A COUPLED p -LAPLACIAN ELLIPTIC SYSTEM: EXISTENCE, UNIQUENESS AND ASYMPTOTIC BEHAVIOR

. We prove uniqueness, existence and asymptotic behavior of positive solutions to the system coupled by p -Laplacian elliptic equations Ω , where ∆ p u = div( |∇ u − 2 ∇ u ) , 1 < p < ∞ , λ 1 and λ 2 are positive parameters, Ω is the open unit ball in R N , N


1.
Introduction. It is well known that p-Laplace equations are quasilinear equations when p = 2(see [22], [12]), and there are many important applications in physics, game theory and image processing (see [14], [5]). In the past few decades, a good many of results have been developed for single p-Laplace equations by different methods, for instance, see [21,23,19,18] and the references cited therein. Specially, in [24], Zhang and Li considered the following p-Laplacian equation where p u = div(|∇u| p−2 ∇u) is the p-Laplacian operator, N < p < ∞, Ω is a smooth bounded domain in R N , N ≥ 1. The authors applied differential equations theory in Banach spaces and dynamics theory to study problem (1), and obtained excellent multiple solutions and sign-changing solutions theorems of p-Laplacian. At the same time, we notice that many authors have paid more attention to existence and uniqueness problems, for example, see Castro, Sankar and Shivaji [2], Lin [13], Hai [10], and Guo [6]. Specially, Guo and Webb [7] considered the following p-Laplacian equation They obtained existence and uniqueness results to problem (2) for large λ if f ≥ 0, (f (x)/x p−1 ) < 0 for x > 0 and f satisfies some p-sublinearity conditions at ∞ and 0. In [11], using sub-supersolution method together with sharp estimates near the boundary, Hai and Shivaji improved the results of (2) in a unit ball under much weaker assumptions than in [7]. Recently, Shivaji, Sim and Son [20], and Chu, Hai and Shivaji [3] generalize the study in [7] from a bounded domain to the exterior domains and obtained some excellent results. Moreover, we notice that various of system problems have become an important area of investigation in recent years. To identify a few, we refer the reader to [4,15,16,17]. In [10], Hai considered the existence and uniqueness of positive solutions to the following elliptic system    ∆u = −λf (v) in Ω, ∆v = −µg (u) in Ω, where Ω is the open ball in R N , f, g : R + → R + , λ and µ are positive parameters. Inspired by the above works, we are interested in the existence and uniqueness of positive radial solutions to the following system    −∆ p z 1 = λ 1 g 1 (z 2 ) in Ω, −∆ p z 2 = λ 2 g 2 (z 1 ) in Ω, z 1 = z 2 = 0 on ∂Ω. ( Here p denotes the singular\degenerate p-Laplace operator pu = div(|∇u| p−2 ∇u), 1 < p < ∞, λ 1 > 0 and λ 2 > 0 are parameters, g 1 and g 2 are continuous nonlinearities, and Ω = {x ∈ R N : |x| < 1}, N ≥ 2.
We also give new existence results for system (3). Our main tool is the eigenvalue theory in cones. However, based on the idea of decoupling method we will investigate composite operators. Besides, the exactly determined intervals of positive parameters λ 1 × λ 2 are established.
The rest of the paper is organized as follows. In Section 2, we present some necessary definitions, Lemmas and theorems that will be used to prove our main results, Theorem 2.4. Section 3 is devoted to proving the existence and uniqueness of positive solution to system (3). In Section 4, we establish the exactly determined intervals of positive parameter λ 1 × λ 2 in which system (3) admits at least one positive solution. Section 5 verifies the existence and asymptotic behavior of positive radial solutions to system (3). Finally, in Section 6, we will give some remarks on our main results.
2. Preliminaries and some lemmas. In order to study the existence of the positive radial solutions for system (3), let us firstly introduce the radial coordinates form of the p-Laplacian operator. Letting r = |x|, and u(r Therefore, the study of positive radial solutions of system (3) is reduced to the study of positive solutions to the following system: where ϕ p (s) = |s| p−2 s, (ϕ p ) −1 = ϕ q , 1 p + 1 q = 1.
Next we mainly analyze the existence of positive solutions for system (4). In order to get our theorems, we let R + = [0, +∞) and g i satisfy (C 0 ) g 1 and g 2 : R + → R + are continuous. (C 1 ) g 1 and g 2 : R + → R + are nondecreasing, C 1 on (0, ∞) and There exist nonnegative numbers a, b, A, D, where ab < 1 and A, D > 0 such that and for a 1 > a and b 1 > b, are nonincreasing for x large. A pair of functions u, v ∈ C[0, 1] ∩ C 1 (0, 1) with φ p (u ), φ p (v ) ∈ C 1 (0, 1) is called to be a positive solution of (4) if u(t), v(t) > 0 for all t ∈ (0, 1), and u and v satisfy (4). Let Define a cone P by Define an operator F : E → E by It is easy to check that F : P → P is completely continuous and the solution of system (4) is equivalent to the fixed point equation Therefore, the task of the present paper is to search nonzero fixed points of F .
The following well-known results are crucial in the proofs of our results. , on page 131) Let P be a cone in a Banach space E and T : P → P be a completely continuous mapping satisfying (a) There exist k ∈ K, k = 1, and a number r > 0 such that all solutions y ∈ P of y = T y + θk, 0 < θ < ∞ satisfy y = r.
(b) There exists R > r such that all solutions z ∈ P of z = θT z, 0 < θ < 1.
Then T has a fixed point x ∈ P, r ≤ x ≤ R.

Uniqueness of positive solution.
In this section, we analyze the uniqueness of fixed point of F for λ 1 λ a 2 and λ b 1 λ 2 sufficiently large. Lemma 3.1. Let h be continuous on R + and C 1 on (0, ∞) such that Let M, ε, r be positive numbers with ε < 1. Then there is a positive constant C such that Using the mean value theorem, there is a c ∈ (γ, 1) such that where C = sup{|yh (y)| : 0 < y ≤ M } ε + r(p − 1) max(ε r(p−1)−1 , 1) sup{|h(y)| : y ≤ M }.
Next we will check the upper and lower estimates for possible positive solutions of system (4).
Suppose that u and v are a pair of positive solutions for system (4). By integrating, we get Next, we can denote by c i , i = 1, 2, . . . , positive constants independent of u, v, λ 1 , λ 2 . Since v is decreasing, we have Similarly we can get By (C 2 ), there are two positive constants K 1 and K 2 such that This together with (5) and (6), shows that Similarly, we can get It follows from (7), (8) and (9) Then by integrating, for t ≥ 1 2 , we get Similarly, Because of u, v being decreasing, this shows the left-side inequalities for u, v in Lemma 3.2.
According to the formulas for u, v, we can see that and By (7) and (11), for large λ 1 λ a 2 and and Note that from (C 2 ) it follows that By combining the equation of u and (15), we can get and then it follows from integrating that Similarly, we can get the upper estimate for v(t). This completes the proof.
Then there is a constant β > 0 such that the system (4) admits a unique positive solution for min(λ 1 λ a 2 , λ 2 λ b 1 ) ≥ β. Proof. We shall check the conditions of Lemma 2.2 to prove the existence of solution.
Because of u, v are nonincreasing and u, v > 0 on (0, 1), the proof is analogous to that of Lemma 3.2 that for some θ ∈ (0, 1). Then, by (12) and (13) we get In addition, if |v| ∞ → ∞, by (C 2 ) and ab < 1 we can see that Thus, it follows that (4) admits one positive solution, and the existence is proved. Next, we shall show that the solution is unique. Suppose that (u, v) and (u 1 , u 2 ) are positive solutions of system (4) and let min{λ 1 λ a 2 , λ 2 λ b 1 , } be large enough such that Lemma 3.2 holds. It follows from Lemma 3.2 that M2 . We assert that α ≥ 1. In fact, we can assume by contradiction that α < 1. Since u, v are decreasing and Let b 1 > b 2 > b, a 1 > a and a 1 b 1 < 1. Then we assert that According to g2(x) ϕp(x b 2 ) is nonincreasing for x 1 and α ≥ α 0 , we can get By Lemma 3.2, we can get where T ∈ ( 1 2 , 1). Since α < 1, then for s ≤ T , we have and since there is a positive number l > 0 such that This follows that s 0 τ N −1 (g 2 (αu) − ϕ p (α b1 )g 2 (u))dτ > 0, s > T when T is sufficiently close to 1. So, we can see that s 0 τ N −1 (g 1 (αu) − ϕ p (α b1 )g 1 (u))dτ > 0, for s > T when T is sufficiently close to 1. So, the proof of (18) is finished.
Substituting (18) into (17) and integrating gets . Applying (7) and Lemma 3.2, for t ≤ T , we get ϕp(x a 1 ) is nonincreasing for x 1, we get where H(T ) = l 0 k * K 2 h a (T )) and l 0 is a positive constant so that This proves that On the other hand, if z > T, then for large λ 1 λ a 2 and λ 2 λ b 1 , and T sufficiently close to 1, it follows from Lemma 3.1 and (19) that z 0 G(α, r)dr ≥ This shows that there is a constant α > α in (0, 1) such that u ≥ αu 1 , which is a contradiction. Thus α ≥ 1 and hence u = u 1 in (0,1). Similarly, we can verify v = v 1 in (0, 1) and so we finish the proof of Theorem 3.3.

4.
New existence results. In this section, we will establish some new existence results of positive solutions for system (4). To achieve this goal, we will define a new cone P and a composite operator T . Γx > 0, then Γ has a proper element on P ∩ ∂D associated with a positive eigenvalue. That is, there exist x 0 ∈ P ∩ ∂D and µ 0 > 0 such that Γx 0 = µ 0 x 0 .
Let E = C[0, 1]. Then E is a real Banach space with the norm · defined by Let J = [0, 1] and P be the cone It is easy to see that P is a normal cone of E.
For v ∈ P , define T i : P → E(i = 1, 2) as It follows from Lemma 3 in [1] that T i (i = 1, 2) maps P into itself. Moreover, T 1 and T 2 are completely continuous by standard arguments. Define a composite operator T = T 1 T 2 , which is also completely continuous from P to itself. So the operator T also maps P into P . Therefore the next task of this paper is to search nonzero fixed points of operator T . Let Theorem 4.2. Suppose that (C 0 ) holds. If 0 < g ∞ i < +∞(i = 1, 2), then there exists β 0 > 0 such that, for every R > β 0 , system (4) admits a pair of positive solutions u R , v R satisfying u R = R for any where λ R andλ R are positive finite numbers.
Since R > β 0 , for any u, v ∈ P ∩ ∂Ω R , we get and So, for any v ∈ P ∩ ∂Ω R , we have Analogously, for u ∈ P ∩ ∂Ω R , we obtain Therefore, we get This gives that For any R > β 0 , Lemma 4.1 yields that operator T admits a proper element u R ∈ P associated with the eigenvalue µ 1R > 0, and u R satisfies u R = R.
For operator T , we can denote v R = T 2 u R , then u R and v R are the solutions of system (4).
Let λ 1R = 1 ϕp(µ 1R ) . Then we get It follows from the proof above that, for any R > β 0 , system (4) has a pair of positive solutions u R and v R with u R ∈ P ∩ ∂Ω R associated with λ 1 = λ 1R > 0. Thus, by (24) and so On the one hand, This verifies that u R = R ≤ ϕ q (l 2 2 (B * ) 2 λ 1R λ 2 ) u R , and so, On the other hand, Analogously, we can show that Therefore, we get and so, We hence get λ 1R λ 2 ∈ [λ R ,λ R ]. This gives the proof.
If we define another composite operator T * = T * 2 T * 1 , where , then there exists β 0 > 0 such that, for every R > β 0 , system (4) admits a pair of positive solutions u R , v R satisfying v R = R for any where λ R andλ R are positive finite numbers.
Proof. Similar to the proof of Theorem 4.2, we can prove Corollary 1.
Proof. Similar to the proof of Theorem 4.2, we can prove Theorem 4.3.
So, for any v ∈ P ∩ ∂Ω R * , we have Analogously, for u ∈ P ∩ ∂Ω R * , we obtain Therefore, we get This gives that For any R * >β 0 , Lemma 4.1 yields that operator T admits a proper element u R * ∈ P associated with the eigenvalue µ 1R * > 0, and u R * satisfies u R * = R * .
For operator T , we denote v R * = T 2 u R * , then u R * and v R * are the solutions of system (4).
Proof. Similar to the proof of Theorem 4.4, we can prove Theorem 4.5.
5. Asymptotic behavior of positive solutions. In this section, we study the asymptotic behavior of positive solutions for system (4).
Let P be defined as (20), and T * 1 and T 2 be respectively defined in (26) and (22). Define a composite operator T 1 = T * 1 T 2 , which is completely continuous from P to itself. So the operator T 1 also maps P into P . We also define another composite operator T 2 = T 2 T * 1 , which has the same meaning as T 1 .
So, by (31), we have From the above estimate and the fixed point theorem of cone expansion and compression of norm type, we deduce that operator T 1 has a fixed point u ∈ P ∩ (Ω R \Ω r ). Denote v = T 2 u, then u and v are the desired solution of system (4).
6. Some remarks. In this section, we offer some remarks and applications on the associated system (4).
Remark 6.1. The present research extends the study in Hai [10] from Laplacian system to p-Laplacian system. Meanwhile, we obtain some new existence results by defining composite operators and using the eigenvalue theory in cones. Moreover, we also analyze the asymptotic behavior of positive solutions to system (4).
Remark 6.2. In this paper, we also generalize the study in Guo [6], Guo and Webb [7], Hai and Shivaji [11], Shivaji, Sim and Son [20], and Chu, Hai and Shivaji [3] from single p-Laplacian equation to coupled p-Laplacian system. Here, we not only get the uniqueness results, but also we obtain some existence results, and we consider the asymptotic behavior of positive solutions.