On the uniqueness of positive solution of an elliptic equation

On the uniqueness of positive solution of an elliptic equation1 M. Delgado and A. Su´ arez2 Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Fac. Matem´aticas, C/ Tarfia s/n C.P. 41012, Univ. Sevilla, Spain e-mail: [email protected] and [email protected] Abstract This work deals with the uniqueness of positive solution for an elliptic equation whose nonlinearity satisfies an specific monotony property. In order to prove the main result, we employ a change of variable used in previous papers and the maximum principle. Key Words. Uniqueness of positive solution, maximum principle. AMS Classification. 35J65, 35B50. 1 Introduction Let Ω ⊂IRNbe a regular domain and f: Ω ×IR 7→ IR a measurable function. We are interested in the classical and positive solutions of the elliptic problem          −∆u=f(x, u) in Ω, u= 0 on ∂Ω. (1.1) 1Supported by the Spanish Ministry of Science and Technology under Grants BFM2000-0797 and BFM2003-06446. 2Author who receives correspondence and proofs of correction 2M. Delgado and A. Su´arez One of the more difficult problem related to (1.1) is proving the uniqueness of solution of (1.1). It is well known that if fis decreasing in uthen there exists at most one solution of (1.1), see for instance [1] and [2]. When for a. e. x∈Ω the map u7→ f(x, u) uis decreasing in (0,∞) (1.2) then, there exists at most one positive solution of (1.1), see [3] and [4]. In this note, we employ an appropriate change of variable (yet used in [5], [6], [7] and [8]) and the strong maximum principle to prove that if there exists a regular, positive and concave function g(see Theorem 2.1 and Proposition 2.2 for the exact conditions on g) such that u7→ f(x, u) g(u)is non-increasing in (0,∞) for a. e. x∈Ω (1.3) then, there exists a unique positive solution. When f(x, u) = a(x)g(u) with a∈L∞(Ω), the uniqueness was studied in [5], [6], [7] and [8]. We refer to [6] where a review of the uniqueness question is made. We would like to remark that although the conditions (1.2) and (1.3) seem rather similar, the techniques for the proofs of uniqueness are quite different. In fact, the proofs of the uniqueness result under (1.2) use the monotonicity of the quotient between f(x, t) and exactly the linear function g(t) = t. Our proof, which allows us to use the monotonicity of the quotient between f(x, t) and a concave function g(t), does not reach the linear function; whereas f(x, t)/g(t) is not necessarily decreasing. In the following section we prove the main result of this work. In the last section we employ a specific example from population dynamics that shows that our result improves and complements that obtained under the condition (1.2). 2 Main result Our main result reads as follows: Theorem 2.1 Assume that there exists a function g∈C1(0,+∞)∩C0([0,+∞)),g(t)>0 for t > 0, such that a) g0is non-increasing and Zr 0 1 g(t)dt < ∞,for r > 0.(2.1) Uniqueness of positive solution 3 b) The map u7→ f(x, u) g(u)is non-increasing in (0,∞)for a. e. x∈Ω.(2.2) Then, there exists at most one positive solution of (1.1). Proof: Consider the change of variable v=Zu 0 1 g(t)dt (2.3) which transforms (1.1) into          −∆v=g0(h(v))|∇v|2+f(x, h(v)) g(h(v)) in Ω, v= 0 on ∂Ω, (2.4) where u=h(v),(2.5) and hsatisfies, from (2.3), h0(t) = g(h(t)). Assume that there exists two positive solutions u16=u2of (1.1). Let Ω1:= {x∈Ω : u1(x)> u2(x)}. Assume that Ω1is not empty. It is clear that u1=u2on ∂Ω1. Thanks to monotonicity of h,v1> v2in Ω1and v1=v2on ∂Ω1, where ui=h(vi)i= 1,2. Consider the function Φ := v1−v2, which is positive in Ω1and Φ = 0 on ∂Ω1. After some calculation, we obtain that Φ verifies −∆Φ −g0(h(v1))|∇v1|2+g0(h(v2))|∇v2|2=µf(x, h(v1)) g(h(v1)) −f(x, h(v2)) g(h(v2)) ¶.(2.6) Since g0is non-increasing, g0(h(v1)) ≤g0(h(v2)); and by (2.2), we get that −∆Φ −g0(h(v1))∇(v1+v2)· ∇Φ≤0, which is a contradiction by the maximum principle. This completes the proof. 2 If we look for positive solutions in a more restrictive set, we can weaken the condition (2.1). Let define P:= {u∈C1 0(Ω) : u(x)≥0, u 6= 0 in Ω}, 4M. Delgado and A. Su´arez whose interior is int(P) = {u∈P:u(x)>0 for all x∈Ω, ∂u/∂n < 0 on ∂Ω}, where ndenotes the outward normal direction. Proposition 2.2 Assume that there exists gas in Theorem 2.1 but verifying lim s→0 s g(s)= 0,(2.7) instead of (2.1). Then, there exists a unique solution in int(P)of (1.1). Proof: Observe first that if u∈int(P), there exist positive constants 0 < k1≤k2such that k1dist(x)≤u(x)≤k2dist(x),(2.8) where dist(x) := dist(x, ∂Ω). Assume that there exists two positive u16=u2of (1.1) with ui∈int(P), i= 1,2 . Let Ω1:= {x∈Ω : u1(x)> u2(x)}. We define now for x∈Ω1 Φ(x) := Zu1(x) u2(x) 1 g(t)dt. First, observe that function Φ is continuous in Ω1and Φ = 0 on ∂Ω1. Indeed, for x∈∂Ω1∩Ω it is clear that Φ(x) = 0. For each x∈Ω1there exists ξ(x) with u2(x)≤ξ(x)≤u1(x) such that Φ(x) = u1(x)−u2(x) g(ξ(x)) ≤Cdist(x) g(ξ(x)) →0,as dist(x)→0, where we have used (2.7) and (2.8). On the other hand, as in the proof of Theorem 2.1, we get that −∆Φ −g0(u1)µ∇u1 g(u1)+∇u2 g(u2)¶· ∇Φ≤0. This last inequality leads to a contradiction to the maximum principle in the same way as in the proof of Theorem 2.1. 2 Remark 2.3 a) Observe that, for example, g(s) = slog2(s)verifies (2.7) but not (2.1). Uniqueness of positive solution 5 b) Conditions on fcan be imposed in order that every non-negative and non-trivial solution of (1.1) belongs to int(P), see for instance [9]. c) The same results hold for second order uniformly elliptic operator of the form L:= − N X i,j=1 aij ∂2 ∂xi∂xj + N X i=1 bi ∂ ∂xi with aij ∈C0(Ω), bi∈C0(Ω), aij =aji, see [12]. d) If gis positive only in (0, R)for some R > 0, and ZR 0 1 g(t)dt < +∞(2.9) then, we deduce a uniqueness result for positive solutions, u, such that kuk∞≤R. 3 Example and comparison In this section we apply our result to the nonlinearity f(x, u) = a(x)uq+b(x)up with different values of qand p, and a, b ∈L∞(Ω). This nonlinearity arises from the study of the population density of a species whose mobility depends upon its density, see [10] and [11]. Some uniqueness results were obtained in [12] and [13]. For this function, the condition (1.2) is equivalent to (q−1)a(x)+(p−1)b(x)up−q<0.(3.1) Now, we distinguish between the different cases: Case q= 1, p < 1: In this case, (3.1) holds if b > 0. Theorem 2.1 complements this result. Indeed, taking g(u) = upwe obtain uniqueness of positive solution for a≤0 and any function b. Case q < 1, p > 1: (3.1) holds, for example, if ais positive and b≤0; apositive and b positive or changes sign and kuk∞small, see [14] and [11]. By Theorem 2.1, there exist at most one positive solution if b≤0 and any function a. 6M. Delgado and A. Su´arez Case q < 1, p < 1: In this case (3.1) is satisfied if, for example, aand bare both positive. In the particular case p=q, (3.1) is equivalent to a+b > 0. By Theorem 2.1 we consider three cases: a) If p < q, then we have uniqueness of positive solution for any function aand b≥0 (taking g(u) = uq) and for any function band a≤0 (taking g(u) = up). b) If p > q, then the result is similar to case a) changing aby band bby a. c) If p=q, then there exists at most one positive solution if a+bis non-negative or changes sign. Observe that if a+bis non-positive, (1.1) does not posses non-negative solution. In the cases p= 1, q < 1 and p < 1, q > 1 similar results to the first and third cases respectively can be obtained interchanging the roles of aand b. Acknowledgements: We are delighted to thank to the referee for his/her careful reading of the manuscript and for several useful remarks and suggestions improving the presentation of the paper. References [1] H. Amann, On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J.,21, 125–146 (1971). [2] H. 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Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci.,33, 35–49 (1977). [11] M. Delgado and A. Su´arez, Positive solutions for the degenerate logistic indefinite superlinear problem: the slow diffusion case, Houston J. of Math.,29, 801–820 (2003). [12] M. Delgado and A. Su´arez, On the structure of the positive solutions of the logistic equation with nonlinear diffusion, J. Math. Anal. Appl.,268, 200–216 (2002). [13] M. Delgado, J. L´opez-G´omez and A. Su´arez, Characterizing the existence of large solutions for a class of sublinear problems with nonlinear diffusion, Adv. Differential Eqns.,7, 1235–1256 (2002). [14] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal.,122, 519–543 (1994).