AN AMBROSETTI-PRODI-TYPE PROBLEM FOR AN ELLIPTIC SYSTEM OF EQUATIONS VIA MONOTONE ITERATION METHOD AND LERAY-SCHAUDER DEGREE THEORY

In this paper we employ the Monotone Iteration Method and the Leray-Schauder Degree Theory to study an IR-parametrized system of elliptic equations. We obtain a curve dividing the plane into two regions. Depending on which region the parameter is, the system will or will not have solutions. This is an Ambrosetti-Prodi-type problem for a system of equations. 0. Introduction Let us consider a bounded domain Ω ⊆ IRN , N ≥ 2, with boundary ∂Ω of class C2,α , 0 < α ≤ 1. In what follows we shall denote the Cartesian product of Hölder spaces as (Ci,α)2 = Ci,α(Ω)×Ci,α(Ω), 0 < α ≤ 1, i = 0, 1, 2 and (Cα)2 = (C0,α(Ω))2. Remark 0.1. (Ci,α)2 ↪→ (Cj,α)2 compactly if i ≥ j ≥ 0. Let us consider the following system of partial differential elliptic equations (S)  −∆u = f(x, u, v) + h(x), Ω, −∆v = g(x, u, v) + l(x), Ω, u = v = 0, ∂Ω, where h, l ∈ Cα(Ω) and f, g ∈ Cα(Ω × IR × IR) are real functions. By a solution of (S) we mean a vector-function U = (u, v) ∈ (C2,α)2 satisfying both equations in (S). Any function h ∈ Cα(Ω) can be uniquely decomposed as h = tφ1 + h1, where φ1 > 0 is an eigenfunction associated to the first eigenvalue λ1 >

According the above observations, system (S) may be IR 2 -parametrized, and for the convenience of notation, be rewritten in the matrix form (S) T −∆U = F (x, U ) + T φ 1 + H 1 (x), Ω, U = 0, ∂Ω and We make use of the following notation: ⊥ our main goal is to find the set of points T ∈ IR 2 for those, system (S) T has or has not solutions.Grosso modo, for this matter, with convenient hypotheses on F which shall be timely introduced, we find a curve Γ = Γ(H 1 ) splitting the plane into two disjoint unbounded domains E and N (IR 2 = E ∪ Γ ∪ N ) such that (i) (S T ) does not have a solution if T ∈ N (ii) (S T ) has at least one solution if T ∈ Γ (iii) (S T ) has at least two solutions if T ∈ E.
The paper is divided into two parts.In the first one we use the Monotone Iteration Method to find the first solution and construct the curve.Although some ideas of the first part already appear in [7], where we solved a kind of this problem using variational techniques, we decided to include them here for completeness of the text.In the second part, we obtain a priori estimates for eventual solutions of (S) T using the Hardy-Sobolev inequality (see [2] or [6]) and apply the Leray-Schauder degree theory to obtain the second solution.Then we find a solution for T on the curve.
In this paper we have the advantage of relaxing the very restrictive hypothesis on the non-linearity F = ∇H ( for some vector-function H ) which is imposed by the variational methods we used in [7].Also relating to [7], another sharper result we found in this paper was the existence of a solution for T on the curve Γ.
In the scalar case, the pioneering paper involving this kind of problem but with a very different formulation is [1].Also in the scalar case, good references for this kind of problems up to 80's may be found in [4] and [8].

FIRST PART
1. Existence of a Region R in the Plane Where (S) T

Has No Solution
Let us order IR 2 with the partial order Z = (z 1 , z 2 ) ≥ 0 if and only if z 1, z 2 ≥ 0.
We shall assume the following growth conditions on the non-linearity F : for all x ∈ Ω and Z ∈ IR 2 , where and these matrices satisfy (1.17) Proof.Suppose U = (u, v) is a solution of (S) T .Multiplying both equations of this system by φ 1 , integrating by parts and using (1.1) and (1.2) we achieve the inequalities Hence, independently of the sign of γ, the pair ( r, s) is in a region composed of the intersection of two half-planes, each of them is bounded above by a straight-line of negative or infinity slope passing through the origin.The set R ⊂ IR 2 is the complement of this region in the original variables r and s.

Existence of Sub and Supersolution
The existence of a solution for T in a certain region of IR 2 is guaranteed by the Monotone Iteration Method.This method requires the notions of sub and supersolution.We now give some definitions and remarks in order to go further.Definition 2.1.A vector-function U ∈ (C 2,α ) 2 is a "subsolution" for the system (S) if A "supersolution" is defined likewise by reversing the inequality.
Definition 2.2.We say that a matrix-function A(x) "satisfies a maximum principle" if for all U satisfying we have U ≥ 0.
Remark 2.1.If A is a cooperative matrix (i.e., A has nonnegative off diagonal terms) the condition where µ < λ 1 and ( ., . ) IR 2 is the usual IR 2 inner product, is a sufficient condition for A to satisfy a maximum principle (see [10]).If A is a symmetric matrix, condition (2.1) is equivalent to condition (1.18).Remark 2.2.If the matrix A has entries in the space C α (Ω) and satisfies a maximum principle, then the linear system with F ∈ (C α ) 2 has a (unique) solution.Indeed, in this case the above system has the uniqueness of solution property and this fact yields readily an (C 2,α ) 2 a priori estimates for its solutions.Hence by Remark 0.1, a Fredholm Alternative is valid for this system.
Let us prove the existence of sub and supersolutions for (S) T .Lemma 2.1.(Existence of subsolutions) Suppose that (1.1), (1.6), (1.7) and (1.8) hold.Then for a given vector-function there exists a subsolution U ∈ (C 2,α ) 2 of the system (S) T such that if U is any supersolution of this system, we have that ∂ν is the outward directional derivative).
Proof.Choose C such that one has strict inequality in (1.1).Assertions (1.6), (1.7) and (1.8) give sufficient and necessary conditions for A to satisfy a Maximum Principle (see [5]).Therefore by Remark 2.2 the system has a unique solution U .By (1.1) it is immediate that U is a subsolution for (S) T .From (1.6), (1.8) and the choice of C, using the Maximum Principle and the Hopf Maximum Principle of the scalar case for both equations in the above system we assure that (2.3) holds.

Lemma 2.2. (Existence of supersolutions) For any given vector-function
such that system (S) T has a positive supersolution U for all T ≤ T o .
Proof.Let us fix N > 0 and define Consider the domains below to be chosen timely Let w be the solution of the equation Hence, w ∈ C 2,α (Ω) and w > 0. From L p estimates and the Sobolev embedding Theorem it is possible to choose Ω 1 such that |w(x)| < N for x ∈ Ω.Now choosing T o = (r o , s o ) with r o and s o sufficiently negative as to have it is readily seen from the definition of m that U = (w, w) > 0 is the wanted supersolution.

The Monotone Iteration Method
This method has the advantage of being constructive and yields easily some important facts (see Corollary 3.1 and Lemma 5.2).Theorem 3.1.(Monotone iteration method) Let U and U be, respectively, ordered sub and supersolution of (S) with U ≤ U .Let A(x) be a matrixfunction with entries in the space C α such that: Proof.By Remark 2.2 the following operator is well defined where W is the (unique) solution of the system (3.2) By (3.1), using the Maximum Principle for systems, we have that T is monotone.Let us define the sequences By the previous arguments, we prove that Iterating the above sequences we achieve that So, there are pointwise convergent subsequences (denoted again by U n and V n ) such that By a standard bootstrap argument we may assume the above convergences in (C 2,α ) 2 .Hence, going to the limit in (3.2) we conclude the first part of the proof.The second one is also proved by iteration.
In this way, if Now by Remark 2.1 and the previous theorem there exists a solution U ∈ (C 2,α ) 2 of system (3.3).The vector-function −U is the other solution.

The Curve
Let us now construct the curve.For this purpose, let us fix and define By Lemma 1.1 and Theorem 3.1, the curve below is well-defined: It is readily seen the following properties of Γ: Γ 1 ) Γ increases in the r direction and decreases in the s direction (ie, t < t implies r(t ) ≤ r(t ) and s(t From all our results so far and from the construction of Γwe are now able to state the final Theorem of the first part.Changing the previous order in IR 2 by the new order U ≥ 0 if and only if u ≥ 0 and v ≤ 0, mutatis mutandis previous definitions and hypotheses, a similar theorem to Theorem 4.1still holds when the matrices A and A are not cooperative, with non-positive off diagonal terms.In this case a similar curve to Γ is found with similar properties. To finish this section, we give Example 4.1.(The Ambrosetti-Prodi problem for the biharmonic equation) Let us consider the biharmonic equation with a, b, t ∈ IR.This equation is equivalent to the elliptic system If the following hypotheses hold: uniformly for all x ∈ Ω and

SECOND PART
In this second part we shall improve Theorem 4.1 by finding another solution if T ∈ E and a solution if T ∈ Γ.

A Priori Bounds
The a priori estimates we are to obtain are needed to the use of the Leray-Schauder Degree.In the following section we apply degree theory in order to find the other solution.In this section we use the ideas of Brèzis-Turner [2], where they made estimates in the scalar case using the Hardy-Sobolev inequality (see [2] or [6] ).In our work the limitations of the growth conditions on the non-linearities f and g ( see (5.16) and (5.17) below ) are due to the restriction of the exponents of the Hardy-Sobolev inequality.Let us denote by U = (u, v) an eventual solution of the system (5.1) The same letter c will denote a generic positive constant.Proof.As in Lemma 1.1, using assumptions (1.1) and (1.2) we obtain where I is the 2 × 2 identity matrix.
Remark 5.1.The constant c appearing in (5.9) -and hence in (5.6)depends upon H but it is independent of U .It is relevant to remark that this constant may be taken uniformly with respect to H varying in a bounded subset of (C α ) 2 .Later we shall make use of this important fact.
For estimating purposes we need the following growth conditions on the non-linearities, for all x ∈ Ω ; η, ξ ∈ IR: (5.17) Remark 5.2.In our estimates, p and q are restricted to be below the socalled critical hyperbola: . However, our estimates include some cases that are not covered in the papers [3] and [9], where a priori estimates are also obtained.It is also important to point out that, contrary the scalar case, in our work the exponents p or q may grow beyond the Brèzis-Turner exponent: Theorem 5.1.(H 1 0 (Ω)-a priori estimate for the positive part) If the assumptions (1.1), (1.2), (1.5), (1.6), (1.7), (1.8), (3.1), (5.10), . . ., (5.16) and (5.17) hold, then there exists a constant c > 0 such that

18)
Proof.There is no loss of generality in assuming that f 1 (x, η) ≥ 0 for η ≥ 0 and f 2 (x, ξ) ≥ 0 for ξ ≥ 0. Indeed, using (5.10) and (1.1) we have = − m 1 in (5.10).Then f 1 satisfies all the hypotheses required for f 1 .The same is done for f 2 .Hence hereafter we shall assume that f 1, f 2 ≥ 0 for η ≥ 0 and ξ ≥ 0, respectively.We may write The above facts with (5.5), (5.6) and (5.10) lead to for some c > 0. Multiplying the first equation of (5.1) by u + , then integrating by parts the resulting expression and using the a priori estimates for the negative parts (5.5), we obtain Now let us estimate I 1 and I 2 , using the Hardy-Sobolev Inequality (see [2]): By virtue of (5.11) and (5.20), for an arbitrary ε > 0 we have where c(ε) = 0(1) as ε → 0 and C ε > 0 (see Lemma 2.4 in [2]).The proof of this inequality follows the same ideas that we are going to use in the following to estimate I 2 .Hence we suppress it here.
As the growth condition on the non-linearities f and g are subcritical, a bootstrap argument enable us to obtain estimates in spaces of more regular functions.Indeed, from Lemma 5.

Computation of Some Topological Degrees
As previously said, in this section, employing Leray-Schauder degree Theory, we shall improve Theorem 4.1 by finding another solution of system (S) if T ∈ E and one solution if T is on the curve Γ.A priori estimates are required for the use of topological degree.So, throughout this section we shall assume all the hypotheses of Theorem 5.1, where a priori bounds were found, with the additional hypothesis that A in (3.1) is cooperative.
First, for each T = r s ∈ IR 2 let us define the operator where V is the unique solution (see Remark 3.1) of the system (6.2) (A is the same matrix that appears in Theorem 3.1).From Remark 0.1 and Remark 2.2, it follows that K T is a compact operator if looked from (C 1,α ) 2 into itself.
With this setting, U is a solution of (S) if, and only if (I − K T )U = 0. Therefore Leray-Schauder degree is applicable to find a solution of the last equation.Lemma 6.1.Let T o ∈ IR 2 and H 1 ∈ N 2 ⊥ be fixed.Then there exists a constant R > 0 such that Proof.T 0 ∈ W t 0 for some t 0 ∈ IR.Fix some T 1 ∈ W t 0 ∩ N and let W t 0 be the closed segment joining the points T 0 and T 1 .From Corollary 5.1 and Remark 5.1, there exists R > 0 such that ||U || (C 1,α ) 2 < R for any eventual solution of (S) T for all T ∈ W t 0 .Since K T is an admissible homotopy joining K T 0 and K T 1 and (S) T 1 has no solution, we obtain (6.3).Proof.T 0 ∈ W t 0 some t 0 ∈ IR.Let Ẇt 0 be the open segment on W t 0 joining T 0 to Γ(t 0 ).Fixed T 1 ∈ Ẇt 0 , it follows from Theorem 4.1 that system (S) T 1 has a solution U = u v which is a supersolution for (S) T 0 .By Lemma 2.1 there exists a subsolution U = u v for (S) T 0 , such that U < U in Ω and and ||U || (C 1,α ) 2 < r} for some r > 0 to be chosen later.
Let us define In a similar way we define a function g.The functions f and g are bounded and non-decreasing .With these functions we can introduce, as in the beginning of this section, the operator K T 0 : (C 1,α ) 2 → (C 2,α ) 2 such that K T 0 U = V where V is the unique solution of the system (6.6)−∆V = A(x)V + F (x, U ) + T 0 φ 1 + H 1 , Ω, V = 0, ∂Ω.

and Z 2 .
, Z 1 ∈ [min x∈Ω The same definition for max x∈Ω U ).Then, system (S) has two solutions U and V in (C 2,α ) 2 (eventually it may occur that

2 ;Theorem 6 . 1 .
U ∈ (C 1,α ) 2 }, use (3.1), use the fact that f and g are non-decreasing, and use the Maximum Principle for systems, it follows that K T 0 (O) ⊆ O. Now take ψ ∈ O and consider the admissible compact homotopy H θ(U ) = θ K T 0 (U )+(1−θ)ψ, 0 ≤ θ ≤ 1.As ψ is a constant mapping, D(I −H 0 , O, 0) = 1 and hence D(I − K T 0 , O, 0) = 1.The proof is finished since K T 0 | O ≡ K T 0 .Consider the assumptions of Theorem 5.1.Then the conclusions of Theorem 4.1 can be strengthened as follows:(i) (S) T has no solution if T ∈ N (ii) (S) T has at least one solution if T ∈ Γ (iii) (S) T has at least two solutions if T ∈ E.Proof.Case (i) is already contained in Theorem 4.1.Case (ii) is consequence of Lemmas 6.1 and 6.2.Indeed take R > 0 in Lemma 6.1 such that R > r.Since D(I − K T 0 , O, 0) = 1 we have a solution U in O.By Lemma 6.1 and excision property of the degree we have that D(I − K T , B R (0)/O, 0) = −1 and then there exists another solution W = U of (S) T in B R (0)/O.In Case (iii), take a sequence (T n ) ⊆ E such that T n → T .By the uniform a priori bound (Corollary 5.1 and Remark 5.1 ) we have that ||U n || (C 1,α ) 2 ≤ c for some c > 0. By the compact embedding Theorem (Remark 0.1) we have (passing eventually to a subsequence) U n → U 0 in (C α ) 2 for some U 0 ∈ (C α ) 2 .