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.

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 $u \in C^{2} (\overline{B_{1}(0)} )$, ∇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:

• (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.

• (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.

• (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 Monge–Ampère equations. Concerning Laplacian equations and Monge–Ampère 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 Monge–Ampère 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 Cîrstea and Rădulescu [3] for coupling Laplacian systems and Wang and An [24] and Zhang and Qi [26] for coupling Monge–Ampère 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 Ó, 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).

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 $\sigma _{k} (\lambda (D^{2} u+\eta \vert \nabla u \vert I ) )$. □

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 φ 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 φ is nontrivial, it is easy to see that φ 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. □

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 Arzela–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 $n\rightarrow +\infty$. □

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. □

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

Assume that α, β, $\alpha _{1}$, $\beta _{1}$, $\alpha _{2}$, 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 $(u,v) \in C^{2} (\overline{B_{1}(0)} )\times C^{2} ( \overline{B_{1}(0)} )$:

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

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.