We now state and prove the main results of this paper.

In order to describe our results conveniently, let us define

${\mathrm{\Phi}}_{p}(x)={|x|}^{p-2}x$,

${\mathrm{\Phi}}_{q}(x)={|x|}^{q-2}x$ and let

${\mathrm{\Phi}}_{{p}^{\prime}}(x)$,

${\mathrm{\Phi}}_{{q}^{\prime}}(x)$ denote inverse functions of

${\mathrm{\Phi}}_{p}$ and

${\mathrm{\Phi}}_{q}$, where

${p}^{\prime}=\frac{p}{p-1}$,

${q}^{\prime}=\frac{q}{q-1}$. Moreover, we define

where $r>0$.

Firstly, we give a nonexistence result of a positive entire radial large solution of system (1.1).

**Theorem 1**
*Suppose that*
*f*
*and*
*g*
*satisfy*
$max\{\underset{s+t\ge 1}{sup}\frac{f(s,t)}{{(s+t)}^{m-1}},\underset{s+t\ge 1}{sup}\frac{g(s,t)}{{(s+t)}^{m-1}}\}<+\mathrm{\infty},$

(2.1)

*and* *a*,

*b* *satisfy the decay conditions* ${\int}_{0}^{\mathrm{\infty}}{\mathrm{\Phi}}_{{p}^{\prime}}(a(t))\phantom{\rule{0.2em}{0ex}}dt<+\mathrm{\infty},\phantom{\rule{2em}{0ex}}{\int}_{0}^{\mathrm{\infty}}{\mathrm{\Phi}}_{{q}^{\prime}}(b(t))\phantom{\rule{0.2em}{0ex}}dt<+\mathrm{\infty},$

(2.2)

*where* $m=min\{p,q\}$, *then problem* (1.1) *has no positive entire radial large solution*.

*Proof* Our proof is by the method of contradiction. That is, we assume that system (1.1) has the positive entire radial large solution

$(u,v)$. From (1.1), we know that

Now, we set

$U(r)=\underset{0\le t\le r}{max}u(r),\phantom{\rule{2em}{0ex}}V(r)=\underset{0\le t\le r}{max}v(r).$

It is easy to see that

$(U,V)$ are positive and nondecreasing functions. Moreover, we have

$U\ge u$,

$V\ge v$ and

$U(r),V(r)\to +\mathrm{\infty}$ as

$r\to +\mathrm{\infty}$. It follows from (2.1) that there exists

${C}_{0}$ such that

$max\{f(s,t),g(s,t)\}<{C}_{0}{(s+t)}^{m-1},\phantom{\rule{1em}{0ex}}s+t\ge 1$

(2.3)

and

$max\{f(s,t),g(s,t)\}<{C}_{0},\phantom{\rule{1em}{0ex}}s+t\le 1.$

(2.4)

Combining (2.3) and (2.4), we can get

$max\{f(s,t),g(s,t)\}<{C}_{0}{(1+s+t)}^{m-1},\phantom{\rule{1em}{0ex}}s+t\ge 0.$

(2.5)

Then we have

$\begin{array}{rcl}f(u(r),v(r))& \le & {C}_{0}{(1+u(r)+u(r))}^{m-1}\\ \le & {C}_{0}{(1+U(r)+V(r))}^{m-1},\phantom{\rule{1em}{0ex}}r\ge 0.\end{array}$

Thus, for all

$r\ge {r}_{0}\ge 0$, we obtain

$\begin{array}{rcl}u(r)& =& u({r}_{0})+{\int}_{{r}_{0}}^{r}{\mathrm{\Phi}}_{{p}^{\prime}}({M}_{1}^{-1}(t){\int}_{0}^{t}{M}_{1}(s)a(s)f(u(s),v(s))\phantom{\rule{0.2em}{0ex}}ds)\phantom{\rule{0.2em}{0ex}}dt\\ \le & u({r}_{0})+C{\int}_{{r}_{0}}^{r}{\mathrm{\Phi}}_{{p}^{\prime}}({M}_{1}^{-1}(t){\int}_{0}^{t}{M}_{1}(s)a(s){(1+U(s)+V(s))}^{m-1}\phantom{\rule{0.2em}{0ex}}ds)\phantom{\rule{0.2em}{0ex}}dt\\ \le & u({r}_{0})+C{\mathrm{\Phi}}_{{p}^{\prime}}\left({(1+U(r)+V(r))}^{m-1}\right){\int}_{{r}_{0}}^{r}{\mathrm{\Phi}}_{{p}^{\prime}}({M}_{1}^{-1}(t){\int}_{0}^{t}{M}_{1}(s)a(s)\phantom{\rule{0.2em}{0ex}}ds)\phantom{\rule{0.2em}{0ex}}dt\\ \le & u({r}_{0})+C(1+U(r)+V(r)){\int}_{{r}_{0}}^{r}{\mathrm{\Phi}}_{{p}^{\prime}}(a(t))\phantom{\rule{0.2em}{0ex}}dt,\end{array}$

where

*C* is a positive constant. Because of

$m=min\{p,q\}$, the last inequality above is valid for

$0<m-1<p-1$. Noticing that (2.2), we choose

${r}_{0}>0$ such that

$max\{{\int}_{{r}_{0}}^{\mathrm{\infty}}{\mathrm{\Phi}}_{{p}^{\prime}}(a(r))\phantom{\rule{0.2em}{0ex}}dr,{\int}_{{r}_{0}}^{\mathrm{\infty}}{\mathrm{\Phi}}_{{q}^{\prime}}(b(r))\phantom{\rule{0.2em}{0ex}}dr\}<\frac{1}{4C}.$

(2.6)

It follows that

${lim}_{r\to \mathrm{\infty}}u(r)={lim}_{r\to \mathrm{\infty}}v(r)=\mathrm{\infty}$, and we can find

${r}_{1}>{r}_{0}$ such that

$\overline{U}(r)=\underset{{r}_{0\le t\le r}}{max}u(t),\phantom{\rule{2em}{0ex}}\overline{V}(r)=\underset{{r}_{0\le t\le r}}{max}v(t),\phantom{\rule{1em}{0ex}}r\ge {r}_{1}.$

(2.7)

Thus, we have

$\overline{U}(r)\le u({r}_{0})+C(1+\overline{U}(r)+\overline{V}(r)){\int}_{{r}_{0}}^{r}{\mathrm{\Phi}}_{{p}^{\prime}}(a(t))\phantom{\rule{0.2em}{0ex}}dt,\phantom{\rule{1em}{0ex}}r\ge {r}_{1}.$

From (2.6), we can get

$\overline{U}(r)\le u({r}_{0})+\frac{(1+\overline{U}(r)+\overline{V}(r))}{4},\phantom{\rule{1em}{0ex}}r\ge {r}_{1},$

that is,

$\overline{U}(r)\le {C}_{1}+\frac{(\overline{U}(r)+\overline{V}(r))}{4},\phantom{\rule{1em}{0ex}}r\ge {r}_{1},$

where

${C}_{1}=C+\frac{1}{4}+u({r}_{0})$,

$r\ge {r}_{1}$. Similarly,

$\overline{V}(r)\le {C}_{2}+\frac{(\overline{U}(r)+\overline{V}(r))}{4},\phantom{\rule{1em}{0ex}}r\ge {r}_{1},$

then we can get

$\overline{U}(r)+\overline{V}(r)\le 2({C}_{1}+{C}_{2}),\phantom{\rule{1em}{0ex}}r\ge {r}_{1},$

(2.8)

which means that *U* and *V* are bounded and so *u* and *v* are bounded, which is a contradiction. It follows that (1.1) has no positive entire radial large solutions. □

**Remark 1** In fact, through a slight change of the proofs of Theorem 1, we can obtain the same result as that of problem (1.1). That is, if

$p,q>2$ and

*f*,

*g* satisfy

$max\{\underset{s+t\ge 1}{sup}\frac{f(s,t)}{s+t},\underset{s+t\ge 1}{sup}\frac{g(s,t)}{s+t}\}<+\mathrm{\infty},$

and *a*, *b* satisfy the decay conditions (2.2), then problem (1.1) still has no positive entire radial large solution.

Secondly, we give existence results of positive entire solutions of system (1.1).

**Theorem 2**
*Suppose that*
$F(\mathrm{\infty})=\mathrm{\infty}.$

*Then system* (1.1)

*has infinitely many positive entire solutions* $(u,v)\in {C}^{2}[0,+\mathrm{\infty})$.

*Moreover*,

*the following hold*:

- (i)
*If* *a* *and* *b* *satisfy* ${H}_{p}a(\mathrm{\infty})={H}_{q}b(\mathrm{\infty})=\mathrm{\infty}$, *then all entire positive solutions of* (1.1) *are large*.

- (ii)
*If* *a* *and* *b* *satisfy* ${H}_{p}a(\mathrm{\infty})<\mathrm{\infty}$, ${H}_{q}b(\mathrm{\infty})<\mathrm{\infty}$, *then all entire positive solutions of* (1.1) *are bounded*.

*Proof* We start by showing that (1.1) has positive radial solutions. To this end, we fix

$c,d>\beta $ and show that the system

$\{\begin{array}{c}{({|{u}^{\prime}|}^{p-2}{u}^{\prime})}^{\prime}+\frac{N-1}{r}({|{u}^{\prime}|}^{p-2}{u}^{\prime})+{m}_{1}(r){|{u}^{\prime}|}^{p-1}=a(r)f(u,v),\hfill \\ {({|{v}^{\prime}|}^{q-2}{v}^{\prime})}^{\prime}+\frac{N-1}{r}({|{v}^{\prime}|}^{q-2}{v}^{\prime})+{m}_{2}(r){|{v}^{\prime}|}^{q-1}=b(r)g(u,v),\hfill \\ {u}^{\prime},{v}^{\prime}\ge 0\phantom{\rule{1em}{0ex}}\text{on}[0,\mathrm{\infty}),\hfill \\ u(0)=d0,\phantom{\rule{2em}{0ex}}v(0)=c0,\hfill \end{array}$

(2.9)

has solutions

$(u,v)$. Thus,

$U(x)=u(|x|)$,

$V(x)=v(|x|)$ are positive solutions of system (1.1). Integrating (2.9), we have

Let

${\{{u}_{k}\}}_{k\ge 0}$ and

${\{{v}_{k}\}}_{k\ge 0}$ be sequences of positive continuous functions defined on

$[0,\mathrm{\infty})$ by

$\{\begin{array}{c}{u}_{0}(r)=d,\phantom{\rule{2em}{0ex}}{v}_{0}(r)=c,\hfill \\ {u}_{k+1}(r)=d+{\int}_{0}^{r}{\mathrm{\Phi}}_{{p}^{\prime}}({M}_{1}^{-1}(t){\int}_{0}^{t}{M}_{1}(s)a(s)f({u}_{k},{v}_{k})\phantom{\rule{0.2em}{0ex}}ds)\phantom{\rule{0.2em}{0ex}}dt,\phantom{\rule{1em}{0ex}}r\ge 0,\hfill \\ {v}_{k+1}(r)=c+{\int}_{0}^{r}{\mathrm{\Phi}}_{{q}^{\prime}}({M}_{2}^{-1}(t){\int}_{0}^{t}{M}_{2}(s)b(s)g({u}_{k},{v}_{k})\phantom{\rule{0.2em}{0ex}}ds)\phantom{\rule{0.2em}{0ex}}dt,\phantom{\rule{1em}{0ex}}r\ge 0.\hfill \end{array}$

Obviously, ${u}_{k}(r)\ge c$, ${v}_{k}(r)\ge d$, ${u}_{0}\le {u}_{1}$, ${v}_{0}\le {v}_{1}$ for all $r\ge 0$. And the monotonicity of *f* and *g* yields ${u}_{1}(r)\le {u}_{2}(r)$, ${v}_{1}(r)\le {v}_{2}(r)$ for $r\ge 0$.

Repeating such arguments, we can deduce that

${u}_{k}(r)\le {u}_{k+1}(r),\phantom{\rule{2em}{0ex}}{v}_{k}(r)\le {v}_{k+1}(r),\phantom{\rule{1em}{0ex}}\text{for}r\ge 0,k\ge 1,$

and

${\{{u}_{k}\}}_{k\ge 0}$,

${\{{v}_{k}\}}_{k\ge 0}$ are nondecreasing sequences on

$[0,\mathrm{\infty})$. Noticing that

$\begin{array}{rcl}{u}_{k+1}^{\prime}(r)& =& {\mathrm{\Phi}}_{{p}^{\prime}}({M}_{1}^{-1}(r){\int}_{0}^{r}{M}_{1}(s)a(s)f({u}_{k}(r),{v}_{k}(r))\phantom{\rule{0.2em}{0ex}}ds)\\ \le & {\mathrm{\Phi}}_{{p}^{\prime}}\left(f({u}_{k}(r),{v}_{k}(r))\right){\mathrm{\Phi}}_{{p}^{\prime}}({M}_{1}^{-1}(r){\int}_{0}^{r}{M}_{1}(s)a(s)\phantom{\rule{0.2em}{0ex}}ds)\\ \le & [{\mathrm{\Phi}}_{{p}^{\prime}}(f({u}_{k+1}+{v}_{k+1},{u}_{k+1}+{v}_{k+1}))+{\mathrm{\Phi}}_{{q}^{\prime}}(g({u}_{k+1}+{v}_{k+1},{u}_{k+1}+{v}_{k+1}))]\\ \times {\mathrm{\Phi}}_{{p}^{\prime}}({M}_{1}^{-1}(r){\int}_{0}^{r}{M}_{1}(s)a(s)\phantom{\rule{0.2em}{0ex}}ds),\end{array}$

and

$\begin{array}{rcl}{v}_{k+1}^{\prime}(r)& =& {\mathrm{\Phi}}_{{q}^{\prime}}({M}_{2}^{-1}(r){\int}_{0}^{r}{M}_{2}(s)b(s)g({u}_{k}(r),{v}_{k}(r))\phantom{\rule{0.2em}{0ex}}ds)\\ \le & {\mathrm{\Phi}}_{{q}^{\prime}}\left(g({u}_{k}(r),{v}_{k}(r))\right){\mathrm{\Phi}}_{{q}^{\prime}}({M}_{2}^{-1}(r){\int}_{0}^{r}{M}_{2}(s)b(s)\phantom{\rule{0.2em}{0ex}}ds)\\ \le & [{\mathrm{\Phi}}_{{p}^{\prime}}(f({u}_{k+1}+{v}_{k+1},{u}_{k+1}+{v}_{k+1}))+{\mathrm{\Phi}}_{{q}^{\prime}}(g({u}_{k+1}+{v}_{k+1},{u}_{k+1}+{v}_{k+1}))]\\ \times {\mathrm{\Phi}}_{{q}^{\prime}}({M}_{2}^{-1}(r){\int}_{0}^{r}{M}_{2}(s)b(s)\phantom{\rule{0.2em}{0ex}}ds),\end{array}$

that is,

$F({u}_{k}(r)+{v}_{k}(r))-F(b+c)\le {H}_{p}a(r)+{H}_{q}b(r),\phantom{\rule{1em}{0ex}}r\ge 0.$

(2.10)

It follows from

${F}^{-1}$ is increasing on

$[0,\mathrm{\infty})$ and (2.10) that

${u}_{k}(r)+{v}_{k}(r)\le {F}^{-1}({H}_{p}a(r)+{H}_{q}b(r)+F(b+c)),\phantom{\rule{1em}{0ex}}r\ge 0.$

(2.11)

And from

$F(\mathrm{\infty})=\mathrm{\infty}$, we know that

${F}^{-1}(\mathrm{\infty})=\mathrm{\infty}$. By (2.11), the sequences

${u}_{k}$ and

${v}_{k}$ are bounded and increasing on

$[0,{c}_{0}]$ for any

${c}_{0}>0$. Thus,

${u}_{k}$ and

${v}_{k}$ have subsequences converging uniformly to

*u* and

*v* on

$[0,{c}_{0}]$. Consequently,

$(u,v)$ is a positive solution of (2.9); therefore,

$(U,V)$ is an entire positive solution of (1.1). Noticing that

$U(0)=c$,

$V(0)=d$ and

$(c,d)\in (0,\mathrm{\infty})\times (0,\mathrm{\infty})$ are chosen arbitrarily, we can obtain that system (1.1) has infinitely many positive entire solutions.

- (i)
If

${H}_{p}a(\mathrm{\infty})={H}_{q}b(\mathrm{\infty})=\mathrm{\infty}$, since

$u(r)\ge c+{\mathrm{\Phi}}_{{p}^{\prime}}(f(c,d)){H}_{p}a(r)$,

$v(r)\ge d+{\mathrm{\Phi}}_{{q}^{\prime}}(g(c,d)){H}_{q}b(r)$ for

$r\ge 0$, we have

$\underset{r\to \mathrm{\infty}}{lim}u(r)=\underset{r\to \mathrm{\infty}}{lim}v(r)=\mathrm{\infty},$

which yields

$(U,V)$ is the positive entire large solution of (1.1).

- (ii)
If

${H}_{p}a(\mathrm{\infty})<\mathrm{\infty}$,

${H}_{q}b(\mathrm{\infty})<\mathrm{\infty}$, then

$u(r)+v(r)\le {F}^{-1}(F(b+c)+{H}_{p}a(\mathrm{\infty})+{H}_{q}b(\mathrm{\infty}))<\mathrm{\infty},$

which implies that $(U,V)$ is the positive entire bounded solution of system (1.1). Thus, the proof of Theorem 2 is finished.

□

**Theorem 3** *If* $F(\mathrm{\infty})<\mathrm{\infty}$,

${H}_{p}a(\mathrm{\infty})<\mathrm{\infty}$,

${H}_{q}b(\mathrm{\infty})<\mathrm{\infty}$,

*and there exist* $c>\beta $,

$d>\beta $ *such that* ${H}_{p}a(\mathrm{\infty})+{H}_{q}b(\mathrm{\infty})<F(\mathrm{\infty})-F(c+d),$

(2.12)

*then system* (1.1)

*has an entire positive radial bounded solution* $(u,v)\in {C}^{1+\theta}([0,\mathrm{\infty}))\times {C}^{1+\theta}([0,\mathrm{\infty}))$ (

*for* $0<\theta <1$)

*satisfying* *Proof* If the condition (2.12) holds, then we have

$\begin{array}{rcl}F({u}_{k}+{v}_{k}(r))& \le & F(c+d)+{H}_{p}a(r)+{H}_{q}b(r)\\ \le & F(c+d)+{H}_{p}a(\mathrm{\infty})+{H}_{q}b(\mathrm{\infty})<F(\mathrm{\infty})<\mathrm{\infty}.\end{array}$

Since

${F}^{-1}$ is strictly increasing on

$[0,\mathrm{\infty})$, we have

${u}_{k}+{v}_{k}(r)\le {F}^{-1}(F(b+c)+{H}_{p}a(\mathrm{\infty})+{H}_{q}b(\mathrm{\infty}))<\mathrm{\infty}.$

The rest of the proof obviously holds from the proof of Theorem 2. The proof of Theorem 3 is now finished. □

**Theorem 4**
- (i)
*If*
${H}_{p}a(\mathrm{\infty})={H}_{q}b(\mathrm{\infty})=\mathrm{\infty},$

*and*
$\underset{s\to \mathrm{\infty}}{lim}\frac{f(s,s)+g(s,s)}{s}=0,$

(2.13)

*then system* (1.1)

*has infinitely many positive entire large solutions*.

- (ii)
*If* ${H}_{p}a(\mathrm{\infty})<\mathrm{\infty}$,

${H}_{q}b(\mathrm{\infty})<\mathrm{\infty}$,

*and* $\underset{s\ge 0}{sup}(f(s,s)+g(s,s))<\mathrm{\infty},$

*then system* (1.1) *has infinitely many positive entire bounded solutions*.

*Proof*
- (i)
It follows from the proof of Theorem 2 that

$\begin{array}{rcl}{u}_{k}(r)& \le & {u}_{k+1}(r)\le {\mathrm{\Phi}}_{{p}^{\prime}}\left(f({u}_{k}(r),{v}_{k}(r))\right){H}_{p}a(r)\\ \le & {\mathrm{\Phi}}_{{p}^{\prime}}\left(f({u}_{k}(r)+{v}_{k}(r),{u}_{k}(r)+{v}_{k}(r))\right){H}_{p}a(r)\end{array}$

(2.14)

and

$\begin{array}{rcl}{v}_{k}(r)& \le & {v}_{k+1}(r)\le {\mathrm{\Phi}}_{{q}^{\prime}}\left(g({u}_{k}(r),{v}_{k}(r))\right){H}_{q}b(r)\\ \le & {\mathrm{\Phi}}_{{q}^{\prime}}\left(g({u}_{k}(r)+{v}_{k}(r),{u}_{k}(r)+{v}_{k}(r))\right){H}_{q}b(r).\end{array}$

(2.15)

Choosing an arbitrary

$R>0$, from (2.14) and (2.15), we can get

$\begin{array}{rcl}{u}_{k}(R)+{v}_{k}(R)& \le & c+d+{\mathrm{\Phi}}_{{p}^{\prime}}\left(f({u}_{k}(r)+{v}_{k}(r),{u}_{k}(r)+{v}_{k}(r))\right){H}_{p}a(r)\\ +{\mathrm{\Phi}}_{{q}^{\prime}}\left(g({u}_{k}(r)+{v}_{k}(r),{u}_{k}(r)+{v}_{k}(r))\right){H}_{q}b(r)\\ \le & c+d+[{\mathrm{\Phi}}_{{p}^{\prime}}\left(f({u}_{k}(r)+{v}_{k}(r),{u}_{k}(r)+{v}_{k}(r))\right)+{\mathrm{\Phi}}_{{q}^{\prime}}(g({u}_{k}(r)\\ +{v}_{k}(r),{u}_{k}(r)+{v}_{k}(r)\left)\right)]({H}_{p}a(r)+{H}_{q}b(r)),\phantom{\rule{1em}{0ex}}k\ge 1,\end{array}$

which implies

$\begin{array}{rcl}1& \le & \frac{c+d}{{u}_{k}(R)+{v}_{k}(R)}\\ +\frac{{\mathrm{\Phi}}_{{p}^{\prime}}(f({u}_{k}(r)+{v}_{k}(r),{u}_{k}(r)+{v}_{k}(r)))+{\mathrm{\Phi}}_{{q}^{\prime}}(g({u}_{k}(r)+{v}_{k}(r),{u}_{k}(r)+{v}_{k}(r)))}{{u}_{k}(R)+{v}_{k}(R)}\\ \times ({H}_{p}a(r)+{H}_{q}b(r)),\phantom{\rule{1em}{0ex}}k\ge 1.\end{array}$

(2.16)

Taking account of the monotonicity of

${({u}_{k}(R)+{v}_{k}(R))}_{k\ge 1}$, there exists

$L(R)=\underset{k\to \mathrm{\infty}}{lim}({u}_{k}(R)+{v}_{k}(R)).$

We claim that

$L(R)$ is finite. Indeed, if not, we let

$k\to \mathrm{\infty}$ in (2.16) and the assumption (2.13) leads to a contradiction. Thus,

$L(R)$ is finite. Since

${u}_{k}$,

${v}_{k}$ are increasing functions, it follows that the map

$L:(0,\mathrm{\infty})\to (0,\mathrm{\infty})$ is nondecreasing and

${u}_{k}(r)+{v}_{k}(r)\le {u}_{k}(R)+{v}_{k}(R)\le L(R),\phantom{\rule{1em}{0ex}}r\in [0,R],k\ge 1.$

Thus, the sequences

${({u}_{k})}_{k\ge 1}$ and

${({v}_{k})}_{k\ge 1}$ are bounded from above on bounded sets. Let

$u(r)=\underset{k\to \mathrm{\infty}}{lim}{u}_{k}(r),\phantom{\rule{2em}{0ex}}v(r)=\underset{k\to \mathrm{\infty}}{lim}{v}_{k}(r),\phantom{\rule{1em}{0ex}}r\ge 0,$

then $(u,v)$ is a positive solution of (2.9).

In order to conclude the proof, we need to show that

$(u,v)$ is a large solution of (2.9). By the proof of Theorem 2, we have

$u(r)\ge c+{\mathrm{\Phi}}_{{p}^{\prime}}(f(c,d)){H}_{p}a(r),\phantom{\rule{2em}{0ex}}v(r)\ge d+{\mathrm{\Phi}}_{{q}^{\prime}}(g(c,d)){H}_{q}b(r),\phantom{\rule{1em}{0ex}}r\ge 0.$

And because

*f* and

*g* are positive functions and

${H}_{p}a(\mathrm{\infty})={H}_{q}b(\mathrm{\infty})=\mathrm{\infty},$

we can conclude that

$(u,v)$ is a large solution of (2.9) and so

$(U,V)$ is a positive entire large solution of (1.1). Thus, any large solution of (2.9) provides a positive entire large solution

$(U,V)$ of (1.1) with

$U(0)=c$ and

$V(0)=d$. Since

$(c,d)\in (0,\mathrm{\infty})\times (0,\mathrm{\infty})$ was chosen arbitrarily, it follows that (1.1) has infinitely many positive entire large solutions.

- (ii)
If

$\underset{s\ge 0}{sup}(f(s,s)+g(s,s))<\mathrm{\infty},$

holds, then by (2.16), we have

$L(R)=\underset{k\to \mathrm{\infty}}{lim}({u}_{k}(R)+{v}_{k}(R))<\mathrm{\infty}.$

Thus,

${u}_{k}(r)+{v}_{k}(r)\le {u}_{k}(R)+{v}_{k}(R)\le L(R),\phantom{\rule{1em}{0ex}}r\in [0,R],k\ge 1.$

Thus, the sequences

${({u}_{k})}_{k\ge 1}$ and

${({v}_{k})}_{k\ge 1}$ are bounded from above on bounded sets. Let

$u(r)=\underset{k\to \mathrm{\infty}}{lim}{u}_{k}(r),\phantom{\rule{2em}{0ex}}v(r)=\underset{k\to \mathrm{\infty}}{lim}{v}_{k}(r),\phantom{\rule{1em}{0ex}}r\ge 0,$

then $(u,v)$ is a positive solution of (2.9).

It follows from (2.14) and (2.15) that $(u,v)$ is bounded, which implies that (1.1) has infinitely many positive entire bounded solutions. □