at least one radial solution for the given $\lambda$ and $\mu$ due to Lemma 2.2.

Similarly, we can also prove $\|T(y_1, y_2)\|\geq\|(y_1, y_2)\|$ if $g_\infty\in(0, \infty)$. This completes the proof. □

Theorem 3.4. Assume (H) holds, $f_0, \, g_0, \, f^\infty, \, g^\infty\in(0, \infty)$, $f_0 > f^\infty$ and $g_0 > g^\infty$. Then, for each $\lambda\in\left(\frac{1}{L_4^kf_0\theta^k}, \frac{1}{L_3^kf^\infty2^k}\right)$ and $\mu\in\left(\frac{1}{L_4^kg_0\theta^k}, \frac{1}{L_3^kg^\infty2^k}\right)$, the $k$-Hessian system (1.1) has at least one radial solution.

Proof. Let $\lambda$ and $\mu$ be given in (3.13). Now, let $\varepsilon > 0$ be chosen, such that

and

For $f_0\in(0, \infty)$, an analogous estimate holds for $g_0\in(0, \infty)$, there exists positive constant $J_7$, for $r\in[\theta, 1-\theta], \, 0 < y_1+y_2\leq J_7$, such that

Therefore, choosing $(y_1, y_2)\in K$ with $\|(y_1, y_2)\| = J_7$, one gets from (3.14)

Thus, if we set

then, for $(y_1, y_2)\in K\cap\partial\Omega_7$, we have

In order to construct $\Omega_{8}$, for $t\geq 0$, we define two new functions

It is easy to see $F(r, t)$ and $G(r, t)$ are nondecreasing about variable $t$. From the definitions of $f^{\infty}$ and $g^{\infty}$ and the expressions of $F$ and $G$, it can be seen that the following two limits are valid according to $f^{\infty}\in(0, \infty)$ and $g^{\infty}\in(0, \infty)$,

Therefore, there is a constant $J_{8} > 2J_{7}$, such that

where $\varepsilon > 0$ satisfying

Set

for any $(y_{1}, y_{2})\subset K\cap\partial\Omega_{8}$, we have

Therefore, we have

Similarly, we can get

Combining (3.16) and (3.17), for $(y_{1}, y_{2})\in K\cap\partial\Omega_{8}$, we can get

System (1.1) has at least one radial solution for the given $\lambda$ and $\mu$ due to Lemma 2.2. □

4.   Examples

Example 4.1. Consider the following 3-Hessian system

where $\Omega = \{x\in \mathbb{R}^{4}:|x| < 1\}$. Similar to the transformation method in Section 2, the corresponding nonlinear terms in (4.1) have the following form

Therefore, the nonlinear terms $f$ and $g$ are continuous functions. It is not difficult to check that $f^0 = g^0 = 0$ and $f_\infty = \infty$. Then, all the conditions of Theorem 3.1 are fulfilled, the conclusion follows, that is, for any positive parameters $\lambda$ and $\mu$, (4.1) has at least one radial solution.

Example 4.2. Consider the following 4-Hessian system

where $\Omega = \{x\in \mathbb{R}^{5}:|x| < 1\}$. Similar to the transformation method in Section 2, the corresponding nonlinear terms of (4.2) have the following form

So, the nonlinear terms $f$ and $g$ are continuous functions. It is not difficult to check that $f_0 = \infty$ and $f^\infty = g^\infty = 0$. Therefore, all the conditions of Theorem 3.2 are satisfied, the conclusion follows, that is, for any positive parameters $\lambda$ and $\mu$, (4.2) has at least one radial solution.

Example 4.3. Consider the following 3-Hessian system

where $\Omega = \{x\in \mathbb{R}^{4}:|x| < 1\}$, the nonlinearities $f$ and $g$ are given by

and

respectively.

In fact, similar to the transformation method in Section 2, the corresponding nonlinear terms in (4.3) have the following form

and

From the expressions of $f$ and $g$, we can calculate

so, $f_\infty > f^0$ and $g_\infty > g^0$. Then all the conditions of Theorem 3.3 hold.

Furthermore, for fixed $\theta = \frac14$,

Therefore, it follows from Theorem 3.3 that (4.3) has at least one radial solution for any $\lambda\in(\frac{1}{50}, \ \frac{37}{200})$ and $\mu\in(\frac{3}{400}, 10)$.

Example 4.4. Consider the following 3-Hessian system

where $\Omega = \{x\in \mathbb{R}^{4}:|x| < 1\}$, the nonlinearities $f$ and $g$ are given by

and

respectively.

As a matter of fact, similar to the transformation method in Section 2, the corresponding nonlinear terms in (4.4) have the following form

and

From the expressions of $f$ and $g$, we can calculate

so, $f_0 > f^\infty$ and $g_0 > g^\infty$. Then all the conditions of Theorem 3.4 are fulfilled.

Furthermore, for fixed $\theta = \frac14$,

Therefore, it follows from Theorem 3.4 that (4.4) has at least one radial solution for any $\lambda\in(\frac{1}{100}, \ \frac{21}{250})$ and $\mu\in(\frac{1}{100}, \ \frac{21}{250})$.

5.   Conclusions

In this paper, a class of $k$-Hessian system with parameter is concerned. we are interested in the range of parameters $\lambda$ and $\mu$ on which there exists at least one radial solution. By using the Guo-Krasnosel'skii fixed point theorem, we find that there is a close relation between the range of parameters and the growth of nonlinearities $f$ and $g$ at the origin and infinity. The results obtained in this paper generalize and develop some of the known results, such as, parts of the results in ^[21,26] from two sides. The one is that the more general form of nonlinearities $f$ and $g$ is investigated here. The other is that there are no existence results for $k$-Hessian system under the conditions that nonlinearities $f$ and $g$ satisfy $k$-superlinear, $k$-sublinear and $k$-asymptotically linear conditions at the origin and infinity. Finally, we also illustrate the obtained results with four examples.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by the NSFC (No. 11801243) and the Innovation Ability Improvement Project of Universities of Gansu (No. 2019B-054), China.

Conflict of interest

The authors declare no conflict of interest.