Theory and Modern Applications

# Existence and nonexistence of entire k-convex radial solutions to Hessian type system

## Abstract

In this paper, a Hessian type system is studied. After converting the existence of an entire solution to the existence of a fixed point of a continuous mapping, the existence of entire k-convex radial solutions is established by the monotone iterative method. Moreover, a nonexistence result is also obtained.

## 1 Introduction

In this paper, we study the existence of entire k-convex radial solutions to the following problem of Hessian type system:

$$\textstyle\begin{cases} \sigma _{k} (\lambda (D^{2} u+\mu \vert \nabla u \vert I ) )=p( \vert x \vert ) f_{1}(u)f_{2}(v), & x \in B_{1}(0), \\ \sigma _{l} (\lambda (D^{2} v+\nu \vert \nabla v \vert I ) )=q( \vert x \vert ) g_{1}(u)g_{2}(v), & x \in B_{1}(0), \\ u=v=0,& x\in \partial B_{1}(0), \end{cases}$$
(1.1)

where $$k,l=1,2,\ldots, N$$, $$\mu , \nu \ge 0$$ are constants, $$B_{1}(0)$$ is the unit ball in $$\mathbb{R}^{N}$$, for any $$N \times N$$ real symmetric matrix A, $$\lambda (A)$$ denotes the eigenvalues of A, $$D^{2} u(x)= ( \frac{\partial ^{2} u(x)}{\partial x_{i} \partial x_{j}} )$$ denotes the Hessian matrix of the function $$u \in C^{2} (\overline{B_{1}(0)} )$$, âˆ‡u denotes the gradient of u, and $$\sigma _{k}(\lambda )=\sum_{1 \leq i_{1}<\cdots <i_{k} \leq N} \lambda _{i_{1}} \cdots \lambda _{i_{k}}$$ denotes the kth elementary symmetric function of $$\lambda = (\lambda _{1},\ldots, \lambda _{N} ) \in \mathbb{R}^{N}$$.

For p, q, $$f_{1}$$, $$f_{2}$$, $$g_{1}$$, $$g_{2}$$, we introduce the following conditions:

1. (H1)

$$p,q\in C([0,1],(0,+\infty ))$$. $$f_{1},f_{2},g_{1},g_{2}\in C((-\infty ,0],[0,+\infty ))$$ are decreasing.

2. (H2)

For any $$a>0$$, the integral $$\int _{-\infty }^{-a} \frac{d\tau }{(f_{1}(\tau )f_{2}(\tau ))^{\frac{1}{k}}+(g_{1}(\tau )g_{2}(\tau ))^{\frac{1}{l}}}$$ is divergent.

3. (H3)

For any $$a>0$$, the integral $$\int _{-a}^{0} \frac{d\tau }{(f_{1}(\tau )f_{2}(\tau ))^{\frac{1}{k}}+(g_{1}(\tau )g_{2}(\tau ))^{\frac{1}{l}}}$$ is divergent.

Denote

$$\Gamma _{k}:= \bigl\{ \lambda \in \mathbb{R}^{N}: \sigma _{j}(\lambda )>0, 1 \leq j \leq k \bigr\} .$$

We say that a function $$u \in C^{2} (\overline{B_{1}(0)} )$$ is k-convex in $$B_{1}(0)$$ if $$\lambda (D^{2} u(x) ) \in \Gamma _{k}$$ for all $$x \in B_{1}(0)$$.

In (1.1), if $$\mu =0$$ and $$f_{2}(v)\equiv 1$$, the first equation in the system becomes the following k-Hessian type equation:

$$\sigma _{k} \bigl(\lambda \bigl(D^{2} u \bigr) \bigr)=p \bigl( \vert x \vert \bigr) f_{1}(u);$$
(1.2)

if $$\mu =\nu =0$$ and $$f_{1}(u)=g_{2}(v)\equiv 1$$, the system becomes the following coupling k-Hessian system:

$$\textstyle\begin{cases} \sigma _{k} (\lambda (D^{2} u ) )=p( \vert x \vert ) f_{2}(v), \\ \sigma _{l} (\lambda (D^{2} v ) )=q( \vert x \vert ) g_{1}(u). \end{cases}$$
(1.3)

Related to k-Hessian equations, if $$k=1$$ the k-Hessian equations become the well-known Laplacian equations, and if $$k=N$$ the k-Hessian equations become the Mongeâ€“AmpÃ¨re equations. Concerning Laplacian equations and Mongeâ€“AmpÃ¨re equations, there are a great number of research papers, see for examples [1, 6, 7, 22] and the references therein. Here we specially mention Keller [15], Osserman [21], and Lair and Wood [17] for Laplacian equations and Cheng and Yau [2] and Laser and McKenna [19] for Mongeâ€“AmpÃ¨re equations. Similar situations occur for coupling k-Hessian system (1.3), although in this case there are not so many research papers. Here we only mention Lair and Wood [18] and CÃ®rstea and RÄƒdulescu [3] for coupling Laplacian systems and Wang and An [24] and Zhang and Qi [26] for coupling Mongeâ€“AmpÃ¨re systems.

For general k-Hessian equation (1.2), when $$p \equiv 1$$ and $$f(u)=u^{\gamma k}$$, $$\gamma >1$$, Jin, Li, and Xu [13] showed the nonexistence of entire k-convex positive solutions. When $$p \equiv 1$$, Ji and Bao [11] gave necessary and sufficient conditions on the existence of entire positive k-convex radial solutions. If we generalize $$p( \vert x \vert )f(u)$$ to $$f(x,u)$$, de Oliveira, do Ã“, and Ubilla obtained the existence of k-convex radial solutions in the case of supercritical nonlinearity by means of variational techniques (see [5] and the references therein for research in this direction). For general k-Hessian equation (1.2) and coupling k-Hessian system (1.3), Zhang and Zhou [27] obtained several results on the existence of entire positive k-convex radial solutions. We refer to the papers of Feng and Zhang [8] and Gao, He, and Ran [9] and the references therein for research on coupling k-Hessian system (1.3).

It is obvious that the k-Hessian type equation

$$\sigma _{k} \bigl(\lambda \bigl(D^{2} u+\mu \vert \nabla u \vert I \bigr) \bigr)=p \bigl( \vert x \vert \bigr) f(u)$$

is a generalization of k-Hessian equation (1.2), but it is a special case of the following fully nonlinear Hessian equation:

$$F \bigl(\lambda \bigl(D^{2} u+A(x,u,\nabla u) \bigr) \bigr)=f(x,u, \nabla u).$$
(1.4)

See Guan and Jiao [10] and Jiang and Trudinger [12] and the references therein for research on fully nonlinear Hessian equation (1.4). Here we also want to mention the work of Dai [4] for similar study.

Inspired by the works above, and as we know that now there are no papers on the problem of k-Hessian type system (1.1), we obtain the following results in this paper.

### Theorem 1.1

Under conditions (H1) and (H2), if $$f_{1}(0)g_{2}(0)\neq0$$ and $$f_{2}(0)+g_{1}(0)\neq0$$, then problem (1.1) admits an entire k-convex radial solution $$(u,v) \in C^{2} (\overline{B_{1}(0)} )\times C^{2} ( \overline{B_{1}(0)} )$$.

### Remark 1.1

In the case of $$f_{1}(0)g_{2}(0)=0$$, if $$f_{1}(0)=g_{2}(0)=0$$, then there is a trivial solution $$(u,v)=(0,0)$$ to problem (1.1); if $$f_{1}(0)=0$$ or $$g_{2}(0)=0$$, then there is a semi-trivial solution $$(u,v)=(0,v)$$ or $$(u,v)=(u,0)$$ to problem (1.1); moreover, the semi-trivial solution may become trivial if $$f_{1}(0)=0$$ with $$g_{1}(0)=0$$ or $$g_{2}(0)=0$$ with $$f_{2}(0)=0$$.

In the case of $$f_{2}(0)+g_{1}(0)=0$$, there is a trivial solution $$(u,v)=(0,0)$$ to problem (1.1).

### Theorem 1.2

Under conditions (H1) and (H3), problem (1.1) admits no entire k-convex radial solution $$(u,v) \in C^{2} (\overline{B_{1}(0)} )\times C^{2} ( \overline{B_{1}(0)} )$$.

### Remark 1.2

In this case, $$f_{1}(0)f_{2}(0)=g_{1}(0)g_{2}(0)=0$$, and there is a trivial solution $$(u,v)=(0,0)$$ to problem (1.1).

## 2 Preliminaries

In this section, we give some preliminary results which will be used to prove the main results in the next section.

### Lemma 2.1

Assume $$\varphi (r) \in C^{2}[0, 1]$$ with $$\varphi ^{\prime }(0)=0$$. Then, for $$u(x)=\varphi (r)$$, there holds that $$u \in C^{2} (\overline{B_{1}(0)} )$$ and

$$\lambda \bigl(D^{2} u+\eta \vert \nabla u \vert I \bigr)= \textstyle\begin{cases} (\varphi ^{\prime \prime }(r)+\eta \varphi ^{\prime }(r), ( \frac{1}{r}+\eta )\varphi ^{\prime }(r),\ldots, ( \frac{1}{r}+\eta )\varphi ^{\prime }(r) ), &r \in (0, 1], \\ (\varphi ^{\prime \prime }(0), \varphi ^{\prime \prime }(0),\ldots, \varphi ^{\prime \prime }(0) ), &r=0, \end{cases}$$

and further

\begin{aligned} &\sigma _{k} \bigl(\lambda \bigl(D^{2} u+\eta \vert \nabla u \vert I \bigr) \bigr) \\ &\quad = \textstyle\begin{cases} C_{N-1}^{k-1} (\varphi ^{\prime \prime }(r)+\eta \varphi ^{\prime }(r)) ( (\frac{1}{r}+\eta )\varphi ^{\prime }(r) )^{k-1}+C_{N-1}^{k} ( (\frac{1}{r}+\eta )\varphi ^{\prime }(r) )^{k}, &r \in (0, 1], \\ C_{N}^{k} (\varphi ^{\prime \prime }(0) )^{k}, &r=0, \end{cases}\displaystyle \end{aligned}

where $$C_{N}^{k}=\frac{N!}{k!(N-k)!}$$.

### Proof

It is immediate that, for $$x \neq 0$$, $$1 \leq i, j \leq N$$,

$$\frac{\partial u(x)}{\partial x_{i}}= \biggl( \frac{\varphi ^{\prime }(r)}{r} \biggr) x_{i}$$

and

$$\frac{\partial ^{2} u(x)}{\partial x_{i} \partial x_{j}}= \biggl( \frac{\varphi ^{\prime \prime }(r)}{r^{2}} \biggr) x_{i} x_{j}- \biggl( \frac{\varphi ^{\prime }(r)}{r^{3}} \biggr) x_{i} x_{j}+ \biggl( \frac{\varphi ^{\prime }(r)}{r} \biggr) \delta _{i j}.$$

Further if define

$$\frac{\partial u(0)}{\partial x_{i}}=0,\qquad \frac{\partial ^{2} u(0)}{\partial x_{i} \partial x_{j}}=\varphi ^{ \prime \prime }(0) \delta _{i j},$$

then $$u \in C^{2} (\overline{B_{1}(0)} )$$.

Now it is easy to show the two equalities for $$\lambda (D^{2} u+\eta \vert \nabla u \vert I )$$ and $$\sigma _{k} (\lambda (D^{2} u+\eta \vert \nabla u \vert I ) )$$.â€ƒâ–¡

### Lemma 2.2

Let $$f\in C(-\infty ,0]$$ be decreasing. Assume that $$\varphi \in C^{0}[0, 1] \cap C^{1}(0, 1]$$ is a solution of the Cauchy problem

$$\textstyle\begin{cases} \varphi ^{\prime }(r)= (\frac{k}{C_{N-1}^{k-1}}\mathrm{e}^{-\psi _{k, \eta }(r)} \int _{0}^{r} \mathrm{e}^{\psi _{k,\eta }(s)} \frac{s^{k-1}p(s)}{(1+\eta s)^{k-1}} f(\varphi (s)) \,d s )^{ \frac{1}{k}},\quad 0< r< 1, \\ \varphi (1)=0, \end{cases}$$

where

$$\psi _{k,\eta }(r)=\frac{k}{C_{N-1}^{k-1}} \bigl(C_{N}^{k} \eta r+C_{N-1}^{k} \ln r \bigr).$$

Then $$\varphi \in C^{2}[0, 1]$$, and it satisfies the problem

$$\textstyle\begin{cases} C_{N-1}^{k-1}\varphi ^{\prime \prime }(r) (\varphi ^{\prime }(r) )^{k-1}r + (C_{N}^{k}\eta r+C_{N-1}^{k} ) ( \varphi ^{\prime }(r) )^{k} =\frac{r^{k}p(r)}{(1+\eta r)^{k-1}} f( \varphi (r)), \quad 0< r< 1, \\ \varphi ^{\prime }(0)=0. \end{cases}$$

Furthermore, if Ï† is nontrivial, i.e., $$\varphi (r)<0$$ for $$0\le r<1$$, then

$$\lambda _{r}:= \biggl(\varphi ^{\prime \prime }(r)+\eta \varphi ^{ \prime }(r), \biggl(\frac{1}{r}+\eta \biggr)\varphi ^{\prime }(r),\ldots, \biggl(\frac{1}{r}+\eta \biggr)\varphi ^{\prime }(r) \biggr) \in \Gamma _{k}$$

for $$0 \leq r<1$$.

### Proof

It is easy to see that $$\varphi (r) \in C^{2}[0, 1]$$.

From

$$\varphi ^{\prime }(r)= \biggl(\frac{k}{C_{N-1}^{k-1}}\mathrm{e}^{-\psi _{k, \eta }(r)} \int _{0}^{r} \mathrm{e}^{\psi _{k,\eta }(s)} \frac{s^{k-1}p(s)}{(1+\eta s)^{k-1}} f \bigl(\varphi (s) \bigr) \,d s \biggr)^{ \frac{1}{k}}$$

we have

$$\bigl(\varphi ^{\prime }(r) \bigr)^{k}=\frac{k}{C_{N-1}^{k-1}} \mathrm{e}^{-\psi _{k,\eta }(r)} \int _{0}^{r} \mathrm{e}^{\psi _{k,\eta }(s)} \frac{s^{k-1}p(s)}{(1+\eta s)^{k-1}} f \bigl(\varphi (s) \bigr) \,d s,$$

and further differentiating with respect to r we have

$$C_{N-1}^{k-1}\varphi ^{\prime \prime }(r) \bigl(\varphi ^{\prime }(r) \bigr)^{k-1}r + \bigl(C_{N}^{k} \eta r+C_{N-1}^{k} \bigr) \bigl( \varphi ^{\prime }(r) \bigr)^{k} =\frac{r^{k}p(r)}{(1+\eta r)^{k-1}} f \bigl( \varphi (r) \bigr).$$

If Ï† is nontrivial, it is easy to see that Ï† is increasing, so for $$0 \leq r<1$$ we conclude $$\varphi (r)<\varphi (1)=0$$, $$f(\varphi (r))>f(\varphi (1))\ge 0$$ and further

$$\sigma _{k} (\lambda _{r} )=f \bigl(\varphi (r) \bigr)>0 \quad \text{for }0 \leq r< 1.$$

By the properties of kth elementary symmetric functions (see for example [20]), we know $$\sigma _{j} (\lambda _{r} )>0$$ for $$1\le j< k$$ and $$0 \leq r<1$$. Therefore we conclude the lemma.â€ƒâ–¡

## 3 Proofs of the main results

In this section, we prove the main results in this paper, i.e., the existence and nonexistence of entire k-convex radial solutions for problem (1.1).

### Proof of Theorem 1.1

From the system

$$\textstyle\begin{cases} C_{N-1}^{k-1}u^{\prime \prime }(r) (u^{\prime }(r) )^{k-1}r + (C_{N}^{k}\mu r+C_{N-1}^{k} ) (u^{\prime }(r) )^{k} =\frac{r^{k}p(r)}{(1+\mu r)^{k-1}} f_{1}(u(r))f_{2}(v(r)), \\ C_{N-1}^{l-1}v^{\prime \prime }(r) (v^{\prime }(r) )^{l-1}r + (C_{N}^{l}\nu r+C_{N-1}^{l} ) (v^{\prime }(r) )^{l} =\frac{r^{l}q(r)}{(1+\nu r)^{l-1}} g_{1}(u(r))g_{2}(v(r)), \end{cases}$$

we get

$$\textstyle\begin{cases} u^{\prime }(r)= (\frac{k}{C_{N-1}^{k-1}}\mathrm{e}^{-\psi _{k,\mu }(r)} \int _{0}^{r} \mathrm{e}^{\psi _{k,\mu }(s)} \frac{s^{k-1}p(s)}{(1+\mu s)^{k-1}} f_{1}(u(s))f_{2}(v(s)) \,d s )^{ \frac{1}{k}}, \\ v^{\prime }(r)= (\frac{l}{C_{N-1}^{l-1}}\mathrm{e}^{-\psi _{l,\nu }(r)} \int _{0}^{r} \mathrm{e}^{\psi _{l,\nu }(s)} \frac{s^{l-1}q(s)}{(1+\nu s)^{l-1}} g_{1}(u(s))g_{2}(v(s)) \,d s )^{ \frac{1}{k}}, \end{cases}$$

furthermore we have

$$\textstyle\begin{cases} u(r)=\int _{1}^{r} (\frac{k}{C_{N-1}^{k-1}}\mathrm{e}^{-\psi _{k, \mu }(t)} \int _{0}^{t} \mathrm{e}^{\psi _{k,\mu }(s)} \frac{s^{k-1}p(s)}{(1+\mu s)^{k-1}} f_{1}(u(s))f_{2}(v(s)) \,d s )^{ \frac{1}{k}}\,dt, \\ v(r)=\int _{1}^{r} (\frac{l}{C_{N-1}^{l-1}}\mathrm{e}^{-\psi _{l, \nu }(t)} \int _{0}^{t} \mathrm{e}^{\psi _{l,\nu }(s)} \frac{s^{l-1}q(s)}{(1+\nu s)^{l-1}} g_{1}(u(s))g_{2}(v(s)) \,d s )^{ \frac{1}{k}}\,dt. \end{cases}$$

Define

$$\mathcal{L}(u,v) (r)= \begin{pmatrix} \int _{1}^{r} (\frac{k}{C_{N-1}^{k-1}}\mathrm{e}^{-\psi _{k,\mu }(t)} \int _{0}^{t} \mathrm{e}^{\psi _{k,\mu }(s)} \frac{s^{k-1}p(s)}{(1+\mu s)^{k-1}} f_{1}(u(s))f_{2}(v(s)) \,d s )^{ \frac{1}{k}}\,dt \\ \int _{1}^{r} (\frac{l}{C_{N-1}^{l-1}}\mathrm{e}^{-\psi _{l,\nu }(t)} \int _{0}^{t} \mathrm{e}^{\psi _{l,\nu }(s)} \frac{s^{l-1}q(s)}{(1+\nu s)^{l-1}} g_{1}(u(s))g_{2}(v(s)) \,d s )^{ \frac{1}{k}}\,dt \end{pmatrix}^{T},$$

then we need only to find a fixed point of $$\mathcal{L}$$. Here we use the monotone iterative method to find such a fixed point.

It is easy to show that $$\mathcal{L}$$ is a mapping from $$C^{2}[0,1]\times C^{2}[0,1]$$ to $$C^{2}[0,1]\times C^{2}[0,1]$$, and it is continuous on $$C[0,1]\times C[0,1]$$.

Let $$\{u_{n}\}$$ and $$\{v_{n}\}$$ be the sequence of continuous functions defined by

$$\textstyle\begin{cases} u_{0}(r)=0, \\ v_{0}(r)=0, \\ u_{n}(r)=\int _{1}^{r} (\frac{k}{C_{N-1}^{k-1}}\mathrm{e}^{-\psi _{k, \mu }(t)} \int _{0}^{t} \mathrm{e}^{\psi _{k,\mu }(s)} \frac{s^{k-1}p(s)}{(1+\mu s)^{k-1}} f_{1}(u_{n-1}(s))f_{2}(v_{n-1}(s)) \,d s )^{\frac{1}{k}}\,dt, \\ v_{n}(r)=\int _{1}^{r} (\frac{l}{C_{N-1}^{l-1}}\mathrm{e}^{-\psi _{l, \nu }(t)} \int _{0}^{t} \mathrm{e}^{\psi _{l,\nu }(s)} \frac{s^{l-1}q(s)}{(1+\nu s)^{l-1}} g_{1}(u_{n-1}(s))g_{2}(v_{n-1}(s)) \,d s )^{\frac{1}{k}}\,dt. \end{cases}$$

It is easy to see that $$u_{n}$$ and $$v_{n}$$ are decreasing on $$[0,1]$$ for $$n>1$$ and by induction $$\{u_{n}\}$$ and $$\{v_{n}\}$$ are decreasing as well, i.e., $$u_{n+1}(r)< u_{n}(r)$$ and $$v_{n+1}(r)< v_{n}(r)$$ for $$0\leq r<1$$ and $$n\ge 1$$.

By condition (H1), for each $$0< r<1$$ and $$n>1$$,

\begin{aligned} 0 < &u_{n}^{\prime }(r) \\ =& \biggl(\frac{k}{C_{N-1}^{k-1}}\mathrm{e}^{-\psi _{k,\mu }(r)} \int _{0}^{r} \mathrm{e}^{\psi _{k,\mu }(s)} \frac{s^{k-1}p(s)}{(1+\mu s)^{k-1}} f_{1} \bigl(u_{n-1}(s) \bigr)f_{2} \bigl(v_{n-1}(s) \bigr) \,d s \biggr)^{\frac{1}{k}} \\ \le &C(N,k,p) \bigl(f_{1} \bigl(u_{n}(r) \bigr)f_{2} \bigl(v_{n}(r) \bigr) \bigr)^{ \frac{1}{k}} \\ \le &C(N,k,p) \bigl(f_{1} \bigl(u_{n}(r)+v_{n}(r) \bigr)f_{2} \bigl(u_{n}(r)+v_{n}(r) \bigr) \,d s \bigr)^{\frac{1}{k}}, \end{aligned}

where $$C(N,k,p)$$ is a constant dependent on N, k, and p.

Similarly,

$$0< v_{n}^{\prime }(r)\le C(N,l,q) \bigl(g_{1} \bigl(u_{n}(r)+v_{n}(r) \bigr)g_{2} \bigl(u_{n}(r)+v_{n}(r) \bigr) \bigr)^{\frac{1}{l}}$$

and further

\begin{aligned} \begin{aligned} 0&< \bigl(u_{n}(r)+v_{n}(r) \bigr)^{\prime } \\ &\le C(N,k,l,p,q) \bigl( \bigl(f_{1} \bigl(u_{n}(r)+v_{n}(r) \bigr)f_{2} \bigl(u_{n}(r)+v_{n}(r) \bigr) \bigr)^{\frac{1}{k}} \\ & \quad{} + \bigl(g_{1} \bigl(u_{n}(r)+v_{n}(r) \bigr)g_{2} \bigl(u_{n}(r)+v_{n}(r) \bigr) \bigr)^{\frac{1}{l}} \bigr), \end{aligned} \end{aligned}
(3.1)

i.e.,

\begin{aligned} 0 < &\frac{ (u_{n}(r)+v_{n}(r) )^{\prime }}{ (f_{1}(u_{n}(r)+v_{n}(r))f_{2}(u_{n}(r)+v_{n}(r)) )^{\frac{1}{k}} + (g_{1}(u_{n}(r)+v_{n}(r))g_{2}(u_{n}(r)+v_{n}(r)) )^{\frac{1}{l}}} \\ \le & C(N,k,l,p,q), \end{aligned}

where $$C(N,l,q)$$ and $$C(N,k,l,p,q)$$ are constants dependent on N, l, q and N, k, l, p, q, respectively.

Integrating from 1 to r, we have

$$\int _{0}^{u_{n}(r)+v_{n}(r)} \frac{d\tau }{ (f_{1}(\tau )f_{2}(\tau ) )^{\frac{1}{k}} + (g_{1}(\tau )g_{2}(\tau ) )^{\frac{1}{l}}}\ge -C(N,k,l,p,q).$$
(3.2)

By condition (H2), denote

$$F(w)= \int _{0}^{w} \frac{d\tau }{ (f_{1}(\tau )f_{2}(\tau ) )^{\frac{1}{k}} + (g_{1}(\tau )g_{2}(\tau ) )^{\frac{1}{l}}},$$

then F is continuous and increasing on $$(-\infty ,0]$$, and it has an inverse function $$F^{-1}$$. From (3.2), we have

$$F^{-1} \bigl(-C(N,k,l,p,q) \bigr)\le u_{n}(r)+v_{(}r) \le 0$$

for $$0\le r\le 1$$ and $$n\ge 1$$.

By condition (H1) and (3.1), we have for $$n\ge 1$$

\begin{aligned} 0 < & \bigl(u_{n}(r)+v_{n}(r) \bigr)^{\prime } \\ \le &C(N,k,l,p,q) \Bigl(\max_{F^{-1}(-C(N,k,l,p,q))\le w \le 0} \bigl( \bigl(f_{1}(w)f_{2}(w) \bigr)^{\frac{1}{k}} + \bigl(g_{1}(w)g_{2}(w) \bigr)^{\frac{1}{l}} \bigr) \Bigr) \\ =&C(N,k,l,p,q, f_{1},f_{2},g_{1},g_{2}), \end{aligned}

where $$C(N,k,l,p,q, f_{1},f_{2},g_{1},g_{2})$$ is a constant dependent on N, k, l, p, q, $$f_{1}$$, $$f_{2}$$, $$g_{1}$$, and $$g_{2}$$. So $$\{u_{n}\}$$ and $$\{v_{n}\}$$ are bounded in $$C^{1}[0,1]$$ and by Arzelaâ€“Ascoli theorem $$\{u_{n}\}$$ and $$\{v_{n}\}$$ have convergent subsequences (still denoted by $$\{u_{n}\}$$ and $$\{v_{n}\}$$) in $$C[0,1]$$. Denote

\begin{aligned}& u(r)=\lim_{n\rightarrow +\infty }u_{n}(r), \\& v(r)=\lim_{n\rightarrow +\infty }v_{n}(r). \end{aligned}

By the continuity of $$\mathcal{L}$$ on $$C[0,1]\times C[0,1]$$, from

$$(u_{n},v_{n})=\mathcal{L}(u_{n-1},v_{n-1}),$$

we conclude that $$(u,v)$$ is a fixed point of $$\mathcal{L}$$ after letting $$n\rightarrow +\infty$$.â€ƒâ–¡

### Proof of Theorem 1.2

We prove by contradiction. Suppose that $$(u,v)$$ is a k-convex radial solution to problem (1.1). Then u and v are decreasing on $$[0,1]$$. For $$0< r< 1$$, by Lemma 2.2 we can get

\begin{aligned}& 0< u^{\prime }(r)\le C(N,k,p) \bigl(f_{1} \bigl(u(r)+v(r) \bigr)g_{2} \bigl(u(r)+v(r) \bigr) \bigr)^{\frac{1}{k}}, \\& 0< v^{\prime }(r)\le C(N,l,q) \bigl(g_{1} \bigl(u(r)+v(r) \bigr)g_{2} \bigl(u(r)+v(r) \bigr) \bigr)^{\frac{1}{l}}. \end{aligned}

So

\begin{aligned} 0 < &\frac{ (u(r)+v(r) )^{\prime }}{ (f_{1}(u(r)+v(r))f_{2}(u(r)+v(r)) )^{\frac{1}{k}} + (g_{1}(u(r)+v(r))g_{2}(u(r)+v(r)) )^{\frac{1}{l}}} \\ \le & C(N,k,l,p,q). \end{aligned}

Integrating for 0 to 1, we have

$$0< \int ^{0}_{u(0)+v(0)} \frac{d\tau }{ (f_{1}(\tau )f_{2}(\tau ) )^{\frac{1}{k}} + (g_{1}(\tau )g_{2}(\tau ) )^{\frac{1}{l}}}\le C(N,k,l,p,q),$$

which contradicts condition (H3). Now we finish the proof.â€ƒâ–¡

At the end of this section, we give some examples for the sake of clearly understanding the results in this paper.

Assume that Î±, Î², $$\alpha _{1}$$, $$\beta _{1}$$, $$\alpha _{2}$$, and $$\beta _{2}$$ are positive.

### Example 3.1

If $$\alpha _{1} +\beta _{1}\le k$$ and $$\alpha _{2} +\beta _{2}\le l$$, then the following problem admits an entire k-convex radial solution $$(u,v) \in C^{2} (\overline{B_{1}(0)} )\times C^{2} ( \overline{B_{1}(0)} )$$:

$$\textstyle\begin{cases} \sigma _{k} (\lambda (D^{2} u+\mu \vert \nabla u \vert I ) )=(1+ \vert x \vert )^{\alpha } (1+ \vert u \vert )^{\alpha _{1}} \vert v \vert ^{\beta _{1}}, & x \in B_{1}(0), \\ \sigma _{l} (\lambda (D^{2} v+\nu \vert \nabla v \vert I ) )=(1+ \vert x \vert )^{\beta } (1+ \vert u \vert )^{\alpha _{2}}(1+ \vert v \vert )^{\beta _{2}}, & x \in B_{1}(0), \\ u=v=0,& x\in \partial B_{1}(0). \end{cases}$$

### Example 3.2

If $$\alpha _{1} +\beta _{1}\ge k$$ and $$\alpha _{2} +\beta _{2}\ge l$$, then the following problem admits no entire k-convex radial solution $$(u,v) \in C^{2} (\overline{B_{1}(0)} )\times C^{2} ( \overline{B_{1}(0)} )$$:

$$\textstyle\begin{cases} \sigma _{k} (\lambda (D^{2} u+\mu \vert \nabla u \vert I ) )=(1+ \vert x \vert )^{\alpha } \vert u \vert ^{\alpha _{1}} \vert v \vert ^{\beta _{1}}, & x \in B_{1}(0), \\ \sigma _{l} (\lambda (D^{2} v+\nu \vert \nabla v \vert I ) )=(1+ \vert x \vert )^{\beta } \vert u \vert ^{\alpha _{2}} \vert v \vert ^{\beta _{2}}, & x \in B_{1}(0), \\ u=v=0,& x\in \partial B_{1}(0). \end{cases}$$

## 4 Conclusion

In this paper, by converting the existence of an entire solution to the existence of a fixed point of a continuous mapping, we establish the existence of entire k-convex radial solutions for a Hessian type system. Moreover the nonexistence of entire k-convex radial solutions is also obtained. In the process of obtaining the existence of entire k-convex radial solutions, we utilize the monotone iterative method. By different fixed point theorems (such as the ones in [14] and [25]) or different methods (such as degree theory in [16] and the regularization method in [23]), we may get different results on Hessian type systems. In our opinion, it is interesting to fulfil this kind of works in the future.

## Availability of data and materials

No data were used to support this study. It is not applicable for my paper.

## References

1. Cavalheiro, A.C.: Existence results for Navier problems with degenerated $$(p,q)$$-Laplacian and $$(p,q)$$-biharmonic operators. Results Nonlinear Anal. 1(2), 74â€“87 (2018)

2. Cheng, S.Y., Yau, S.T.: On the existence of a complete KÃ¤hler metric on noncompact complex manifolds and the regularity of Feffermanâ€™s equation. Commun. Pure Appl. Math. 33, 507â€“544 (1980)

3. CÃ®rstea, F.C., RÄƒdulescu, V.: Entire solutions blowing up at infinity for semilinear elliptic systems. J.Â Math. Pures Appl. 81, 827â€“846 (2002)

4. Dai, L.M.: Existence and nonexistence of subsolutions for augmented Hessian equations. Discrete Contin. Dyn. Syst. 40(1), 579â€“596 (2020)

5. de Oliveira, J.F., doÂ Ã“, J.M., Ubilla, P.: Existence for a k-Hessian equation involving supercritical growth. J.Â Differ. Equ. 267, 1001â€“1024 (2019)

6. Enache, C., Porru, G.: AÂ note on Mongeâ€“Ampere equation in $$\mathbb{R}^{2}$$. Results Math. 76(1), Article ID 29 (2021)

7. Feng, M.Q.: Convex solutions of Mongeâ€“AmpÃ¨re equations and systems: existence, uniqueness and asymptotic behavior. Adv. Nonlinear Anal. 10(1), 371â€“399 (2021)

8. Feng, M.Q., Zhang, X.M.: AÂ coupled system of k-Hessian equations. Math. Methods Appl. Sci. 44(9), 7377â€“7394 (2021)

9. Gao, C.H., He, X.Y., Ran, M.J.: On a power-type coupled system of k-Hessian equations. Quaest. Math. https://doi.org/10.2989/16073606.2020.1816586

10. Guan, B., Jiao, H.: Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds. Calc. Var. Partial Differ. Equ. 54, 2693â€“2712 (2015)

11. Ji, X., Bao, J.: Necessary and sufficient conditions on solvability for Hessian inequalities. Proc. Am. Math. Soc. 138, 175â€“188 (2010)

12. Jiang, F., Trudinger, N.S.: On the Dirichlet problem for general augmented Hessian equations. J.Â Differ. Equ. 269, 5204â€“5227 (2020)

13. Jin, Q., Li, Y., Xu, H.: Nonexistence of positive solutions for some fully nonlinear elliptic equations. Methods Appl. Anal. 12, 441â€“450 (2005)

14. KarapÄ±nar, E.: AÂ fixed point theorem without a Picard operator. Results Nonlinear Anal. 4(3), 127â€“129 (2021)

15. Keller, J.B.: On solutions of $$\triangle u=f(u)$$. Commun. Pure Appl. Math. 10, 503â€“510 (1957)

16. Kim, I.S.: Semilinear problems involving nonlinear operators of monotone type. Results Nonlinear Anal. 2(1), 25â€“35 (2019)

17. Lair, A.V., Wood, A.W.: Large solutions of semilinear elliptic problems. Nonlinear Anal. 37, 805â€“812 (1999)

18. Lair, A.V., Wood, A.W.: Existence of entire large positive solutions of semilinear elliptic systems. J.Â Differ. Equ. 164, 380â€“394 (2000)

19. Lazer, A.C., McKenna, P.J.: On singular boundary value problems for the Mongeâ€“AmpÃ¨re operator. J.Â Math. Anal. Appl. 197, 341â€“362 (1996)

20. Lieberman, G.: Second Order Parabolic Differential Equations. World Scientific, New Jersey (1996)

21. Osserman, R.: On the inequality $$\triangle u \geq f(u)$$. Pac. J. Math. 7, 1641â€“1647 (1957)

22. Ourraou, A.: Existence and uniqueness of solutions for Steklov problem with variable exponent. Adv. Theory Nonlinear Anal. Appl. 5(1), 158â€“166 (2021)

23. Phuong, N.D., Luc, N.H., Long, L.D.: Modified quasi boundary value method for inverse source problem of the bi-parabolic equation. Adv. Theory Nonlinear Anal. Appl. 4(3), 132â€“142 (2020)

24. Wang, F., An, Y.: Triple nontrivial radial convex solutions of systems of Mongeâ€“AmpÃ¨re equations. Appl. Math. Lett. 25, 88â€“92 (2012)

25. Zhang, C., Chen, J.: Convergence analysis of variational inequality and fixed point problems for pseudo-contractive mapping with Lipschitz assumption. Results Nonlinear Anal. 2(3), 102â€“112 (2019)

26. Zhang, Z., Qi, Z.: On a power-type coupled system of Mongeâ€“AmpÃ¨re equations. Topol. Methods Nonlinear Anal. 46, 717â€“729 (2015)

27. Zhang, Z., Zhou, S.: Existence of entire positive k-convex radial solutions to Hessian equations and systems with weights. Appl. Math. Lett. 50, 48â€“55 (2015)

## Acknowledgements

The author is grateful to the referees for their helpful remarks and suggestions.

## Funding

IÂ am supported by the National Natural Science Foundation of China (Grant No. U2031142).

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The author contributed independently to the manuscript and read and approved the final manuscript.

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Correspondence to Jixian Cui.

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Cui, J. Existence and nonexistence of entire k-convex radial solutions to Hessian type system. Adv Differ Equ 2021, 462 (2021). https://doi.org/10.1186/s13662-021-03601-8