EXISTENCE OF NONZERO POSITIVE SOLUTIONS OF SYSTEMS OF SECOND ORDER ELLIPTIC BOUNDARY VALUE PROBLEMS∗

Existence of nonzero positive solutions of systems of second order elliptic boundary value problems under sublinear conditions is obtained. The methodology is to establish a new result on existence of nonzero positive solutions of a fixed point equation in real Banach spaces by using the wellknown theory of fixed point index for compact maps defined on cones, where the fixed point equation involves composition of a compact linear operator and a continuous nonlinear map. The conditions imposed on the nonlinear maps involve the spectral radii of the compact linear operators. Moreover, the nonlinear maps are not required to be increasing in ordered Banach spaces.


Introduction
We study existence of nonzero positive solutions of systems of second order elliptic equations Lz i (x) = g i (x)f i (x, z(x)) on Ω, i ∈ I n := {1, · · · , n} (1.1) the principal eigenvalues of the corresponding linear systems (see [19, question (c) in section 4.2]). There have been some results on the above question in more general settings under the sublinear cases. Hai and Wang [12] proved that (1.2) has a nonzero positive solution for each λ ∈ (0, ∞) under the following sublinear condition: (i) lim |z|1→0 f i0 (z) |z| 1 = ∞ for some i 0 ∈ I n and f i0 (z) > 0 for z ∈ R n + and where |z| 1 = n i=1 |z i | and p-Laplacian systems are considered (see [12,Theorem 1.2]). Recently, Lan [15] used the theory of fixed point index for compact maps defined on cones [1] to prove that (1.1) with g i ≡ 1 has a nonzero positive solution under a sublinear condition which contains, as a special case, the following condition: where µ 1 is the largest characteristic value of the linear system corresponding to (1.1) with g i ≡ 1 and |z| = max{|z i | : i ∈ I n }. The above result improves the results in [12]. However, it is essential in [15] to require g i ≡ 1 and thus, the approach used in [15] can not be used to treat the case when g i = 1. We refer to [9,10,11] for the existence and uniqueness of elliptic systems related to (1.2) under some sublinear conditions and to [4,5,6,10,16,17,18,20,23,24] for the study of such systems under the superlinear cases and some related problems for similar systems.
In this paper, we improve results in [15] and allow g i = 1. The largest characteristic values involved in our results depend on g = (g 1 , · · · , g n ). Our approach is to change (1.1) into a special form of the following general fixed point equation where L is a compact linear operator and F is a continuous nonlinear map defined on cones in real Banach spaces. We shall establish a new result on the existence of nonzero positive solutions of (1.3) by utilizing the well-known theory of fixed point index for compact maps. The existence of one or several solutions of (1.3) was studied by Amann [2], where F is an increasing map defined on an order interval. Our result does not require F to be increasing and we impose suitable conditions on the nonlinear map F which involve the spectral radius and the principle eigenvalue of the compact linear operator L. These conditions imposed on F correspond to the sublinear conditions in applications. As illustrations, we apply our result to (1.1) with some specific nonlinearities.

Nonzero positive solutions of fixed point equations in ordered Banach spaces
In this section, we consider existence of nonzero positive solutions of fixed point equations of the form z = LF z := Az, (2.1) where L is a linear operator and F is a nonlinear map defined in real Banach spaces.
The existence of one or several solutions of (2.1) was studied in ordered Banach spaces by Amman [2], where F was assumed to be an increasing map defined on order intervals. In the following we do not assume that F is increasing, but impose suitable conditions on F . In the applications given in the following sections, these conditions on F become the sublinear conditions involving the principal eigenvalues of the linear operators L.
Recall that a nonempty closed convex subset P in a real Banach space X is called a cone if δP ⊂ P for each δ ≥ 0 and P ∩ (−P ) = {0}. A cone P defines a partial order ≤ in X by A cone P is said to be reproducing if X = P − P , to be total if X = P − P and to be normal if there exists σ > 0 such that 0 ≤ x ≤ y implies x ≤ σ y . σ is called the normality constant of P . We refer to [1] for other cones.
Recall that a real number λ is called an eigenvalue of the linear operator L : X → X if there exists ϕ ∈ X \ {0} such that λϕ = Lϕ. The reciprocals of eigenvalues are called characteristic values of L. The radius of the spectrum of L in X, denoted by r(L), is given by the well-known spectral radius formula where L is the norm of L. Let ρ > 0 and let P ρ = {x ∈ P : x < ρ}, P ρ = {x ∈ P : x ≤ ρ} and ∂P ρ = {x ∈ P : x = ρ}.
We need some results from the theory of fixed point index for compact maps defined on cones in X (see [1,8]). Recall that a map A : D ⊂ X → X is said to be compact if A is continuous and A(S) is compact for each subset S ⊂ D.
Lemma 2.1. Assume that A : P ρ → P is a compact map. Then the following results hold.
Let X, Y be real Banach spaces with norms · and · Y and with cones P and P Y , respectively. We denote by the partial order in Y induced by P Y .
We always assume that the following conditions hold.
where L| X is the restriction of L on X. By (H 0 ) and (H 1 ), L : X → X is a compact linear operator such that L(P ) ⊂ P . We write .
The following result gives conditions which ensure that the fixed point index of A is zero.
Proof. It is clear that under the hypotheses (H 1 )-(H 2 ) the map A defined in (2.1) maps P into P and is compact. We prove that In fact, if not, there exist z ∈ ∂P ρ0 and ν > 0 such that Hence, we have τ 1 ≥ (µ 1 + ε)τ 1 /µ 1 > τ 1 , a contradiction. It follows from (2.5) and Lemma 2.1 (1) that i P (A, P ρ0 ) = 0. Condition (E) of Lemma 2.2 requires r(L) to be a positive eigenvalue of L with a positive eigenvector.
The following result is the well-known Krein-Rutman theorem which requires P to be total and shows that if r(L) > 0, then (E) holds (see [1,Theorem 3.1] or [14,22]). Lemma 2.3. Assume that P is a total cone in X and L : X → X is a compact linear operator such that L(P ) ⊂ P and r(L) > 0. Then there exists an eigenvector ϕ ∈ P \ {0} such that ϕ = µ 1 Lϕ.
In some applications, it is not easy to show r(L) > 0 by using the spectral radius formula (2.2) directly. The following result provides sufficient conditions which ensure that r(L) > 0 and will be used in section 3.
Proposition 2.1. Let P be a total cone in X and let L : X → X be a compact linear operator such that L(P ) ⊂ P . Assume that there exist u ∈ P − P with −u ∈ P , m ∈ N and α > 0 such that Then Lemma 2.2 (E) holds.
The following result provides conditions which ensure that the fixed point index of A is 1, where r(L) is not required to be an eigenvalue of L, but P needs to be normal.
Lemma 2.4. Assume that P is a normal cone, r(L) ∈ (0, ∞) and the following condition holds.

Nonzero positive solutions of systems of elliptic boundary value problems
In this section we study existence of nonzero positive solutions of systems of second order elliptic boundary value problems of the form subject to the following boundary condition where z(x) = (z 1 (x), · · · , z n (x)), L is a strongly uniformly elliptic differential operator and B is a first order boundary operator. When n = 1, (3.1) was studied in [1,2] and the references therein, where g i ≡ 1 and f i satisfies suitable monotonicity conditions. We refer to [3,7] and the references therein for the study of systems similar to (3.1).
Recently, Lan [15] studied existence of nonzero positive solutions of system (1.1) with g i ≡ 1 under a sublinear condition using the theory of fixed point index for compact maps defined on cone [1]. However, the approach used in [15] can not be used to treat the case when g i = 1. We shall apply results obtained in the above section to obtain results on existence of nonzero positive solutions of system (1.1), where g i is not required to be 1. Following If m ≥ 2, the boundary ∂Ω of Ω is assumed to be an (m − 1)dimensional C 2+µ -manifold for some µ ∈ (0, 1) such that Ω lies locally on one side of ∂Ω.
Recall that a second order elliptic differential operator L defined by is called to be strongly uniformly elliptic if a kj , b k , c ∈ Cμ(Ω) for k, j ∈ I m and c(x) ≥ 0 for x ∈ Ω; a kj (x) = a jk (x) for x ∈ Ω and k, j ∈ I m , and there exists µ 0 > 0 such that m k,j=1 If m = 1, the first order boundary operator B is where v is an outward pointing, nowhere tangent vector field on ∂Ω of C 1+µ , ∂u/∂v denotes the directional derivative of u with respective to v, and δ and b satisfy one of the following conditions: (i) δ = 0 and b ≡ 1 (Dirichlet boundary operator); (ii) δ = 1, b ≡ 0 and c ≡ 0 on Ω (Neumann boundary operator) or (iii) δ = 1, b ∈ C 1+µ (∂Ω), b(x) ≥ 0 and b ≡ 0 on ∂Ω (Regular oblique derivative boundary operator).
For every v ∈ Cμ(Ω), we denote by T * v the unique solution of (3.6). It is known that T * : Cμ(Ω) → C 2+μ (Ω) is a bounded and surjective linear operator and has a unique extension, denote by T , to C(Ω). We write It is known that e is an interior point of the positive cone in C(Ω): The following result gives the properties of T which are contained in [1, Theorem 4.2] and [2, Lemma 5.3].

Lemma 3.2. T : C(Ω) → C 1 (Ω) ⊂ C(Ω) is a compact linear operator such that
We always assume the following conditions on g i and f i : For each i ∈ I n , (C 2 ) f i : Ω × R n + → R + is continuous. We denote by C(Ω; R n ) the Banach space of continuous functions from Ω into R n with norm z = max{ z i : i ∈ I n }, where z(x) = (z 1 (x), · · · , z n (x)) for x ∈ Ω.
We shall use the standard positive cone in C(Ω; R n ) defined by It is well known that P is a normal and reproducing cone in C(Ω; R n ). We define L : . By (C 1 ), g i u ∈ Cμ(Ω) for u ∈ C(Ω). Hence, we have for u ∈ C(Ω), The following result gives the properties of L defined in (3.9). Proof. (i) By Lemma 3.2, for each i ∈ I n , L i : C(Ω) → C(Ω) is compact and L i (C + (Ω)) ⊂ C + (Ω). The results follows.
This, together with Lemma 2.1 implies that (ii) holds.
We define a Nemytskii operator F : P → P by It is easy to see that (3.1) is equivalent to the following fixed point equation: Now, we give our main result of this section.
Theorem 3.1. Assume that the following conditions hold.
Then (3.1) has a nonzero positive solution in P .
Lz(x) = g(x)f (x, z(x)) on Ω, Bz(x) = 0 on ∂Ω (3.12) and g and f satisfy (C 1 ) and (C 2 ). Let The following result shows that when n = 1, the conditions: ((f i ) 0 ) ρ0 and (f ∞ i ) ρ1 in Theorem 3.1 can be replaced by suitable stronger limit conditions. Corollary 3.1. Assume that the following condition holds.
As illustration, we consider existence of nonzero positive solutions of systems of second order elliptic boundary value problems of the form Lz i (x) = g i (x) a i 1 + |z(x)| αi z i (x) for x ∈ Ω and i ∈ I n (3.13) subject to the boundary condition (3.2). Proof. For each i ∈ I n , we define a function f i : Ω × R n + → R + by f i (x, z) = a i 1 + |z| αi z i .