Sharp conditions for the existence of infinitely many positive solutions to $q$-$k$-Hessian equation and systems

1.   Introduction and main results

The aims of this presentation are to investigate the existence of positive $q$-$k$-convex radial solutions to the $q$-$k$-Hessian equation

and systems

and

where $k\in \{1, \cdot\cdot\cdot, N\}$, $D_{i}(|D\omega|^{q-2}D_{j}\omega)$ denotes the element of row $i$ and column $j$ in the matrix $(D_{i}(|D\omega|^{q-2}D_{j}\omega))_{ij = 1, \cdots, N}$, $q\geq 2$, $\mathfrak{R}\leq\infty$ and

In this article, we assume that $H, L, f, g, f_{1}, f_{2}, g_{1}, g_{2}$ satisfy

$\mathbf{(H_{1})}$ $H, L\in C([0, \infty), (0, \infty))$;

$\mathbf{(H_{2})}$ $f, g\in C([0, \infty), [0, \infty))$ are increasing on $[0, \infty)$;

$\mathbf{(H_{3})}$ $f_{1}, f_{2}, g_{1}, g_{2}\in C([0, \infty), [0, \infty))$ are increasing on $[0, \infty)$.

Let $\mathcal {M}$ be a $N$-order symmetric real matrix and

where $\lambda = (\lambda_{1}, \cdots, \lambda_{N})$ and $\lambda_{1}, \cdots, \lambda_{N}$ are the eigenvalues of the matrix, $\mathcal {M}$. Trudinger and Wang in ^[1] first introduced the operator $S_{k}(D_{i}(|D\omega|^{q-2}D_{j}\omega))$ to establish the local integral estimates for the gradients of $k$-convex functions in the study of the weak continuity of the associated $k$-Hessian measure with respect to convergence in measure. If $k = 1$, this operator becomes the well-known $q$-Laplacian operator; if $q = 2$, it is the $k$-Hessian operator, and it is the Laplacian operator provided $k = 1$. In particular, if $k = N$ and $q = 2$, it is the famous Monge–Ampère operator.

We first review the following: Laplacian equation

The study of the existence, uniqueness, and asymptotic behavior of (1.4) has a long story. If $\Omega\subseteq \mathbb{R}^{2}$ is a bounded domain with $C^{2}$-boundary, Bieberbach ^[2] in 1916 first studied the existence, uniqueness, and asymptotic behavior of classical boundary blow-up solutions to Eq (1.4) with $f(u) = e^{u}$. In 1943, Rademacher ^[3], using the ideas of Bieberbach, proved that the results still hold for $N = 3$. If $\Omega = \mathbb{R}^{N}$, Wittich ^[4] in 1944 proved that if $N = 2$ and $f(u) = e^{u}$, then (1.4) has no entire solution. In 1951, Haviland ^[5] showed that Eq (1.4) with $\Omega = \mathbb{R}^{N}$ has no entire solution for $N = 3$ if and only if

In 1955, under some additional conditions, Walter ^[6] generalized the above result to the $N$-dimension case. In 1957, Keller ^[7] and Osserman ^[8] obtained two very famous theorems:

(ⅰ) If $\Omega$ is a bounded domain, then (1.4) has an entire subsolution if and only if $f$ satisfies (1.5);

(ⅱ) If $\Omega = \mathbb{R}^{N}$, then (1.4) has an entire subsolution if and only if $f$ satisfies

After the works of Keller ^[7] and Osserman ^[8], the conditions (1.5) and (1.6) and their generalizations are all called Keller–Osserman conditions by many authors in the literature. When $H$ satisfies $\mathbf{(H_{1})}$, Lair ^[9] first using (1.6) studied existence of the radial solution to (1.1) with $q = 2$ and $k = 1$. Then, Lair and Mohammed ^[10] proved the existence and nonexistence of nonnegative entire large solutions to a class of semilinear elliptic equations of mixed type. When $H$ and $L$ satisfy $\mathbf{(H_{1})}$, Lair ^[11] consider the following system:

where $N\geq 3$ and $\alpha, \beta$ are positive constants. The author showed that if $\alpha \beta < 1$, then (1.7) has an entire large solution if and only if

and

For some related insights on semilinear elliptic equations, we refer readers to ^{[12,13,14,15,16,17]}.

In fact, the condition (1.6) and its generalization are usually used to study the existence of entire solutions to some nonlinear elliptic equations. In 1997, Naito and Usami ^[18] showed that the $q$-Laplacian equation

has a positive entire subsolution $u\in C^{1}(\mathbb{R}^{N})$ with $|\nabla u|^{p-2}\nabla u\in C^{1}(\mathbb{R}^{N})$ if and only if $f$ satisfies the Keller–Osserman condition

In 2010, Filippucci et al. ^[19] proved the more general equation

has a nonnegative, entire, unbounded subsolution if and only if $f$ satisfies the Keller–Osserman condition

where $\theta\in [0, q-1)$.

Next, we review the $k$-Hessian equation

In 2005, Jin et al. ^[20] proved that if $f(u) = u^{\gamma}$ with $\gamma > k$, then (1.8) has no entire subsolution. In 2010, Ji and Bao ^[21] made an important contribution to this problem, i.e., they showed that Eq (1.8) has an entire $k$-convex positive subsolution if and only if $f$ satisfies the Keller–Osserman condition

If $f$ is a continuous and nondecreasing function on $\mathbb{R}$ and has a positive lower bound, Dai ^[22] in 2020 generalized the work of Ji and Bao ^[21] to a more general Hessian-type equation. When $H$, $L$, $f$, $g$ satisfy $\mathbf{(H_{1})}$–$\mathbf{(H_{2})}$, $q = 2$, $k\in\{1, \cdots, N\}$ and $\mathfrak{R} = \infty$, Zhang and Zhou ^[23] in 2015 studied the existence of radial solutions to (1.1) and (1.2) by using the following integral conditions:

Moreover, under some additional conditions, they also considered the existence of entire positive bounded radial solutions when (1.10) is false. In 2021, Bhattacharya and Mohammed ^[24] investigated a class of $k$-Hessian equations with lower-order terms on unbounded domains. Especially, they obtained the Phragmén–Lindelöf and Liouville type results. Let $\mathfrak{R} = \infty$ and $q = 2$, and $H$ and $f$ satisfy $\mathbf{(H_{1})}$–$\mathbf{(H_{2})}$. If $H$ further satisfies

and there exists some positive constant $s_{0}$ such that

Zhang and Xia ^[25] in 2023 showed that Eq (1.1) (with $q = 2$) has a large radial convex solution if and only if (1.9) holds. A similar result of existence was also obtained in ^[26]. When $f(u)$ is replaced by $b(x)f(u)$ in (1.8), where $b\in C(\mathbb{R}^{N})$ is positive in $\mathbb{R}^{N}$ and $f\in C^{1}(0, \infty)$ is a nonnegative, nondecreasing function, $f(0) = 0$ and

Li and Bao ^[27] in 2024 showed a necessary and sufficient condition for the existence of nonradial, entire large solutions. Moreover, they also studied the asymptotic behavior of entire solutions at infinity. With regard to the other works of Monge–Ampère type equation (system), we refer readers to ^{[28,29,30,31,32]}. For more general Hessian type equation (system), we refer readers to ^{[33,34,35,36,37,38,39]}.

Now, let us return to (1.1)–(1.3). As far as we know, the $q$-$k$ Hessian equation (system) has rarely been investigated in previous literature. When $H(|\cdot|)\equiv1$ in $\mathscr{D}_{\mathfrak{R}}$ with $\mathfrak{R} = \infty$, the sufficient and necessary condition for the existence of the entire subsolution to Eq (1.1) was given via the Keller–Osserman condition

by Bao and Feng ^[40] for $q\geq2$ and $k\in\{1, \cdots, N\}$. Recently, the results in ^[23] were generalized by Fan et al. ^[41] and Kan and Zhang ^[42] to the cases of $q$-$k$ Hessian equation and system. In particular, Kan and Zhang ^[42] showed that if $H$, $L$, $f$, and $g$ satisfy $\mathbf{(H_{1})}$–$\mathbf{(H_{2})}$ and $\mathfrak{R} = \infty$, then (1.1) has an entire positive $q$-$k$-convex radial solution provided $f$ satisfies

and (1.2) has an entire positive $q$-$k$-convex radial solution provided $f$ and $g$ satisfy

Fan et al. ^[41] showed that if $H, L$ satisfy $\mathbf{(H_{1})}$, $f_{1}, f_{2}, g_{1}, g_{2}$ satisfy $\mathbf{(H_{3})}$, and $\mathfrak{R} = \infty$, then (1.3) has a radial solution provided $f_{1}, f_{2}, g_{1}, g_{2}$ satisfy

Especially under some additional conditions, they further investigated the result of the existence in entire bounded solutions when (1.15) holds. Recently, by using (1.12), the result of existence to (1.1) was investigated by Feng and Zhang in ^[43]. Specifically, they showed that if $\mathfrak{R} = 1$, $H$ satisfies $\mathbf{(H_{1})}$ and the following condition

$\mathbf{(C_{1})}$ there are two positive constants $d_{1}, d_{2}$ and some function $L\in \Lambda$ such that

where $\Lambda$ denotes the set of functions $L$ that satisfy $L\in C^{1}(0, \infty)$, $L > 0$, $L' < 0$,

$f$ satisfies the following conditions:

$\mathbf{(C_{2})}$ $f\in C(0, \infty)$ is positive and increasing and is local Lipschitz on $(0, \infty)$; moreover, $f$ satisfies (1.12);

$\mathbf{(C_{3})}$ let $c_{0}$ be a positive constant,

and

then Eq (1.1) has innumerable radial $q$-$k$-convex boundary blow-up solutions that are positive in $\mathscr{D}_{\mathfrak{R}}$. For further insights on $q$-Mange–Ampère equation and $q$-$k$ Hessian type equation, we refer the readers to ^[44,45].

Inspired by the above works, in this paper, we prove the existence of innumerable positive $q$-$k$-convex radial solutions (including boundary blow-up solutions) to Eq (1.1), the systems (1.2), and (1.3) by using the Keller–Osserman conditions (1.12),

and

respectively. We omit the hypothesis $\mathbf{(C_{3})}$ in ^[43] and our assumptions on $f$ and $H$ are weaker than the ones in ^[43]. Moreover, we note the conditions (1.12), (1.16), and (1.17) are strictly weaker than the conditions (1.13)–(1.15), respectively (the reasons are given by Remark 2.4 and Proposition 3.4).

2.   The main results

Theorem 2.1. Let $H, f$ satisfy $\mathbf{(H_{1})}$–$\mathbf{(H_{2})}$ and (1.12) hold, then for any $a_{0}\in \mathbb{R}^{+}$, Eq (1.1) has a radial $q$-$k$-convex positive solution $\omega$ satisfying

where

$\mathcal {T}_{0}^{-1}$ is the inverse of $\mathcal {T}_{0}$ given by

In particular, if $\mathcal {H}(\mathfrak{R}) = \infty$, then $\omega(\mathfrak{R}) = \infty$.

Remark 2.2. If $\mathfrak{R} < \infty$, then $\mathcal {H}(\mathfrak{R}) = \infty$ is equivalent to

Theorem 2.3. Let $\mathfrak{R} = \infty$, $H, f$ satisfy $\mathbf{(H_{1})}$–$\mathbf{(H_{2})}$ and (1.12) be false. If $\mathscr{H}(s) > 0 \mathit{\mbox{for}}s\in (0, \infty)$ and there exists some positive constant $s_{0}$ such that

where $\mathscr{H}$ is given by (1.11). Then (1.1) has no radial $q$-$k$-convex positive large solution.

Remark 2.4. From Proposition 3.4 (see page 8), we see that if (1.13) holds, then (1.12) holds. But, the converse of the result is not true. A basic example is

By a simple calculation, we see that

This implies that ${\int}_{1}^{\infty}(f(\tau))^{-1/(q-1)k}d\tau < \infty$ and (1.12) holds. So, the condition (1.12) is strictly weaker than (1.13).

Theorem 2.5. Let $H, L, f$, and $g$ satisfy $\mathbf{(H_{1})}$–$\mathbf{(H_{2})}$ and (1.16) hold, then for any $a_{0}\in \mathbb{R}^{+}$, (1.2) has a radial $q$-$k$-convex positive solution $(\omega, \upsilon)$ satisfying

where $\mathcal {H}$ is given by (2.2) and

$\mathcal {T}^{-1}_{1}$ is the inverse of $\mathcal {T}_{1}$ given by

In particular, if $\mathcal {H}(\mathfrak{R}) = \infty$, then $\omega(\mathfrak{R}) = \infty$; if $\mathcal {L}(\mathfrak{R}) = \infty$, then $\upsilon(\mathfrak{R}) = \infty$.

Theorem 2.6. Let $H, L$ satisfy $\mathbf{(H_{1})}$, $f_{1}, f_{2}, g_{1}$, and $g_{2}$ satisfy $\mathbf{(H_{3})}$, then for any $a_{0}\in \mathbb{R}^{+}$, (1.3) has a radial $q$-$k$-convex positive solution $(\omega, \upsilon)$ satisfying

where $\mathcal {H}$, $\mathcal {L}$, and $\mathfrak{B}$ are given by (2.2), (2.5), and (2.6), and $\mathcal {T}_{2}^{-1}$ is the inverse of $\mathcal {T}_{2}$ given by

In particular, if $\mathcal {H}(\mathfrak{R}) = \infty$, then $\omega(\mathfrak{R}) = \infty$; if $\mathcal {L}(\mathfrak{R}) = \infty$, then $\upsilon(\mathfrak{R}) = \infty$.

Remark 2.7. By the same argument as Remark 2.4, we see that (1.16) is strictly weaker than (1.14), and (1.17) is strictly weaker than (1.15).

3.   Preliminary results

Definition 3.1. The $q$-$k$-convex function in $\mathscr{D}_{\mathfrak{R}}$ is defined as below: if

where $\Gamma_{k}: = \big\{\lambda\in \mathbb{R}^{N}: S_{i}(\lambda) > 0, \, i = 1, \cdots, k\big\}$. Especially, if $\omega\in \Phi^{2, k}(\mathscr{D}_{\mathfrak{R}})\cap C^{2}(\mathscr{D}_{\mathfrak{R}})$, then $\omega$ is the $k$-convex function.

By Lemmas 1 and 2 and Corollary 1 of Fan et al. in ^[41], we obtain the following lemma:

Lemma 3.2. Let $H, L, f$, and $g$ satisfy $\mathbf{(H_{1})}$–$\mathbf{(H_{2})}$, $a_{0}$ be a positive constant, and $\zeta_{0}, \zeta, \eta \in C^{0}[0, R)\cap C^{1}(0, R)$ satisfy the following equation and system:

and

then $\zeta_{0}, \zeta, \eta\in C^{2}(0, \mathfrak{R})\cap C^{1}[0, \mathfrak{R})$ satisfy $\zeta_{0}(0) = a_{0}, \, \zeta'_{0}(0) = 0$,

and $\omega_{0}(x) = \zeta_{0}(s)$ and $(\omega(x), \upsilon(x)) = (\zeta(s), \eta(s))$ are, respectively, the radial $q$-$k$-convex solutions to the Eq (1.1) and system (1.2).

Remark 3.3. In Lemma 3.2, if $f(\eta(s))$ is replaced by $f_{1}(\eta(s))f_{2}(\zeta(s))$ and $g(\zeta(s))$ is replaced by $g_{1}(\zeta(s))g_{2}(\eta(s))$ in (3.1) and (3.3), where $f_{1}, f_{2}, g_{1}, g_{2}$ are given by $\mathbf{(H_{3})}$, then by Lemmas 1 and 2 of Fan et al. ^[41] we see that this conclusion still holds.

Proposition 3.4. Let $h\in C([0, \infty), [0, \infty))$ be increasing on $(0, \infty)$. If

Proof. The proof is divided into two steps.

Step 1. We show that for any positive constant $M > 0$, there exists $t_{*} > 0$ such that for any $t\geq t_{*}$,

Otherwise, there exist a positive constant $c_{0} > 0$ and an increasing sequence $\{t_{i}\}_{i = 0}^{\infty}$ of real numbers satisfying $\lim_{i\rightarrow \infty}t_{i} = \infty$ and $2t_{i-1}\leq t_{i}$, $i = 1, 2, \cdot\cdot\cdot$, such that $\frac{h(t_{i})}{t_{i}^{(q-1)k}}\leq c_{0}.$ This, together with

shows that

This is a contradiction. So, the first step is finished.

Step 2. By (3.4), we see that

So, we obtain

The proof is finished. □

4.   Proof of Theorem 2.1

Proof. Let $\mathcal {T}_{0}$ be given by (2.4). Since

we can obtain that $\mathcal {T}_{0}$ has the inverse $\mathcal {T}_{0}^{-1}$, which is increasing on $[0, \infty)$ with

We consider the following initial value problem:

Problem (4.2) is equivalent to the integral equation

Now, by constructing some iterative approximation sequence, we prove the existence of $q$-$k$-convex solutions to problem (4.2). We assume that $\{\omega_{m}\}$ is the sequence of positive continuous functions defined by

The conditions $\mathbf{(H_{1})}$–$\mathbf{(H_{2})}$ imply that

and

So, we see that $\omega_{m}$ is a positive increasing function and $\{\omega_{m}\}$ is an increasing sequence. These facts, together with (4.2), imply that for any $s\in (0, \mathfrak{R})$, we have

and

For any $\mathfrak{R_{0}}\in(0, \mathfrak{R})$, we set

This fact, combined with (4.4), shows that

Moreover, by direct calculation, we see that

Integrating (4.6) from $\tau$ $(\tau\in (0, \mathfrak{R_{0}}))$ to $s$ and letting $\tau\rightarrow0$, we obtain

Furthermore, we arrive at

where $\mathfrak{A}$ is given by (2.3). It is clear that $\mathcal {T}_{0}(\omega_{m})\leq\mathfrak{A}(\mathfrak{R_{0}})\mathfrak{R_{0}} \mbox{ on }[0, \mathfrak{R_{0}}].$ It follows from (4.1) that

This implies that $\{\omega_{m}\}$ is a uniformly bounded sequence on $[0, \mathfrak{R_{0}}]$ for any $\mathfrak{R_{0}}\in [0, \mathfrak{R})$. On the other hand, it follows from (4.7) and (4.8) that $\{\omega_{m}'\}$ is also uniformly bounded on $[0, \mathfrak{R_{0}}]$. We conclude by Arzela–Ascoli's theorem that there is a subsequence of $\{\omega_{m}\}$, denoted by itself, such that $\omega_{m}\rightarrow \omega$ on $[0, \mathfrak{R_{0}}]$. The arbitrariness of $\mathfrak{R_{0}}$ and Lemma 3.2 imply that $\omega$ is a positive $q$-$k$-convex solution to problem (4.2). It follows from (4.3) and (4.8) that (2.1) holds. The proof is finished. □

5.   Proof of Theorem 2.3

Proof. Suppose $\omega$ is a positive $q$-$k$-convex radial large solution. We will derive a contradiction. By Lemma 3.2, we see that

i.e.,

Furthermore, we have

Since $\omega$ satisfies Eq (3.2), we obtain by (5.1) that

where $\mathscr{H}$ is given by (1.11). Multiplying both sides of the above inequality by $\omega'(s)$, we have

i.e.,

Integrating this inequality from $s_{0}$ to $s$, we obtain

This implies that

which is a contradiction to (1.12). □

6.   Proof of Theorem 2.5

Proof. Let $\mathcal {T}_{1}$ be given by (2.7). It is clear that

It follows that $\mathcal {T}_{1}$ has the inverse $\mathcal {T}_{1}^{-1}$, which is increasing on $[0, \infty)$ with

We consider the following system:

System (6.1) is equivalent to the integral system

By a similar argument as in the proof of Theorem 2.1, we construct the iterative approximation sequence $\{(\omega_{m}, \upsilon_{m})\}$ as below:

and

From $\mathbf{(H_{1})}$–$\mathbf{(H_{2})}$, we obtain

and

So, we see that $\omega_{m}$ and $\upsilon_{m}$ are positive increasing functions, and $\{\omega_{m}\}$ and $\{\upsilon_{m}\}$ are increasing sequences. Furthermore, we have that for any $s\in (0, \mathfrak{R})$, there hold

and

For an arbitrary $\mathfrak{R_{0}}\in(0, \mathfrak{R})$, we define

These facts, combined with (6.2) and (6.3), show that

and

Moreover, by direct calculation, we see that

Integrating (6.5) and (6.6) from $\tau$ $(\tau\in (0, \mathfrak{R_{0}}))$ to $s$ and letting $\tau\rightarrow0$, we obtain

and

Furthermore, we arrive at

where $\mathfrak{B}$ is given by (2.6). It is clear that

The rest of the proof is similar to the one in Theorem 2.1, so we omit it here. The proof is finished. □

7.   Proof of Theorem 2.6

Proof. Let $\mathcal {T}_{2}$ be given by (2.8). We have

It is clear that $\mathcal {T}_{2}$ has the inverse $\mathcal {T}_{2}^{-1}$, which is increasing on $[0, \infty)$ with

As the proof of Theorem 2.5, we consider the system:

System (7.1) is equivalent to the integral system:

where $s\in [0, \mathfrak{R})$. As the proof of Theorem 2.5, we construct the iterative approximation sequence $\{(\omega_{m}, \upsilon_{m})\}$ as below:

and

From $\mathbf{(H_{1})}$ and $\mathbf{(H_{3})}$, we have

and

So, we have that $\omega_{m}$ and $\upsilon_{m}$ are increasing functions, and $\{\omega_{m}\}$ and $\{\upsilon_{m}\}$ are increasing sequences. Furthermore, we obtain that for any $s\in (0, \mathfrak{R})$, there hold

and

The above facts imply that for any $\mathfrak{R_{0}}\in(0, \mathfrak{R})$, we have

and

where $\mathfrak{H}_{\mathfrak{R_{0}}}$ and $\mathfrak{L}_{\mathfrak{R_{0}}}$ are defined as shown in (6.4). Moreover, by a direct calculation, we see that (6.7) holds here. Integrating (7.2) and (7.3) from $\tau$ $(\tau\in (0, \mathfrak{R_{0}}))$ to $s$ and letting $\tau\rightarrow0$, we obtain

and

Furthermore, we have

where $\mathfrak{B}$ is given by (2.6). It is clear that

The rest of the proof is similar to the one in Theorem 2.1, so we omit it here. The proof is finished. □

Use of AI tools declaration

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

Acknowledgments

The author was supported by NSF of Shandong Province, China (Grant No. ZR2021MA007).

Conflict of interest

The authors declare there is no conflict of interest.