Existence and uniqueness of positive solution of a logistic equation with nonlinear gradient term

EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTION OF A LOGISTIC EQUATION WITH NONLINEAR GRADIENT TERM DAVID RUIZ AND ANTONIO SU´ AREZ Abstract. The main goal of this work is to study the existence and uniqueness of positive solution of a logistic equation including a nonlinear gradient term. In particular, we use local and global bifurcation together with some a-priori estimates. To prove uniqueness, the sweeping method of Serrin is employed. 1. Introduction In this paper we study the existence and possible uniqueness of positive solutions of the problem: −∆u=λu −up± |∇u|qin Ω, u= 0 on ∂Ω, (P±) where Ω ⊂RNis a bounded and regular domain and 1 < p, q. The constant λ∈Rwill be regarded as a bifurcation parameter. When the nonlinear gradient term does not appear, the above equation is the classical logistic one, which has been extensively studied in the literature. It is very well-known that there exists a positive solution if, and only if, λ>λ1, where λ1denotes the principal eigenvalue of the laplacian. In such case, this solution is unique and stable. However, when the gradient term is included, the equation is less known. In the particular case λ= 0 and under a blow-up Dirichlet condition, it was studied in [4], [12] and [18]. In these papers, the solution is obtained as limit of a sequence of solutions of the corresponding nonhomogeneous Dirichlet problems. In each step, the authors use [17], Theorem 8.3, page 301, and so q≤2 must be imposed. Some other papers consider similar problems with critical growth (q= 2), in which a convenient change of variables works (see, for instance, [20, 21] and [23]). 2000 Mathematics Subject Classification. 35B32, 35B45, 35J25, 35J60. Key words and phrases. Elliptic logistic equation, bifurcation, a-priori bounds, nonlinear gradient term. The first author has been supported by the Spanish Ministry of Science and Technology under Grant BFM2002-02649 and by J. Andaluc´ıa (FQM 116). The second author has been supported by the Spanish Ministry of Science and Technology under Grant BFM2003- 06446. 1 2 DAVID RUIZ AND ANTONIO SU´ AREZ Generally speaking, equations of the form (1) −∆u=f(x, u, ∇u) in Ω, u= 0 on ∂Ω, have been very studied in the literature, see the classical works [3, 5, 6, 8, 10, 15, 22]. Most of these are concerned with a-priori estimates, and from those they get existence by using topological methods or sub-super solution techniques. In order to obtain our existence results, we use bifurcations methods (local and global) and a-priori bounds in C1. It is well-known that these a-priori estimates hold for q≤2. However, as a special feature of our problem, we can even provide a C1estimate for (P−) and any q > 1. In order to do that, we first find a supersolution, greater than any solution uof (P−), and from this we can estimate the maximum of the function w=|∇u|2/2 (see, for instance, [26], Chapter 5 and [23]). In general, the uniqueness of elliptic equations is a hard question as was remarked in [11]. For the equation (1) one can easily obtain uniqueness from the maximum principle if fis decreasing in u(see [13], Theorem 8.1). Clearly, the function f(u) = λu−up−|∇u|qis not decreasing in ufor λ > 0, and these results cannot be applied. Anyway, u= 0 is always a solution: we are looking for uniqueness of positive solutions. We present here a result of uniqueness of positive solutions for (1) which generalizes a classical one for semilinear equations, see for instance [2, 7, 14], [16] (Theorems 7.14, 7.15), [24] (page 39) and the references therein. Our proof makes use of the sweeping method of Serrin ([25], page 12), as in [16, 24]. To the best of our knowledge, the uniqueness result stated in Theorem 4.1 is completely new. Generally speaking, we show that in the case (P−) the results about existence and uniqueness are similar to those for the semilinear equation. However, in the case (P+) the presence of the gradient may have an important influence (depending on the values of pand q, see Theorem 3.3). In some case there will not be uniqueness of positive solution. An outline of the work is as follows: in Section 2 we prove the existence and non-existence results using bifurcation and a-priori estimates. In Section 3 the general result of uniqueness is established. 2. Preliminaries In this section we state the exact version of the maximum principle that will be used throughout the paper. The results we present are classical: we include them for the sake of clarity. We begin with a the generalized weak maximum principle. A more general statement and the proof can be found in [13], Theorem 8.1. Theorem 2.1. Let Ωbe a bounded smooth domain in RN,ci, d : Ω →R be L∞functions (i= 1 . . . n), d(x)≥0. Suppose that u∈H1 0(Ω) is a weak POSITIVE SOLUTION OF A LOGISTIC EQUATION 3 solution of the inequality: (2) −∆u+ N X i=1 ci(x)ui+d(x)u≥0 Then, u≥0. We now state a version of the strong maximum principle for C1weak solutions: Theorem 2.2. Let Ωbe a bounded smooth domain in RN,ci, d : Ω →R are L∞functions (i= 1 . . . n), d(x)≥0. Suppose that u∈C1(Ω) is a weak solution of the inequality (2). Assume also that uis not constantly equal to zero, and u(x) = 0 ∀x∈∂Ω. Then, u(x)>0∀x∈Ωand ∂u ∂η (x)<0∀x∈∂Ω, where ηis the unit outward normal to ∂Ωat x. Proof. The proof follows that of Lemma 3.4 of [13]. Since the statement we have chosen does not match exactly with that of [13], we reproduce the proof in detail. There are several steps: Step 1: Define B=B(y, R)⊂RNan open ball, and suppose that u verifies inequality (2) in B,u(x)>0∀x∈B,u(x0) = 0 for some x0∈∂B. We claim that ∂u ∂η (x0)<0, where ηis the unit outward normal to Bat x0. We can assume y= 0. Let α > 0 be a constant to be determined later, and define v(x) = e−α|x|2−e−αR2defined in the annulus A={x∈RN: R/2≤ |x| ≤ R}. We compute: −∆v+ N X i=1 ci(x)vi+d(x)v= e−α|x|2 −4|x|2α2+ 2αN −2α N X i=1 ci(x)xi+d(x)h1−e−α(R2−|x|2)i!≥ (3) e−α|x|2−4(R/2)2α2+ 2αN −2αMR −M where Mis a constant such that M > kdkL∞,M > PN i=1 kcik2 L∞1/2. We now choose αlarge enough such that (3) is positive for any x∈A. Clearly, v(x) = 0 ≤u(x) for |x|=R. Since uis positive in the ball, we can take ε > 0 small enough such that ε v(x)< u(x) for |x|=R/2. Now, apply Theorem 2.1 to the function u−ε v in A, to conclude that u(x)≥ε v(x)∀x∈A. Recall now that u(x0) = v(x0) = 0, to conclude that ∂u ∂η (x0)< ε ∂v ∂η (x0)<0 4 DAVID RUIZ AND ANTONIO SU´ AREZ Step 2: u(x)>0 for any x∈Ω. By Theorem 2.1, we have that u≥0. In order to prove the strict inequality, we reason by contradiction. Take Ω0={x∈Ω : u(x) = 0}, Ω+={x∈Ω : u(x)>0}. Observe that both previous sets are nonempty. Choose y∈Ω+such that R=d(y, Ω0)< d(y, ∂Ω). Then, B(y, R)⊂Ω+ and there exists x0∈∂B ∩Ω0. Obviously uattains a minimum at x0, and hence ∇u(x0) = 0. We arrive then to a contradiction with Step 1. Step 1: ∂u ∂η (x)<0∀x∈∂Ω. Take x0∈∂Ω, and Ban interior sphere at x0. Thanks to Step 2, u(x)>0 for x∈B, and u(x0) = 0. We conclude by applying Step 1.  In the previous theorem the condition d(x)≥0 is needed. In next Theorem we consider any L∞function d(x) but assuming that the function uis nonnegative (which is no longer provided by Theorem 2.1). Theorem 2.3. Let Ωbe a bounded smooth domain in RN, and ci, d : Ω →R are L∞functions (i= 1 . . . n). Suppose that u∈C1(Ω) is a weak solution of the inequality (2). Assume also that u(x)≥0but is not constantly equal to zero, and u(x) = 0 ∀x∈∂Ω. Then, u(x)>0∀x∈Ωand ∂u ∂η (x)<0∀x∈∂Ω, where ηis the unit outward normal to ∂Ωat x. Proof. We denote d+(x) = max{d(x),0}. Since uis nonnegative, it follows: −∆u+ N X i=1 ci(x)ui+d+(x)u≥ −∆u+ N X i=1 ci(x)ui+d(x)u≥0 Now we conclude the proof by applying Theorem 2.2 to the operator: L[z] = −∆z+ N X i=1 ci(x)zi+d+(x)z  3. Bifurcation of positive solution First, we need some notations: λ1and ϕ1stand for the principal eigenvalue of −∆ subject to homogeneous Dirichlet boundary condition and its positive eigenfunction associated, respectively. We denote by E:= C1(Ω) and P={u∈E:u(x)≥0∀x∈Ω}its positive cone. We look for usolution belonging to P. Observe, that if u∈Eis solution of (P±), by the elliptic regularity u∈C3,α(Ω) for some α∈(0,1). On the other hand, if uis a nontrivial solution in P, we have that −∆u∓ |∇u|q−2∇u· ∇u−(λ−up−1)u= 0, POSITIVE SOLUTION OF A LOGISTIC EQUATION 5 and so, Theorem 2.3 implies that u(x)>0 for any x∈Ω, and also that ∂u ∂η (x)<0, where x∈∂Ω and ηis the unit outward normal to ∂Ω at x. That is to say, u∈int(P); in such case, we will say that uis positive. Our first result provides us with the existence of a continuum of positive solutions of (P±). We also obtain an a-priori bound in L∞for any solution of (P±) in E. Proposition 3.1. There exists an unbounded continuum of positive solution C ⊂ R×Ebifurcating from the trivial solution at λ=λ1. Moreover, if (λ, u)is a solution of (P±),u6= 0, then (4) λ > 0,and kuk∞≤λ1/(p−1). Proof. Observe that (P±) can be written as u=K(λu +N(u)) ,in E, where K:= (−∆)−1:C(Ω) →Eunder homogeneous Dirichlet boundary conditions, and: N:E→C(Ω) ,N(u) := (−up± |∇u|q) Observe that if f∈C(Ω), then u=K(f) belongs to W2,p for all p > 1 (see [1]), and hence u∈C1,γ. Because of that, the operator Kis compact. The maximum principle implies also that is strongly positive. The nonlinear operator Nis continuous and bounded. Moreover, since p > 1, q > 1, we have that N(u) = o(kukE) as u→0; we can apply for instance Theorem 6.5.5 of [19] and conclude the existence of an unbounded continuum Cin R×Pof positive solution of (P±) emanating from (λ1,0). On the other hand, suppose that λ≤0 and u∈Pis a solution of (P±). Then, L[u] = 0, where Lis defined as: L[z] = −∆z∓ |∇u|q−2∇u· ∇z−(λ−up−1)z= 0, Since (λ−up−1)≤0, we can apply Theorem 2.1 to −uto conclude that u≤0. Therefore, u= 0. Finally, if xM∈Ω is such that u(xM) = maxx∈Ωu(x), then λu(xM)−u(xM)p− |∇u(xM)|q≥0, and now, taking into account that ∇u(xM) = 0, we get (4).  The next result characterizes the existence and uniqueness of positive solution of (P−). Theorem 3.2. Consider the case (P−). For all p, q > 1there exists a positive solution if, and only if, λ>λ1. Moreover, if λ>λ1there exists a unique positive solution of (P−)which is linearly asymptotically stable. Proof. If uis a positive solution of (P−), then multiplying the equation by ϕ1and integrating by parts, we have (λ1−λ)ZΩ uϕ1=Z(−up− |∇u|q)ϕ1, 6 DAVID RUIZ AND ANTONIO SU´ AREZ which implies that λ>λ1. Now, by Proposition 3.1 the proof of existence for λ>λ1concludes if we find a priori estimates. Specifically, we claim that if (λ, u) is a positive solution of (P−) and λ∈I⊂Rcompact, then there exists C > 0 such that (5) kukE≤C. We prove the result for all q > 1 by estimating first the derivative in the boundary, and afterwards in the interior of Ω. First of all, we claim that if uis a solution of (P−) then (6) u≤θλ, where θλis the unique positive solution of the logistic equation (7) −∆u=λu −upin Ω, u= 0 on ∂Ω. Indeed, if uis a solution of (P−) then uis a subsolution of (7). Clearly, a large positive constant is a supersolution of (7). The sub-super solution method yields a solution of (7) greater than u, and since θλis the unique positive solution of (7), (6) is verified. Hence, it follows that (8) ∂θλ ∂η ≤∂u ∂η <0 on ∂Ω. This gives us an a-priori bound of ∇uon the boundary. In order to estimate the gradient in the interior of Ω, define: w:= 1 2|∇u|2. It is not hard to show that ∆w= 2w(pup−1−λ) + q|∇u|q−2∇u· ∇w+ N X i,j=1 (∂2u ∂xi∂xj )2. Assume that the maximum of wis attained at xM∈Ω, then using that ∆w(xM)≤0 and ∇w(xM) = 0, we get (9) 2w(xM)(λ−pup−1(xM)) ≥ N X i,j=1 (∂2u ∂xi∂xj (xM))2. On the other hand, there exists C > 0 such that N X i,j=1 (∂2u ∂xi∂xj (xM))2≥C(∆u(xM))2=C(−λu(xM)+u(xM)p+|∇u(xM)|q)2, By taking into account the L∞bound of u, there exist positive constants C1, C2>0 such that (10) 2w(xM)(λ−pup−1(xM)) ≥C1w(xM)q−C2. POSITIVE SOLUTION OF A LOGISTIC EQUATION 7 Inequalities (8), (10) imply the a-priori estimate in C1of the solutions of (P−) for any λfixed. The uniqueness of positive solution follows from Theorem 4.1, whose statement and proof are postponed to the next section. Let u0>0 be a positive solution of (P−). We now plan to prove that u0is asymptotically stable. It is well-known (see, for instance, [24]) that it suffices to show that the first eigenvalue of the linearized around of u0is positive, i.e., the first eigenvalue of the following problem: (11) L[v] = σv in Ω, v= 0 on ∂Ω, where L[v] := −∆v+q|∇u0|q−2∇u0· ∇v+ (pup−1 0−λ)v. For that, it suffices to find a positive supersolution uof L, that is, a positive function usuch that L[u]≥0 in Ω and u≥0 on ∂Ω with some inequality strict. Take u=u0, then L[u] = (q−1)|∇u0|q+ (p−1)up 0>0 in Ω and u= 0 on ∂Ω. The proof is completed.  Observe that for (P−) the bifurcation is always supercritical for any value of pand q. This is different in the case (P+). Theorem 3.3. Consider the case (P+). (1) With respect to the local bifurcation, we obtain: (a) If p > q (resp. p < q) the bifurcation is subcritical (resp. supercritical). (b) If p=q, and (λn, un)is a solution of (P+)such that λn→λ1 and kunkE→0as n→ ∞, then 0<(λn−λ1)ZΩ ϕp+1 1−ZΩ |∇ϕ1|qϕ1. whenever this last term is different from zero. (2) If q≤2, then there exists at least a positive solution for λ>λ1. Proof. It is not difficult to show that we are in conditions of apply the Crandall-Rabinowitz’s Theorem, see [9], and so if Yis any closed subspace of Esuch that E= [Span ϕ1]⊕Y, there exist ε > 0 and two continuous functions λ: (−ε, ε)7→ Rand u: (−ε, ε)7→ Y with (12) λ(s) = λ1+µ(s), u(s) = s(ϕ1+v(s)), s ∈(−ε, ε), and µ(0) = v(0) = 0 and in a neighborhood of (λ1,0) all the solutions are of the form (λ(s), u(s)). Introducing this expression in the equation (P+) and 8 DAVID RUIZ AND ANTONIO SU´ AREZ applying the Fredholm alternative, we get µ(s) = sp−1ZΩ (ϕ1+v(s))pϕ1−sq−1ZΩ (∇ϕ1+∇v(s))qϕ1 ZΩ (ϕ1+v(s))ϕ1 , whence we deduce the paragraph (1). In order to prove paragraph (2), recall that (P+) has no solutions different from zero if λ≤0 (Theorem 3.1). Moreover, since q≤2 the nonlinearity f(x, ξ, η) = λξ −ξp+|η|qsatisfies that |f(x, ξ, η)| ≤ c(|ξ|)(1 + |η|2) for (x, ξ, η)∈Ω×R×RN, and so, by Proposition 2 of [3], it follows that kukE≤γ(kuk∞)≤C, where γ:R+7→ Ris an increasing function and Ca positive constant. Thus, the result follows by Proposition 3.1 and the fact that there exist no solution for λ≤0, see (4).  Corollary 3.4. If q≤2and p>q, there exist at least two positive solutions of (P+)in (λ1−δ, λ1)for some δ > 0small. Proof. Define C ⊂ R+×int(P) the connected set of solutions bifurcating from (λ1,0). As we mentioned in the proof of Theorem 3.3, in a neighborhood U of (λ1,0) all solutions of (P+) are of the form (λ(s), u(s)) where λ(s)< λ1 and u(s) are defined in (12). We can assume that this neighborhood is given by: U= (λ1−δ, λ1+δ)×(B(0, ε)∩P) for some ε > 0, δ > 0. By taking smaller δand εif necessary, we can assume that (P+) has no solutions (λ, u) such that λ∈(λ1−δ, λ1+δ), kuk=ε. Recall that Cdoes not cross the λ= 0 line, and cannot blow up for finite λ, since we got a priori estimates. Then, the unbounded continuum Chas an unbounded projection [λ, +∞) on the real λ-axis with λ > 0. Reasoning by contradiction, suppose that there exists δ0∈(0, δ) such that (P+) has a unique positive solution for λ=λ1−δ. Define: S1={(λ, u)∈R+×P:λ∈[λ1−δ0, λ1],kukE=ε} S2={(λ, u)∈R+×P− {0}:λ=λ1,kukE≤ε} S3={(λ, u)∈R+×P:λ=λ1−δ0,kukE≥ε} S=S1∪ S2∪ S3 Clearly, Sseparates R+×P−{0}in two connected components, and C∩S = ∅. Moreover, we have that the branch (λ(s), u(s)) is in one of these components, whereas any solution (λ, u)∈ C with λ>λ1is in the other one. This is a contradiction with the fact that Cis connected.  POSITIVE SOLUTION OF A LOGISTIC EQUATION 9 4. A uniqueness result In this section we prove the uniqueness of positive solution of (1) under a certain condition on f, extending the result of [2, 7, 14, 16] to the case of quasilinear equations. Theorem 4.1. Consider the problem: (13) −∆u=f(x, u, ∇u)in Ω,u= 0 on ∂Ω, where f: Ω ×R+ 0×Rn→Ris a continuous function, locally Lipschitz with respect to (u, η)∈R+ 0×Rn. Suppose that fverifies also the following condition: (14) For any x∈Ω,(u, η)∈R+×Rn, the function t7→ f(x, tu, tη) tis strictly decreasing in t∈R+# Then, there exists at most one non-negative nonzero solution of problem (13) in E. Proof. First, let us show that (14) implies that f(x, 0,0) ≥0 for any x∈Ω. Otherwise, if there exists x0∈Ω such that f(x0,0,0) <0, we would have that: lim t→0+ f(x0, tu, tη) t=−∞ for any (u, η)∈R+×Rn. This would contradict (14). We now prove that any non-negative nonzero solution of (13) must be positive, that is, must belong to int(P). Given any u∈Pa solution, we can write: −∆u=f(x, u, ∇u)−f(x, u, 0) + f(x, u, 0) −f(x, 0,0) + f(x, 0,0) = f(x, u, ∇u)−f(x, u, 0) |∇u|2∇u· ∇u+f(x, u, 0) −f(x, 0,0) uu+f(x, 0,0). Since u,∇uare both bounded and fis locally Lipschitz, the functions: f(x, u, ∇u)−f(x, u, 0) |∇u|2∇u , f(x, u, 0) −f(x, 0,0) u are uniformly bounded. Therefore, uis a solution of the problem L[u] = f(x, 0,0), where Lis a linear operator defined as: L[z] = −∆z−f(x, u, ∇u)−f(x, u, 0) |∇u|2∇u· ∇z−f(x, u, 0) −f(x, 0,0) uz. Recall now that uvanishes in ∂Ω and is nonnegative and different from zero. Theorem 2.3 implies that u∈int(P). We now prove uniqueness by using the sweeping method of Serrin. Suppose that u,vare two nonnegative solutions of (13), both different from zero. By direct computation (using (14)) we conclude that the functions su, with s∈(0,1), are strict subsolutions of (13). Define the set: D={s∈[0,1] : su(x)≤v(x) for any x∈Ω}