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Existence and nonexistence of entire positive solutions for $(p,q)$Laplacian elliptic system with a gradient term
 Zhong Bo Fang^{1}Email author and
 SuCheol Yi^{2}
https://doi.org/10.1186/16872770201318
© Fang and Yi; licensee Springer. 2013
 Received: 10 August 2012
 Accepted: 9 January 2013
 Published: 5 March 2013
Abstract
This work is concerned with the entire positive solutions for a $(p,q)$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.
Keywords
 Elliptic System
 Quasiconformal Mapping
 Nondecreasing Function
 Slow Decay
 Fast Decay
1 Introduction
where $N>2$, $p\ge 2$, $q\ge 2$, the nonlinearities $f,g:[0,\mathrm{\infty})\times [0,\mathrm{\infty})\to (0,\mathrm{\infty})$ are positive, continuous and nondecreasing functions for each variable, ${m}_{1}(x)$ and ${m}_{2}(x)$ are continuous functions, and the potentials $a,b\in C({R}^{N})$ are cpositive functions (or circumferentially positive) in a domain $\mathrm{\Omega}\subset {R}^{N}$ which are nonnegative in Ω and satisfy the following:

If ${x}_{0}\in \mathrm{\Omega}$ and $a({x}_{0})=0$, then there exists a domain ${\mathrm{\Omega}}_{0}$ such that ${x}_{0}\in {\mathrm{\Omega}}_{0}\subset \mathrm{\Omega}$ and $a(x)>0$ for all $x\in \partial {\mathrm{\Omega}}_{0}$.
Problem (1.1) arises in the theory of quasiregular and quasiconformal mappings, stochastic control and nonNewtonian fluids, etc. In the nonNewtonian theory, the quantity $(p,q)$ is a characteristic of the medium. Media with $(p,q)>(2,2)$ are called dilatant fluids, while $(p,q)<(2,2)$ are called pseudoplastics. If $(p,q)=(2,2)$, 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 $(u,v)\in {C}^{1+\theta}({R}^{N})\times {C}^{1+\theta}({R}^{N})$ for $\theta \in (0,1)$ solving problem (1.1) in the weak sense and $u(x)\to \mathrm{\infty}$, $v(x)\to \mathrm{\infty}$ as $x\to \mathrm{\infty}$.
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 $(p,q)$Laplacian system (1.1) with $f(u,v)=\phi (v)$, $g(u,v)=\psi (u)$. 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 ${m}_{1}(x){\mathrm{\nabla}u}^{\alpha}$, ${m}_{2}(x){\mathrm{\nabla}u}^{\alpha}$ for $0<\alpha <p1$.
2 Main results and proof
We now state and prove the main results of this paper.
where $r>0$.
Firstly, we give a nonexistence result of a positive entire radial large solution of system (1.1).
where $m=min\{p,q\}$, then problem (1.1) has no positive entire radial large solution.
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. □
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).
 (i)
If a and b satisfy ${H}_{p}a(\mathrm{\infty})={H}_{q}b(\mathrm{\infty})=\mathrm{\infty}$, then all entire positive solutions of (1.1) are large.
 (ii)
If a and b satisfy ${H}_{p}a(\mathrm{\infty})<\mathrm{\infty}$, ${H}_{q}b(\mathrm{\infty})<\mathrm{\infty}$, then all entire positive solutions of (1.1) are bounded.
Obviously, ${u}_{k}(r)\ge c$, ${v}_{k}(r)\ge d$, ${u}_{0}\le {u}_{1}$, ${v}_{0}\le {v}_{1}$ for all $r\ge 0$. And the monotonicity of f and g yields ${u}_{1}(r)\le {u}_{2}(r)$, ${v}_{1}(r)\le {v}_{2}(r)$ for $r\ge 0$.
 (i)If ${H}_{p}a(\mathrm{\infty})={H}_{q}b(\mathrm{\infty})=\mathrm{\infty}$, since $u(r)\ge c+{\mathrm{\Phi}}_{{p}^{\prime}}(f(c,d)){H}_{p}a(r)$, $v(r)\ge d+{\mathrm{\Phi}}_{{q}^{\prime}}(g(c,d)){H}_{q}b(r)$ for $r\ge 0$, we have$\underset{r\to \mathrm{\infty}}{lim}u(r)=\underset{r\to \mathrm{\infty}}{lim}v(r)=\mathrm{\infty},$
 (ii)If ${H}_{p}a(\mathrm{\infty})<\mathrm{\infty}$, ${H}_{q}b(\mathrm{\infty})<\mathrm{\infty}$, then$u(r)+v(r)\le {F}^{1}(F(b+c)+{H}_{p}a(\mathrm{\infty})+{H}_{q}b(\mathrm{\infty}))<\mathrm{\infty},$
which implies that $(U,V)$ is the positive entire bounded solution of system (1.1). Thus, the proof of Theorem 2 is finished.
□
The rest of the proof obviously holds from the proof of Theorem 2. The proof of Theorem 3 is now finished. □
 (i)If${H}_{p}a(\mathrm{\infty})={H}_{q}b(\mathrm{\infty})=\mathrm{\infty},$
 (ii)If ${H}_{p}a(\mathrm{\infty})<\mathrm{\infty}$, ${H}_{q}b(\mathrm{\infty})<\mathrm{\infty}$, and$\underset{s\ge 0}{sup}(f(s,s)+g(s,s))<\mathrm{\infty},$
then system (1.1) has infinitely many positive entire bounded solutions.
 (i)It follows from the proof of Theorem 2 that$\begin{array}{rcl}{u}_{k}(r)& \le & {u}_{k+1}(r)\le {\mathrm{\Phi}}_{{p}^{\prime}}\left(f({u}_{k}(r),{v}_{k}(r))\right){H}_{p}a(r)\\ \le & {\mathrm{\Phi}}_{{p}^{\prime}}\left(f({u}_{k}(r)+{v}_{k}(r),{u}_{k}(r)+{v}_{k}(r))\right){H}_{p}a(r)\end{array}$(2.14)
then $(u,v)$ is a positive solution of (2.9).
 (ii)If$\underset{s\ge 0}{sup}(f(s,s)+g(s,s))<\mathrm{\infty},$
then $(u,v)$ is a positive solution of (2.9).
It follows from (2.14) and (2.15) that $(u,v)$ is bounded, which implies that (1.1) has infinitely many positive entire bounded solutions. □
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
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