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Existence of entire radial solutions to Hessian type system


In this paper, the Dirichlet problem of 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 radial solutions is established by Schaefer fixed point theorem.


In this paper, we study the existence of entire radial solutions to the following Dirichlet problem of a Hessian type system:

$$ \textstyle\begin{cases} \sigma _{k_{j}} (\lambda (D^{2} u_{j}+\mu _{j} \vert \nabla u_{j} \vert I ) )=f_{j}( \vert x \vert ,u), &1\le j\le m, x \in B_{1}(0), \\ u=(0,0,\ldots ,0),& x\in \partial B_{1}(0), \end{cases} $$

where \(u=(u_{1},u_{2},\ldots ,u_{m})\). For each integer j satisfying \(1\le j\le m\le N\), where m is a fixed integer, the integer \(k_{j}\) satisfies \(1\le k_{j}\le N\), \(\mu _{j}\) is a nonnegative constant. \(B_{1}(0)\) is the unit ball in \(\mathbb{R}^{N}\). For any \(N \times N\) real symmetric matrix A, \(\lambda (A)\) denotes the eigenvalues of A. \(D^{2} w(x)= ( \frac{\partial ^{2} u(x)}{\partial x_{i} \partial x_{j}} )_{1 \le i,j\le N}\) denotes the Hessian matrix of the function \(w \in C^{2} (\overline{B_{1}(0)} )\), w denotes the gradient of w, and \(\sigma _{k_{j}}(\lambda )=\sum_{1 \leq i_{1}<\cdots <i_{k_{j}} \leq N} \lambda _{i_{1}} \cdots \lambda _{i_{k_{j}}}\) denotes the \(k_{j}\)th elementary symmetric function of \(\lambda = (\lambda _{1}, \lambda _{2},\ldots , \lambda _{N} ) \in \mathbb{R}^{N}\).

For \(f_{j}\), \(1\le j\le m\), we introduce the following conditions:

  1. (H1)

    \(f_{j}\in C([0,1]\times (-\infty ,0]^{m},[0,+\infty ))\) and \(f(r,w)=0\) if and only if \(w=(0,0,\ldots ,0)\).

  2. (H2)

    \(\vert f_{j}(r,w) \vert \le L(r) \vert w \vert ^{\alpha}\), where \(r\in [0,1]\), \(w\in (-\infty ,0]^{m}\), \(L\in C[0,1]\) and the constant α satisfies

    $$ 0\le \alpha < \min_{1\le j\le m} \{k_{j} \}.$$


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

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 problem (1.1), if \(m=1\), the equation becomes the following k-Hessian type equation:

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

which is a generalization of the k-Hessian equation

$$ \sigma _{k} \bigl(\lambda \bigl(D^{2} u \bigr) \bigr)=f \bigl( \vert x \vert ,u\bigr), $$

but a special case of the following fully nonlinear Hessian equation:

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

We refer to Guan and Jiao [8] and Jiang and Trudinger [10] and the references therein for research on fully nonlinear Hessian equation (1.3). See also Dai [4] for a similar study. For k-Hessian equation (1.2), it is well known that if \(k=1\), it becomes the famous Laplacian equation. Laplacian equations attract a great deal of attention, and until now there have been a vast number of research papers on this kind of equations. Here we specially mention Keller [12] and Osserman [16] for Laplacian equations. If \(k=N\), k-Hessian equation (1.2) becomes the well-known Monge–Ampère equation. Nowadays, the research on Monge–Ampère equations is still a hot topic. Here we want to mention Cheng and Yau [1] and Laser and McKenna [14] for Monge–Ampère equations. For general k-Hessian equation (1.2), when \(f(|x|,u)=f(u)\), Ji and Bao [9] gave necessary and sufficient conditions on the existence of entire positive k-convex radial solutions; when \(f(|x|,u)=u^{\gamma k}\), \(\gamma >1\), Jin, Li, and Xu [11] showed the nonexistence of entire k-convex positive solutions. Because of the work of Jin, Li, and Xu, to establish the existence of solutions for problem (1.1) we study under condition (H2), which can be treated as a sublinear condition. By a similar method, we can also treat the linear case but under a bit more complex condition than condition (H2). If we generalize \(f(|x|,u)\) 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).

In problem (1.1), if \(m>1\), the system becomes a coupling k-Hessian type system. There are not so many research papers on coupling k-Hessian type systems. Here we only mention Lair and Wood [13] and Cîrstea and Rădulescu [2] for coupling Laplacian systems (i.e., in the case \(k=1\), \(m=2\), and \(\mu _{1}=\mu _{2}=0\)) and Wang and An [18] and Zhang and Qi [19] for coupling Monge–Ampère systems (i.e., in the case \(k=N\), \(m=2\), and \(\mu _{1}=\mu _{2}=0\)). For a general coupling k-Hessian system, if \(m=2\) and \(f_{1}(|x|,u,v)=p(|x|)f(v)\), \(f_{2}(|x|,u,v)=q(|x|)g(u)\), the system becomes

$$ \textstyle\begin{cases} \sigma _{k} (\lambda (D^{2} u+\mu \vert \nabla u \vert I ) )=p( \vert x \vert ) f(v), \\ \sigma _{l} (\lambda (D^{2} v+\nu \vert \nabla u \vert I ) )=q( \vert x \vert ) g(u). \end{cases} $$

We refer to the papers of Feng and Zhang [6] and Gao, He, and Ran [7] and the references therein for research on coupling k-Hessian system (1.4) when \(\mu =\nu =0\). By the way, on general k-Hessian equation (1.2) and general coupling k-Hessian system (1.4) when \(\mu =\nu =0\), Zhang and Zhou [20] obtained several results on the existence of entire positive k-convex radial solutions. In [3], we studied the coupling k-Hessian system (1.4) and obtained the existence and nonexistence of entire k-convex radial solutions. In the process of obtaining the existence of entire k-convex radial solutions, we utilized the monotone iterative method, and so we require the monotonicity of f and g. In the present paper, we want to remove the requirement for monotonicity and utilize a method different from the monotone iterative method. As a cost, we only treat the so called sublinear case. We study the Dirichlet problem of k-Hessian type system (1.1) and obtain the following result in this paper.

Theorem 1.1

Under conditions (H1) and (H2), problem (1.1) admits an entire radial solution \(u \in (C^{2} (\overline{B_{1}(0)} ) )^{m}\) and the jth component of u is \(k_{j}\)-convex.

The paper is organized as follows. In Sect. 2, we introduce basic concepts and results. In Sect. 3, we prove the existence of entire radial solutions of problem (1.1), which is the main part of this paper.


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

Lemma 2.1

Let the integer j be fixed and satisfy \(1\le j\le m\). Assume \(\varphi _{j}(r) \in C^{2}[0, 1]\) with \(\varphi _{j}^{\prime}(0)=0\). Then, for \(u_{j}(x)=\varphi _{j}(r)\), there holds that \(u_{j} \in C^{2} (\overline{B_{1}(0)} )\) and

$$ \lambda \bigl(D^{2} u_{j}+\mu _{j} \vert \nabla u_{j} \vert I \bigr)=\textstyle\begin{cases} (\varphi _{j}^{\prime \prime}(r)+\mu _{j}\varphi _{j}^{\prime}(r), (\frac{1}{r}+\mu _{j} )\varphi _{j}^{\prime}(r), \ldots , (\frac{1}{r}+\mu _{j} )\varphi _{j}^{\prime}(r) ), \\ \quad r\in (0, 1], \\ (\varphi _{j}^{\prime \prime}(0), \varphi _{j}^{\prime \prime}(0), \ldots , \varphi _{j}^{\prime \prime}(0) ), \quad r=0, \end{cases} $$

and further

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

where \(\mu _{j} \ge 0\) is a constant and \(C_{N}^{k_{j}}=\frac{N!}{(k_{j})!(N-k_{j})!}\).

See [3] for the proof.

Lemma 2.2

Denote \(\varphi =(\varphi _{1},\varphi _{2},\ldots ,\varphi _{m})\). Assume that \(\varphi \in (C[0, 1])^{m} \cap (C^{1}(0, 1])^{m}\) is a solution of the Cauchy problem

$$ \textstyle\begin{cases} \varphi _{j}^{\prime}(r)= (\frac{k_{j}}{C_{N-1}^{k_{j}-1}} \textit{e}^{-\psi _{k_{j},\mu _{j}}(r)} \int _{0}^{r} \textit{e}^{\psi _{k_{j}, \mu _{j}}(s)} \frac{s^{k_{j}-1}f_{j}(s,\varphi (s))}{(1+\mu _{j} s)^{k-1}}\,ds )^{\frac{1}{k_{j}}},\quad 0< r< 1, \\ \varphi (1)=(0,0,\ldots ,0), \end{cases} $$


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

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

$$ \textstyle\begin{cases} C_{N-1}^{k_{j}-1}\varphi _{j}^{\prime \prime}(r) (\varphi _{j}^{ \prime}(r) )^{k_{j}-1}r + (C_{N}^{k_{j}}\mu _{j} r+C_{N-1}^{k_{j}} ) (\varphi _{j}^{\prime}(r) )^{k_{j}} = \frac{r^{k_{j}}f_{j}(r,\varphi (r))}{(1+\mu _{j} r)^{k_{j}-1}}, \\ \quad 1\le j\le m, 0< r< 1, \\ \varphi ^{\prime}(0)=(0,0,\ldots ,0). \end{cases} $$

Furthermore, if \(\varphi _{j}\) is nontrivial, i.e., \(\varphi _{j}(r)<0\) for \(0\le r<1\), then

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

for \(0 \leq r<1\).


It is immediate that \(\varphi (r) \in (C^{2}[0, 1])^{m}\).


$$ \varphi _{j}^{\prime}(r)= \biggl(\frac{k_{j}}{C_{N-1}^{k_{j}-1}} \text{e}^{-\psi _{k_{j},\mu _{j}}(r)} \int _{0}^{r} \text{e}^{\psi _{k_{j}, \mu _{j}}(s)} \frac{s^{k_{j}-1}f_{j}(s,\varphi (s))}{(1+\mu _{j} s)^{k_{j}-1}}\,ds \biggr)^{\frac{1}{k_{j}}},$$

we can get

$$ \bigl(\varphi _{j}^{\prime}(r) \bigr)^{k_{j}}= \frac{k_{j}}{C_{N-1}^{k_{j}-1}}\text{e}^{-\psi _{k_{j},\mu _{j}}(r)} \int _{0}^{r} \text{e}^{\psi _{k_{j},\mu _{j}}(s)} \frac{s^{k_{j}-1}f_{j}(s,\varphi (s))}{(1+\mu _{j} s)^{k_{j}-1}}\,ds,$$

and further, by differentiating with respect to r, we have

$$ C_{N-1}^{k_{j}-1}\varphi _{j}^{\prime \prime}(r) \bigl( \varphi _{j}^{ \prime}(r) \bigr)^{k_{j}-1}r + \bigl(C_{N}^{k_{j}}\mu _{j} r+C_{N-1}^{k_{j}} \bigr) \bigl(\varphi _{j}^{\prime}(r) \bigr)^{k_{j}} = \frac{r^{k_{j}}f_{j}(r,\varphi (r))}{(1+\mu _{j} r)^{k_{j}-1}}.$$

If \(\varphi _{j}\) is nontrivial, for \(0 \leq r<1\), we conclude that \(\varphi _{j}(r)<\varphi _{j}(1)=0\), \(f_{j}(r,\varphi (r))>0\) and further

$$ \sigma _{k_{j}} (\lambda _{r} )=f_{j}\bigl(r, \varphi (r)\bigr)>0 \quad \text{for }0 \leq r< 1.$$

By the properties of \(k_{j}\)th elementary symmetric functions (see for example [15]), we know \(\sigma _{i} (\lambda _{r} )>0\) for \(1\le i< k_{j}\) and \(0 \leq r<1\), from which we conclude the result. □

Proof of the main result

In this section, we show the existence of entire k-convex radial solutions for problem (1.1) by means of Schaefer fixed point theorem (see [17]).

Theorem 3.1

(Schaefer fixed point theorem)

Let X be a Banach space and \(T: X\rightarrow X\) be a compact operator. If the set \(E=\{u \textit{ in }X: u=\lambda Tu\textit{ for some } 0\le \lambda \le 1 \}\) is bounded, then T has at least a fixed point in X.

For each j satisfying \(1\le j\le m\), from

$$ C_{N-1}^{k_{j}-1}u_{j}^{\prime \prime}(r) \bigl(u_{j}^{\prime}(r) \bigr)^{k_{j}-1}r + \bigl(C_{N}^{k_{j}}\mu _{j} r+C_{N-1}^{k_{j}} \bigr) \bigl(u_{j}^{\prime}(r) \bigr)^{k_{j}} = \frac{r^{k_{j}}f_{j}(r,u(r))}{(1+\mu _{j} r)^{k_{j}-1}}, $$

we get

$$ u_{j}^{\prime}(r)= \biggl(\frac{k_{j}}{C_{N-1}^{k_{j}-1}} \text{e}^{- \psi _{k_{j},\mu _{j}}(r)} \int _{0}^{r} \text{e}^{\psi _{k_{j},\mu _{j}}(s)} \frac{s^{k_{j}-1}f_{j}(s,u(s))}{(1+\mu _{j} s)^{k_{j}-1}}\,ds \biggr)^{ \frac{1}{k_{j}}}, $$

and further

$$ u_{j}(r)= \int _{1}^{r} \biggl(\frac{k_{j}}{C_{N-1}^{k_{j}-1}} \text{e}^{- \psi _{k_{j},\mu _{j}}(t)} \int _{0}^{t} \text{e}^{\psi _{k_{j},\mu _{j}}(s)} \frac{s^{k_{j}-1}f_{j}(s,u(s))}{(1+\mu _{j} s)^{k_{j}-1}}\,ds \biggr)^{ \frac{1}{k_{j}}}\,dt. $$

Define an operator \({\mathcal{L}}= (({\mathcal{L}})_{1},({\mathcal{L}})_{2},\ldots ,({\mathcal{L}})_{m} )\) by

$$ \bigl({\mathcal{L}}(u) \bigr)_{j}(r)= \int _{1}^{r} \biggl( \frac{k_{j}}{C_{N-1}^{k_{j}-1}} \text{e}^{-\psi _{k_{j},\mu _{j}}(t)} \int _{0}^{t} \text{e}^{\psi _{k_{j},\mu _{j}}(s)} \frac{s^{k_{j}-1}f_{j}(s,u(s))}{(1+\mu _{j} s)^{k_{j}-1}}\,ds \biggr)^{ \frac{1}{k_{j}}}\,dt, $$

then we need only to find a fixed point of \(\mathcal{L}\). As we want to use Schaefer fixed point theorem to find such a fixed point, we should first check that the conditions of Schaefer fixed point theorem are satisfied.

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

Proposition 3.1

Under conditions (H1) and (H2), \({\mathcal{L}}\) is compact on \((C[0,1])^{m}\).


Let A be a bounded subset of \((C[0,1])^{m}\), i.e.,

$$ \max_{1\le j\le m}\max_{0\le r\le 1} \bigl\vert u_{j}(r) \bigr\vert \le M$$

for some constant \(M>0\). As \(f_{j}\) is continuous on \([0,1]\times [-M,0]^{m}\), there exists a constant \(\bar{M}>0\) such that

$$ \bigl\vert f_{j}(r,w) \bigr\vert \le \bar{M}, \quad \forall (r, w)\in [0,1]\times [-M,0]^{m}, 1\le j\le m.$$

Then, for any \(u=(u_{1},u_{2},\ldots ,u_{m})\in A\), we have

$$\begin{aligned} 0 < & \bigl({\mathcal{L}}(u) \bigr)_{j}^{\prime}(r) \\ =& \biggl(\frac{k_{j}}{C_{N-1}^{k_{j}-1}}\text{e}^{-\psi _{k_{j},\mu _{j}}(r)} \int _{0}^{r} \text{e}^{\psi _{k_{j},\mu _{j}}(s)} \frac{s^{k_{j}-1}f_{j}(s,u(s))}{(1+\mu _{j} s)^{k_{j}-1}}\,ds \biggr)^{ \frac{1}{k_{j}}} \\ \le & \biggl(\frac{\bar{M} k_{j}}{C_{N-1}^{k_{j}-1}} \biggr)^{ \frac{1}{k_{j}}} \\ =&C(m,M,N,k_{j},p_{j},f_{j}), \end{aligned}$$

where \(C(m,M,N,k_{j},p_{j},f_{j})\) is a constant dependent on m, MN, \(k_{j}\), \(p_{j}\), \(f_{j}\). Therefore \({\mathcal{L}}(A)\) is equicontinuous on \([0,1]^{m}\).

On the other hand, we have

$$\begin{aligned} \bigl\vert \bigl({\mathcal{L}}(u) \bigr)_{j}(r) \bigr\vert =& \biggl\vert \int _{1}^{r} \bigl({\mathcal{L}}(u) \bigr)_{j}^{\prime}(s)\,ds \biggr\vert \\ \le & \int _{r}^{1} \bigl\vert \bigl({\mathcal{L}}(u) \bigr)_{j}^{\prime}(s) \bigr\vert \,ds \\ \le &C(m,M,N,k_{j},p_{j},f_{j}), \end{aligned}$$

from which we conclude that \({\mathcal{L}}(A)(r)\) is uniformly bounded on \([0,1]^{m}\), i.e., \({\mathcal{L}}(A)\) is bounded in \((C[0,1])^{m}\).

By Arzela–Ascoli theorem, \({\mathcal{L}}(A)\) is a sequentially compact subset of \((C[0,1])^{m}\) and further \({\mathcal{L}}\) is compact on \((C[0,1])^{m}\). □


$$ E=\bigl\{ u\in \bigl(C[0,1]\bigr)^{m}: u=\lambda {\mathcal{L}}(u)\text{ for some } 0\le \lambda \le 1\bigr\} .$$

Proposition 3.2

Under conditions (H1) and (H2), the set E is bounded in \((C[0,1])^{m}\).


Assume that \(u=\lambda {\mathcal{L}}(u)\) for some \(0\le \lambda \le 1\). By condition (H2), for \((r,w)\in [0,1]\times (-\infty ,0]^{m}\), we get

$$ \bigl\vert f_{j}(r,w) \bigr\vert \le L(r) \vert w \vert ^{\alpha},$$


$$\begin{aligned} \bigl\vert u_{j}(r) \bigr\vert =& \bigl\vert \lambda \bigl({\mathcal{L}}(u) \bigr)_{j}(r) \bigr\vert \\ =& \biggl\vert \lambda \int _{1}^{r} \biggl(\frac{k_{j}}{C_{N-1}^{k_{j}-1}} \text{e}^{-\psi _{k_{j},\mu _{j}}(t)} \int _{0}^{t} \text{e}^{\psi _{k_{j}, \mu _{j}}(s)} \frac{s^{k_{j}-1}f_{j}(s,u(s))}{(1+\mu _{j} s)^{k_{j}-1}}\,ds \biggr)^{ \frac{1}{k_{j}}}\,dt \biggr\vert \\ \le &C(m,L,N,k_{j},p_{j})\max_{0\le r\le 1} \bigl\vert u(r) \bigr\vert ^{ \frac{\alpha}{k_{j}}}. \end{aligned}$$

Suppose that \(k_{j_{0}}=\min_{1\le j\le m} \{k_{j} \}\). Taking the maximum on \([0,1]\), we have

$$ \max_{0\le r\le 1} \bigl\vert u_{j}(r) \bigr\vert ^{k_{j}}\le C(m,L,N,k_{j},p_{j}) \max _{0\le r\le 1} \bigl\vert u(r) \bigr\vert ^{\alpha},$$

and further

$$\begin{aligned} \max_{0\le r\le 1} \bigl\vert u(r) \bigr\vert ^{k_{j_{0}}} =&\max_{0\le r \le 1} \Biggl(\sum _{j=1}^{m} \bigl\vert u_{j}(r) \bigr\vert ^{2} \Biggr)^{ \frac{k_{j_{0}}}{2}} \\ \le & \Biggl(\sum_{j=1}^{m} \max _{0\le r\le 1} \bigl\vert u_{j}(r) \bigr\vert \Biggr)^{k_{j_{0}}} \\ \le &C(m,k_{j}) \Biggl(\sum_{j=1}^{m} \max_{0\le r \le 1} \bigl\vert u_{j}(r) \bigr\vert ^{k_{j_{0}}} \Biggr) \\ \le &C(m,k_{j}) \Biggl(1+ \Biggl(\sum_{j=1}^{m} \max_{0 \le r\le 1} \bigl\vert u_{j}(r) \bigr\vert ^{k_{j}} \Biggr) \Biggr) \\ \le &C(m,L,N,k_{j},p_{j}) \Bigl(1+ \Bigl(\max _{0\le r\le 1} \bigl\vert u(r) \bigr\vert ^{ \alpha} \Bigr) \Bigr) \\ \le &C(\alpha ,\varepsilon ,m,L,N,k_{j},p_{j})+ \varepsilon \max_{0\le r\le 1} \bigl\vert u(r) \bigr\vert ^{k_{j_{0}}} \end{aligned}$$

in view of Young’s inequality. Taking \(\varepsilon =\frac{1}{2}\), we can get

$$ \max_{0\le r\le 1} \bigl\vert u(r) \bigr\vert \le C(\alpha ,m,L,N,k_{j},p_{j}),$$

from which we conclude that E is bounded in \((C[0,1])^{m}\). □

At last we prove the main result of this paper.

Proof of Theorem 1.1

First, in view of Propositions 3.1 and 3.2, we conclude that \(\mathcal{L}\) admits a fixed point in \((C[0,1])^{m}\) by Schaefer fixed point theorem. Second, by Lemmas 2.1 and 2.2, the fixed point u of \(\mathcal{L}\) in \((C[0,1])^{m}\) is in fact an entire radial solution in \((C^{2} (\overline{B_{1}(0)} ) )^{m}\) of problem (1.1) and the jth component of u is \(k_{j}\)-convex. □

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

Example 3.1

If \(0\le \alpha < N\) in condition (H2), then the following problem admits an entire convex radial solution \((u,v) \in C^{2} (\overline{B_{1}(0)} )\times C^{2} ( \overline{B_{1}(0)} )\):

$$ \textstyle\begin{cases} \det (\lambda (D^{2} u+\mu \vert \nabla u \vert I ) )=f( \vert x \vert ,u,v), & x \in B_{1}(0), \\ \det (\lambda (D^{2} v+\nu \vert \nabla v \vert I ) )=g( \vert x \vert ,u,v), & x \in B_{1}(0), \\ u=v=0,& x\in \partial B_{1}(0). \end{cases} $$

Example 3.2

If \(0\le \alpha <1\) in condition (H2), then the following problem admits an entire k-convex radial solution \(u=(u_{1},u_{2},\ldots ,u_{N})\in (C^{2} ( \overline{B_{1}(0)} ) )^{N}\):

$$ \textstyle\begin{cases} \operatorname{trace} (\lambda (D^{2} u_{1}+\mu _{1} \vert \nabla u_{1} \vert I ) )=f_{1}( \vert x \vert ,u), & x \in B_{1}(0), \\ \sigma _{2} (\lambda (D^{2} u_{2}+\mu _{2} \vert \nabla u_{2} \vert I ) )=f_{2}( \vert x \vert ,u), & x \in B_{1}(0), \\ \cdots &\cdots \\ \sigma _{N-1} (\lambda (D^{2} u_{N-1}+\mu _{N-1} \vert \nabla u_{N-1} \vert I ) )=f_{N-1}( \vert x \vert ,u), & x \in B_{1}(0), \\ \det (\lambda (D^{2} u_{N}+\mu _{N} \vert \nabla u_{N} \vert I ) )=f_{N}( \vert x \vert ,u), & x \in B_{1}(0), \\ u=(0,0,\ldots ,0),& x\in \partial B_{1}(0). \end{cases} $$

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  1. Cheng, S.Y., Yau, S.T.: On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation. Commun. Pure Appl. Math. 33, 507–544 (1980)

    Article  Google Scholar 

  2. Cîrstea, F.C., Rădulescu, V.: Entire solutions blowing up at infinity for semilinear elliptic systems. J. Math. Pures Appl. 81, 827–846 (2002)

    MathSciNet  Article  Google Scholar 

  3. Cui, J.X.: Existence and nonexistence of entire k-convex radial solutions to Hessian type system. Adv. Differ. Equ. 2021, 462 (2021)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  5. de Oliveira, J.F., do Ó, J.M., Ubilla, P.: Existence for a k-Hessian equation involving supercritical growth. J. Differ. Equ. 267, 1001–1024 (2019)

    MathSciNet  Article  Google Scholar 

  6. Feng, M.Q., Zhang, X.M.: A coupled system of k-Hessian equations. Math. Methods Appl. Sci. 44(9), 7377–7394 (2021)

    MathSciNet  Article  Google Scholar 

  7. Gao, C.H., He, X.Y., Ran, M.J.: On a power-type coupled system of k-Hessian equations. Quaest. Math.

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  10. Jiang, F., Trudinger, N.S.: On the Dirichlet problem for general augmented Hessian equations. J. Differ. Equ. 269, 5204–5227 (2020)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  12. Keller, J.B.: On solutions of \(\triangle u=f(u)\). Commun. Pure Appl. Math. 10, 503–510 (1957)

    Article  Google Scholar 

  13. Lair, A.V., Wood, A.W.: Existence of entire large positive solutions of semilinear elliptic systems. J. Differ. Equ. 164, 380–394 (2000)

    MathSciNet  Article  Google Scholar 

  14. Lazer, A.C., McKenna, P.J.: On singular boundary value problems for the Monge–Ampère operator. J. Math. Anal. Appl. 197, 341–362 (1996)

    MathSciNet  Article  Google Scholar 

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

    Book  Google Scholar 

  16. Osserman, R.: On the inequality \(\triangle u \geq f(u)\). Pac. J. Math. 7, 1641–1647 (1957)

    Article  Google Scholar 

  17. Schaefer, H.: Über die method der a priori-Schranken. Math. Ann. 126, 415–416 (1955)

    Article  Google Scholar 

  18. Wang, F., An, Y.: Triple nontrivial radial convex solutions of systems of Monge–Ampère equations. Appl. Math. Lett. 25, 88–92 (2012)

    MathSciNet  Article  Google Scholar 

  19. Zhang, Z., Qi, Z.: On a power-type coupled system of Monge–Ampère equations. Topol. Methods Nonlinear Anal. 46, 717–729 (2015)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

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College of Science, Qiqihar University, Qiqihar 161006, P.R. China.


The author is supported by the National Natural Science Foundation of China (Grant No. U2031142).

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Cui, J. Existence of entire radial solutions to Hessian type system. Bound Value Probl 2022, 31 (2022).

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  • k-convex radial solution
  • Existence
  • Hessian type system