Open Access

Existence and nonexistence of entire positive solutions for ( p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq1_HTML.gif-Laplacian elliptic system with a gradient term

Boundary Value Problems20132013:18

DOI: 10.1186/1687-2770-2013-18

Received: 10 August 2012

Accepted: 9 January 2013

Published: 5 March 2013

Abstract

This work is concerned with the entire positive solutions for a ( p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq1_HTML.gif-Laplacian elliptic system of equations with a gradient term. We find the sufficient condition for nonexistence of entire large positive solutions and existence of infinitely many entire solutions, which are large or bounded.

1 Introduction

In this paper, we consider a class of ( p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq1_HTML.gif-Laplacian elliptic system of equations with a gradient term
{ div ( | u | p 2 u ) + m 1 ( | x | ) | u | p 1 = a ( | x | ) f ( u , v ) , in  R N , div ( | v | p 2 v ) + m 2 ( | x | ) | v | q 1 = b ( | x | ) g ( u , v ) , in  R N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equ1_HTML.gif
(1.1)

where N > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq2_HTML.gif, p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq3_HTML.gif, q 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq4_HTML.gif, the nonlinearities f , g : [ 0 , ) × [ 0 , ) ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq5_HTML.gif are positive, continuous and nondecreasing functions for each variable, m 1 ( | x | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq6_HTML.gif and m 2 ( | x | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq7_HTML.gif are continuous functions, and the potentials a , b C ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq8_HTML.gif are c-positive functions (or circumferentially positive) in a domain Ω R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq9_HTML.gif which are nonnegative in Ω and satisfy the following:

  • If x 0 Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq10_HTML.gif and a ( x 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq11_HTML.gif, then there exists a domain Ω 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq12_HTML.gif such that x 0 Ω 0 Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq13_HTML.gif and a ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq14_HTML.gif for all x Ω 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq15_HTML.gif.

Problem (1.1) arises in the theory of quasiregular and quasiconformal mappings, stochastic control and non-Newtonian fluids, etc. In the non-Newtonian theory, the quantity ( p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq1_HTML.gif is a characteristic of the medium. Media with ( p , q ) > ( 2 , 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq16_HTML.gif are called dilatant fluids, while ( p , q ) < ( 2 , 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq17_HTML.gif are called pseudoplastics. If ( p , q ) = ( 2 , 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq18_HTML.gif, they are Newtonian fluids.

We are concerned only with the entire positive solutions of problem (1.1). An entire large (or explosive) solution of problem (1.1) means a pair of functions ( u , v ) C 1 + θ ( R N ) × C 1 + θ ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq19_HTML.gif for θ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq20_HTML.gif solving problem (1.1) in the weak sense and u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq21_HTML.gif, v ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq22_HTML.gif as | x | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq23_HTML.gif.

In recent years, existence and nonexistence of entire solutions for the semilinear elliptic system
{ u + f ( x , u , v ) = 0 , in  R N , v + f ( x , u , v ) = 0 , in  R N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equa_HTML.gif
have been studied by many authors; see [13] and the references therein. For example, Ghergu and Radulescu [1], Lair and Wood [2], Kawano and Kusano [3] discussed the entire solutions under proper conditions. For other works for a single equation, we refer to [4, 5] and the references therein. Moreover, a comprehensive discussion on entire solutions for a large class of semilinear systems
{ u = a ( | x | ) f ( v ) , in  R N , v = b ( | x | ) g ( u ) , in  R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equb_HTML.gif
can be found in Ghergu and Radulescu [1]. Later, Yang [6] extended their results to a class of quasilinear elliptic systems. To our best knowledge, problem (1.1) of equations with a gradient term has not been sufficiently investigated. Only a few papers have dealt with this problem (1.1). In [7], Ghergu and Radulescu studied the existence of blow-up solutions for the system
{ u + | u | = a ( | x | ) f ( v ) , in  Ω , v + | v | = b ( | x | ) g ( u ) , in  Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equc_HTML.gif
They proved that boundary blow-up solutions fail to exist if f and g are sublinear, whereas this result holds if Ω = R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq24_HTML.gif is bounded and a, b are slow decay at infinity. They also showed the existence of infinitely blow-up solutions in Ω = R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq24_HTML.gif if a, b are of fast decay and f, g satisfy a sublinear-type growth condition at infinity. In [8], Cirstea and Radulescu studied a related problem. Recently, Zhang and Liu [9] studied the semilinear elliptic systems with a gradient term
{ u + | u | = a ( | x | ) f ( u , v ) , in  R N , v + | v | = b ( | x | ) g ( u , v ) , in  R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equd_HTML.gif

and obtained the sufficient condition of nonexistence and existence of positive entire solutions. Furthermore, for the single equation with a gradient term, we read [1012] and the references therein.

Motivated by the results of the above cited papers, we study the nonexistence and existence of positive entire solutions for system (1.1) deeply, and the results of the semilinear systems are extended to the quasilinear ones. In [13], the authors studied the existence and nonexistence of entire large positive solutions of ( p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq1_HTML.gif-Laplacian system (1.1) with f ( u , v ) = φ ( v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq25_HTML.gif, g ( u , v ) = ψ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq26_HTML.gif. However, they obtained different results under the suitable conditions. In this paper, our main purpose is to establish new results under new conditions for system (1.1). Roughly speaking, we find that the entire large positive solutions fail to exist if f, g are sublinear and a, b have fast decay at infinity, while f, g satisfy some growth conditions at infinity, and a, b are of slow decay or fast decay at infinity, then the system has many infinitely entire solutions, which are large or bounded. Unfortunately, it remains unknown whether an analogous result holds for system (1.1) with different gradient power m 1 ( | x | ) | u | α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq27_HTML.gif, m 2 ( | x | ) | u | α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq28_HTML.gif for 0 < α < p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq29_HTML.gif.

2 Main results and proof

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

In order to describe our results conveniently, let us define Φ p ( x ) = | x | p 2 x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq30_HTML.gif, Φ q ( x ) = | x | q 2 x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq31_HTML.gif and let Φ p ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq32_HTML.gif, Φ q ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq33_HTML.gif denote inverse functions of Φ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq34_HTML.gif and Φ q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq35_HTML.gif, where p = p p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq36_HTML.gif, q = q q 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq37_HTML.gif. Moreover, we define
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Eque_HTML.gif

where r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq38_HTML.gif.

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 { sup s + t 1 f ( s , t ) ( s + t ) m 1 , sup s + t 1 g ( s , t ) ( s + t ) m 1 } < + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equ2_HTML.gif
(2.1)
and a, b satisfy the decay conditions
0 Φ p ( a ( t ) ) d t < + , 0 Φ q ( b ( t ) ) d t < + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equ3_HTML.gif
(2.2)

where m = min { p , q } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq39_HTML.gif, 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq40_HTML.gif. From (1.1), we know that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equf_HTML.gif
Now, we set
U ( r ) = max 0 t r u ( r ) , V ( r ) = max 0 t r v ( r ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equg_HTML.gif
It is easy to see that ( U , V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq41_HTML.gif are positive and nondecreasing functions. Moreover, we have U u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq42_HTML.gif, V v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq43_HTML.gif and U ( r ) , V ( r ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq44_HTML.gif as r + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq45_HTML.gif. It follows from (2.1) that there exists C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq46_HTML.gif such that
max { f ( s , t ) , g ( s , t ) } < C 0 ( s + t ) m 1 , s + t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equ4_HTML.gif
(2.3)
and
max { f ( s , t ) , g ( s , t ) } < C 0 , s + t 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equ5_HTML.gif
(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 , s + t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equ6_HTML.gif
(2.5)
Then we have
f ( u ( r ) , v ( r ) ) C 0 ( 1 + u ( r ) + u ( r ) ) m 1 C 0 ( 1 + U ( r ) + V ( r ) ) m 1 , r 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equh_HTML.gif
Thus, for all r r 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq47_HTML.gif, we obtain
u ( r ) = u ( r 0 ) + r 0 r Φ p ( M 1 1 ( t ) 0 t M 1 ( s ) a ( s ) f ( u ( s ) , v ( s ) ) d s ) d t u ( r 0 ) + C r 0 r Φ p ( M 1 1 ( t ) 0 t M 1 ( s ) a ( s ) ( 1 + U ( s ) + V ( s ) ) m 1 d s ) d t u ( r 0 ) + C Φ p ( ( 1 + U ( r ) + V ( r ) ) m 1 ) r 0 r Φ p ( M 1 1 ( t ) 0 t M 1 ( s ) a ( s ) d s ) d t u ( r 0 ) + C ( 1 + U ( r ) + V ( r ) ) r 0 r Φ p ( a ( t ) ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equi_HTML.gif
where C is a positive constant. Because of m = min { p , q } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq39_HTML.gif, the last inequality above is valid for 0 < m 1 < p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq48_HTML.gif. Noticing that (2.2), we choose r 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq49_HTML.gif such that
max { r 0 Φ p ( a ( r ) ) d r , r 0 Φ q ( b ( r ) ) d r } < 1 4 C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equ7_HTML.gif
(2.6)
It follows that lim r u ( r ) = lim r v ( r ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq50_HTML.gif, and we can find r 1 > r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq51_HTML.gif such that
U ¯ ( r ) = max r 0 t r u ( t ) , V ¯ ( r ) = max r 0 t r v ( t ) , r r 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equ8_HTML.gif
(2.7)
Thus, we have
U ¯ ( r ) u ( r 0 ) + C ( 1 + U ¯ ( r ) + V ¯ ( r ) ) r 0 r Φ p ( a ( t ) ) d t , r r 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equj_HTML.gif
From (2.6), we can get
U ¯ ( r ) u ( r 0 ) + ( 1 + U ¯ ( r ) + V ¯ ( r ) ) 4 , r r 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equk_HTML.gif
that is,
U ¯ ( r ) C 1 + ( U ¯ ( r ) + V ¯ ( r ) ) 4 , r r 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equl_HTML.gif
where C 1 = C + 1 4 + u ( r 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq52_HTML.gif, r r 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq53_HTML.gif. Similarly,
V ¯ ( r ) C 2 + ( U ¯ ( r ) + V ¯ ( r ) ) 4 , r r 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equm_HTML.gif
then we can get
U ¯ ( r ) + V ¯ ( r ) 2 ( C 1 + C 2 ) , r r 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equ9_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq54_HTML.gif and f, g satisfy
max { sup s + t 1 f ( s , t ) s + t , sup s + t 1 g ( s , t ) s + t } < + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equn_HTML.gif

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 ( ) = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equo_HTML.gif
Then system (1.1) has infinitely many positive entire solutions ( u , v ) C 2 [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq55_HTML.gif. Moreover, the following hold:
  1. (i)

    If a and b satisfy H p a ( ) = H q b ( ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq56_HTML.gif, then all entire positive solutions of (1.1) are large.

     
  2. (ii)

    If a and b satisfy H p a ( ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq57_HTML.gif, H q b ( ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq58_HTML.gif, 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 > β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq59_HTML.gif and show that the system
{ ( | u | p 2 u ) + N 1 r ( | u | p 2 u ) + m 1 ( r ) | u | p 1 = a ( r ) f ( u , v ) , ( | v | q 2 v ) + N 1 r ( | v | q 2 v ) + m 2 ( r ) | v | q 1 = b ( r ) g ( u , v ) , u , v 0 on  [ 0 , ) , u ( 0 ) = d > 0 , v ( 0 ) = c > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equ10_HTML.gif
(2.9)
has solutions ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq60_HTML.gif. Thus, U ( x ) = u ( | x | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq61_HTML.gif, V ( x ) = v ( | x | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq62_HTML.gif are positive solutions of system (1.1). Integrating (2.9), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equp_HTML.gif
Let { u k } k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq63_HTML.gif and { v k } k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq64_HTML.gif be sequences of positive continuous functions defined on [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq65_HTML.gif by
{ u 0 ( r ) = d , v 0 ( r ) = c , u k + 1 ( r ) = d + 0 r Φ p ( M 1 1 ( t ) 0 t M 1 ( s ) a ( s ) f ( u k , v k ) d s ) d t , r 0 , v k + 1 ( r ) = c + 0 r Φ q ( M 2 1 ( t ) 0 t M 2 ( s ) b ( s ) g ( u k , v k ) d s ) d t , r 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equq_HTML.gif

Obviously, u k ( r ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq66_HTML.gif, v k ( r ) d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq67_HTML.gif, u 0 u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq68_HTML.gif, v 0 v 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq69_HTML.gif for all r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq70_HTML.gif. And the monotonicity of f and g yields u 1 ( r ) u 2 ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq71_HTML.gif, v 1 ( r ) v 2 ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq72_HTML.gif for r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq70_HTML.gif.

Repeating such arguments, we can deduce that
u k ( r ) u k + 1 ( r ) , v k ( r ) v k + 1 ( r ) , for  r 0 , k 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equr_HTML.gif
and { u k } k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq63_HTML.gif, { v k } k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq64_HTML.gif are nondecreasing sequences on [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq65_HTML.gif. Noticing that
u k + 1 ( r ) = Φ p ( M 1 1 ( r ) 0 r M 1 ( s ) a ( s ) f ( u k ( r ) , v k ( r ) ) d s ) Φ p ( f ( u k ( r ) , v k ( r ) ) ) Φ p ( M 1 1 ( r ) 0 r M 1 ( s ) a ( s ) d s ) [ Φ p ( f ( u k + 1 + v k + 1 , u k + 1 + v k + 1 ) ) + Φ q ( g ( u k + 1 + v k + 1 , u k + 1 + v k + 1 ) ) ] × Φ p ( M 1 1 ( r ) 0 r M 1 ( s ) a ( s ) d s ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equs_HTML.gif
and
v k + 1 ( r ) = Φ q ( M 2 1 ( r ) 0 r M 2 ( s ) b ( s ) g ( u k ( r ) , v k ( r ) ) d s ) Φ q ( g ( u k ( r ) , v k ( r ) ) ) Φ q ( M 2 1 ( r ) 0 r M 2 ( s ) b ( s ) d s ) [ Φ p ( f ( u k + 1 + v k + 1 , u k + 1 + v k + 1 ) ) + Φ q ( g ( u k + 1 + v k + 1 , u k + 1 + v k + 1 ) ) ] × Φ q ( M 2 1 ( r ) 0 r M 2 ( s ) b ( s ) d s ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equt_HTML.gif
it follows that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equu_HTML.gif
Then we can get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equv_HTML.gif
that is,
F ( u k ( r ) + v k ( r ) ) F ( b + c ) H p a ( r ) + H q b ( r ) , r 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equ11_HTML.gif
(2.10)
It follows from F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq73_HTML.gif is increasing on [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq65_HTML.gif and (2.10) that
u k ( r ) + v k ( r ) F 1 ( H p a ( r ) + H q b ( r ) + F ( b + c ) ) , r 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equ12_HTML.gif
(2.11)
And from F ( ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq74_HTML.gif, we know that F 1 ( ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq75_HTML.gif. By (2.11), the sequences u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq76_HTML.gif and v k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq77_HTML.gif are bounded and increasing on [ 0 , c 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq78_HTML.gif for any c 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq79_HTML.gif. Thus, u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq76_HTML.gif and v k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq77_HTML.gif have subsequences converging uniformly to u and v on [ 0 , c 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq78_HTML.gif. Consequently, ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq60_HTML.gif is a positive solution of (2.9); therefore, ( U , V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq80_HTML.gif is an entire positive solution of (1.1). Noticing that U ( 0 ) = c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq81_HTML.gif, V ( 0 ) = d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq82_HTML.gif and ( c , d ) ( 0 , ) × ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq83_HTML.gif are chosen arbitrarily, we can obtain that system (1.1) has infinitely many positive entire solutions.
  1. (i)
    If H p a ( ) = H q b ( ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq84_HTML.gif, since u ( r ) c + Φ p ( f ( c , d ) ) H p a ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq85_HTML.gif, v ( r ) d + Φ q ( g ( c , d ) ) H q b ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq86_HTML.gif for r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq70_HTML.gif, we have
    lim r u ( r ) = lim r v ( r ) = , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equw_HTML.gif
     
which yields ( U , V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq87_HTML.gif is the positive entire large solution of (1.1).
  1. (ii)
    If H p a ( ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq88_HTML.gif, H q b ( ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq58_HTML.gif, then
    u ( r ) + v ( r ) F 1 ( F ( b + c ) + H p a ( ) + H q b ( ) ) < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equx_HTML.gif
     

which implies that ( U , V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq41_HTML.gif is the positive entire bounded solution of system (1.1). Thus, the proof of Theorem 2 is finished.

 □

Theorem 3 If F ( ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq89_HTML.gif, H p a ( ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq88_HTML.gif, H q b ( ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq58_HTML.gif, and there exist c > β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq90_HTML.gif, d > β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq91_HTML.gif such that
H p a ( ) + H q b ( ) < F ( ) F ( c + d ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equ13_HTML.gif
(2.12)
then system (1.1) has an entire positive radial bounded solution ( u , v ) C 1 + θ ( [ 0 , ) ) × C 1 + θ ( [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq92_HTML.gif (for 0 < θ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq93_HTML.gif) satisfying
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equy_HTML.gif
Proof If the condition (2.12) holds, then we have
F ( u k + v k ( r ) ) F ( c + d ) + H p a ( r ) + H q b ( r ) F ( c + d ) + H p a ( ) + H q b ( ) < F ( ) < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equz_HTML.gif
Since F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq73_HTML.gif is strictly increasing on [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq65_HTML.gif, we have
u k + v k ( r ) F 1 ( F ( b + c ) + H p a ( ) + H q b ( ) ) < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equaa_HTML.gif

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

Theorem 4
  1. (i)
    If
    H p a ( ) = H q b ( ) = , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equab_HTML.gif
     
and
lim s f ( s , s ) + g ( s , s ) s = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equ14_HTML.gif
(2.13)
then system (1.1) has infinitely many positive entire large solutions.
  1. (ii)
    If H p a ( ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq88_HTML.gif, H q b ( ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq58_HTML.gif, and
    sup s 0 ( f ( s , s ) + g ( s , s ) ) < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equac_HTML.gif
     

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

Proof
  1. (i)
    It follows from the proof of Theorem 2 that
    u k ( r ) u k + 1 ( r ) Φ p ( f ( u k ( r ) , v k ( r ) ) ) H p a ( r ) Φ p ( f ( u k ( r ) + v k ( r ) , u k ( r ) + v k ( r ) ) ) H p a ( r ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equ15_HTML.gif
    (2.14)
     
and
v k ( r ) v k + 1 ( r ) Φ q ( g ( u k ( r ) , v k ( r ) ) ) H q b ( r ) Φ q ( g ( u k ( r ) + v k ( r ) , u k ( r ) + v k ( r ) ) ) H q b ( r ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equ16_HTML.gif
(2.15)
Choosing an arbitrary R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq94_HTML.gif, from (2.14) and (2.15), we can get
u k ( R ) + v k ( R ) c + d + Φ p ( f ( u k ( r ) + v k ( r ) , u k ( r ) + v k ( r ) ) ) H p a ( r ) + Φ q ( g ( u k ( r ) + v k ( r ) , u k ( r ) + v k ( r ) ) ) H q b ( r ) c + d + [ Φ p ( f ( u k ( r ) + v k ( r ) , u k ( r ) + v k ( r ) ) ) + Φ q ( g ( u k ( r ) + v k ( r ) , u k ( r ) + v k ( r ) ) ) ] ( H p a ( r ) + H q b ( r ) ) , k 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equad_HTML.gif
which implies
1 c + d u k ( R ) + v k ( R ) + Φ p ( f ( u k ( r ) + v k ( r ) , u k ( r ) + v k ( r ) ) ) + Φ q ( g ( u k ( r ) + v k ( r ) , u k ( r ) + v k ( r ) ) ) u k ( R ) + v k ( R ) × ( H p a ( r ) + H q b ( r ) ) , k 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equ17_HTML.gif
(2.16)
Taking account of the monotonicity of ( u k ( R ) + v k ( R ) ) k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq95_HTML.gif, there exists
L ( R ) = lim k ( u k ( R ) + v k ( R ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equae_HTML.gif
We claim that L ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq96_HTML.gif is finite. Indeed, if not, we let k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq97_HTML.gif in (2.16) and the assumption (2.13) leads to a contradiction. Thus, L ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq96_HTML.gif is finite. Since u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq76_HTML.gif, v k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq77_HTML.gif are increasing functions, it follows that the map L : ( 0 , ) ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq98_HTML.gif is nondecreasing and
u k ( r ) + v k ( r ) u k ( R ) + v k ( R ) L ( R ) , r [ 0 , R ] , k 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equaf_HTML.gif
Thus, the sequences ( u k ) k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq99_HTML.gif and ( v k ) k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq100_HTML.gif are bounded from above on bounded sets. Let
u ( r ) = lim k u k ( r ) , v ( r ) = lim k v k ( r ) , r 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equag_HTML.gif

then ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq60_HTML.gif is a positive solution of (2.9).

In order to conclude the proof, we need to show that ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq60_HTML.gif is a large solution of (2.9). By the proof of Theorem 2, we have
u ( r ) c + Φ p ( f ( c , d ) ) H p a ( r ) , v ( r ) d + Φ q ( g ( c , d ) ) H q b ( r ) , r 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equah_HTML.gif
And because f and g are positive functions and
H p a ( ) = H q b ( ) = , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equai_HTML.gif
we can conclude that ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq60_HTML.gif is a large solution of (2.9) and so ( U , V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq80_HTML.gif is a positive entire large solution of (1.1). Thus, any large solution of (2.9) provides a positive entire large solution ( U , V ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq80_HTML.gif of (1.1) with U ( 0 ) = c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq101_HTML.gif and V ( 0 ) = d https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq102_HTML.gif. Since ( c , d ) ( 0 , ) × ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq103_HTML.gif was chosen arbitrarily, it follows that (1.1) has infinitely many positive entire large solutions.
  1. (ii)
    If
    sup s 0 ( f ( s , s ) + g ( s , s ) ) < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equaj_HTML.gif
     
holds, then by (2.16), we have
L ( R ) = lim k ( u k ( R ) + v k ( R ) ) < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equak_HTML.gif
Thus,
u k ( r ) + v k ( r ) u k ( R ) + v k ( R ) L ( R ) , r [ 0 , R ] , k 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equal_HTML.gif
Thus, the sequences ( u k ) k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq99_HTML.gif and ( v k ) k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq100_HTML.gif are bounded from above on bounded sets. Let
u ( r ) = lim k u k ( r ) , v ( r ) = lim k v k ( r ) , r 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equam_HTML.gif

then ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq40_HTML.gif is a positive solution of (2.9).

It follows from (2.14) and (2.15) that ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq60_HTML.gif is bounded, which implies that (1.1) has infinitely many positive entire bounded solutions. □

Declarations

Acknowledgements

The first and second authors were supported by the National Science Foundation of Shandong Province of China (ZR2012AM018) and Changwon National University in 2013, respectively. The authors would like to express their sincere gratitude to the anonymous reviewers for their insightful and constructive comments.

Authors’ Affiliations

(1)
School of Mathematical Sciences, Ocean University of China
(2)
Department of Mathematics, Changwon National University

References

  1. Ghergu M, Radulescu V Oxford Lecture Ser. Math. Appl. 17. In Singular Elliptic Equations: Bifurcation and Asymptotic Analysis. Oxford University Press, London; 2008.
  2. Lair AV, Wood AW: Existence of entire large positive solutions of semilinear elliptic systems. J. Differ. Equ. 2000, 164: 380-394. 10.1006/jdeq.2000.3768MATHMathSciNetView Article
  3. Kawano N, Kusano T: On positive entire solutions of a class of second order semilinear elliptic systems. Math. Z. 1984, 186(3):287-297. 10.1007/BF01174883MATHMathSciNetView Article
  4. Lair AV, Shaker AW: Entire solution of a singular semilinear elliptic problem. J. Math. Anal. Appl. 1996, 200: 498-505. 10.1006/jmaa.1996.0218MATHMathSciNetView Article
  5. Marcus M, Veron L: Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1997, 14: 237-274. 10.1016/S0294-1449(97)80146-1MATHMathSciNetView Article
  6. Yang Z: Existence of entire explosive positive radial solutions for a class of quasilinear elliptic systems. J. Math. Anal. Appl. 2003, 288: 768-783. 10.1016/j.jmaa.2003.09.027MATHMathSciNetView Article
  7. Ghergu M, Radulescu V: Explosive solutions of semilinear elliptic systems with gradient term. Rev. R. Acad. Cienc. Ser. a Mat. 2003, 97(3):437-445.MathSciNet
  8. Cirstea F, Radulescu V: Entire solutions blowing up at infinity for semilinear elliptic systems. J. Math. Pures Appl. 2002, 81: 827-846.MATHMathSciNetView Article
  9. Zhang X, Liu L: The existence and nonexistence of entire positive solutions of semilinear elliptic systems with gradient term. J. Math. Anal. Appl. 2010, 371: 300-308. 10.1016/j.jmaa.2010.05.029MATHMathSciNetView Article
  10. Guo Z: Existence and uniqueness of positive radial solutions for a class of quasilinear elliptic equations. Appl. Anal. 1992, 47: 173-190. 10.1080/00036819208840139MATHMathSciNetView Article
  11. Ghergu M, Niculescu C, Radulescu V: Explosive solutions of elliptic equations with absorption and nonlinear gradient term. Proc. Indian Acad. Sci. Math. Sci. 2002, 112: 441-451. 10.1007/BF02829796MATHMathSciNetView Article
  12. Hamydy A: Existence and uniqueness of nonnegative solutions for a boundary blow-up problem. J. Math. Anal. Appl. 2010, 371: 534-545. 10.1016/j.jmaa.2010.05.053MATHMathSciNetView Article
  13. Hamydy A, Massar M, Tsouli N:Blow-up solutions to a ( p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq104_HTML.gif-Laplacian system with gradient term. Appl. Math. Lett. 2012, 25(4):745-751. 10.1016/j.aml.2011.10.014MATHMathSciNetView Article

Copyright

© Fang and Yi; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.