Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 16, pp. 1–10. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu NONEXITENCE OF NONTRIVIAL SOLUTIONS TO DIRICHLET PROBLEMS FOR THE FRACTIONAL LAPLACIAN JOSÉ CARMONA, ALEXIS MOLINO Abstract. In this article we prove that there are no nontrivial solutions to the Dirichlet problem for the fractional Laplacian (−∆)su = f(u) in Ω, u = 0 in RN \ Ω, where Ω ⊂ RN (N ≥ 1) is a bounded domain, and f is locally Lipschitz with non-positive primitive F (t) = ∫ t 0 f(τ)dτ . 1. Introduction In this work, we investigate the nonexistence of nontrivial bounded solutions for the Dirichlet problem for the fractional Laplacian (−∆)su = f(u) in Ω, u = 0 in RN \ Ω, (1.1) where Ω ⊂ RN (N ≥ 1) is a bounded domain with C1,1 regular boundary, ∂Ω, and f : R→ R is a locally Lipschitz sign-changing function. Throughout this article, the fractional Laplacian operator (−∆)s (also called, Riesz fractional Laplacian) is formally defined by (−∆)su = C(N, s) P.V. ∫ RN u(x)− u(y) |x− y|N+2s dy, s ∈ (0, 1), where C(N, s) is the positive constant given by C(N, s) = s22sΓ ( 2s+N 2 ) πN/2Γ(1− s) , (1.2) Γ denotes the Gamma function, and P.V. stands for the principal value of the integral P.V. ∫ RN u(x)− u(y) |x− y|N+2s dy = lim ε→0 ∫ RN\Bε(x) u(x)− u(y) |x− y|N+2s dy, where Bε(x) is the ball of radius ε centered at x. 2020 Mathematics Subject Classification. 35J05, 35J15, 35J25. Key words and phrases. Fractional Laplacian; Dirichlet problem; nonexistence of solutions. ©2023. This work is licensed under a CC BY 4.0 license. Submitted March 30, 2022 Published February 17, 2023. 1 2 J. CARMONA, A. MOLINO EJDE-2023/16 To establish the concept of solution to problem (1.1) we consider the usual Sobolev fractional spaces Hs(RN ) = { u ∈ L2(RN ) : u(x)− u(y) |x− y|N/2+s ∈ L2 ( RN × RN )} , Hs 0(Ω) = {u ∈ Hs(RN ) : u ≡ 0, a.e. RN \ Ω}. Let us recall that Hs 0(Ω) is a Hilbert space with the scalar product 〈u, v〉Hs 0 (Ω) = C(N, s) 2 ∫ RN ∫ RN (u(x)− u(y))(v(x)− v(y)) |x− y|N+2s dx dy, u, v ∈ Hs 0(Ω). We denote the induced norm by ‖u‖2Hs 0 (Ω) = C(N, s) 2 ∫ RN ∫ RN (u(x)− u(y))2 |x− y|N+2s dx dy, u ∈ Hs 0(Ω). For a detailed study on the different approaches to fractional Sobolev spaces, see [3]. In addition, a more extensively study of non-local operators, of which the fractional Laplacian is a particular case, can be found in the survey [16]. Multiplying equation (1.1) by v ∈ Hs 0(Ω) and integrating in RN , we obtain C(N, s) 2 ∫ RN ∫ RN (u(x)− u(y))(v(x)− v(y)) |x− y|N+2s dy dx = ∫ RN f(u(x))v(x)dx. Therefore, we say that u ∈ Hs 0(Ω) is a weak solution to problem (1.1) if 〈u, v〉Hs 0 (Ω) = ∫ RN f(u(x))v(x)dx, (1.3) for every test function v ∈ Hs 0(Ω). We deal with bounded solutions (solutions from now on) of problem (1.1). A direct consequence of [17, Corollary 1.6] ensures that weak bounded solutions to problem (1.1) belong to Cs(RN )∩C2s+ε(Ω), whenever ∂Ω is C1,1. As a consequence, weak bounded solutions are classical solutions to problem (1.1) in the sense that the fractional Laplacian operator can be pointwise evaluated in Ω. It follows immediately, taking u as a test function in (1.3), that a necessary condition for the existence of a solution u to (1.1) is ‖u‖2Hs 0 (Ω) = ∫ RN f(u)u. (1.4) In particular, if the hypothesis f(t)t ≤ 0, for all t ∈ R, (1.5) holds, then the unique solution to problem (1.1) is the trivial one. The main motivation for this work comes from the interest in finding sufficient conditions in the nonlinear term f , beyond (1.5), that guarantee the nonexistence of a nontrivial solution to (1.1). This represents a challenging problem even in the case of the Laplace operator, where (1.1) becomes −∆u = f(u) in Ω, u = 0 on ∂Ω. (1.6) Nonexistence results for (1.6) are usually deduced from the Pohozaev identity [12]∫ Ω (2NF (u)− (N − 2)uf(u)) dx = ∫ ∂Ω (∂u ∂ν )2 (x · ν) dσ(x), EJDE-2023/16 NONEXITENCE OF NONTRIVIAL SOLUTIONS 3 where ν is the unit outward normal to ∂Ω at x and F (t) = ∫ t 0 f(τ)dτ . Indeed, for star-shaped domains with respect to 0 (i.e., x · ν(x) > 0 on ∂Ω) and f(s) = |s|p−1s with p > 1, Pohozaev identity leads to nonexistence of nontrivial solutions for supercritical values of p, i.e. p ≥ N+2 N−2 , N > 2. However, existence of solution is known in the supercritical regime when Ω is not star-shaped [9]. A second nonexistence of nontrivial solution result is deduced from the Pohozaev identity when Ω is star-shaped and F satisfies F (t) = ∫ t 0 f(τ)dτ ≤ 0, for all t ∈ R. (1.7) A typical example satisfying (1.7) is f(t) = λ sin t, with λ < 0. By similarity with the power case, in [15], the author conjectured that, when Ω is not star-shaped and −λ is large enough, a nontrivial solution may exist. Some partial results were obtained in [4, 5, 6, 7, 8, 14]. Finally, it was shown in [10] that this conjecture is false by proving, for general domains Ω, that (1.6) admits only the trivial solution when (1.7) is satisfied. To the best of our knowledge, there are no such results in the case of non-local operators. This is the main goal of this paper and the main result we obtain is the following. Theorem 1.1. Let Ω ⊂ RN (N ≥ 1) be a C1,1 bounded domain, f : R → R a locally Lipschitz function satisfying (1.7). Then, u ≡ 0 is the unique solution to problem (1.1). Let us recall the Pohožaev identity for solutions to (1.1) due to [18] which states that, for N > 2s, (2s−N) ∫ Ω uf(u) dx+ 2N ∫ Ω F (u) dx = Γ(1 + s)2 ∫ ∂Ω ( u δs )2 (x · ν) dσ, (1.8) where δ(x) = dist(x, ∂Ω). Observe that, as in the local case, this equality implies Theorem 1.1 when Ω is starshaped and N ≥ 2s. Indeed, in this case, the right hand side is non-negative, and taking into account (1.4) we obtain the following inequality∫ Ω F (u) dx = Γ(1 + s)2 2N ∫ ∂Ω ( u δs )2 (x · ν) dσ + N − 2s 2N ∫ Ω uf(u) dx ≥ 0. Moreover, the equality holds only when u ≡ 0. In the present work, we prove that condition (1.7) ensures the nonexistence of nontrivial solutions to problem (1.1) with no additional hypotheses on the geometry of Ω or the dimension N . For the proof of Theorem 1.1 we rely mostly on two results: on one hand, we are inspired in the result for problem (1.6) carried out by the second author in [10] and, on the other hand, on [2] for the existence of an increasing solution for a certain type of non-local ordinary differential equation. This articleis organized as follows. In Section 2, we establish Maximum Principles to the fractional Laplacian. In Section 3, we prove Theorem 1.1, and in Section 4, we summarize in conclusion the main findings though some examples. Finally, in the Appendix we prove a technical Lemma used in the previous section. 4 J. CARMONA, A. MOLINO EJDE-2023/16 2. Maximum Principle This section is devoted to the Maximum Principle for the fractional Laplacian operator. Specifically, we prove a version of the well-known Serrin’s Sweeping Prin- ciple (see e.g. [13]). We denote by λ1 > 0 the first eigenvalue with associated nonnegative eigenfunc- tion ϕ1, i.e. it is satisfied that (−∆)sϕ1(x) = λ1ϕ1(x) in Ω, ϕ1(x) = 0 in RN \ Ω. First, we state and prove the Strong Maximum Principle for fractional Laplacian operators, [1], with the intention of making this section self-contained. Proposition 2.1 (Strong Maximum Principle). Let Ω ⊂ RN (N ≥ 1) be a bounded domain. Consider the function m : Ω → (−λ1,∞) and u ∈ Hs(RN ) satisfying the following inequality pointwise (−∆)su(x) +m(x)u(x) ≥ 0 in Ω, u ≥ 0 in RN \ Ω. (2.1) Then u ≥ 0 in RN . Furthermore, either u ≡ 0 in RN or u > 0 in Ω. Remark 2.2. As a direct consequence, it follows that if u satisfies the hypotheses of Proposition 2.1 and there exists x0 ∈ Ω such that u(x0) = 0, then u ≡ 0 in RN . Proof of Proposition 2.1. To prove that u ≥ 0 in RN we observe that (2.1) is true for u−(x) = min{u(x), 0}. Thus, multiplying by the first eigenfunction ϕ1 and integrating in RN we obtain 0 ≤ ∫ RN ( (−∆)su−(x) +m(x)u−(x) ) ϕ1(x)dx = C(N, s) 2 ∫ RN ∫ RN (u−(x)− u−(y))(ϕ1(x)− ϕ1(y)) |x− y|N+2s dy dx + ∫ RN m(x)u−(x)ϕ1(x)dx = ∫ RN (λ1 +m(x))u−(x)ϕ1(x)dx ≤ 0. Thus, u− ≡ 0 and we have that u is non-negative. We assume now that u is non-trivial in RN . Then, the set A = {x ∈ RN : u(x) > 0} has non-zero measure. To prove that u > 0 in Ω we argue by contradiction. Suppose that there exists x0 ∈ Ω such that u(x0) = 0. Evaluating x0 in inequality (2.1) we obtain the following contradiction 0 ≤ (−∆)su(x0) +m(x0)u(x0) = C(N, s) ∫ RN −u(y) |x0 − y|N+2s dy = C(N, s) ∫ A −u(y) |x0 − y|N+2s dy < 0. � Next result is known as Serrin’s Sweeping Principle and it has been shown to hold for the Laplacian operator (and other uniformly elliptic operators), see the EJDE-2023/16 NONEXITENCE OF NONTRIVIAL SOLUTIONS 5 pioneering works [11] and [19]. To our knowledge, this result is new in the field of fractional Laplacian operators. Proposition 2.3 (Sweeping Principle). Let Ω ⊂ RN (N ≥ 1) be a bounded domain, a : Ω → R and f be a globally Lipschitz function, with Lipschitz constant L > 0 such that a(x) + L > −λ1. Assume that {vλ}, λ ∈ R, is a one-parameter family of lower semicontinuous functions in Hs(RN ) such that the application λ → vλ is continuous (uniformly in x ∈ RN ) and, for every λ ∈ R, vλ satisfies pointwise the inequalities (−∆)svλ(x) + a(x)vλ(x) ≥ f(vλ(x)) in Ω, vλ(x) ≥ 0 in RN \ Ω. Assume also that u ∈ Hs(RN ) is an upper semicontinuous function satisfying point- wise the inequalities (−∆)su(x) + a(x)u(x) ≤ f(u(x)) in Ω, u(x) ≤ 0 in RN \ Ω. Moreover, let us suppose that u(x) 6≡ vλ(x) for every λ ∈ R and x ∈ RN \ Ω, and that there exists λ0 ∈ R such that u(x) ≤ vλ0 (x) in RN . Then u(x) ≤ vλ(x), for all λ ∈ R, x ∈ RN . Remark 2.4. This result shows that for a subsolution u to be u ≤ vλ for every λ ∈ R, being vλ a familiy of supersolutions continuous respect to λ, it suffices that u ≤ vλ0 , for some value λ0 ∈ R. Proof of Proposition 2.3. First, let us define the following subset A = {λ ∈ R : u(x) ≤ vλ(x), for all x ∈ RN}. Since λ0 ∈ A, this subset is not empty. Obviously as vλ is continuous with respect to λ, A is closed. In order to prove that A is open, consider λ̄ ∈ A and, by hypotheses, we have that (−∆)svλ̄(x) + a(x)vλ̄(x) ≥ f(vλ̄(x)) in Ω. Now we define wλ̄ = vλ̄ − u ≥ 0, which satisfies (−∆)swλ̄(x) + a(x)wλ̄(x) ≥ f(vλ̄(x))− f(u(x)), x ∈ Ω. We add L(vλ̄ − u) to both sides of this inequality and we obtain (−∆)swλ̄ + (a(x) + L)wλ̄ ≥ f(vλ̄(x))− f(u(x)) + L(vλ̄ − u) ≥ 0. Therefore, wλ̄ ∈ Hs(RN ) is a lower semicontinuous function which satisfies inequal- ity (2.1) with m(x) ≡ a(x) + L for all x ∈ RN . By the Strong Maximum Principle (Proposition 2.1), this implies that either wλ̄ > 0 in Ω or wλ̄ ≡ 0 in RN (which is not possible since by hypothesis wλ̄ 6≡ 0 in RN \ Ω). In particular u(x) < vλ̄(x), for all x ∈ Ω̄. As a consequence, since vλ̄ − u is lower semicontinuous, there is x∗ ∈ Ω̄ such that inf x∈Ω̄ |vλ̄(x)− u(x)| ≥ |vλ̄(x∗)− u(x∗)| > ε > 0. Thus, since the application λ → vλ is uniformly continuous respect to x, there exists δ > 0 such that u(x) < vκ(x), for all x ∈ RN and κ ∈ (λ̄− δ, λ̄+ δ). This proves that A is open and this finally leads us to confirm that A = R. � 6 J. CARMONA, A. MOLINO EJDE-2023/16 3. Proof of the main result To prove the main result we use the Sweeping Principle for the fractional Lapla- cian operator (Proposition 2.3), and the following result from [2, Theorem 2.4, Remark 2.5]. Theorem 3.1. Let f̃ be any Lipschitz function in [−1, 1] such that f̃(−1) = f̃(1) = 0 and F̃ (t) < F̃ (−1) = F̃ (1) for all t ∈ (−1, 1), where F̃ (t) = ∫ t 0 f̃(τ)dτ . Then there exists ṽ solution of (−∂tt)sṽ(t) = f̃(ṽ(t)), t ∈ R, with ṽ′(t) > 0 for all t ∈ R and limt→±∞ ṽ(t) = ±1. Here, (−∂tt)s is the one dimensional fractional Laplacian (−∆)s. Next we prove Theorem 1.1. It should be noticed that, for this purpose, Theorem 3.1 will allow us to overcame the main difficulties in adapting the proof used in [10] to the fractional Laplacian operator framework. Proof of Theorem 1.1. Obviously u ≡ 0 is solution to problem (1.1) since (1.7) implies that f(0) = 0. Thus, arguing by contradiction, we suppose that there exists u ∈ Hs 0(Ω) being a nontrivial solution to (1.1). In this case, v = −u satisfies the equation (−∆)sv = −f(−v) in Ω, v = 0 in RN \ Ω, and the function −f(−τ) satisfies the same hypotheses of Theorem 1.1. Therefore, since the maximum value of either u or −u is positive, without loss of generality, we may assume that u∞ := max x∈Ω̄ u(x) > 0. Since the value of f(τ) is irrelevant for τ > u∞, we also assume that limτ→∞ f(τ) = −∞ and that function f is globally Lipschitz, with Lipschitz constant L > 0. On the other hand, we claim that f(u∞) > 0. Indeed, otherwise f(u∞) ≤ 0 and, since (−∆)su∞ = 0, we have the inequality (−∆)su∞ + Lu∞ ≥ f(u∞) + Lu∞, in Ω. (3.1) Moreover, since u solves (1.1), we also have that (−∆)su+ Lu = f(u) + Lu, in Ω. (3.2) Subtracting (3.2) from (3.1), and using that f is L-Lipschitz, we obtain (−∆)s(u∞ − u) + L(u∞ − u) ≥ f(u∞) + Lu∞ − f(u)− Lu ≥ 0, in Ω. Since u∞ − u > 0 in RN \ Ω, we deduce, from the Strong Maximum Principle (Proposition 2.1), that u∞ > u(x) in Ω, which is a contradiction. Therefore, as f(u∞) > 0, we can assume that there are τ1 and τ2 positive constants such that τ1 < u∞ < τ2 and f(τ) > 0, for all τ ∈ (τ1, τ2), and f(τ1) = 0. (3.3) Even more, since limτ→∞ f(τ) = −∞ and the value of f(τ) is irrelevant for s > u∞, we can modify the function f , being still L-Lipschitz, and choose τ2 such that f(τ2) = F (τ2) = 0 and f(τ) < 0, for τ > τ2. (3.4) EJDE-2023/16 NONEXITENCE OF NONTRIVIAL SOLUTIONS 7 Now, we set τ̄ = max{τ ∈ R : τ < τ1, F (τ) = 0}, and we observe that τ̄ ≥ 0 and, since F satisfies (1.7), f(τ̄) = 0. We define now g(t) = τ̄+τ2 2 + τ2−τ̄ 2 t and the auxiliary function f̃(t) = 2 τ2 − τ̄ f ( g(t) ) . Note that f̃(−1) = f̃(1) = 0 and f̃ is a Lipschitz function in [−1, 1]. Even more, since F (τ2) = F (τ̄) = 0, it follows that F̃ (t) < F̃ (−1) = F̃ (1), for all t ∈ (−1, 1). Then, by using Theorem 3.1, there exists a function ṽ which is solution to problem (−∂tt)sṽ(t) = f̃(ṽ(t)), t ∈ R, with ṽ′ > 0 and limt→±∞ ṽ(t) = ±1. Let us define w̃(t) = g(ṽ(t)), t ∈ R, which satisfies the equation (−∂tt)sw̃(t) = f(w̃(t)), t ∈ R. Furthermore, w̃ is increasing (w̃′ > 0), since ṽ and g are also increasing functions. In addition, limt→−∞ w̃(t) = τ̄ ≥ 0 and limt→∞ w̃(t) = τ2. In particular, w̃ is uniformly continuous. For every λ ∈ R, consider the family of parametric functions vλ(x) = w̃(x1 + λ) > 0, for all x = (x1, . . . , xN ) ∈ RN . Clearly, the application λ→ vλ is continuous, since w̃ is uniformly continuous and vλ(x)→ τ2, as λ→∞, for all x ∈ Ω. (3.5) Also, it satisfies (see Lemma 5.1) (−∆)svλ(x) = f(vλ(x)) in Ω, vλ(x) > 0 in RN \ Ω, for every λ ∈ R. Furthermore, vλ > u(x) = 0 in RN \ Ω and, due to (3.5), there exists λ0 >> 0 such that u(x) ≤ vλ0(x) in Ω (since u∞ < τ2). Then, by using the Sweeping Principle (Proposition 2.3), u(x) ≤ vλ(x) for every λ ∈ R and every x ∈ RN . In particular, u(x) ≤ inf λ∈R vλ(x) = inf t∈R w̃(t) = τ̄ , for all x ∈ RN , which contradicts τ̄ < τ1 < u∞. � 4. Conclusion In this section we summarize the main consequences of Theorem 1.1 through a series of corollaries. On the one hand we are concerned with nonlinear eigenvalue problems (−∆)su = λu− g(u) in Ω, u = 0 in RN \ Ω. (4.1) Existing methods for proving nonexistence of nontrivial solutions for some values of the parameter λ requires to multiply by a convenient test function and integrate. 8 J. CARMONA, A. MOLINO EJDE-2023/16 Theorem 1.1 provides an alternative by imposing conditions on λ to assure that (1.7) is satisfied with f(t) = λt− g(t). Corollary 4.1. Let Ω ⊂ RN (N ≥ 1) be a C1,1 bounded domain, g : R → R a locally Lipschitz function and assume that λ ≤ 2 ∫ t 0 g(τ)dτ t2 , for all t ∈ R. Then, u ≡ 0 is the unique solution to problem (4.1). We include here some particular choice of functions g(t) leading to a simpler condition on λ in the above result. Corollary 4.2. Let Ω ⊂ RN (N ≥ 1) be a C1,1 bounded domain and α, λ ∈ R with λ ≤ min{0,−α}. Then, u ≡ 0 is the unique solution to problem (−∆)su = λu+ α sin(u) in Ω, u = 0 in RN \ Ω. Corollary 4.3. Let Ω ⊂ RN (N ≥ 1) be a C1,1 bounded domain and λ ∈ R with λ ≤ 1. Then u ≡ 0 is the unique solution to the problem (−∆)su = λu− 2ueu 2 in Ω, u = 0 in RN \ Ω. On the other hand we include some applications to obtain a priori bounds for positive solutions to (1.1). We observe that when f(0) = 0 then nonnegative solutions to problem (1.1) are solutions to (−∆)su = f(u+) in Ω, u = 0 in RN \ Ω, where u+ = max{u, 0}. Thus, the following corollaries show how Theorem 1.1 also provides a priori estimates of positive solutions to (1.1) when (1.7) is satisfied in a positive interval. Corollary 4.4. Let Ω ⊂ RN (N ≥ 1) be a C1,1 bounded domain, f : R → R a locally Lipschitz function with f(0) = 0 and F (t) = ∫ t 0 f(τ)dτ ≤ 0, for all t > 0. Then, u ≡ 0 is the unique nonnegative solution to problem (1.1). Corollary 4.5. Let Ω ⊂ RN (N ≥ 1) be a C1,1 bounded domain, f : R → R a locally Lipschitz function with f(0) = 0 and t0 > 0 such that F (t) = ∫ t 0 f(τ)dτ ≤ 0, for all t ∈ (0, t0), and assume that there exists a positive solution u to problem (1.1). Then ‖u‖∞ ≥ t0. EJDE-2023/16 NONEXITENCE OF NONTRIVIAL SOLUTIONS 9 5. Appendix Lemma 5.1. Let u ∈ Hs(RN ) defined as u(x) := v(x1) for a certain function v, for all x = (x1, . . . , xN ) ∈ RN . Then (−∆) s u(x) = (−∂x1x1) s v(x1). Proof. By definition, (−∆) s u(x) = C(N, s) P.V. ∫ RN u(x)− u(y) |x− y|N+2s dy = C(N, s) P.V. ∫ RN v(x1)− v(y1) |x− y|N+2s dy = C(N, s) P.V. ∫ R (v(x1)− v(y1) |x1 − y1|1+2s ∫ RN−1 |x1 − y1|1+2s |x− y|N+2s dyN · · · dy2 ) dy1, (5.1) with C(N, s) defined by (1.2). Now, relabeling the last integral expression as IN and computing the integral∫ R |x1 − y1|1+2s |x− y|N+2s dyN = √ π Γ ( N−1 2 + s ) Γ ( N 2 + s ) · |x1 − y1|1+2s ((x1 − y1)2 + · · ·+ (xN−1 − yN−1)2) N−1 2 +s , we obtain the recursive sequence IN = √ π Γ ( N−1 2 + s ) Γ ( N 2 + s ) IN−1. A simple computation leads us to the explicit expression IN = π N−1 2 Γ( 1 2 + s) Γ(N2 + s) = C(1, s) C(N, s) . Hence, replacing in (5.1), we obtain that (−∆) s u(x) = C(1, s) P.V. ∫ R v(x1)− v(y1) |x1 − y1|1+2s dy1 = (−∂x1x1) s v(x1). � Acknowledgements. The authors were partially supported by Grant PID2021- 122122NB-I00 funded by MCIN/AEI/10.13039/501100011033 by the “ERDF A way of making Europe”, and by grant P18-FR-667 funded by Junta de Andalućıa, Consejeŕıa de Transformación Económica, Industria, Conocimiento y Universidades Unión Europea. J. Carmona was supported by Junta de Andalućıa FQM194 and CDTIME. A. Molino was supported by grant UAL2020-FQM-B2046 (UAL/ CTE- ICU/FEDER) and FQM-116. 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Anal., 213 (2014), No. 2, 587-628. [19] D. H. Sattinger; Topics in stability and bifurcation theory. Lecture Notes in Mathematics, Vol. 309. Springer-Verlag, Berlin-New York, 1973. José Carmona Departamento de Matemáticas, Universidad de Almeŕıa, Facultad de Ciencias Experi- mentales, Ctra. de Sacramento sn. 04120 La Cañada de San Urbano. Almeŕıa, Spain Email address: jcarmona@ual.es Alexis Molino Departamento de Matemáticas, Universidad de Almeŕıa, Facultad de Ciencias Experi- mentales, Ctra. de Sacramento sn. 04120 La Cañada de San Urbano. Almeŕıa, Spain Email address: amolino@ual.es 1. Introduction 2. Maximum Principle 3. Proof of the main result 4. Conclusion 5. Appendix Acknowledgements References