Entire positive $k$-convex solutions to $k$-Hessian type equations and systems

1.   Introduction

Consider the existence of entire positive $k$-convex solutions to the following $k$-Hessian type equation

and system

where $k\in\{1, 2, \dots, n\}, \ \alpha\geq 0$ is a constant, $I$ is the identity function and $p, q$ are continuous functions on $[0, +\infty)$. Letting $D^2u = (\frac{\partial^2 u}{\partial x_i \partial x_j})$ denote the Hessian of $u \in C^2(\mathbb{R}^n)$ and $\lambda_{i}\ (i\in\{1, 2, \dots, n\})$ denote the eigenvalues of $D^2u$, then

where $\delta_{j_1, j_2, ..., j_k}^{i_1, i_2, ..., i_k}$ is the generalized Kronecker symbol, is the $k$-Hessian type operator. When $\alpha = 0$, $S_k(D^2u)$ is the standard $k$-Hessian operator.

Denote

We call a function $u \in C^2(\mathbb{R}^n)$ $k$-convex in $\mathbb{R}^n$ if $\lambda(D^2u(x)+\alpha I) \in \Gamma_k$ for all $x \in \mathbb{R}^n$.

In particular,

The $k$-Hessian equation is fully nonlinear PDEs for $k\neq 1$ (see Urbas ^[1] and Wang ^[2]), and there are many important applications in fluid mechanics, geometric problems and other applied subjects. Many authors have demonstrated increasing interest in $k$-Hessian equations by different methods, for instance, see (^{[3,4,5,6,7,8,9]}) and the references cited therein(^{[10,11,12,13,14,15]}). In particular, problem $(E)$ reduces to the problems studied by Keller ^[16] and Osserman ^[17] when $k = 1$, $p(|x|) = 1$ on $\mathbb{R}^n$ and $f:[0, \infty) \rightarrow [0, \infty)$ is continuous and increasing. The authors studied a necessary and sufficient condition

for the existence of entire large positive radial solutions to $(E)$. When $k = 1$, $f(u) = u^{\gamma}\ (\gamma \in (0, 1])$ and $p:[0, \infty) \rightarrow [0, \infty)$ is continuous, Lair and Wood ^[18] showed that $(E)$ admits infinitely many entire large positive radial solutions if and only if

For the case $k = 1$, system $(S)$ reduces to the following problem

Lair and Wood ^[19] analyzed the existence and nonexistence of entire positive radial solutions to Eq (1.1) when $f(v) = v^{\beta}$, $g(u) = u^{\gamma}\ (0 < \beta \le \gamma)$. For the further results, we can see ^{[20,21,22,23]} and the reference therein.

When $\alpha = 0$, Zhang and Zhou ^[24] considered the existence of entire positive $k$-convex solutions to problem $(E)$ and system $(S)$.

For the case $k = n$, Zhang and Liu ^[25] studied the existence of entire radial large solutions for a Monge-Ampère type equation

and system

Their results have been improved by Covei ^[26].

Recently, when $p(|x|)\equiv1$ on $\mathbb{R}^n$, Dai ^[27] showed that there exists a subsolution $u \in C^2(\mathbb{R}^n)$ of $(E)$ if and only if

holds.

Motivated by these works mentioned above, in this paper we will obtain some new results on the existence of entire positive $k$-convex radial solutions for equation $(E)$ and system $(S)$. The arguments are based upon a new monotone iteration scheme.

Let $\alpha_0$ denote a positive constant. In the following, we always suppose that

We give the following conditions:

$(f1)$ $f$, $g$ :[0, $\infty$)$\rightarrow$ $(\alpha_0, \infty$) are continuous and nondecreasing;

$(f2)$ $p$, $q$ :[0, $\infty$)$\rightarrow$ $(0, \infty$) are continuous and nondecreasing.

Define

where

For an arbitrary $a > 0$, we also define

and we see that

and $H_{1a}, H_{2a}$ admit the inverse functions $H_{1a}^{-1}$ and $H_{2a}^{-1}$ on $[0, H_{1a}(\infty))$ and $[0, H_{2a}(\infty))$ respectively.

The main results of this paper can be stated as follows.

Theorem 1.1. Suppose that $(f1)$ and $(f2)$ hold. If $\alpha = 0$, then Eq $(E)$ admits an entire positive $k$-convex radial solution $u \in C^2(\mathbb{R}^n)$ satisfying

Moreover, if $P(\infty) = \infty$ and $H_{1a}(\infty) = \infty$, then $\lim\limits_{r \rightarrow \infty}u(r) = \infty$; if $P(\infty) < H_{1a}(\infty) < \infty$, then $u$ is bounded.

If $\alpha > 0$ and $p(|x|)\ge 1$, then Eq $(E)$ admits an entire positive $k$-convex radial solution $u \in C^2(\mathbb{R}^n)$ satisfying

If further suppose $H_{1a}(\infty) = \infty$, then $\lim\limits_{r \rightarrow \infty}u(r) = \infty$.

Theorem 1.2. Suppose that $(f1)$ and $(f2)$ hold. If $\alpha = 0$, then system $(S)$ admits an entire positive $k$-convex radial solution $(u, v) \in C^2(\mathbb{R}^n) \times C^2(\mathbb{R}^n)$ satisfying

Moreover, if $P(\infty) = \infty = Q(\infty)$ and $H_{2a}(\infty) = \infty$, then $\lim\limits_{r \rightarrow \infty}u(r) = \lim\limits_{r \rightarrow \infty}v(r) = \infty$; if $P(\infty)+Q(\infty) < H_{2a}(\infty) < \infty$, then$u$ and $v$ are bounded.

If $\alpha > 0$ and $p(|x|)\ge 1$, $q(|x|)\ge 1$, then system $(S)$ admits an entire positive $k$-convex radial solution $(u, v) \in C^2(\mathbb{R}^n) \times C^2(\mathbb{R}^n)$ satisfying

If further suppose $H_{2a}(\infty) = \infty$, $\lim\limits_{r \rightarrow \infty}u(r) = \lim\limits_{r \rightarrow \infty}v(r) = \infty.$

2.   Preliminary lemmas

For convenience, we give some lemmas for the radial functions before proving the main results.

Let $r = |x| = \sqrt{x_1^2+...+x_n^2}$ and $B_R: = \{x\in \mathbb{R}^n:|x| < R\}$ for $R \in (0, \infty]$.

Lemma 2.1. (Lemma 2.1, ^[25]) Suppose that $\varphi \in C^2[0, R)$ with $\varphi'(0) = 0$. Then, for $u(x) = \varphi(r)$, we have $u(x) \in C^2(B_R)$, and the eigenvalues of $D^2u+\alpha I$ are

and so

By Lemma 2.1, we can conclude that $u(x) = \varphi(r)$ is a $C^2$ radial solution of $(E)$ if and only if $\varphi(r)$ satisfies

Lemma 2.2. Suppose that $(f1)$ and $(f2)$ hold. For any positive number $a$, let $\varphi \in C[0, R) \cap C^1(0, R)$ be a solution of the Cauchy problem

Then $\varphi \in C^2[0, R)$, and it satisfies (2.1) with $\varphi'(0) = 0$.

Proof. Firstly, we have

Since

This shows that $\varphi(r) \in C^1[0, R)$.

Secondly,

It is easy to know that $\varphi(r) \in C^2(0, R)$ for $r \in (0, R)$. By calculating,

Hence, $\varphi(r) \in C^2[0, R)$. And by direct calculation, we can prove that $\varphi(r)$ satisfies (2.1).

Remark 2.3. When $p(r) \equiv 1$, Lemma 2.2 is consistent with Lemma 2.3 in ^[25].

Lemma 2.4. (Lemma 2.2, ^[25]) Suppose that $(f1), (f2)$ hold and $\varphi(r) \in C^2[0, R)$ satifies (2.1) with $\varphi'(0) = 0$. Then $\varphi'(r) \ge 0$ and $\varphi''(r)+\alpha > 0$.

Proof. From (2.1), we have

Noticing that $\varphi'(0) = 0$ and intergrating from $0$ to $r$, combining with (1.7), we have

If $\alpha = 0$, then we can easily prove that $\varphi'(r) > 0$; if $\alpha > 0$ and $p(|x|) > 1$, then we have

On the other hand, by calculating, for $0 < s < r$, we have

This gives the proof of Lemma 2.4.

Remark 2.5. When $p(r) \equiv 1$, Lemma 2.4 is consistent with Lemma 2.2 in ^[25].

3.   Proof of the main results

In this section, we prove Theorems 1.1 and 1.2.

Proof of Theorem 1.1. Firstly, we consider the equations

and

Apparently, solutions in $C[0, \infty)$ to (3.3) are solutions in $C[0, \infty) \cap C^1(0, \infty)$ to (3.2).

Let $\{u_m\}_{m \ge 1}$ be the sequences of positive continuous functions defined on $[0, \infty)$ by

Obviously, for all $r \ge 0$ and $m \in \mathbb{N}$, we have

Therefore, $u_m(r) \ge a$, and $u_0(r) < u_1(r)$. Since $(f1)$ holds, we have $u_1(r) < u_2(r)$ for $r \ge 0$. According to the above reasons, we obtain that the sequences $\{u_m\}$ is increasing on $[0, \infty)$. Also, we obtain by $(f1)$ and $(f2)$ that for each $r > 0$

Therefore,

This shows that

and

It follows that the sequences $\{u_m\}$, $\{u_m'\}$ are bounded on $[0, R_0]$ for an arbitrary $R_0 > 0$. By Arzel$\grave{a}$-Ascoli theorem, $\{u_m\}$ has subsequences converging uniformly to $u$ on $[0, R_0]$. Since $\{u_m\}$ is increasing on $[0, \infty)$, we see that $\{u_m\}$ itself converges uniformly to $u$ on $[0, R_0]$. By arbitrariness of $R_0$ and Lemma 2.2, we get that $u$ is an entire positive $k$-convex radial solution to $(E)$, and $u$ satisfies

If $\alpha = 0$, by (3.6), it is easy to obtain that if $P(\infty) = \infty$ and $H_{1a}(\infty) = \infty$, then $\lim\limits_{r \rightarrow \infty}u(r) = \infty$; if $P(\infty) < H_{1a}(\infty) < \infty$, then $u$ is bounded. If $\alpha > 0$, combining the fact that $p(|x|)\ge 1$, $\alpha_0 > \frac{n}{k}(C_n^k)^{\frac{1}{k}}\alpha$ and $H_{1a}(\infty) = \infty$, it is obvious that $\lim\limits_{r \rightarrow \infty}u(r) = \infty$. This finishes the proof of Theorem 1.1.

Remark 3.1. Theorem 1.1 generalizes Theorem 1.1 with $\alpha > 0$ in ^[24]. In the case $\alpha > 0$, since $P(\infty) = \infty$ for the positivity of $u$, it is difficult to ensure if there is bounded positive entire solution of $(E)$.

Proof of Theorem 1.2. Consider the following systems

and

Let $\{u_m\}_{m \ge 1}$ and $\{v_m\}_{m \ge 1}$ be the sequences of positive continuous functions defined on $[0, \infty)$ by

Similarly, for all $r \ge 0$ and $m \in \mathbb{N}$, when $m \ge 1$, we have

Therefore, $u_m(r) \ge \frac{a}{2}$, $v_m(r) \ge \frac{a}{2}$ and $v_0(r) < v_1(r)$. Since $f$, $g$ are continuous and nondecreasing, we have $u_1(r) < u_2(r)$, $\forall r \ge 0$, and $v_1(r) < v_2(r)$, $\forall r \ge 0$. According to the above reasons, we obtain that the sequences $\{u_m\}$ and $\{v_m\}$ are increasing on $[0, \infty)$.

Moreover, for $r > 0$, by $(f1)$ and $(f2)$, one can prove that

and

Therefore,

which shows that

and

It so follows that the sequences $\{u_m\}$, $\{u_m'\}$ and $\{v_m\}$, $\{v_m'\}$ are bounded on $[0, R_0]$ for an arbitrary $R_0 > 0$. By Arzel$\grave{a}$-Ascoli theorem, $\{u_m\}$ and $\{v_m\}$ have subsequences converging uniformly to $u$ and $v$ respectively on $[0, R_0]$. Since $\{u_m\}$, $\{v_m\}$ are increasing on $[0, \infty)$, we see that $\{u_m\}$ itself converges uniformly to $u$ on $[0, R_0]$, so is $\{v_m\}$. By arbitrariness of $R_0$ and Lemma 2.2, we get that $(u, v)$ is an entire positive $k$-convex radial solution to $(S)$.

The rest proof is similar to that of Theorem 1.1. So we omit it here.

4.   Conclusions

In this paper, we use a new monotone iteration scheme to obtain some new existence results of entire positive solutions for a $k$-Hessian type equation and system.

Acknowledgments

S. Bai, X. Zhang and M. Feng are partially supported by the Beijing Natural Science Foundation of China (1212003).

Conflict of interest

The authors declare there is no conflict of interest.