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EXISTENCE AND UNIQUENESS OF SOLUTION FOR A NONHOMOGENEOUS NONLOCAL PROBLEM CAMIL S.Z. REDWAN,JOAÕ R.SANTOS JUŃIOR,AND ANTONIOSUA´REZ 7 1 0 Abstract. In this paper we investigate a class of elliptic problems involving a nonlocal 2 Kirchhoff type operator with variable coefficients and data changing its sign. Under n appropriated conditions on the coefficients, we have shown existence and uniqueness of a solution. J 1 1. Introduction ] In this paper we are concerned with uniqueness of nontrivial classic solution to the following P class of nonlocal elliptic equations A . h a(x)+b(x) u2dx ∆u= h(x) in Ω, at (P) − Ω|∇ | u = 0 on ∂Ω, m (cid:26) (cid:0) R (cid:1) [ where Ω IRN, N 2, is a bounded domain with smooth boundary, a,b C0,γ(Ω), γ (0,1), ⊂ ≥ ∈ ∈ are positive functions with a(x) a > 0, b(x) b > 0 and h C0,γ(Ω) is given. 1 ≥ 0 ≥ 0 ∈ v When functions a,b are positive constants, problem (P) is the N-dimensional stationary 7 version of a hyperbolic problem proposed in [5] to model small transversal vibrations of an 6 2 elastic string with fixed ends which is composed by a homogeneous material. Such equation is 0 a more realistic model than that provided by the classic D’Alembert’s wave equation because 0 takes account the changing in the length of the string during the vibrations. The hyperbolic . 1 Kirchhoff problem (with a,b constants) began receiving special attention mainly after that 0 in [6] the author used an approach of functional analysis to attack it. 7 1 At least in our knowledge, the first work in studying uniqueness questions to problem (P) : with a,b constants was [2]. It is an immediate consequence of Theorem 1 in [2] that if h is a v i Hölder continuous nonnegative (nonzero) function then problem (P), with a,b constants, has a X unique positive solution. In [2] functions h sign changing are not considered. In the case that r a a,barenotconstant, problem(P)isyetmorerelevantinanapplicationspointofaviewbecause its unidimensional version models small transversal vibrations of an elastic string composed by non-homogeneous materials (see [4], section 2). In [4] (see Theorem 1) the authors proved that for each h L∞(Ω) (h 0) given, problem (P) admits at least a nontrivial solution. ∈ 6≡ Moreover, same article tells us that if h has defined sign (h 0 or h 0) such a solution ≤ ≥ is unique. Unfortunately, since their approach was based in a monotonicity argument which does not work when h is a sign changing function, they were not able to say anything about uniqueness in this case. Indeed, at least in our knowledge, actually the uniqueness of solution to problem (P) in the general case is an open problem. 2000 Mathematics Subject Classification. 35J15, 35J25, 35Q74. Key words and phrases. Kirchhoff typeequation, sublinear problem, topological method. Camil S. Z. Redwan was partially supported by Fapespa, Brazil. Joaõ R. Santos Juńior was partially supported by CNPq-Proc. 302698/2015-9, Brazil. Antonio Sua´rez has been partially supported by MTM2015- 69875-P (MINECO/FEDER, UE) and CNPq-Proc. 400426/2013-7. 1 2 C.S.Z.REDWAN,J.R.SANTOSJR.,ANDA.SUA´REZ Atthisarticlewehaveobtainedsufficientconditionsonthequotienta/btoensureuniqueness ofsolutionwhenfunctionh,given,changesitssign. Themainresultsofthispaperareasfollows Theorem 1.1. If there exists θ > 0 such that a/b= θ in Ω, then for each h C0,γ(Ω) problem ∈ (P) has a unique solution. Theorem 1.2. Let a,b C2,γ(Ω), h C0,γ(Ω) is sign changing and suppose c = a/b not ∈ ∈ constant. (i) If ∆c 2 c2/c in Ω, then, for each h C0,γ(Ω) given, problem (P) has a unique ≥ |∇ | ∈ nontrivial classic solution. (ii) If ∆c < 2 c2/c in some open Ω Ω then, for each h C0,γ(Ω) given, problem (P) 0 |∇ | ⊂ ∈ has a unique nontrivial classic solution, provided that c c ∞ M |∇ | 3/2, √λ c2 ≤ 1 L where λ is the first eigenvalue of laplacian operator with Dirichlet boundary condition, 1 c =min c(x), c = max c(x) and c = max c(x). L x∈Ω M x∈Ω |∇ |∞ x∈Ω|∇ | First theorem above generalizes Theorem 1 in [2] because it is true to functions h sign changing or not. The second theorem above complements Theorem 1 in [4]. The paper is organized as follows. In Section 2 we present some abstracts results, notations and definitions. In Section 3 we investigated a nonlocal eigenvalue problem which seems to be closely related with uniqueness questionstoproblem(P). InSection4weproveTheorems1.1and1.2. Moreover, analternative proof of the existence and uniqueness result in [4] is supplied. 2. Preliminaries In this section we state some results and fix notations used along of paper. Definition 2.1. We say that a function h is signed in Ω if h 0 in Ω or h 0 in Ω. ≥ ≤ Definition 2.2. An application Ψ : E F defined in Banach spaces is locally invertible in → u E if there are open sets A u in E and B Ψ(u) in F such that Ψ: A B is a bijection. ∈ ∋ ∋ → If Ψ is locally invertible in any point u E it is said that Ψ: E F is locally invertible. ∈ → Definition 2.3. Let M,N be metric spaces. We say that a map Ψ : M N is proper if → Ψ−1(K) = u M :Ψ(u) K is compact in M for all compact set K N. { ∈ ∈ } ⊂ Below we enunciate the classic local and global inverse function theorems, whose proofs can be found, for instance, in [1]. Theorem 2.4 (Local Inverse Theorem). Let E,F be two Banach spaces. Suppose Ψ ∈ C1(E,F) and Ψ′(u) : E F is a isomorphism. Then Ψ is locally invertible at u and its → local inverse, Ψ−1, is also a C1-function. Theorem 2.5 (Global Inverse Theorem). Let M,N be two metric spaces and Ψ C(M,N) ∈ a proper and locally invertible function on all of M. Suppose that M is arcwise connected and N is simply connected. Then Ψ is a homeomorphism from M onto N. Next, we state another classical result which will be used in our arguments and whose proof can be found, for instance, for a more general class of problems, in [3]. EXISTENCE AND UNIQUENESS OF SOLUTION FOR A NONHOMOGENEOUS NONLOCAL PROBLEM 3 Proposition 2.6. Let m L∞(Ω), m(x)> 0 in a set of positive measure. Then, problem ∈ div(A(x) u) = λm(x)u in Ω, (2.1) − ∇ u = 0 on ∂Ω, (cid:26) where A L∞(Ω) and A(x) m for some positive constant m, has a smallest positive ∈ ≥ eigenvalue λ (m) which is simple and corresponding eigenfunctions do not change sign in Ω. 1 Throughout this paper X is the Banach space X = u C2,γ(Ω): u= 0 on ∂Ω { ∈ } with norm u = u +max[Dβu] , k kX k kC2(Ω) |β|=2 γ where γ (0,1), β = (β ,...,β ) INN, β = β +...+β , 1 N 1 N ∈ ∈ | | Dβu(x) Dβu(y) u = Dβu and [Dβu] = sup | − |. k kC2(Ω) k kC(Ω) γ x,y∈Ω x y γ 0≤X|β|≤2 x6=y | − | Moreover Y will denote the Banach space C0,γ(Ω) with norm f = f +[f] , k kY k kC(Ω) γ where f = max f(x). k kC(Ω) x∈Ω| | Hereafter same symbol C denotes different positive constants. 3. A nonlocal eigenvalue problem In this section we are interested in studying the following nonlocal eigenvalue problem u c div ∇ = λ div ∇ u in Ω, (EP) − c+ u2 − (c+ u2)2  (cid:18) |∇ |2(cid:19) (cid:26) (cid:20) |∇ |2 (cid:21)(cid:27) u= 0 on ∂Ω,  where Ω IRN is bounded smooth domain, λ is a positive parameter and c C2(Ω) is a  ⊂ ∈ positive (not constant) function. As we will see in the next section, problem (EP) arises naturally when one studies questions of uniqueness to the problem (P). Before to state the main results of this section, we observe that Lemma 3.1. The set c := α > 0 : div ∇ > 0 in some open Ω Ω A − (c+α)2 0 ⊂ (cid:26) (cid:20) (cid:21) (cid:27) is not empty if, and only if, there is an open Ωˆ IRN such that ⊂ c2 (3.1) ∆c <2|∇ | in Ωˆ. c Proof. Differentiating we get c 1 2 (3.2) div ∇ = ∆c+ c2. − (c+α)2 −(c+α)2 (c+α)3|∇ | (cid:20) (cid:21) Now, note that c (3.3) div ∇ > 0 in some open Ω − (c+α)2 0 (cid:20) (cid:21) 4 C.S.Z.REDWAN,J.R.SANTOSJR.,ANDA.SUA´REZ if, and only if, c2 (3.4) ∆c< 2 |∇ | in Ω . 0 (c+α) It is clear that the existence of a positive number α satisfying (3.4) is equivalent to inequality in (3.1). (cid:3) Remark 1. In previous Lemma we have shown also that = if, and only if, A ∅ c2 (3.5) ∆c 2|∇ | in Ω. ≥ c Certainly, there are many positive functions c C2(Ω) verifying (3.5). For instance, setting ∈ c = δe+1, where 0 < δ min 1/(4 e2 ),1/(2e ) and ≤ { |∇ |∞ | |∞ } ∆e = 1 in Ω, (3.6) e = 0 on ∂Ω, (cid:26) we conclude that c > 0 and satisfies (3.5). Remark 2. An interesting question when (3.1) holds is about the topology of set . In this A direction, the proof of Lemma 3.1 allows us to say that contains ever a neighborhood (0,α ). 0 A Now we are ready to claim the following result. Theorem 3.2. Suppose that (3.1) holds. For each α , problem (EP) has a unique solution ∈A (λ ,u ) such that λ > 0, u > 0 and u 2 = α. α α α α |∇ α|2 Proof. From Lemma 3.1, = . Since c C2(Ω) and b > 0 in Ω, it follows from Proposition A 6 ∅ ∈ 2.6 that, for each α , the eigenvalue problem ∈ A u c div ∇ = λ div ∇ u in Ω, (Pα) − c+α − (c+α)2  (cid:18) (cid:19) (cid:26) (cid:20) (cid:21)(cid:27) u= 0 on ∂Ω,  has a positive smallest eigenvalue λ whose associated eigenspace V is unidimensional and its  α α eigenfunctions have defined sign. Choosing u V such that u > 0 and u2 = α, the result follows. ∈ α |∇ |2 (cid:3) Remark 3. In particular, if (3.1) holds then c 1 u 2 (3.7) u2 div ∇ dx = |∇ α| dx, α . α − (c+α)2 λ c+α ∀ ∈ A ZΩ (cid:26) (cid:20) (cid:21)(cid:27) α ZΩ Corollary 3.3. Suppose (3.1). For each α , the following inequality holds ∈ A √λ (c +α)2 1 L λ , α ≥ 2 c (c +α) ∞ M |∇ | where λ is the first eigenvalue of laplacian operator with Dirichlet boundary condition, 1 c = min c(x), c = max c(x) and c = max c(x). L x∈Ω M x∈Ω |∇ |∞ x∈Ω|∇ | Proof. From Remark 3, we get u 2 α |∇ | dx c+α (3.8) λ = ZΩ . α c u2 div ∇ dx α − (c+α)2 ZΩ (cid:26) (cid:20) (cid:21)(cid:27) EXISTENCE AND UNIQUENESS OF SOLUTION FOR A NONHOMOGENEOUS NONLOCAL PROBLEM 5 Observe that u 2 α α (3.9) |∇ | dx . c+α ≥ c +α ZΩ M Moreover, by using the Divergence Theorem, 2 c u u dx u2 div ∇c dx = 2 uα∇uα∇cdx |∇ |∞ZΩ α|∇ α| . α − (c+α)2 (c+α)2 ≤ (c +α)2 ZΩ (cid:26) (cid:20) (cid:21)(cid:27) ZΩ L From Hölder and Poincaré inequalities, we conclude that c 2 c α (3.10) u2 div ∇ dx |∇ |∞ . α − (c+α)2 ≤ √λ (c +α)2 ZΩ (cid:26) (cid:20) (cid:21)(cid:27) 1 L From (3.8), (3.9) and (3.10) we have √λ (c +α)2 1 L λ , α ≥ 2 c (c +α) ∞ M |∇ | for all α . (cid:3) ∈ A 4. Uniqueness results In order to apply Theorem 2.5 we define operator Ψ :X Y by → Ψ(u) = a(x)+b(x) u2dx ∆u. |∇ | (cid:18) ZΩ (cid:19) In the sequel, we will denote M x, u2 = a(x) + b(x) u2dx for short, where |∇ |2 Ω|∇ | u2 = u2dx. The proof of main results of this paper will be divided in various |∇ |2 Ω|∇ | (cid:0) (cid:1) R propositions. R Proposition 4.1. Operator Ψ : X Y is proper. → Proof. Itissufficienttoprovethatif h Y isasequenceconvergingtoh Y and u X n n { }⊂ ∈ { } ⊂ is another sequence with Ψ(u )= h then u has a convergent subsequencein X. For this, n n n − { } note that the equality Ψ(u )= h is equivalent to n n − h n (4.1) ∆u = . − n M x, u 2 |∇ n|2 Observe that hn/M .,|∇un|22 ∈ Y because hn(cid:0)∈ Y, M (cid:1).,|∇un|22 ∈ Y and M x,|∇un|22 ≥ a . 0 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) Moreover, h (x) n (4.2) h /a , n IN. (cid:13)(cid:13)M x,|∇un|22 (cid:13)(cid:13)C(Ω) ≤ k nkC(Ω) 0 ∀ ∈ (cid:13) (cid:13) (cid:13) (cid:0) (cid:1)(cid:13) From h (cid:13)h , (4.2) a(cid:13)nd from boundedness of h in Y follows that k nkC(Ω) ≤ k nkY { n} h /M x, u 2 is bounded in C(Ω). Thus, the continuous embedding from C1,γ(Ω) n |∇ n|2 into C(Ω) and equality in (4.1) tell us that u is bounded in C1,γ(Ω) (see Theorem 0.5 (cid:8) (cid:0) (cid:1)(cid:9) { n} in [1]). Finally, by compact embedding from C1,γ(Ω) into C1(Ω), we conclude that there exists u C1(Ω) such that, passing to a subsequence, ∈ (4.3) u u in C1(Ω). n → 6 C.S.Z.REDWAN,J.R.SANTOSJR.,ANDA.SUA´REZ Last convergence leads to (4.4) u (x)2 u(x)2 uniformly in x Ω. n |∇ | → |∇ | ∈ Whence (4.5) u 2 u2. |∇ n|2 → |∇ |2 In the follows, we show that h n (4.6) C, (cid:13)(cid:13)M .,|∇un|22 (cid:13)(cid:13)Y ≤ (cid:13) (cid:13) for some positive constant C. In(cid:13)fact(cid:0), since (cid:1)h(cid:13) Y and M ., u 2 Y, with (cid:13) { (cid:13)n} ⊂ |∇ n|2 ⊂ M(x,t) a > 0 for all t 0, a straightforward calculation shows us that 0 ≥ ≥ (cid:8) (cid:0) (cid:1)(cid:9) h 1 n h M ., u 2 + M ., u 2 [h ] . "M .,|∇un|22 #γ ≤ a20 (cid:16)k nkC(Ω)(cid:2) (cid:0) |∇ n|2(cid:1)(cid:3)γ (cid:13) (cid:0) |∇ n|2(cid:1)(cid:13)C(Ω) n γ(cid:17) From h(cid:0) ,[h(cid:1)] C, (cid:13) (cid:13) k nkC(Ω) n γ ≤ (4.7) M ., u 2 [a] +[b] u 2 [a] +C[b] |∇ n|2 γ ≤ γ γ|∇ n|2 ≤ γ γ and (cid:2) (cid:0) (cid:1)(cid:3) (4.8) M ., u 2 a + b u 2 a +C b |∇ n|2 C(Ω) ≤ k kC(Ω) k kC(Ω)|∇ n|2 ≤ k kC(Ω) k kC(Ω) it follows tha(cid:13)t (cid:0) (cid:1)(cid:13) (cid:13) (cid:13) h C C C2 n [a] +C[b] + a +C b = a + b . "M .,|∇un|22 #γ ≤ a20 (cid:16) γ γ k kC(Ω) k kC(Ω)(cid:17) a20k kY a20 k kY Being h (cid:0)/M x, (cid:1)u 2 bounded in C(Ω), the last inequality proves the assertion in (4.6). n |∇ n|2 By (4.1), (4.6) and Theorem 0.5 in [1], sequence u is bounded in X. By compact n (cid:8) (cid:0) (cid:1)(cid:9) { } embedding from X in C2(Ω), passing to a subsequence, we get (4.9) u u in C2(Ω). n → By (4.9), passing to the limit in n in (4.1) we have → ∞ h (4.10) ∆u= . − M x, u2 |∇ |2 Last equality and Theorem 0.5 in [1] allow us to(cid:0)conclud(cid:1)e that u X. ∈ Finally, by linearity of laplacian, we have h h n (4.11) ∆(u u)= . − n − M x, u 2 − M x, u2 |∇ n|2 |∇ |2 From (4.11) and Theorem 0.5 in [1] we con(cid:0)clude that(cid:1)un (cid:0)u in X. (cid:1) (cid:3) → Proposition 4.2. Let a,b C0,γ(Ω) and u X. If ∈ ∈ b(x)u∆u (4.12) dx = 1/2 M(x, u2) 6 ZΩ |∇ |2 holds then Ψ is locally invertible in u. EXISTENCE AND UNIQUENESS OF SOLUTION FOR A NONHOMOGENEOUS NONLOCAL PROBLEM 7 Proof. We are interested in using Theorem 2.4 to prove this Lemma. It is standard to show that Ψ C1(X,Y) and ∈ Ψ′(u)v = 2b(x)∆u u vdx+M(x, u 2)∆v. ∇ ∇ |∇ |2 ZΩ Remain us proving that Ψ′(u) : X Y is an isomorphism. It is clear that if u = 0 there is → nothing to prove. Now, if u= 0, observes that Ψ′(u) is an isomorphism if, and only if, for each 6 g Y given, there is a unique v X such that Ψ′(u)v = g, this is ∈ ∈ − (4.13) M(x, u2)∆v = g(x)+2b(x)∆u u vdx. − |∇ |2 ∇ ∇ ZΩ From Divergence Theorem, equation in (4.13) is equivalent to (4.14) M(x, u2)∆v = g(x) 2b(x)∆u u∆vdx. − |∇ |2 − ZΩ Consequently, Ψ′(u) is an isomorphism if, and only if, for each g Y given, there is a unique ∈ v X such that ∈ 2b(x)∆u u∆vdx g(x) (4.15) ∆v = Ω . M(x, u2) − M(x, u2) R|∇ |2 |∇ |2 To study equation (4.15) we define the mapping T :Y Y by → 2b(x)∆u uwdx g(x) (4.16) T(w) = Ω M(x, u2) − M(x, u2) |R∇ |2 |∇ |2 and we note that, since for each w Y problem ∈ ∆z = w(x) in Ω, (LP) z = 0 on ∂Ω, (cid:26) has a unique solution z X, looking for solutions of (4.15) is equivalent to find fixed points of ∈ T. Denoting t = uwdx, it follows that w is a fixed point of T if, and only if, Ω R 2b(x)∆u g(x) (4.17) w = T(w) = t . M(x, u2) − M(x, u2) |∇ |2 |∇ |2 Therefore w is a fixed point of T if, and only if, 2b(x)∆u g(x) 2b(x)∆u g(x) T t = t . M(x, u2) − M(x, u2) M(x, u2) − M(x, u2) (cid:18) |∇ |2 |∇ |2 (cid:19) |∇ |2 |∇ |2 From (4.16), we get 2b(x)∆u 2b(x)∆u g(x) 2b(x)∆u u t dx = t . M(x, u2) M(x, u2) − M(x, u2) M(x, u2) |∇ |2 ZΩ (cid:20) |∇ |2 |∇ |2 (cid:21) |∇ |2 Since b > 0 and ∆u 0 (because u= 0), T admits a fixed point if, and only if, 6≡ 6 2b(x)∆u g(x) 2 u t dx = t, M(x, u2) − M(x, u2) ZΩ (cid:20) |∇ |2 |∇ |2 (cid:21) namely, 2b(x)u∆u g(x)u (4.18) t dx 1 = 2 dx. M(x, u2) − M(x, u2) (cid:20)ZΩ |∇ |2 (cid:21) ZΩ |∇ |2 8 C.S.Z.REDWAN,J.R.SANTOSJR.,ANDA.SUA´REZ Equality (4.18) say us that if (4.12) occurs then T has a unique fixed point w given by 2b(x)∆u g(x) w = t , M(x, u2) − M(x, u2) |∇ |2 |∇ |2 with g(x)u 2b(x)u∆u t = 2 dx/ dx 1 . M(x, u2) M(x, u2) − ZΩ |∇ |2 (cid:20)ZΩ |∇ |2 (cid:21) (cid:3) Remark 4. Equality (4.18) shows us that Ψ′(u) : X Y is not surjective if → b(x)u∆u dx = 1/2. M(x, u2) ZΩ |∇ |2 In fact, in this case, functions g Y such that ∈ g(x)u dx = 0 M(x, u2) 6 ZΩ |∇ |2 are not in the range of Ψ′(u). Actually, itis possibleto getthesameresultof (existence and)uniquenessprovidedin [4]for signed functions as a consequence of Global Inverse Theorem and previous Proposition. This is exactly the content of next corollary. Corollary 4.3. For each signed function h Y given, problem (P) has a unique solution. ∈ Proof. First of all, we define the sets P = u X :∆u 0 X 1 { ∈ ≥ } ⊂ and P = h Y : h 0 Y. 2 { ∈ ≥ } ⊂ Consider P ( P ) and P ( P ) as metric spaces whose metrics are induced from X and 1 1 2 2 ∪ − ∪ − Y, respectively. It is clear that P ( P ) is arcwise connected (because P and P are convex sets and 1 1 1 1 ∪ − − P ( P ) = 0 ) closed in X. On the other hand, since P ( P ) is the union of the closed 1 1 2 2 ∩ − { } ∪ − cone of nonnegative functions of Y with the closed cone of nonpositive functions of Y, follows that P ( P ) is simply connected. 2 2 ∪ − From Ψ(P ) P and Ψ( P ) ( P ), it follows that Ψ is well defined from P ( P ) to 1 2 1 2 1 1 ⊂ − ⊂ − ∪ − P ( P ). 2 2 ∪ − Moreover, being Ψ proper from X to Y (see Proposition 4.1) and P ( P ) and P ( P ) 1 1 2 2 ∪ − ∪ − are closed metric spaces in X and Y, respectively, it follows that Ψ is proper from P ( P ) 1 1 ∪ − to P ( P ). 2 2 ∪ − Note that if u P (resp. P ) then, as u is (the unique) solution to problem 1 1 ∈ − ∆u= ∆u in Ω, (4.19) u= 0 on ∂Ω. (cid:26) Follows from maximum principle that u 0 (resp. u 0). Whence, we have ≤ ≥ b(x)u∆u dx 0, u P ( P ). M(x, u2) ≤ ∀ ∈ 1∪ − 1 ZΩ |∇ |2 Therefore, from Proposition 4.2, Ψ: P ( P ) P ( P ) is locally invertible. The result 1 1 2 2 follows now from Global Inverse Theore∪m.− → ∪ − (cid:3) EXISTENCE AND UNIQUENESS OF SOLUTION FOR A NONHOMOGENEOUS NONLOCAL PROBLEM 9 Next corollary does not ensureuniqueness of solution for problem (P) when function h given is sign changing, but it tells us that there is a unique solution with “little variation” if h Y ∈ given (signed or not) has “little variation”. Corollary 4.4. There are positive constants ε,δ such that for each h Y with h < ε, Y ∈ k k problem (P) has a unique solution u with u < δ. X k k Proof. It is sufficient to note that when u= 0 the integral in previous proposition is null. (cid:3) We are now ready to prove Theorem 1.1. Proof of Theorem 1.1. Since X and Y are Banach spaces then X is arcwise connected and Y is simply connected. Moreover, from Proposition 4.1, operator Ψ is proper and by Divergence Theorem b(x)u∆u 1 u2 dx = u∆udx = |∇ |2 < 0, u X. M(x, u2) θ+ u2 −θ+ u2 ∀ ∈ ZΩ |∇ |2 |∇ |2 ZΩ |∇ |2 The result follows directly from Proposition 4.2 and Global Inverse Theorem. (cid:3) Nextpropositionprovidesusasufficientconditiononfunctionsaandbforthat(4.12)occurs when a/b is not constant. Proposition 4.5. Let a,b C2,γ(Ω) and c = a/b. ∈ (i) If ∆c 2 c2/c in Ω, then ≥ |∇ | b(x)u∆u (4.20) dx 0, u X. M(x, u2) ≤ ∀ ∈ ZΩ |∇ |2 (ii) If ∆c < 2 c2/c in some open Ω Ω then 0 |∇ | ⊂ b(x)u∆u (4.21) dx < 1/2, u X, M(x, u2) ∀ ∈ ZΩ |∇ |2 provided that c c ∞ M |∇ | 3/2. √λ c2 ≤ 1 L Proof. Putting b in evidence in the integral (4.12), we get b(x)u∆u u∆u dx = dx, M(x, u2) c+ u2 ZΩ |∇ |2 ZΩ |∇ |2 where c =c(x) = a(x)/b(x). From Divergence Theorem, we have u∆u u dx = udx. c+ u2 − ∇ c+ u2 ∇ ZΩ |∇ |2 ZΩ (cid:18) |∇ |2(cid:19) Since u 1 u = u c, ∇ c+ u2 c+ u2∇ − (c+ u2)2∇ (cid:18) |∇ |2(cid:19) |∇ |2 |∇ |2 we conclude that b(x)u∆u u2 u u c dx = |∇ | dx+ ∇ ∇ dx M(x, u2) − c+ u2 (c+ u2)2 ZΩ |∇ |2 ZΩ |∇ |2 ZΩ |∇ |2 u2 1 (u2) c = |∇ | dx+ ∇ ∇ dx. − c+ u2 2 (c+ u2)2 ZΩ |∇ |2 ZΩ |∇ |2 10 C.S.Z.REDWAN,J.R.SANTOSJR.,ANDA.SUA´REZ Using again the Divergence Theorem b(x)u∆u u2 1 c (4.22) dx = |∇ | dx+ u2 div ∇ dx. M(x, u2) − c+ u2 2 − (c+ u2)2 ZΩ |∇ |2 ZΩ |∇ |2 ZΩ (cid:26) (cid:20) |∇ |2 (cid:21)(cid:27) (i) At this case, from Lemma 3.1 (see also Remark 1), = and, consequently, for each A ∅ u X we have ∈ c u2 div ∇ dx 0. − (c+ u2)2 ≤ ZΩ (cid:26) (cid:20) |∇ |2 (cid:21)(cid:27) Whence, by (4.22), b(x)u∆u dx 0, u X. M(x, u2) ≤ ∀ ∈ ZΩ |∇ |2 (ii) In this case = . If u X is such that u 2 we saw already that A6 ∅ ∈ |∇ |2 6∈ A b(x)u∆u dx 0. M(x, u2) ≤ ZΩ |∇ |2 Now, if u X is such that u2 then, from (4.22) and Proposition 3.2 (see also Remark ∈ |∇ |2 ∈ A 3), we obtain b(x)u∆u 1 u2 (4.23) dx 1 |∇ | dx, M(x, u2) ≤ 2λ − c+α ZΩ |∇ |2 (cid:18) α (cid:19)ZΩ where α := u2. If α is such that 1/2 λ then, by (4.23), b(x)u∆u/M(x, u2)dx 0. |∇ |2 ≤ α Ω |∇ |2 ≤ Finally, if 0 < λ < 1/2 follows from α = u2 and from Corollary 3.3 that, α |∇ |2 R b(x)u∆u c (c +α) ∞ M (4.24) dx < |∇ | 1 =:g(α). M(x, u2) √λ (c +α)2 − ZΩ |∇ |2 1 L We have that g(0) = c c /√λ c2 1 and |∇ |∞ M 1 L− c (c 2c α) g′(α) = |∇ |∞ L − M − < 0, α> 0. √λ (c +α)3 ∀ 1 L Therefore g is decreasing and, from (4.24), we conclude that if c c 3 ∞ M |∇ | √λ c2 ≤ 2 1 L then (4.21) holds. (cid:3) Bellow we give the proof of our main uniqueness result to problem (P) which covers sign changing functions. Proof of Theorem 1.2. It follows directly from Proposition 4.1, Proposition 4.2, Proposition 4.5 and Global Inverse Theorem. (cid:3) Theorems1.1and1.2seemtoindicatethatinthecasethathissignchangingtheuniqueness of solution to the problem (P) is, in some way, related with the variation of a/b. In any way, remains open the question to know what happens with the number of solutions of (P) in the case that h is sign changing, ∆c < 2 c2/c in some open Ω Ω and c c /√λ c2 is |∇ | 0 ⊂ |∇ |∞ M 1 L large.