SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Highly Accessed Research

Existence and nonexistence of entire positive solutions for ( p , q ) -Laplacian elliptic system with a gradient term

Zhong Bo Fang1* and Su-Cheol Yi2

Author affiliations

1 School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, P.R. China

2 Department of Mathematics, Changwon National University, Changwon, 641-773, Republic of Korea

For all author emails, please log on.

Citation and License

Boundary Value Problems 2013, 2013:18  doi:10.1186/1687-2770-2013-18

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/18


Received:10 August 2012
Accepted:9 January 2013
Published:5 March 2013

© 2013 Fang and Yi; licensee Springer

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.

Abstract

This work is concerned with the entire positive solutions for a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M1">View MathML</a>-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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M1">View MathML</a>-Laplacian elliptic system of equations with a gradient term

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M4">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M7">View MathML</a>, the nonlinearities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M8">View MathML</a> are positive, continuous and nondecreasing functions for each variable, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M10">View MathML</a> are continuous functions, and the potentials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M11">View MathML</a> are c-positive functions (or circumferentially positive) in a domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M12">View MathML</a> which are nonnegative in Ω and satisfy the following:

• If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M14">View MathML</a>, then there exists a domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M15">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M17">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M18">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M1">View MathML</a> is a characteristic of the medium. Media with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M20">View MathML</a> are called dilatant fluids, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M21">View MathML</a> are called pseudoplastics. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M22">View MathML</a>, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M23">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M24">View MathML</a> solving problem (1.1) in the weak sense and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M26">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M27">View MathML</a>.

In recent years, existence and nonexistence of entire solutions for the semilinear elliptic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M28">View MathML</a>

have been studied by many authors; see [1-3] 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M29">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M30">View MathML</a>

They proved that boundary blow-up solutions fail to exist if f and g are sublinear, whereas this result holds if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M31">View MathML</a> is bounded and a, b are slow decay at infinity. They also showed the existence of infinitely blow-up solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M31">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M33">View MathML</a>

and obtained the sufficient condition of nonexistence and existence of positive entire solutions. Furthermore, for the single equation with a gradient term, we read [10-12] 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M1">View MathML</a>-Laplacian system (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M36">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M38">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M39">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M41">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M43">View MathML</a> denote inverse functions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M45">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M47">View MathML</a>. Moreover, we define

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M49">View MathML</a>.

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

Theorem 1Suppose thatfandgsatisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M50">View MathML</a>

(2.1)

anda, bsatisfy the decay conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M51">View MathML</a>

(2.2)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M52">View MathML</a>, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M53">View MathML</a>. From (1.1), we know that

Now, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M55">View MathML</a>

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M56">View MathML</a> are positive and nondecreasing functions. Moreover, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M59">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M60">View MathML</a>. It follows from (2.1) that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M61">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M62">View MathML</a>

(2.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M63">View MathML</a>

(2.4)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M64">View MathML</a>

(2.5)

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M65">View MathML</a>

Thus, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M66">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M67">View MathML</a>

where C is a positive constant. Because of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M52">View MathML</a>, the last inequality above is valid for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M69">View MathML</a>. Noticing that (2.2), we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M70">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M71">View MathML</a>

(2.6)

It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M72">View MathML</a>, and we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M73">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M74">View MathML</a>

(2.7)

Thus, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M75">View MathML</a>

From (2.6), we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M76">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M77">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M79">View MathML</a>. Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M80">View MathML</a>

then we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M81">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M82">View MathML</a> and f, g satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M83">View MathML</a>

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 2Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M84">View MathML</a>

Then system (1.1) has infinitely many positive entire solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M85">View MathML</a>. Moreover, the following hold:

(i) Ifaandbsatisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M86">View MathML</a>, then all entire positive solutions of (1.1) are large.

(ii) Ifaandbsatisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M88">View MathML</a>, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M89">View MathML</a> and show that the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M90">View MathML</a>

(2.9)

has solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M91">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M93">View MathML</a> are positive solutions of system (1.1). Integrating (2.9), we have

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M95">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M96">View MathML</a> be sequences of positive continuous functions defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M97">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M98">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M102">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M103">View MathML</a>. And the monotonicity of f and g yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M105">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M103">View MathML</a>.

Repeating such arguments, we can deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M107">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M96">View MathML</a> are nondecreasing sequences on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M97">View MathML</a>. Noticing that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M111">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M112">View MathML</a>

it follows that

Then we can get

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M115">View MathML</a>

(2.10)

It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M116">View MathML</a> is increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M97">View MathML</a> and (2.10) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M118">View MathML</a>

(2.11)

And from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M119">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M120">View MathML</a>. By (2.11), the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M121">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M122">View MathML</a> are bounded and increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M123">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M124">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M121">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M122">View MathML</a> have subsequences converging uniformly to u and v on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M123">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M91">View MathML</a> is a positive solution of (2.9); therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M129">View MathML</a> is an entire positive solution of (1.1). Noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M131">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M132">View MathML</a> are chosen arbitrarily, we can obtain that system (1.1) has infinitely many positive entire solutions.

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M133">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M135">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M103">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M137">View MathML</a>

which yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M138">View MathML</a> is the positive entire large solution of (1.1).

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M88">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M141">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M56">View MathML</a> is the positive entire bounded solution of system (1.1). Thus, the proof of Theorem 2 is finished.

 □

Theorem 3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M88">View MathML</a>, and there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M147">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M148">View MathML</a>

(2.12)

then system (1.1) has an entire positive radial bounded solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M149">View MathML</a> (for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M150">View MathML</a>) satisfying

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M152">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M116">View MathML</a> is strictly increasing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M97">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M155">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M156">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M157">View MathML</a>

(2.13)

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

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M88">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M160">View MathML</a>

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

Proof

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M161">View MathML</a>

(2.14)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M162">View MathML</a>

(2.15)

Choosing an arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M163">View MathML</a>, from (2.14) and (2.15), we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M164">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M165">View MathML</a>

(2.16)

Taking account of the monotonicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M166">View MathML</a>, there exists

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M167">View MathML</a>

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M168">View MathML</a> is finite. Indeed, if not, we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M169">View MathML</a> in (2.16) and the assumption (2.13) leads to a contradiction. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M168">View MathML</a> is finite. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M122">View MathML</a> are increasing functions, it follows that the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M173">View MathML</a> is nondecreasing and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M174">View MathML</a>

Thus, the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M175">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M176">View MathML</a> are bounded from above on bounded sets. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M177">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M91">View MathML</a> is a positive solution of (2.9).

In order to conclude the proof, we need to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M91">View MathML</a> is a large solution of (2.9). By the proof of Theorem 2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M180">View MathML</a>

And because f and g are positive functions and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M181">View MathML</a>

we can conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M91">View MathML</a> is a large solution of (2.9) and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M129">View MathML</a> is a positive entire large solution of (1.1). Thus, any large solution of (2.9) provides a positive entire large solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M129">View MathML</a> of (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M185">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M186">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M187">View MathML</a> was chosen arbitrarily, it follows that (1.1) has infinitely many positive entire large solutions.

(ii) If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M188">View MathML</a>

holds, then by (2.16), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M189">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M190">View MathML</a>

Thus, the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M175">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M176">View MathML</a> are bounded from above on bounded sets. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M193">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M53">View MathML</a> is a positive solution of (2.9).

It follows from (2.14) and (2.15) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M91">View MathML</a> is bounded, which implies that (1.1) has infinitely many positive entire bounded solutions. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

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.

References

  1. Ghergu, M, Radulescu, V: 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.. 164, 380–394 (2000). Publisher Full Text OpenURL

  3. Kawano, N, Kusano, T: On positive entire solutions of a class of second order semilinear elliptic systems. Math. Z.. 186(3), 287–297 (1984). Publisher Full Text OpenURL

  4. Lair, AV, Shaker, AW: Entire solution of a singular semilinear elliptic problem. J. Math. Anal. Appl.. 200, 498–505 (1996). Publisher Full Text OpenURL

  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. 14, 237–274 (1997). Publisher Full Text OpenURL

  6. Yang, Z: Existence of entire explosive positive radial solutions for a class of quasilinear elliptic systems. J. Math. Anal. Appl.. 288, 768–783 (2003). Publisher Full Text OpenURL

  7. Ghergu, M, Radulescu, V: Explosive solutions of semilinear elliptic systems with gradient term. Rev. R. Acad. Cienc. Ser. a Mat.. 97(3), 437–445 (2003)

  8. Cirstea, F, Radulescu, V: Entire solutions blowing up at infinity for semilinear elliptic systems. J. Math. Pures Appl.. 81, 827–846 (2002)

  9. Zhang, X, Liu, L: The existence and nonexistence of entire positive solutions of semilinear elliptic systems with gradient term. J. Math. Anal. Appl.. 371, 300–308 (2010). Publisher Full Text OpenURL

  10. Guo, Z: Existence and uniqueness of positive radial solutions for a class of quasilinear elliptic equations. Appl. Anal.. 47, 173–190 (1992). Publisher Full Text OpenURL

  11. Ghergu, M, Niculescu, C, Radulescu, V: Explosive solutions of elliptic equations with absorption and nonlinear gradient term. Proc. Indian Acad. Sci. Math. Sci.. 112, 441–451 (2002). Publisher Full Text OpenURL

  12. Hamydy, A: Existence and uniqueness of nonnegative solutions for a boundary blow-up problem. J. Math. Anal. Appl.. 371, 534–545 (2010). Publisher Full Text OpenURL

  13. Hamydy, A, Massar, M, Tsouli, N: Blow-up solutions to a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/18/mathml/M196">View MathML</a>-Laplacian system with gradient term. Appl. Math. Lett.. 25(4), 745–751 (2012). Publisher Full Text OpenURL