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}^{p2}x, {\mathrm{\Phi}}_{q}(x)={x}^{q2}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}{p1}, {q}^{\prime}=\frac{q}{q1}. 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)}^{m1}},\underset{s+t\ge 1}{sup}\frac{g(s,t)}{{(s+t)}^{m1}}\}<+\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)}^{m1},\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)}^{m1},\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))}^{m1}\\ \le & {C}_{0}{(1+U(r)+V(r))}^{m1},\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))}^{m1}\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))}^{m1}\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<m1<p1. 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}}^{p2}{u}^{\prime})}^{\prime}+\frac{N1}{r}({{u}^{\prime}}^{p2}{u}^{\prime})+{m}_{1}(r){{u}^{\prime}}^{p1}=a(r)f(u,v),\hfill \\ {({{v}^{\prime}}^{q2}{v}^{\prime})}^{\prime}+\frac{N1}{r}({{v}^{\prime}}^{q2}{v}^{\prime})+{m}_{2}(r){{v}^{\prime}}^{q1}=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}
it follows that
Then we can get
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. □