Existence and nonexistence of entire positive solutions for ( p , q ) http://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

  • Zhong Bo Fang1Email author and

    Affiliated with

    • Su-Cheol Yi2

      Affiliated with

      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 ) http://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 ) http://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 , http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq2_HTML.gif, p 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq3_HTML.gif, q 2 http://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 , ) http://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 | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq6_HTML.gif and m 2 ( | x | ) http://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 ) http://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 http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq13_HTML.gif and a ( x ) > 0 http://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 http://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 ) http://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 ) http://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 ) http://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 ) http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq19_HTML.gif for θ ( 0 , 1 ) http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq21_HTML.gif, v ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq22_HTML.gif as | x | http://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 , http://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 http://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 . http://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 http://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 http://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 http://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 ) http://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 ) http://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 ) http://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 http://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 http://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 http://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 http://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 http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq32_HTML.gif, Φ q ( x ) http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq34_HTML.gif and Φ q http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq36_HTML.gif, q = q q 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq37_HTML.gif. Moreover, we define
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Eque_HTML.gif

      where r > 0 http://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 } < + , http://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 < + , http://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 } http://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 ) http://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
      http://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 ) . http://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 ) http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq42_HTML.gif, V v http://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 ) + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq44_HTML.gif as r + http://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 http://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 http://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 . http://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 . http://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 . http://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 http://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 , http://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 } http://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 http://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 http://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 . http://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 ) = http://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 http://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 . http://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 . http://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 , http://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 , http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq52_HTML.gif, r r 1 http://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 , http://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 , http://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 http://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 } < + , http://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 ( ) = . http://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 , + ) http://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 ( ) = http://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 ( ) < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq57_HTML.gif, H q b ( ) < http://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 http://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 , http://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 ) http://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 | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq61_HTML.gif, V ( x ) = v ( | x | ) http://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
      http://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 http://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 http://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 , ) http://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 . http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq66_HTML.gif, v k ( r ) d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq67_HTML.gif, u 0 u 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq68_HTML.gif, v 0 v 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq69_HTML.gif for all r 0 http://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 ) http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq72_HTML.gif for r 0 http://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 , http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq63_HTML.gif, { v k } k 0 http://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 , ) http://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 ) , http://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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equt_HTML.gif
      it follows that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equu_HTML.gif
      Then we can get
      http://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 . http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq73_HTML.gif is increasing on [ 0 , ) http://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 . http://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 ( ) = http://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 ( ) = http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq76_HTML.gif and v k http://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 ] http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq79_HTML.gif. Thus, u k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq76_HTML.gif and v k http://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 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq78_HTML.gif. Consequently, ( u , v ) http://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 ) http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq81_HTML.gif, V ( 0 ) = d http://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 , ) http://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 ( ) = http://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 ) http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq86_HTML.gif for r 0 http://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 ) = , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equw_HTML.gif
         
      which yields ( U , V ) http://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 ( ) < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq88_HTML.gif, H q b ( ) < http://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 ( ) ) < , http://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 ) http://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 ( ) < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq89_HTML.gif, H p a ( ) < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq88_HTML.gif, H q b ( ) < http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq90_HTML.gif, d > β http://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 ) , http://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 , ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq92_HTML.gif (for 0 < θ < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq93_HTML.gif) satisfying
      http://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 ( ) < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equz_HTML.gif
      Since F 1 http://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 , ) http://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 ( ) ) < . http://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 ( ) = , http://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 , http://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 ( ) < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq88_HTML.gif, H q b ( ) < http://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 ) ) < , http://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 ) http://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 ) . http://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 http://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 , http://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 . http://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 http://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 ) ) . http://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 ) http://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 http://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 ) http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq76_HTML.gif, v k http://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 , ) http://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 . http://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 http://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 http://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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equag_HTML.gif

      then ( u , v ) http://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 ) http://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 . http://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 ( ) = , http://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 ) http://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 ) http://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 ) http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_IEq101_HTML.gif and V ( 0 ) = d http://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 , ) http://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 ) ) < , http://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 ) ) < . http://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 . http://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 http://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 http://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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-18/MediaObjects/13661_2012_Article_280_Equam_HTML.gif

      then ( u , v ) http://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 ) http://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

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      © Fang and Yi; licensee Springer. 2013

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