© Wuhan University 2025

0 Introduction

In this paper, we study existence of entire convex radial solutions to the system

$\{\begin{array}{l}\mathrm{d}\mathrm{e}{\mathrm{t}}^{\mathrm{1}/n}(\mathrm{\Delta }u\mathit{I}-D^{\mathrm{2}}u)=p(|\mathit{x}|)f(v),\mathit{x}\in {\mathbb{R}}^n,\\ \mathrm{d}\mathrm{e}{\mathrm{t}}^{\mathrm{1}/n}(\mathrm{\Delta }v\mathit{I}-D^{\mathrm{2}}v)=q(|\mathit{x}|)g(u),\mathit{x}\in {\mathbb{R}}^n.\end{array}$(1)

Here ${\mathbb{R}}^n$ is $n$-dimensional Euclidean space and $n\ge \mathrm{2}$, $D^{\mathrm{2}}u$ denotes the Hessian matrix of $u$,$\mathrm{\Delta }u={\displaystyle \sum _{i=\mathrm{1}}^n}\frac{{\partial }^{\mathrm{2}}u}{\partial {{\mathit{x}}_i}^{\mathrm{2}}}$ denotes the Laplacian of $u$, $\mathit{I}$ is the identity matrix of order $n$, and we assume that $p$, $q$ , $f$ and $g$ satisfy the following conditions:

$({\mathrm{H}}_{\mathrm{1}})p,q:[\mathrm{0},+\mathrm{\infty })\to (\mathrm{0},+\mathrm{\infty })$ are continuous;

$({\mathrm{H}}_{\mathrm{2}})f,g:[\mathrm{0},+\mathrm{\infty })\to [\mathrm{0},+\mathrm{\infty })$ are continuous and non-decreasing.

It is well known that for the following equation

$\mathrm{\Delta }u=f(u),\mathit{x}\in {\mathbb{R}}^n,$(2)

where $f$ is a positive monotone increasing continuous function on $\mathbb{R}$, Keller^[1] and Osserman^[2] presented the famous Keller-Osserman condition:

${{\int }_{}^{\mathrm{\infty }}\left({\int }_{\mathrm{0}}^{t}f(s)\mathrm{d}s\right)}^{-\frac{\mathrm{1}}{\mathrm{2}}}\mathrm{d}t=\mathrm{\infty },$(3)

where we omit the lower limit to admit an arbitrary positive number. That is (2) has a sub-solution if and only if (3) holds. Ji and Bao^[3] generalized the Laplace operator to the $k$-Hessian operator,

${\sigma }_k^{\frac{\mathrm{1}}{k}}(\lambda (D^{\mathrm{2}}u))=f(u),\mathit{x}\in {\mathbb{R}}^n.$

Bao and Feng^[4] generalized further to the $p$-$k$-Hessian equation. Bao et al^[5] studied the Keller-Osserman type condition for $k$-Yamabe type equations. Dai^[6] investigated more general augmented Hessian equations of the following type and established generalized Keller-Osserman type condition,

${\sigma }_k^{\frac{\mathrm{1}}{k}}(\lambda (D^{\mathrm{2}}u+\alpha \mathit{I}))=f(u),\mathit{x}\in {\mathbb{R}}^n.$

Ji et al^[7] studied a kind of $k$-Hessian equation with gradient terms and the following real $n-\mathrm{1}$ Monge-Ampère equation in Ref. [8]:

$\mathrm{d}\mathrm{e}{\mathrm{t}}^{\mathrm{1}/n}(\mathrm{\Delta }u\mathit{I}-D^{\mathrm{2}}u)=b(\mathit{x})f(u),\mathit{x}\in {\mathbb{R}}^n.$(4)

The author considered three cases respectively: $b=\mathrm{1}$, $b$ is a spherically symmetric function, and $b$ is a non-spherically symmetric function. The equation (4) originates from Gauduchon's conjecture^[9] in complex geometry, which is a crucial conjecture in geometric analysis. Capuzzo Dolcetta et al^[10] studied the Keller-Osserman type condition for degenerate second-order elliptic operators.

There is also a lot of work going to with system, for example,

$\{\begin{array}{l}\mathrm{\Delta }u=p(|\mathit{x}|)v^{\alpha },\mathit{x}\in {\mathbb{R}}^n,\\ \mathrm{\Delta }v=q(|\mathit{x}|)u^{\gamma },\mathit{x}\in {\mathbb{R}}^n.\end{array}$(5)

Here $n\ge \mathrm{3},\mathrm{0}<\alpha \le \gamma .$ Lair and Wood^[11] studied the existence and nonexistence of entire positive radial solutions to the system. For the following system

$\{\begin{array}{l}\mathrm{d}\mathrm{e}\mathrm{t}(D^{\mathrm{2}}u)=f(-v),\mathit{x}\in B_{\mathrm{1}}(\mathrm{0}),\\ \mathrm{d}\mathrm{e}\mathrm{t}(D^{\mathrm{2}}v)=f(-u),\mathit{x}\in B_{\mathrm{1}}(\mathrm{0}),\\ u=v=\mathrm{0},\mathit{x}\in \partial B_{\mathrm{1}}(\mathrm{0}),\end{array}$(6)

Wang^[12], Wang and An^[13] studied the existence of convex radial solutions to (6), where $B_{\mathrm{1}}(\mathrm{0})$ denotes the unit ball. Zhang and Zhou^[14] considered the existence of entire positive radial solutions to the following system,

$\{\begin{array}{l}{\sigma }_k(D^{\mathrm{2}}u)=p(|\mathit{x}|)f(v),\mathit{x}\in {\mathbb{R}}^n,\\ {\sigma }_k(D^{\mathrm{2}}v)=q(|\mathit{x}|)g(u),\mathit{x}\in {\mathbb{R}}^n.\end{array}$

Mi and Ji^[15] extended the $k$-Hessian equations to the augmented $p$-$k$-Hessian systems and Cui^[16] proved the existence and nonexistence of entire radial solutions to the $k$-Hessian type system with gradient terms. For more research on k-Hessian systems, refer to Refs. [17-20] and other relevant references.

We first introduce a fundamental lemma from Ref. [8].

Lemma 1   Let $r=|\mathit{x}|={\left({\displaystyle \sum _{i=\mathrm{1}}^n}{{\mathit{x}}_i}^{\mathrm{2}}\right)}^{\mathrm{1}/\mathrm{2}},$$v(r)\in C^{\mathrm{2}}[\mathrm{0},+\mathrm{\infty })$ be radially symmetric with $v^{\text{'}}(\mathrm{0})=\mathrm{0}.$ Then for $u(\mathit{x})=v(r),$we have $u(\mathit{x})\in C^{\mathrm{2}}({\mathbb{R}}^n)$ and

$\mathrm{d}\mathrm{e}{\mathrm{t}}^{\mathrm{1}/n}(\mathrm{\Delta }u\mathit{I}-D^{\mathrm{2}}u)=\{\begin{array}{l}\frac{n-\mathrm{1}}{r}{v}^{\text{'}}(r){\left(v^{″}(r)+\frac{n-\mathrm{2}}{r}{v}^{\text{'}}(r)\right)}^{n-\mathrm{1}},r\in (\mathrm{0},+\mathrm{\infty }),\\ {\left((n-\mathrm{1})v^{″}(\mathrm{0})\right)}^n,r=\mathrm{0}.\end{array}$

Moreover, $v(r)$ is a radial solution of (4) if and only if $v(r)$ satisfies the following ordinary differential equation

$\frac{n-\mathrm{1}}{r}{v}^{\text{'}}(r){\left(v^{″}(r)+\frac{n-\mathrm{2}}{r}{v}^{\text{'}}(r)\right)}^{n-\mathrm{1}}=b^n(r)f^n(v(r)),r\in [\mathrm{0},+\mathrm{\infty }).$

This equation is equivalent to the following equation

$v^{\text{'}}(r)=\frac{r^{\mathrm{2}-n}}{n-\mathrm{1}}{\left({\int }_{\mathrm{0}}^{r}n{s}^{n-\mathrm{1}}{p}^{\frac{n}{n-\mathrm{1}}}(s)f^{\frac{n}{n-\mathrm{1}}}(v(s))\mathrm{d}s\right)}^{\frac{n-\mathrm{1}}{n}},r\in [\mathrm{0},+\mathrm{\infty }).$

Subsequently, we introduce the main result of this paper. For system (1), we denote

$P(\mathrm{\infty }):=\underset{r\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}P(r),Q(\mathrm{\infty }):=\underset{r\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}Q(r),H_a(\mathrm{\infty }):=\underset{r\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{H}_a(r);$

$P(r)={\int }_{\mathrm{0}}^r\left[\frac{t^{\mathrm{2}-n}}{n-\mathrm{1}}{\left({\int }_{\mathrm{0}}^{t}n{s}^{n-\mathrm{1}}{p}^{\frac{n}{n-\mathrm{1}}}(s)\mathrm{d}s\right)}^{\frac{n-\mathrm{1}}{n}}\right]\mathrm{d}t,r\ge \mathrm{0};Q(r)={\int }_{\mathrm{0}}^r\left[\frac{t^{\mathrm{2}-n}}{n-\mathrm{1}}{\left({\int }_{\mathrm{0}}^{t}n{s}^{n-\mathrm{1}}{q}^{\frac{n}{n-\mathrm{1}}}(s)\mathrm{d}s\right)}^{\frac{n-\mathrm{1}}{n}}\right]\mathrm{d}t,r\ge \mathrm{0};$

$H_a(r)={\int }_a^r\frac{\mathrm{d}\tau }{f(\tau )+g(\tau )},r\ge a.$

Theorem 1   Suppose $({\mathrm{H}}_{\mathrm{1}})$ and $({\mathrm{H}}_{\mathrm{2}})$ hold,${\mathrm{H}}_a(\mathrm{\infty })=\mathrm{\infty },$then (1) has one entire positive convex radial solution $u,v\in C^{\mathrm{2}}({\mathbb{R}}^n)$. Moreover,

(i) if $P(\mathrm{\infty })+Q(\mathrm{\infty })<\mathrm{\infty },$ then $u$ and $v$ are bounded;

(ii) if $P(\mathrm{\infty })=Q(\mathrm{\infty })=\mathrm{\infty },$ then $\underset{r\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}u(r)=\mathrm{\infty },$ $\underset{r\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}v(r)=\mathrm{\infty }.$

Theorem 2   Suppose $({\mathrm{H}}_{\mathrm{1}})$ and $({\mathrm{H}}_{\mathrm{2}})$ hold, $P(\mathrm{\infty })+Q(\mathrm{\infty })<H_a(\mathrm{\infty })=\mathrm{\infty },$then (1) has one entire positive convex radial solution $u,v\in C^{\mathrm{2}}({\mathbb{R}}^n)$. Moreover,

$\frac{a}{\mathrm{2}}+f\left(\frac{a}{\mathrm{2}}\right)P(r)\le u(r)\le H_a^{-\mathrm{1}}(P(r)+Q(r)),\forall r\ge \mathrm{0};\frac{a}{\mathrm{2}}+g\left(\frac{a}{\mathrm{2}}\right)Q(r)\le v(r)\le H_a^{-\mathrm{1}}(P(r)+Q(r)),\forall r\ge \mathrm{0}.$

Remark 1   In a similar way, we can obtain the existence of entire radial solution to the following general system

$\{\begin{array}{l}\mathrm{d}\mathrm{e}{\mathrm{t}}^{\mathrm{1}/n}(\mathrm{\Delta }u\mathit{I}-D^{\mathrm{2}}u)=p(|\mathit{x}|)f_{\mathrm{1}}(v)f_{\mathrm{2}}(u),\mathit{x}\in {\mathbb{R}}^n,\\ \mathrm{d}\mathrm{e}{\mathrm{t}}^{\mathrm{1}/n}(\mathrm{\Delta }v\mathit{I}-D^{\mathrm{2}}v)=q(|\mathit{x}|)g_{\mathrm{1}}(v)g_{\mathrm{2}}(u),\mathit{x}\in {\mathbb{R}}^n.\end{array}$(7)

1 Proof of the Theorems

By Lemma 1, in order to find a radial solution of (1), we only need to prove the existence of solutions to the following system

$\{\begin{array}{l}\frac{n-\mathrm{1}}{r}{u}^{\text{'}}(r){\left(u^{″}(r)+\frac{n-\mathrm{2}}{r}{u}^{\text{'}}(r)\right)}^{n-\mathrm{1}}=p^n(r)f^n(v(r)),r>\mathrm{0},\\ \frac{n-\mathrm{1}}{r}{v}^{\text{'}}(r){\left(v^{″}(r)+\frac{n-\mathrm{2}}{r}{v}^{\text{'}}(r)\right)}^{n-\mathrm{1}}=q^n(r)g^n(u(r)),r>\mathrm{0}.\end{array}$

This is equivalent to the following system

$\{\begin{array}{l}u(r)=\frac{a}{\mathrm{2}}+{\int }_{\mathrm{0}}^r\left[\frac{t^{\mathrm{2}-n}}{n-\mathrm{1}}{\left({\int }_{\mathrm{0}}^{t}n{s}^{n-\mathrm{1}}{p}^{\frac{n}{n-\mathrm{1}}}(s)f^{\frac{n}{n-\mathrm{1}}}(v(s))\mathrm{d}s\right)}^{\frac{n-\mathrm{1}}{n}}\right]\mathrm{d}t,r\ge \mathrm{0},\\ v(r)=\frac{a}{\mathrm{2}}+{\int }_{\mathrm{0}}^r\left[\frac{t^{\mathrm{2}-n}}{n-\mathrm{1}}{\left({\int }_{\mathrm{0}}^{t}n{s}^{n-\mathrm{1}}{q}^{\frac{n}{n-\mathrm{1}}}(s)g^{\frac{n}{n-\mathrm{1}}}(u(s))\mathrm{d}s\right)}^{\frac{n-\mathrm{1}}{n}}\right]\mathrm{d}t,r\ge \mathrm{0}.\end{array}$

Let $\{u_m\}$ and $\{v_m\}$ be the sequences of positive continuous functions defined on $[\mathrm{0},+\mathrm{\infty })$ by

$\{\begin{array}{l}{u}_{\mathrm{0}}(r)=v_{\mathrm{0}}(r)=\frac{a}{\mathrm{2}},\\ u_m(r)=\frac{a}{\mathrm{2}}+{\int }_{\mathrm{0}}^r\left[\frac{t^{\mathrm{2}-n}}{n-\mathrm{1}}{\left({\int }_{\mathrm{0}}^{t}n{s}^{n-\mathrm{1}}{p}^{\frac{n}{n-\mathrm{1}}}(s)f^{\frac{n}{n-\mathrm{1}}}(v_{m-\mathrm{1}}(s))\mathrm{d}s\right)}^{\frac{n-\mathrm{1}}{n}}\right]\mathrm{d}t,r\ge \mathrm{0},\\ v_m(r)=\frac{a}{\mathrm{2}}+{\int }_{\mathrm{0}}^r\left[\frac{t^{\mathrm{2}-n}}{n-\mathrm{1}}{\left({\int }_{\mathrm{0}}^{t}n{s}^{n-\mathrm{1}}{q}^{\frac{n}{n-\mathrm{1}}}(s)g^{\frac{n}{n-\mathrm{1}}}(u_{m-\mathrm{1}}(s))\mathrm{d}s\right)}^{\frac{n-\mathrm{1}}{n}}\right]\mathrm{d}t,r\ge \mathrm{0}.\end{array}$

By $({\mathrm{H}}_{\mathrm{1}})$ and $({\mathrm{H}}_{\mathrm{2}})$, we obtain $\forall r\ge \mathrm{0},$

$\frac{a}{\mathrm{2}}\le u_{\mathrm{1}}(r)\le u_{\mathrm{2}}(r)\le \cdots ,\frac{a}{\mathrm{2}}\le v_{\mathrm{1}}(r)\le v_{\mathrm{2}}(r)\le \cdots ,$

and

$\begin{array}{l}{u}_m^{\text{'}}(r)=\frac{r^{\mathrm{2}-n}}{n-\mathrm{1}}{\left({\int }_{\mathrm{0}}^{r}n{s}^{n-\mathrm{1}}{p}^{\frac{n}{n-\mathrm{1}}}(s)f^{\frac{n}{n-\mathrm{1}}}(v_{m-\mathrm{1}}(s))\mathrm{d}s\right)}^{\frac{n-\mathrm{1}}{n}}\le f(v_m(r))\frac{r^{\mathrm{2}-n}}{n-\mathrm{1}}{\left({\int }_{\mathrm{0}}^{r}n{s}^{n-\mathrm{1}}{p}^{\frac{n}{n-\mathrm{1}}}(s)\mathrm{d}s\right)}^{\frac{n-\mathrm{1}}{n}}=f(v_m(r))P^{\text{'}}(r)\\ \le \left[f(u_m(r)+v_m(r))+g(u_m(r)+v_m(r))\right]P^{\text{'}}(r),\end{array}$

$\begin{array}{l}{v}_m^{\text{'}}(r)=\frac{r^{\mathrm{2}-n}}{n-\mathrm{1}}{\left({\int }_{\mathrm{0}}^{r}n{s}^{n-\mathrm{1}}{q}^{\frac{n}{n-\mathrm{1}}}(s)g^{\frac{n}{n-\mathrm{1}}}(u_{m-\mathrm{1}}(s))\mathrm{d}s\right)}^{\frac{n-\mathrm{1}}{n}}\le g(u_m(r))\frac{r^{\mathrm{2}-n}}{n-\mathrm{1}}{\left({\int }_{\mathrm{0}}^{r}n{s}^{n-\mathrm{1}}{q}^{\frac{n}{n-\mathrm{1}}}(s)\mathrm{d}s\right)}^{\frac{n-\mathrm{1}}{n}}=g(u_m(r))Q^{\text{'}}(r)\\ \le \left[f(u_m(r)+v_m(r))+g(u_m(r)+v_m(r))\right]Q^{\text{'}}(r).\end{array}$

Hence

$u_m^{\text{'}}(r)+v_m^{\text{'}}(r)\le \left[f(u_m(r)+v_m(r))+g(u_m(r)+v_m(r))\right]\left[P^{\text{'}}(r)+Q^{\text{'}}(r)\right].$

We have

$\begin{array}{l}{\int }_a^{u_m(r)+v_m(r)}\frac{\mathrm{d}\tau }{f(\tau )+g(\tau )}\le P(r)+Q(r),r>\mathrm{0},\\ H_a(u_m(r)+v_m(r))\le P(r)+Q(r),\end{array}$

consequently,

$u_m(r)+v_m(r)\le H_a^{-\mathrm{1}}(P(r)+Q(r)),\forall r\ge \mathrm{0}.$(8)

Hence the sequences $\{u_m\}$ and $\{v_m\}$ are bounded on $[\mathrm{0},R]$ for arbitrary $R>\mathrm{0}.$ $\forall m\in \mathbb{N},\forall x,y\in [\mathrm{0},R],$

$\begin{array}{l}|u_m(x)-u_m(y)|\\ =\left|{\int }_{\mathrm{0}}^{\mathit{x}}\left[\frac{t^{\mathrm{2}-n}}{n-\mathrm{1}}{\left({\int }_{\mathrm{0}}^{t}n{s}^{n-\mathrm{1}}{p}^{\frac{n}{n-\mathrm{1}}}(s)f^{\frac{n}{n-\mathrm{1}}}(v_{m-\mathrm{1}}(s))\mathrm{d}s\right)}^{\frac{n-\mathrm{1}}{n}}\right]\mathrm{d}t-{\int }_{\mathrm{0}}^y\left[\frac{t^{\mathrm{2}-n}}{n-\mathrm{1}}{\left({\int }_{\mathrm{0}}^{t}n{s}^{n-\mathrm{1}}{p}^{\frac{n}{n-\mathrm{1}}}(s)f^{\frac{n}{n-\mathrm{1}}}(v_{m-\mathrm{1}}(s))\mathrm{d}s\right)}^{\frac{n-\mathrm{1}}{n}}\right]\mathrm{d}t\right|\\ =\left|{\int }_{\mathit{x}}^y\left[\frac{t^{\mathrm{2}-n}}{n-\mathrm{1}}{\left({\int }_{\mathrm{0}}^{t}n{s}^{n-\mathrm{1}}{p}^{\frac{n}{n-\mathrm{1}}}(s)f^{\frac{n}{n-\mathrm{1}}}(v_{m-\mathrm{1}}(s))\mathrm{d}s\right)}^{\frac{n-\mathrm{1}}{n}}\right]\mathrm{d}t\right|=\underset{r\in [\mathrm{0},R]}{\mathrm{m}\mathrm{a}\mathrm{x}}p(r)f(v_{m-\mathrm{1}}(R))\left|{\int }_{\mathit{x}}^y\left[\frac{t^{\mathrm{2}-n}}{n-\mathrm{1}}{\left({\int }_{\mathrm{0}}^{t}n{s}^{n-\mathrm{1}}\mathrm{d}s\right)}^{\frac{n-\mathrm{1}}{n}}\right]\mathrm{d}t\right|\\ =\underset{r\in [\mathrm{0},R]}{\mathrm{m}\mathrm{a}\mathrm{x}}p(r)f(v_{m-\mathrm{1}}(R))\frac{\mathrm{1}}{\mathrm{2}(n-\mathrm{1})}|y^{\mathrm{2}}-x^{\mathrm{2}}|\le \frac{R\underset{r\in [\mathrm{0},R]}{\mathrm{m}\mathrm{a}\mathrm{x}}p(r)f(v_{m-\mathrm{1}}(R))}{n-\mathrm{1}}|x-y|.\end{array}$

Hence $\{u_m\}$ (and $\{v_m\}$ ) is equicontinuous on $[\mathrm{0},R]$ for arbitrary $R>\mathrm{0}.$

By Arzelà-Ascoli theorem, $\{u_m\}$ and $\{v_m\}$ have subsequences $\{u_{m_k}\}$ and $\{v_{m_k}\}$ converge uniformly to $u$ and $v$ on $[\mathrm{0},R]$, respectively. Since $\{u_m\}$ and $\{v_m\}$ are monotonically increasing on $[\mathrm{0},\mathrm{\infty })$, $\{u_m\}$ and $\{v_m\}$ converge uniformly to $u$ and $v$ on $[\mathrm{0},R]$, respectively. By the arbitrariness of $R$, we obtain $(u,v)$ is an entire positive $k$-convex radial solution of (1).

(i)$H_a^{}(\mathrm{\infty })=\mathrm{\infty }.$

If $P(\mathrm{\infty })+Q(\mathrm{\infty })<\mathrm{\infty },$ by (8), we have $u(r)+v(r)\le H_a^{-\mathrm{1}}(P(\mathrm{\infty })+Q(\mathrm{\infty })).$So $u$ and $v$ are bounded.

If $P(\mathrm{\infty })=Q(\mathrm{\infty })=\mathrm{\infty },$ then

$u(r)\ge u_{\mathrm{1}}(r)=\frac{a}{\mathrm{2}}+f\left(\frac{a}{\mathrm{2}}\right){\int }_{\mathrm{0}}^r\left[\frac{t^{\mathrm{2}-n}}{n-\mathrm{1}}{\left({\int }_{\mathrm{0}}^{t}n{s}^{n-\mathrm{1}}{p}^{\frac{n}{n-\mathrm{1}}}(s)\mathrm{d}s\right)}^{\frac{n-\mathrm{1}}{n}}\right]\mathrm{d}t=\frac{a}{\mathrm{2}}+f\left(\frac{a}{\mathrm{2}}\right)P(r),\forall r\ge \mathrm{0}.$(9)

$v(r)\ge v_{\mathrm{1}}(r)=\frac{a}{\mathrm{2}}+g\left(\frac{a}{\mathrm{2}}\right){\int }_{\mathrm{0}}^r\left[\frac{t^{\mathrm{2}-n}}{n-\mathrm{1}}{\left({\int }_{\mathrm{0}}^{t}n{s}^{n-\mathrm{1}}{q}^{\frac{n}{n-\mathrm{1}}}(s)\mathrm{d}s\right)}^{\frac{n-\mathrm{1}}{n}}\right]\mathrm{d}t=\frac{a}{\mathrm{2}}+g\left(\frac{a}{\mathrm{2}}\right)Q(r),\forall r\ge \mathrm{0}.$(10)

Let $r\to \mathrm{\infty },$ we have $\underset{r\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}u(r)=\mathrm{\infty },\underset{r\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}v(r)=\mathrm{\infty }.$ Thus, the proof of Theorem 1 is completed.

(ii)$P(\mathrm{\infty })+Q(\mathrm{\infty })<H_a(\mathrm{\infty })=\mathrm{\infty }.$

By (8), (9), and (10), we obtain

$\frac{a}{\mathrm{2}}+f\left(\frac{a}{\mathrm{2}}\right)P(r)\le u(r)\le H_a^{-\mathrm{1}}(P(r)+Q(r)),\forall r\ge \mathrm{0};\frac{a}{\mathrm{2}}+g\left(\frac{a}{\mathrm{2}}\right)Q(r)\le v(r)\le H_a^{-\mathrm{1}}(P(r)+Q(r)),\forall r\ge \mathrm{0}.$

This is precisely what Theorem 2 aims to prove.

References