Some nonlinear heterogeneous problems with nonlocal reaction term3

Some nonlinear heterogeneous problems with nonlocal reaction term 3

ˆa1 , Manuel Delgado2 and Antonio Sua ´ rez2 , Francisco Julio S.A. Corre 1. Universidade Federal de Campina Grande Centro de Ciˆencias e Tecnologia Unidade Acadˆemica de Matem´atica e Estat´ıstica CEP:58.109-970, Campina Grande - PB - Brazil 2. Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico Fac. de Matem´aticas, Univ. de Sevilla Calle Tarfia s/n, 41012-Sevilla, Spain E-mail addresses: [email protected], [email protected], [email protected]

Abstract In this paper we analyze the existence, uniqueness or multiplicity and stability of positive solutions to some nonlinear heterogeneous problems with nonlocal reaction term and linear diffusion. We employ mainly bifurcation and sub-supersolutions methods.

Key Words. Population dynamic, nonlocal terms, nonlinear diffusion AMS Classification. 35K57, 35B32, 92B05.

3

The authors have been supported by the Spanish Ministry of Science and Technology under Grant

MTM2006-07932 and the Spanish Ministry of Science and Innovation under Grant MTM2009-12367.

2

F. J. Corrˆea, M. Delgado, and A. Su´arez

Abbreviated version of the title: Non-local heterogeneous problems .

Corresponding author: Antonio Su´ arez Fern´ andez, Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico Fac. de Matem´ aticas, Univ. de Sevilla Calle Tarfia s/n, 41012-Sevilla, Spain E-mail address: [email protected] Phone: (+34) 95 455 68 34, Fax:(+34) 95 455 28 98.

Non-linear heterogeneous problems with nonlocal reaction term

1

3

Introduction

In the last years nonlocal terms have been included in population dynamics models, see [11]. One of the most classical prototype is the logistic equation. In this paper we analyze basically this equation incorporating a nonlocal reaction term. Specifically, we analyze the following problem    Z   p q  in Ω, −∆u = u λ + a(x) b(x)u    Ω   u>0         u=0

in Ω,

(1.1)

on ∂Ω,

where Ω is a bounded and regular domain of IRN , N ≥ 1, a, b ∈ C(Ω), b ≥ 0, b 6≡ 0, λ ∈ IR,

0 < q ≤ 1,

p > 0,

and a verifies either a > 0 or a < 0. Problems like (1.1) appear quite often in some models related with population dynamics. Here u represents the population of some species inhabiting a region Ω surrounded by an inhospitable area, since the population is subject to homogeneous Dirichlet boundary conditions. Equation (1.1) appears when one makes the change of variable wm = u in the nonlinear diffusion equation    Z   r m   −∆w = w λ + a(x) b(x)w in Ω, Ω

    w=0

(1.2)

on ∂Ω,

with m ≥ 1 and r > 0; and so q = 1/m and p = r/m. The real parameter m represents the velocity of diffusion, the rate of movement of the species from high-density regions to low-density ones. In this context, m > 1 means that the diffusion is slower than in the linear case (m = 1), which seems to give more realistic models, see [12]. The term m > 1 was introduced in [12], see also [15], by describing the dynamics of biological population whose mobility depends upon their density. In this context, λ is the growth rate of the species, a(x) describes the limiting effects of crowding when a < 0 and the intraspecific cooperation when a > 0. The presence of nonlocal reaction term in equation (1.2) means,

4

F. J. Corrˆea, M. Delgado, and A. Su´arez

from the biological point of view, that the crowding effect depends not only on their own point in space but also depends on the entire population, see [11]. In this sense, the R nonlocal term Ω b(x)wr can be interpreted as a weighted average of u at all the domain. Equation (1.1) has been studied previously. In order to state the results, let us introduce some notations: for a continuous function f ∈ C(Ω) we write f > 0 if f ≥ 0 and f 6≡ 0 in Ω; and f  0 if f (x) > 0 for all x ∈ Ω. On the other hand, we denote by fL := min f (x), x∈Ω

fM := max f (x). x∈Ω

Finally, for a regular and bounded domain D ⊂ IRN , λD 1 stands for the principal eigenvalue of the operator −∆ in D under homogeneous Dirichlet boundary conditions. In the case D = Ω we write λ1 := λΩ 1. When q = p = 1 and a = −1 in [2] the authors proved the existence, uniqueness and stability of positive solution when λ > λ1 . In this case, the solution can be explicitly built, it is proportional to a positive eigenfunction associated to λ1 . More recently, and again with q = p = 1 but a a function such that a ≤ a0 < 0, in [4] the authors proved the existence and uniqueness of positive solution for λ > λ1 . In this paper, the authors used bifurcation methods to prove the results. See also [6] for a related problem. When q < 1 and a and b are constants the problem was studied in [7] using a fixed point argument. In this case, a complete picture of the set of positive solutions was given there. In this paper, we study (1.1) when q = 1 and q < 1, a > 0 and a < 0 completing and improving the previous works. We would like to point some facts. The fixed point argument used in [7] can not be adapted to this case because to the heterogeneities of the functions a and b. On the other hand, although in the case q = 1 we could use a bifurcation argument as in [4], the a priori bound obtained in this reference can not be obtained in the cases a > 0 and a < 0. So, in the case q = 1 we have used a different argument based on the study of certain eigenvalues problems. Finally, in order to study the equation when q < 1 we use the bifurcation method, used previously in nonlocal problems in [8], [13], [4], [11] and [6]. On the other hand, the stability of the positive solution is a hard problem, see [9]. Moreover, in the case q < 1 when we linearize around a positive solution, we arrive at eigenvalue problem with singular potentials. We overcome these difficulties in Section 2. Our first result is:

Non-linear heterogeneous problems with nonlocal reaction term

5

Theorem 1.1. Assume q = 1. a) Assume that a < 0 and define Ω0 := int {x ∈ Ω : a(x) = 0} and assume that Ω0 is a subdomain of Ω. Then, there exists a positive solution of (1.1) if and only if 0 λ ∈ (λ1 , λΩ 1 ) 0 where λΩ 1 = ∞ if Ω0 = ∅. Moreover, the solution is unique and

lim kuλ k∞ = 0,

λ↓λ1

lim kuλ k∞ = +∞. Ω

λ↑λ1 0

b) Assume that a > 0. Then, there exists a positive solution of (1.1) if and only if λ < λ1 . Moreover, the solution is unique and unstable and lim kuλ k∞ = 0,

λ↓λ1

lim kuλ k∞ = +∞.

λ→−∞

With respect to the nonlinear diffusion case, 0 < q < 1, we get: Theorem 1.2. Assume a < 0 and 0 < q < 1. Then, there exists a positive solution of (1.1) if and only if λ > 0. Moreover, lim kuλ k∞ = 0.

λ→0

Furthermore, there exist 0 < λ < λ < ∞ such that if some of the following conditions holds: a) p + q < 1 and λ ≥ λ or; b) p + q = 1 and |aL | small or; c) p + q > 1 and λ ≤ λ; problem (1.1) possesses a unique strictly positive solution. Finally, before we state the third main result we need some notations. Consider a ∈ C(Ω), a > 0 and denote by ωa the unique positive solution of     −∆u = a(x)uq in Ω,    u=0

on ∂Ω,

(1.3)

6

F. J. Corrˆea, M. Delgado, and A. Su´arez

and Z A :=

b(x)ωap .

(1.4)

Ω

Now, we can state the result Theorem 1.3. Assume a > 0 and 0 < q < 1. a) Assume p + q < 1. Then, there exists a value λ < 0 such that there exists a positive solution of (1.1) if and only if λ ≥ λ. Moreover, if λ ≥ 0 the solution is strictly positive, unique and stable. Furthermore, lim kuλ k∞ = +∞.

λ→+∞

b) Assume p + q = 1. (a) If A < 1 there exists a positive solution of (1.1) if and only if λ > 0. The solution is strictly positive, unique and stable and lim kuλ k∞ = 0,

λ→0

lim kuλ k∞ = +∞.

λ→+∞

(b) If A = 1 there exists a positive solution of (1.1) if and only if λ = 0. Moreover, there exist infinite solutions and any positive solution is neutrally stable. (c) If A > 1 there exists a positive solution of (1.1) if and only if λ < 0. Moreover, lim kuλ k∞ = 0,

λ→0

lim kuλ k∞ = +∞.

λ→−∞

c) Assume p + q > 1. Then there exists a value λ > 0 such that (1.1) possesses a positive solution if and only if λ ≤ λ. Moreover, lim kuλ k∞ = +∞.

λ→−∞

See Figures 1 and 2 where we have represented the set of positive solutions when q = 1 and q < 1, respectively. Figure 1 shows the case q = 1 (see Theorem 1.1): Case 1 and 2 represent the cases 0 a < 0, Ω0 = ∅ and Ω0 6= ∅ respectively; in this case we have denoted by λ01 := λΩ 1 . Finally,

Case 3 represents when a > 0. Figure 2 shows the case q < 1 (see Theorems 1.2 and 1.3): Case 1 represents the solutions of (1.1) when a < 0 and when a > 0, p + q = 1 and A < 1. Case 2 shows the case

Non-linear heterogeneous problems with nonlocal reaction term

u

7

u

u

Case 1

Case 2

Case 3

Figure 1: Bifurcation diagrams for equation (1.1) when q = 1. a > 0, p + q = 1 and A = 1. Case 3 represents the case a > 0, p + q = 1 and A > 1. Case a > 0 and p + q < 1 is drawn in Case 4. Finally, Case 5 represents a > 0 and p + q > 1. An outline of the paper is: In Section 2 we study the eigenvalue problem associated to the linearization around a positive solution of (1.1). In Section 3 we show that the subsupersolution method works for this kind of equations. In Section 4 we use the bifurcation method to show the existence of an unbounded continuum of positive solution of (1.1) for q < 1. Finally, in Section 5 we prove the main results of the paper.

2

Eigenvalue problems

In this section we study a nonlocal and singular eigenvalue problem, which appears when one linearizes around a positive solution of (1.1). Specifically, we study the following problem  Z     −∆u + m(x)u − a(x) b(x)u = σu in Ω, Ω

    u=0

(2.1)

on ∂Ω,

where m ∈ C 1 (Ω), a ∈ C(Ω) and b ∈ C 1 (Ω) and verify: for some α ∈ (−1, 1) and β < 1 (Hm) |∂i m|d(x, ∂Ω)2−α are bounded for all x ∈ Ω and i = 1, ..., N ; (Hb) there exists K > 0 such that b(x) ≤ Kd(x, ∂Ω)−β .

8

F. J. Corrˆea, M. Delgado, and A. Su´arez

u

u

u

Case 1

Case 2

Case 3

u

u

Case 4

Case 5

Figure 2: Bifurcation diagrams for equation (1.1) when q < 1. The next results were proved in [7]: Theorem 2.1. Assume that m verifies (Hm), a ∈ C 1 (Ω) ∩ C(Ω), is a non-negative and non-trivial function, b ∈ C 1 (Ω) is a non-negative and non-trivial function and verifies (Hb). Then, there exists a principal eigenvalue of (2.1), denoted by λ1 (−∆ + m; a; b), which has an associated positive eigenfunction ϕ1 ∈ C 2 (Ω) ∩ C01,δ (Ω) for some δ ∈ (0, 1). Moreover, λ1 (−∆ + m; a; b) is simple, and it is the unique eigenvalue having an associated eigenfunction without change of sign. In the following result we give a criteria to ascertain the sign of λ1 (−∆ + m; a; b). Proposition 2.2.

a) Assume that there exists a positive function u ∈ C 2 (Ω)∩C01,δ (Ω),

δ ∈ (0, 1), such that Z −∆u + m(x)u − a(x)

b(x)u > 0 Ω

in Ω.

Non-linear heterogeneous problems with nonlocal reaction term

9

Then, λ1 (−∆ + m; a; b) > 0. b) Assume that there exists a positive function u ∈ C 2 (Ω) ∩ C01,δ (Ω), δ ∈ (0, 1), such that Z −∆u + m(x)u − a(x)

b(x)u < 0

in Ω.

Ω

Then, λ1 (−∆ + m; a; b) < 0.

3

The sub-supersolution method

Although the sub-supersolution method has been used in reaction-diffusion equations with nonlocal terms, see for instance [10] and [5], we generalize these results for continuous reaction terms and without monotonicity assumption. Consider a continuous operator B : L∞ (Ω) 7→ IR and f : Ω × IR2 7→ IR a continuous function and the general problem     −∆u = f (x, u, B(u)) in Ω,    u=0

(3.1)

on ∂Ω.

Definition 3.1. We say that the pair (u, u), with u, u ∈ H 1 (Ω) ∩ L∞ (Ω), is a pair of sub-supersolution of (3.1) if a) u ≤ u in Ω and u ≤ 0 ≤ u on ∂Ω, b) −∆u−f (x, u, B(u)) ≤ 0 ≤ −∆u−f (x, u, B(u))

in the weak sense for all u ∈ [u, u].

The main result in this section is: Theorem 3.2. Assume that there exists a pair of sub-supersolution of (3.1). Then, there exists a solution u ∈ H 1 (Ω) ∩ L∞ (Ω) of (3.1) such that u ∈ [u, u].

10

F. J. Corrˆea, M. Delgado, and A. Su´arez

Proof. Define the truncation operator      u(x) if u(x) ≥ u(x),     T u(x) := u(x) if u(x) ≤ u(x) ≤ u(x),         u(x) if u(x) ≤ u(x),

(3.2)

and the Nemytskii operator F : L∞ (Ω) 7→ L∞ (Ω) given by F (u)(x) := f (x, T (u)(x), B(T (u))). It is clear that F is continuous and bounded, because there exists M > 0 such that kF (u)k∞ ≤ M

for all u ∈ L∞ (Ω).

Consider the problem −∆w = f (x, T (u), B(T (u)))

in Ω,

w=0

on ∂Ω.

(3.3)

We can define the operator T by u 7→ w := T (u) being w the unique solution of (3.3). By the Schauder Fixed Point Theorem there exists u ∈ L∞ (Ω) such that u = T (u), and then −∆u = f (x, T (u), B(T (u)))

in Ω,

u=0

on ∂Ω.

Now, we show that u ∈ [u, u], which implies that u is solution of (3.1). Indeed, denoting z := u − u and using the definition of supersolution for T (u), we get −∆z ≥ f (x, u, B(T (u))) − f (x, T (u), B(T (u))). Multiplying by (u − u)− we obtain the result.

4

Bifurcation from the trivial solution

In this section we will show that when q < 1, a bifurcation from the trivial solution of (1.1) occurs at λ = 0, independently of the sign of a. For that, we denote by X := C0 (Ω) = {u ∈ C(Ω) : u = 0 on ∂Ω}

and Bρ := {u ∈ X : kuk∞ < ρ}.

Non-linear heterogeneous problems with nonlocal reaction term

11

First we define + q



Z

b(x)(u )

λ + a(x)

f (λ, x, u) := (u )

+ p

 .

Ω

Finally, we also define the map Kλ (u) := u − (−∆)−1 (f (λ, x, u)).

Kλ : X 7→ X;

Now, it is clear that that u is a non-negative solution of (1.1) if, and only if, u is a zero of the map Kλ . We will use the Leray-Schauder degree of Kλ on Bρ with respect to zero, denoted by deg(Kλ , Bρ ), and the index of the isolated zero u0 of Kλ , denoted by i(Kλ , u0 ). The main result of this section is: Theorem 4.1. Assume 0 < q < 1 and a ∈ C(Ω). The value λ = 0 is the only bifurcation point from the trivial solutions for (1.1). Moreover, there exists an unbounded continuum (closed and connected set) C0 of non-negative solutions of (1.1) unbounded in IR × X emanating from (λ, u) = (0, 0). In order to prove this result, we compute the index of the trivial solution for λ < 0 and λ > 0. Lemma 4.2. If λ < 0, then i(Kλ , 0) = 1. Proof. Fix λ < 0 and define the map H1 : [0, 1] × X 7→ X;

H1 (t, u) := (−∆)−1 (tf (λ, x, u)).

We claim that there exists δ > 0 such that u 6= H1 (t, u)

for u ∈ B δ , u 6= 0 and t ∈ [0, 1].

Indeed, suppose on the contrary that there exist sequences un ∈ X\{0} and tn ∈ [0, 1] with kun k∞ → 0 such that un = H1 (tn , un ). We know that un ≥ 0 in Ω. Since kun k∞ → 0 and λ < 0, there exists n0 ∈ IN such that for n ≥ n0 , it holds −∆un ≤ 0

in Ω,

un = 0

on ∂Ω,

12

F. J. Corrˆea, M. Delgado, and A. Su´arez

which is impossible, in view of the maximum principle. Taking now ε ∈ (0, δ], the homotopy defined by H1 is admissible and so, i(Kλ , 0) = deg(Kλ , Bε ) = deg(I − H1 (1, ·), Bε ) = deg(I − H1 (0, ·), Bε ) = = deg(I, Bε ) = 1.

Lemma 4.3. If λ > 0, then i(Kλ , 0) = 0. Proof. Fix λ > 0 and φ ∈ X, φ > 0. We define the map H2 : [0, 1] × X 7→ X;

H2 (t, u) := (−∆)−1 (f (λ, x, u) + tφ).

We will show that there exists δ > 0 such that u 6= H2 (t, u)

for all u ∈ B δ , u 6= 0 and t ∈ [0, 1].

Suppose the contrary: there exist sequences un ∈ X \ {0} and tn ∈ [0, 1] with kun k∞ → 0 such that un = H2 (tn , un ). Since tn φ ≥ 0, multiplying by u− n , we obtain that un ≥ 0. Moreover since λ > 0 and kun k∞ → 0, by the strong maximum principle un  0. We fix M ≥ λ1 . Since kun k∞ → 0 and λ > 0, there exists n0 ∈ IN such that for n ≥ n0 we get   Z q p −∆un = un λ + a(x) b(x)un + tn φ > M un + tn φ, Ω

and then, (−∆ − M )un > 0. So, multiplying this inequality by ϕ1 , a positive eigenfunction associated to λ1 , we conclude that λ1 > M . This is impossible. This proves that the homotopy defined by H2 is admissible. Then, if we take ε ∈ (0, δ] we have i(Kλ , 0) = deg(Kλ , Bε ) = deg(I − H2 (0, ·), Bε ) = deg(I − H2 (1, ·), Bε ) = 0. This last equality is true because the problem −∆u = uq (λ + a(x)

R

p Ω b(x)u )

+ φ has no

solution in B ε . Recall that we have shown that u 6= H2 (1, u) for all u ∈ B δ , u 6= 0.

Non-linear heterogeneous problems with nonlocal reaction term

13

Proof of Theorem 4.1: The fact that λ = 0 is a bifurcation point follows by Lemma 4.2 and Lemma 4.3. Now, even though our map Kλ does not satisfy exactly the hypotheses of Theorem 1.3 in [16], the proof can be modified to obtain the result, see Theorem 3.1 in [1] and Theorem 4.4 in [3], and we can conclude the existence of a continuum of solutions of (1.1) such that meets (0, 0) either infinity or (λ0 , 0) with λ0 6= 0. We can discard this last possibility. Indeed, from Lemma 4.2, (1.1) does not have bifurcation points in (−∞, 0). Assume that there exists a sequence of solutions (λn , un ) such that λn → λ0 > 0 and kun k∞ → 0. We take M ≥ λ1 , so there exists n0 ∈ IN such that uqn (λn + a(x)

Z

b(x)upn ) > M un

for all n ≥ n0 .

Ω

As in the proof of Lemma 4.3, we obtain that λ1 − M > 0, a contradiction, and so (1.1) does not have bifurcation points in (0, +∞). Hence, the existence of an unbounded in IR × X continuum of solutions of (1.1) follows.



The following trivial result will be used constantly in the work, recall that ωa is defined in (1.3). Lemma 4.4. Suppose a > 0, λ > 0 and 0 < q < 1 and let u be a positive solution of     −∆u = λa(x)uq in Ω,    u=0

(4.1)

on ∂Ω.

Then, 1

u = λ 1−q ωa . On the other hand, if u is a positive supersolution (resp. subsolution) of (4.1) then 1

u ≥ λ 1−q ωa

1

(resp. u ≤ λ 1−q ωa )

With respect to the bifurcation direction we get: Proposition 4.5. Assume a > 0 and 0 < q < 1. a) If p + q < 1, then the bifurcation is subcritical. b) If p + q > 1, then the bifurcation is supercritical.

in Ω.

14

F. J. Corrˆea, M. Delgado, and A. Su´arez c) Let p + q = 1 and consider (λ, uλ ) a positive solution of (1.1). Then, if A = 1 (resp. A < 1) (resp. A > 1) then λ = 0 (resp. λ > 0) (resp. λ < 0).

Proof. Assume p + q < 1 and that there exists a sequence (λn , un ) of positive solutions to (1.1) with λn ≥ 0 and kun k∞ → 0. Since λn ≥ 0 we get that −∆un ≥ a(x)uqn tn , where Z tn =

b(x)upn .

Ω

Then, by Lemma 4.4 un ≥ t1/(1−q) ωa , n

(4.2)

and hence Z

b(x)upn

(1−p−q)/(1−q)

Z ≥

Ω

b(x)ωap = A,

(4.3)

Ω

an absurdum since kun k∞ → 0. For the case p + q > 1 and λn ≤ 0, with a similar reasoning we get Ckun kp(1−q−p)/(1−q) ≤ ∞

Z

b(x)upn

(1−p−q)/(1−q)

Z ≤

Ω

b(x)ωap = A,

Ω

again an absurdum. Finally assume p + q = 1. Take λ ≥ 0, then by (4.2) we obtain Z uλ ≥ Ω

b(x)upλ

1/(1−q) ωa ,

and then using that p + q = 1 we have Z 1= Ω

b(x)upλ

(1−p−q)/(1−q) ≥ A.

This completes the proof.

5

Proof of the main results

Before giving the proofs of the main results, we prove the stability results. Proposition 5.1. Assume a > 0.

(4.4)

Non-linear heterogeneous problems with nonlocal reaction term

15

a) If q = 1 any positive solution of (1.1) is unstable for any p > 0. b) Assume q < 1. If p + q < 1 and λ ≥ 0 or p + q = 1 and λ > 0, then, any positive solution of (1.1) is stable. If p + q = 1 and λ = 0 any positive solution of (1.1) is neutrally stable. Proof. Consider a positive solution u0 of (1.1). We have to calculate the sign of the principal eigenvalue of 
R R    −∆ξ − quq−1 λ + a(x) Ω b(x)up0 ξ − uq0 pa(x) Ω b(x)up−1 0 0 ξ = σξ in Ω,    ξ=0

on ∂Ω.

First, observe that in any case, if u0 is a positive solution of (1.1), then u0 ∈ int(P ), where int(P ) := {u ∈ C01 (Ω) : u(x) > 0 ∀x ∈ Ω, ∂u/∂n < 0

on ∂Ω},

being n the unit outward vector field to ∂Ω. Then, there exist constants 0 < k1 < k2 such that k1 d(x, ∂Ω) ≤ u0 ≤ k2 d(x, ∂Ω). So, we can apply the results of Section 2. We are going to apply Proposition 2.2 with u = u0 or u = u0 . Then, we have to ascertain the sign of Z q u0 (λ(1 − q) + (1 − p − q)a(x) b(x)up0 ). Ω

This expression is negative for q = 1, positive if q < 1, p + q < 1 and λ ≥ 0, or p + q = 1 and λ > 0 and zero if p + q = 1 and λ = 0.

5.1

Proof of Theorem 1.1:

Assume q = 1 and a < 0 and consider λ(t) := λ1 (−∆ − ta)

for t ≥ 0.

It is well-known that λ(t) is increasing, λ(0) = λ1 and 0 lim λ(t) = λΩ 1 ,

t→+∞

0 with λΩ 1 = +∞ if Ω0 = ∅, see for instance Theorem 6.2 in [14].

16

F. J. Corrˆea, M. Delgado, and A. Su´arez 0 Take λ ∈ (λ1 , λΩ 1 ), there exists a unique value t0 (λ) > 0 such that λ1 (−∆ − t0 a) = λ

and t0 (λ) → 0

0 t0 (λ) → +∞ as λ → λΩ 1 .

as λ → λ1 and

(5.1)

We prove now the existence and uniqueness of positive solution on (1.1). Observe that if u is a positive solution of (1.1) then Z

b(x)up ) = λu

−∆u + u(−a Ω

and so, Z λ = λ1 (−∆ − a

b(x)up ).

(5.2)

Ω

Hence, Z

b(x)up = t0

and u = Kϕ0

Ω

being ϕ0 = ϕ0 (λ) the positive eigenfunction associated to λ1 (−∆−t0 a) such that kϕ0 k∞ = 1 and K > 0. Then t0

Kp = Z

.

b(x)ϕp0

Ω

It is clear that the existence and uniqueness follow. On the other hand, from (5.2) we get that Z

b(x)up ) > λ1 ,

λ = λ1 (−∆ − a Ω

and Z λ = λ1 (−∆ − a Ω

0 b(x)up ) < λΩ 1 .

Finally, from (5.1), taking into account that uλ = K(λ)ϕ0 (λ), and using that ϕ0 → ϕ1 in H01 (Ω) as λ → λ1 , ϕ1 is a positive eigenfunction associated to λ1 , we get that lim kuλ k∞ = 0.

λ→λ1

On the other hand, K p (λ) = Z

t0 (λ)

Ω

t0 (λ) ≥Z →∞ b(x)ϕp0 b(x) Ω

Non-linear heterogeneous problems with nonlocal reaction term

17

0 as λ → λΩ 1 , and then we obtain that

lim kuλ k∞ = +∞. Ω

λ→λ1 0

Assume now that a > 0. Then, in this case λ(t) := λ1 (−∆ − ta) is decreasing and limt→+∞ λ(t) = −∞. Then, take λ < λ1 , there exists a unique t0 (λ) > 0 such that λ1 (−∆ − t0 a) = λ and t0 (λ) → 0

as λ → λ1 and

t0 (λ) → +∞ as λ → −∞.

(5.3)

We can reason now exactly as in the above case. The stability result follows by Proposition 5.1.

5.2

Proof of Theorem 1.2:

Assume now that 0 < q < 1 and a < 0. Thanks to the Theorem 4.1 we know the existence of a unbounded continuum of positive solutions C0 ⊂ IR × X bifurcating from the trivial solution at λ = 0. Moreover, since a < 0 then if λ ≤ 0 (1.1) does not possess positive solution. Finally, observe that for any solution (λ, uλ ) of (1.1) we have that −∆uλ ≤ λuqλ and so by Lemma 4.4 uλ ≤ λ1/(1−q) ω1 ,

(5.4)

that is, a priori bound. This shows the existence of positive solution for all λ > 0. Of course, from (5.4) we deduce that limλ→0 kuλ k∞ = 0. On the other hand, let uλ a positive solution of (1.1). Then, using (5.4) we get   Z Z −∆uλ = uqλ λ + a(x) b(x)upλ ≥ uqλ λ(1 + a(x)λ(p+q−1)/(1−q) b(x)ω1p ). Ω

Ω

Then, if p + q > 1 and λ small we conclude that −∆uλ > 0, and then uλ  0. Analogously in the other cases. We prove now the uniqueness of strictly positive solution of (1.1). Assume that there exist two strictly positive solutions of (1.1), u and v, and suppose for example that Z Z b(x)up > b(x)v p . Ω

Ω

It is clear that u is a subsolution of the equation Z q −∆w = w (λ + a(x) b(x)v p ) in Ω, Ω

w=0

on ∂Ω.

18

F. J. Corrˆea, M. Delgado, and A. Su´arez

and so, since v is a solution of the equation, we get that u < v, an absurdum. On the other hand, assume that Z

b(x)up =

Ω

Z

b(x)v p .

Ω

Then, it is clear that u ≡ v. This completes the proof.

5.3



Proof of Theorem 1.3:

Assume 0 < q < 1 and a > 0. First, again by Theorem 4.1 there exists a unbounded continuum of positive solutions C0 ⊂ IR × X bifurcating from the trivial solution at λ = 0. Case p + q < 1: We know by Proposition 4.5 that the bifurcation direction is subcritical. Now, we prove that there does not exist positive solution for λ very negative. Assume the contrary, and so there exists a sequence of solutions (λn , un ) of (1.1) such that λn → −∞. Denote by xn ∈ Ω such that un (xn ) := maxx∈Ω un (x). Then, Z λn + a(xn ) b(x)upn ≥ 0, Ω

and hence Z

b(x)upn → +∞.

(5.5)

Ω

On the other hand, since λn < 0 we get, see (4.4), Z

b(x)upn

(1−p−q)/(1−q) ≤ A,

Ω

an absurdum with (5.5). We now prove the a priori bound in compact intervals in λ. Assume that there exists a sequence of positive solutions (λn , un ) of (1.1) such that λn → λ0 and kun k∞ → +∞. Denote by Un :=

un . kun k∞

It is clear that Un verifies −∆Un = Unq

λn kun k1−q ∞

+

Z

a(x) 1−p−q kun k∞

! b(x)Unp

Ω

and then passing to the limit, Un → U in C 2 (Ω) and U is such that −∆U = 0

in Ω, U = 0

on ∂Ω,

,

Non-linear heterogeneous problems with nonlocal reaction term

19

and so U ≡ 0, an absurdum due to kU k∞ = 1. Define now λ := inf{λ ∈ IR : (1.1) has a positive solution}. We know that λ > −∞. Take now λ > λ and uλ a solution of (1.1) with λ = λ. We are going to use Section 3. Indeed, it is clear that the pair (u, u) = (uλ , Ke) is a pair of sub-supersolution of (1.1) for K large enough and e the unique positive solution of     −∆e = 1 in Ω,    e=0

on ∂Ω.

This proves the existence of positive solution for all λ ∈ [λ, ∞). Now, we prove the uniqueness of positive solution of (1.1) for λ ≥ 0. We use a similar argument to the used in [4]. Assume there exist two positive solutions u, v. Thanks to the strong maximum principle u, v ∈ int(P ). Then, there exists t0 > 0 such that t0 u ≤ v. Define now, t∗ := sup{t > 0 : such that tu ≤ v}. We claim that t∗ ≥ 1 and then u ≤ v and the result follows. We show the claim. Assume that t∗ < 1 and consider w := v − t∗ u ≥ 0. Then, using that t∗ < 1 and that p + q < 1 we get that Z

+ a(x)

p

∗ q

Z

b(x)v ) − t u (λ + a(x) b(x)up ) Z Ω = uq (λ((t∗ )q − t∗ ) + ((t∗ )p+q − t∗ )a(x) b(x)up ) > 0,

−∆w =

v q (λ

Ω

(5.6)

Ω

and then by the strong maximum principle we get that w ∈ int(P ), an absurdum with the maximality of t∗ . Finally we show that kuλ k → ∞ as λ → +∞. Observe that u = εϕ1 , ε > 0 and ϕ1 the positive eigenfunction associated to λ1 such that kϕ1 k∞ = 1, is sub-solution of (1.1) provided of ε ≤ λ1/1−q . Then, by the uniqueness λ1/(1−q) ϕ1 ≤ uλ .

(5.7)

20

F. J. Corrˆea, M. Delgado, and A. Su´arez

The stability follows by Proposition 5.1. This concludes the proof. Case p + q > 1: We know that the bifurcation direction is supercritical. Now, we prove that there does not exist positive solution for λ very large. Assume that there exists a sequence (λn , un ) of positive solutions of (1.1) and λn → +∞. First observe that since a > 0 we have that, see (5.7), un ≥ λ1/(1−q) ω1 , n and so Z tn :=

b(x)upn → +∞ as λn → ∞.

Ω

On the other hand, since un is a positive solution of −∆u = uq (λn + a(x)tn ) 1/(1−q)

and tn

ωa is a subsolution of this equation, we get that tn1/(1−q) ωa ≤ un

and then, tp/(1−q) A ≤ tn n an absurdum because p + q > 1 and tn → +∞. We prove now the a priori bound. Assume that for some sequence of positive solutions (λn , un ) such that λn → λ0 and kun k∞ → ∞. Then, it is clear that Z tn = b(x)upn → ∞, Ω

and again we arrive at an absurdum with a similar argument to the used before. Define now λ := inf{λ ∈ IR : (1.1) has a positive solution}. We know that λ < +∞. Take now λ < λ. Then, using again the Theorem 3.2, it is clear that the pair (u, u) = (εϕ1 , uλ ) is a pair of sub-supersolution of (1.1) for ε small enough and ϕ1 a positive eigenfunction associated to λ1 , where uλ is a solution of (1.1) at λ = λ. Hence, there exits a positive solution of (1.1) if λ ≤ λ. Now, we show that kuλn k∞ → ∞ as λn → −∞. Denote by xn ∈ Ω such that uλn (xn ) = kuλn k∞ . Then, Z b(x)uλn ≥ 0,

λn + a(xn ) Ω

Non-linear heterogeneous problems with nonlocal reaction term

21

and so, Z b(x)uλn → ∞, Ω

and therefore kuλn k∞ → ∞. Case p + q = 1: Assume that A < 1. Then by Proposition 4.5 we know that λ > 0. It remains to show the a priori bound. Assume that there exists a sequence of positive solutions (λn , un ) of (1.1) such that λn → λ0 > 0 and kun k∞ → +∞. Denote by Un :=

un . kun k∞

Again it is clear that Un verifies −∆Un = Unq (

λn 1−q kun k∞

Z

b(x)Unp ),

+ a(x) Ω

and then passing to the limit, Un → U in C 2 (Ω) such that q

Z

−∆U = U a(x)

b(x)U p

in Ω,

U =0

on ∂Ω,

Ω

and so A = 1, an absurdum. We can reason similarly in the other cases. For the uniqueness we can follow the lines used in the case p + q < 1 using (5.6) and λ > 0.

References [1] S. Alama, Semilinear elliptic equations with sublinear indefinite nonlinearities, Adv. in Differential Equations, 4 (1999), 813-842. [2] W. Allegretto, and A. Barabanova, Existence of positive solutions of semilinear elliptic equations with nonlocal terms, Funkcial. Ekvac., 40 (1997), 395–409. [3] D. Arcoya, J. Carmona, and B. Pellacci, Bifurcation for some quasi-linear operators, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 733–765. [4] S. M. Bouguima, M. Kada, and J. A. Montero, Bifurcation of positive solutions for a population dynamics model with nonlocal terms, preprint. [5] M. Chipot, and F. J. S. A. Corrˆea, Boundary layer solutions to functional elliptic equations, Bull. Braz. Math. Soc., New Series 40 (2009), 1-13.

22

F. J. Corrˆea, M. Delgado, and A. Su´arez

[6] F. A. Davidson, and N. Dodds, Existence of positive solutions due to nonlocal interactions in a class of nonlinear boundary value problems, Methods Appl. Anal., 14 (2007), 15–27. [7] F. J. S.A. Corrˆea, M. Delgado, and A. Su´arez, Some nonlocal population models with nonlinear diffusion, submitted. [8] P. Freitas, Bifurcation and stability of stationary solutions of nonlocal scalar reactiondiffusion equations, J. Dynam. Differential Equations, 6 (1994), 613–629. [9] P. Freitas, Nonlocal reaction-diffusion equations, Differential equations with applications to biology (Halifax, NS, 1997), 187–204, Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, 1999. [10] P. Freitas, and M. P. Vishnevskii, Stability of stationary solutions of nonlocal reaction-diffusion equations in m-dimensional space, Differential Integral Equations, 13 (2000), 265–288. [11] J. Furter, and M. Grinfeld, Local vs. nonlocal interactions in population dynamics, J. Math. Biol., 27 (1989), 65–80. [12] M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49. [13] J. L´ opez-G´ omez, On the structure and stability of the set of solutions of a nonlocal problem modeling Ohmic heating, J. Dynam. Differential Equations, 10 (1998), 537– 566. [14] J. L´ opez-G´ omez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Differential Eqns., 127 (1996), 263-294. [15] T. Namba, Density-dependent dispersal and spatial distribution of a population, J. Theor. Biol., 86 (1980), 351-363. [16] P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.