INVERTIBILITY OF LINEAR
NON-COOPERATIVE ELLIPTIC SYSTEMS
Julián López-Gómez
and
Rosa M. Pardo
Departamento de Matemática Aplicada
Universidad Complutense de Madrid
28040-Madrid, SPAIN
Abstract. In this work we give a sufficient condition for the invertibility of a linear
weakly coupled non-cooperative elliptic system under general boundary conditions.
Then, we use this result to show the uniqueness of the coexistence state in a large
class of Lotka-Volterra predator-prey models with diffusion and transport effects.
1. Introduction
In this work we analyze the eigenvalue problem
L1 0
u
0
−α(x)
u
u
=
+τ
0 L2
v
β(x)
0
v
v
in (a, b),
(1.1)
Bp1 ,a u = Bq1 ,b u = Bp2 ,a v = Bq2 ,b v = 0 ,
where a, b ∈ R, a < b, and Lj are second order differential operators of the form
Lj = −aj (x)
d2
d
+ bj (x)
+ cj (x) ,
2
dx
dx
j = 1, 2,
(1.2)
with aj , bj , cj ∈ C[a, b], aj (x) > 0 for all x ∈ [a, b], and pj , qj ∈ [0, ∞], j = 1 , 2.
Given p, q ∈ [0, ∞] we have denoted
Bp,a w =
w(a) − p w 0 (a)
,
1+p
Bq,b w =
w(b) + q w 0 (b)
,
1+q
(1.3)
Key words and phrases. Coexistence states, non-cooperative elliptic systems, maximum principle, principal eigenvalue, Lotka-Volterra predator-prey models..
Typeset by AMS-TEX
Typeset by AMS-TEX
2
JULIÁN LÓPEZ-GÓMEZ
AND
ROSA M. PARDO
If p = 0, then B0,a is the Dirichlet operator. In this case we set B0,a = Da . Similarly,
B0,b = Db . We also consider the Neumann operators
B∞,a w := Na w := −w 0 (a) ,
B∞,b w := Nb w := w 0 (b) .
The main result of this paper (Theorem 3.1 of Section 3) shows that τ = 0 is never
an eigenvalue of (1.1) (in (C[a, b])2) if the following two conditions are satisfied:
(A1) The operators (L1 ; Bp1 ,a , Bq1 ,b ) and (L2 ; Bp2 ,a , Bq2 ,b ) satisfy the strong
maximum principle;
(A2) α ≥ 0, β ≥ 0, and { x ∈ (a, b) : α(x) = 0 }, { x ∈ (a, b) : β(x) = 0 } have
empty interior.
This result is known for Dirichlet boundary conditions, [9], [3]. To generalize
it to the case of general boundary conditions we first characterize the maximum
principle in terms of the positivity of the principal eigenvalue. This characterization
is known in the case of Dirichlet boundary conditions [9], [2], [7], and in the context
of weakly coupled cooperative elliptic systems, [8], but it is new in our general
setting. In Section 2 we characterize the strong maximum principle by means of the
existence of a strict positive supersolution and use it to get some basic comparison
results between principal eigenvalues. In Section 3 we show the invertibility result
and in Section 4 we use it to show the uniqueness of the coexistence state for a
class of Lotka-Volterra predator-prey models with one-dimensional diffusion and
convection subject to general boundary conditions. The existence of coexistence
states can be characterized by using the theory of [4] and [6]. Our results are far
from known because of the lack of a priori bounds for the positive solutions of
such systems. We conclude the paper discussing the stability of the coexistence
states. If some coexistence state loses stability when some of the coefficients of
the model varies, then a couple of complex conjugate imaginary eigenvalues of the
linearization across the imaginary axis in the complex plain. Our techniques provide
us with the uniqueness for a wide class of one-dimensional reaction diffusion systems
of predator-prey type.
2. The maximum principle and some properties of principal eigenvalues
Let a, b ∈ R, a < b, and set I := (a, b). In this section we consider a second order
differential operator L of the form
d2
d
L = − a(x) 2 + b(x)
+ c(x) ,
dx
dx
where a , b , c ∈ C[a, b], a(x) > 0, x ∈ [a, b], and p, q ∈ [0, ∞], and we deal with the
eigenvalue problem
Lϕ = τ ϕ
in I ,
Bp,a ϕ = Bq,b ϕ = 0 .
Problem (2.1) possesses a unique principal eigenvalue. We shall denote it by
σ1I [L; Bp,a , Bq,b ] .
(2.1)
INVERTIBILITY OF LINEAR NON-COOPERATIVE ELLIPTIC SYSTEMS
3
By a principal eigenvalue we mean an eigenvalue to a positive eigenfunction. Its
existence and uniqueness can be obtained applying the classical maximum principle
and the Krein-Rutman theorem to the operator L+C, under homogeneous boundary
conditions Bp,a = 0, Bq,b = 0, where C is any constant satisfying C + c > 0, [1,
Theorem 4.3]. Then,
σ1I [L; Bp,a , Bq,b ] = σ1I [L + C; Bp,a , Bq,b ] − C .
Moreover, the principal eigenfunction is unique up to multiplicative constants. The
main result of this section is the following characterization of the maximum principle by means of the sign of the principal eigenvalue. This result is well known
for Dirichlet boundary conditions, [9], [7], [2], but it is new for general boundary
conditions.
Theorem 2.1. The following conditions are equivalent:
(i) σ1I [L; Bp,a , Bq,b ] > 0.
(ii) There exists ψ ∈ C 1 [a, b] ∩ C 2 (a, b) such that ψ(x) > 0 for all x ∈ I and
Lψ ≥ 0
in I ,
Bp,a ψ ≥ 0 ,
Bq,b ψ ≥ 0 ,
(2.2)
with some of these inequalities strict, i.e. the problem
Lw = 0
in I ,
Bp,a w = Bq,b w = 0 ,
(2.3)
has a strict positive supersolution. We will simply say that (L; B p,a , Bq,b ) possesses
a strict positive supersolution.
(iii) The operator (L; Bp,a , Bq,b ) satisfies the strong maximum principle, i.e. if
w ∈ C 1 [a, b] ∩ C 2 (a, b) satisfies
Lw ≥ 0
in I ,
Bp,a w ≥ 0 ,
Bq,b w ≥ 0 ,
(2.4)
then w ≥ 0 in I. Moreover, if some of these inequalities is strict, then w 0 in the
sense that w(x) > 0 for all x ∈ I, w 0 (a) > 0 if w(a) = 0, and w 0 (b) < 0 if w(b) = 0.
Proof. (i) ⇒ (ii). Assume that σ1I [L; Bp,a , Bq,b ] > 0. Let ϕ 0 denote its principal
eigenfunction. Then, L ϕ > 0 and so ϕ is a strict positive supersolution of (2.3).
(ii) ⇒ (iii). Let ψ be a positive strict supersolution of (2.3). Consider w ∈
C 1 [a, b] ∩ C 2 (a, b) satisfying (2.4). To prove that w ≥ 0 we argue by contradiction.
Suppose that there exists x1 ∈ (a, b) such that w(x1 ) < 0. Then, the minimum
t ≥ 0 for which vt := w + t ψ ≥ 0 in I is positive. Denote it by µ > 0. We claim
that vµ (x) > 0 for all x ∈ (a, b). On the contrary assume that vµ (x2 ) = 0 for
some x2 ∈ (a, b). Then, since (L + c+ − c) vµ ≥ L vµ ≥ 0 in I, it follows from [10,
Chap. 1, Theorem 3] that vµ ≡ 0. Thus, w = −µ ψ and hence, using (2.2) and
(2.4), we find that 0 ≤ Lw = −µ Lψ ≤ 0 in I, 0 ≤ Bp,a w = −µ Bp,a ψ ≤ 0 and
0 ≤ Bq,b w = −µ Bq,b ψ ≤ 0. Therefore, Lψ = 0 in I, Bp,a ψ = 0 and Bq,b ψ = 0,
which is impossible, because we are assuming that some of the inequalities of (2.2)
4
JULIÁN LÓPEZ-GÓMEZ
AND
ROSA M. PARDO
is strict. This contradiction shows that vµ (x) > 0 for all x ∈ I. By the definition
of µ we find that either vµ (a) = 0, or vµ (b) = 0. Moreover, due to [10, Chap. 1,
Theorem 4], vµ0 (a) > 0 if vµ (a) = 0, and vµ0 (b) < 0 if vµ (b) = 0. To complete the
proof we distinguish several different cases according to the boundary conditions.
(a) Dirichlet boundary conditions: p = q = 0. In this case, it follows from [11,
Theorem 2] that some of the following options occurs. Either there exists β < 0
such that w = β ψ in I, or w ≡ 0 in I, or w(x) > 0 for all x ∈ I. As we are assuming
that w(x1 ) < 0, necessarily w = β ψ in I for some β < 0. Thus,
0 ≤ L w = β L ψ ≤ 0 , 0 ≤ D a w = β Da ψ ≤ 0 , 0 ≤ D b w = β Db ψ ≤ 0 .
Therefore, L ψ = 0 and Da ψ = Db ψ = 0. As ψ is a strict positive supersolution,
this is a contradiction. This contradiction shows that w ≥ 0.
(b) Robin boundary conditions: (p, q) ∈ (0, ∞)2 . We already know that either
vµ (a) = 0 and vµ0 (a) > 0, or vµ (b) = 0 and vµ0 (b) < 0. Thus, either Bp,a vµ < 0 or
Bq,b vµ < 0. On the other hand, from (2.2), (2.4) and the definition of vµ we get
Bp,a vµ = Bp,a w + µ Bp,a ψ ≥ 0 ,
Bq,b vµ = Bq,b w + µ Bq,b ψ ≥ 0 .
(2.5)
This contradiction shows that w ≥ 0.
(c) Neumann boundary conditions: p = ∞, q = ∞. The same argument used
in case (b) shows that either Na vµ = −vµ0 (a) < 0, or Nb vµ = vµ0 (b) < 0. This
contradicts (2.5) and shows that w ≥ 0.
(d) Dirichlet-Robin boundary conditions: p = 0, q ∈ (0, ∞). If vµ (b) = 0 and
vµ0 (b) < 0, then Bq,b vµ < 0 which contradicts (2.5). Hence, vµ (b) > 0, vµ (a) = 0
and vµ0 (a) > 0. Due to the minimality of µ, there exists a sequence (xk ), k ≥ 1, in
(a, b) such that
w(xk ) + (µ −
1
1
)ψ(xk ) = vµ (xk ) − ψ(xk ) < 0,
k
k
k ≥ 1.
(2.6)
Let x∗ be a limit point of some convergent subsequence of (xk ), again denoted
by (xk ). We already know that vµ (x) > 0 for all x ∈ (a, b]. Thus, if we assume
that x∗ > a, then passing to the limit as k → ∞ in (2.6) we get 0 < vµ (x∗ ) =
w(x∗ ) + µ ψ(x∗ ) ≤ 0, which is impossible. Thus, x∗ = a and (2.6) yields
(µ −
1
)ψ(xk ) < −w(xk ) = w(a) − w(xk ) ≤ M (xk − a) = M d(xk ) ,
k
(2.7)
for some constant M > 0, where d(x) = min{|x − a|, |x − b|}. To obtain (2.7) we
have used w(a) = 0. This follows from 0 = vµ (a) = w(a) + µ ψ(a) ≥ 0, taking into
account that Da w = w(a) ≥ 0 and that Da ψ = ψ(a) ≥ 0. Now, it follows from [9,
Lemma 1] that vµ (x) > δ d(x), x ' a, for some δ > 0. Therefore, we find from (2.7)
that
1
M
vµ (xk ) − ψ(xk ) > δ (1 −
) d(xk ) > 0
k
µk −1
INVERTIBILITY OF LINEAR NON-COOPERATIVE ELLIPTIC SYSTEMS
5
for k sufficiently large. This relation contradicts to (2.6) and completes the proof
of w ≥ 0. The above argument can be easily adapted to cover the case Robin at a
and Dirichlet at b. This technique was also used to prove Theorem 2 of [11].
(e) Dirichlet-Neumann boundary conditions: p = 0, q = ∞. The same
argument as in case (d) shows that w ≥ 0.
(f ) Neumann-Robin boundary conditions: p = ∞, q ∈ (0, ∞). From (2.2),
(2.4) and the definition of vµ we find that
Na vµ = −vµ0 (a) ≥ 0 ,
Bq,b vµ =
vµ (b) + q vµ0 (b)
≥ 0.
1+q
(2.8)
In particular, vµ0 (a) ≤ 0. Therefore, vµ (b) = 0 and vµ0 (b) < 0. This contradicts
the second relation of (2.8) and shows that w ≥ 0. A similar argument shows that
w ≥ 0 in case Robin-Neumann.
We have already seen that w ≥ 0 in [a, b]. If we assume that some of the inequalities
of (2.4) is strict, then w > 0 in I. Moreover, (L + c+ − c) w ≥ Lw ≥ 0 in (a, b).
Thus, it follows from [10, Chap.1, Th. 3] that w(x) > 0 for all x ∈ (a, b). Finally,
[10, Chap.1, Th. 4] completes the proof of (ii) ⇒ (iii).
(iii) ⇒ (i). We argue by contradiction assuming that σ1I [L; Bp,a , Bq,b ] ≤ 0. Let ϕ 0 be its associated principal eigenfunction. Then, L (−ϕ) = σ1I [L; Bp,a , Bq,b ] (−ϕ) ≥
0 in I, Bp,a (−ϕ) = 0 and Bq,b (−ϕ) = 0. Therefore, (iii) implies ϕ ≤ 0. This contradiction shows σ1I [L; Bp,a , Bq,b ] > 0 and completes the proof of the theorem. We now use this characterization to get some comparison results between principal eigenvalues. First, we give the following of min-max characterization of
σ1I [L; Bp,a , Bq,b ].
Theorem 2.2. Let Pa,b denote the set of functions ψ ∈ C 1 [a, b] ∩ C 2 (a, b) such
that ψ(x) > 0 for all x ∈ [a, b], Bp,a ψ > 0 and Bq,b ψ > 0. Then,
σ1I [L; Bp,a , Bq,b ] = sup
inf
ψ∈Pa,b x∈I
Lψ(x)
.
ψ(x)
(2.9)
Proof. Let λ < σ1I [L; Bp,a , Bq,b ] be. Then σ1I [L − λ; Bp,a , Bq,b ] > 0 and it follows
from Theorem 2.1 that the unique solution of
(L − λ) φ = 1
in I ,
Bp,a φ = Bq,b φ = 1 ,
satisfies φ(x) > 0 for all x ∈ [a, b]. In particular,
λ < inf
x∈I
Lψ(x)
Lφ(x)
≤ sup inf
.
φ(x)
ψ∈Pa,b x∈I ψ(x)
Since this inequality holds for any λ < σ1I [L; Bp,a , Bq,b ] we find that
σ1I [L; Bp,a , Bq,b ] ≤ sup
inf
ψ∈Pa,b x∈I
Lψ(x)
.
ψ(x)
6
JULIÁN LÓPEZ-GÓMEZ
AND
ROSA M. PARDO
To complete the proof we argue by contradiction. Suppose that
σ1I [L; Bp,a , Bq,b ] < sup
inf
ψ∈Pa,b x∈I
Lψ(x)
.
ψ(x)
Then, there exist ε > 0 and ψ ∈ Pa,b such that
σ1I [L; Bp,a , Bq,b ] + ε < inf
x∈I
Lψ(x)
.
ψ(x)
Hence,
(L − σ1I [L; Bp,a , Bq,b ] − ε)ψ > 0
and so, ψ is a s.p. supersolution of (L − σ1I [L; Bp,a , Bq,b ] − ε; Bp,a , Bq,b ). Thus, due
to Theorem 2.1 we obtain that
σ1I [L − σ1I [L; Bp,a , Bq,b ] − ε; Bp,a , Bq,b ] = −ε > 0 .
This contradiction completes the proof.
Most of the assertions of next theorem are special cases of Lemma 15.5 and
Proposition 17.5 of [5], because Hölder continuity of the coefficients are not needed
in the proofs. By the sake of completeness we include an alternative proof of all
these properties with none additional regularity requirement.
Theorem 2.3. The following properties of the principal eigenvalue hold:
(a) Let Vj ∈ C[a, b], j = 1 , 2, with V1 < V2 . Then,
σ1I [L + V1 ; Bp,a , Bq,b ] < σ1I [L + V2 ; Bp,a , Bq,b ] .
(b) The following estimates are satisfied
σ1I [L; Bp,a , Nb ] < σ1I [L; Bp,a , Bq,b ] < σ1I [L; Bp,a , Db ] , (p, q) ∈ [0, ∞] × (0, ∞) ,
σ1I [L; Na , Bq,b ] < σ1I [L; Bp,a , Bq,b ] < σ1I [L; Da , Bq,b ] , (p, q) ∈ (0, ∞) × [0, ∞] .
(c) If a < b < c, then
(a,c)
σ1
(a,c)
σ1
(a,b)
[L; Bp,a , Db ] ,
(b,c)
[L; Db , Bq,c ] .
[L; Bp,a , Bq,c ] < σ1
[L; Bp,a , Bq,c ] < σ1
Proof. (a) Let ϕ be the pr. eig. associated with σ1I [L + V1 ; Bp,a , Bq,b ]. Then,
(L + V2 − σ1I [L + V1 ; Bp,a , Bq,b ]) ϕ = (V2 − V1 ) ϕ > 0 in I ,
Bp,a ϕ = Bq,b ϕ = 0 ,
INVERTIBILITY OF LINEAR NON-COOPERATIVE ELLIPTIC SYSTEMS
7
and so ϕ is a s.p. supersolution of (L + V2 − σ1I [L + V1 ; Bp,a , Bq,b ]; Bp,a , Bq,b ). Thus,
it follows from Theorem 2.1 that
σ1I [L + V2 − σ1I [L + V1 ; Bp,a , Bq,b ]; Bp,a , Bq,b ] > 0 .
This completes the proof of Part (a).
(b) Pick up (p, q) ∈ [0, ∞] × (0, ∞) and denote by ψ to the principal eigenfunction
associated with σ1I [L; Bp,a , Nb ]. It is clear that ψ(b) ≥ 0. Since ψ 0 (b) = 0, if ψ(b) =
0, then the uniqueness of the Cauchy problem associated with L − σ1I [L; Bp,a , Nb ]
implies that ψ ≡ 0. Thus, ψ(b) > 0 and hence Bq,b ψ = ψ(b) > 0. So, ψ is a
strict positive supersolution of (L − σ1I [L; Bp,a , Nb ]; Bp,a , Bq,b ) Therefore, we find
from Theorem 2.1 that
σ1I [L − σ1I [L; Bp,a , Nb ]; Bp,a , Bq,b ] > 0 .
This completes the proof of the first inequality in the statement of Part (b). To
prove the second one we denote by ϕ the principal eigenfunction associated with
σ1I [L; Bp,a , Bq,b ]. Then, ϕ(b) + q ϕ0 (b) = 0. Moreover, ϕ(b) ≥ 0. Thus, if we assume
that ϕ(b) = 0, then ϕ0 (b) = 0 and the uniqueness of the Cauchy problem implies
ϕ ≡ 0, which is not possible. Thus, ϕ(b) = Db ϕ > 0. Hence, ϕ is a strict positive
supersolution of (L − σ1I [L; Bp,a , Bq,b ]; Bp,a , Db ) and therefore Theorem 2.1 implies
σ1I [L − σ1I [L; Bp,a , Bq,b ]; Bp,a , Db ] > 0 .
This completes the proof of the second inequality of Part (b). Some similar arguments complete the proof of Part (b).
(a,c)
(c) Let ψ denote the pr. eig. associated with σ1 [L; Bp,a , Bq,c ]. Then, Db ψ =
ψ(b) > 0 and so the function ψ is a strict positive supersolution of the operator
(a,c)
(L − σ1 [L; Bp,a , Bq,b ]; Bp,a , Db ). Due to Theorem 2.1 we find that
(a,b)
σ1
(a,c)
[L − σ1
[L; Bp,a , Bq,b ]; Bp,a , Db ] > 0 .
This completes the proof of the first inequality. This argument can be adapted to
complete the proof of the theorem. 3. Invertibility of linear non-cooperative elliptic systems
In this section we show that under conditions (A1) and (A2) (u, v) = (0, 0) is the
unique solution of
L1 u = −α(x) v
in I := (a, b) ,
(3.1a)
L2 v = β(x) u
Bp1 ,a u = Bq1 ,b u = Bp2 ,a v = Bq2 ,b v = 0 .
To precise, the following result holds.
(3.1b)
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JULIÁN LÓPEZ-GÓMEZ
AND
ROSA M. PARDO
Theorem 3.1. Suppose (A1) and (A2) of Section 1. Then, (u, v) = (0, 0) is the
unique solution of (3.1) in (C[a, b])2 .
Remark 3.2. Any solution (u, v) ∈ (C[a, b])2 of (3.1a) is a classical solution, i.e.
(u, v) ∈ (C 2 [a, b])2 .
To prove Theorem 3.1 we use two general properties of the solutions of (3.1a).
Lemma 3.3. Assume that (3.1a) has a solution (u, v) with u 6= 0 and v 6= 0. Then
the set of zeros of u and v is discrete in [a, b]. In particular, the sets {x ∈ (a, b) :
u(x) = 0 } and {x ∈ (a, b) : v(x) = 0 } have empty interior.
Proof. Let (u, v) be a solution of (3.1a) with u 6= 0 and v 6= 0. First, we show that
the zeros of u(x) do not accumulate at some x0 ∈ [a, b]. We argue by contradiction.
Assume that the zeros of u accumulate at some x0 ∈ [a, b]. Then u(x0 ) = u0 (x0 ) = 0
and necessarily either v(x0 ) 6= 0, or v 0 (x0 ) 6= 0, because of the uniqueness of the
Cauchy problem at x0 for the first order system associated with (3.1a). In any case
v(x) has constant sign in (x0 − ε, x0 ) ∩ [a, b] and (x0 , x0 + ε) ∩ [a, b]; not necessarily
the same if v(x0 ) = 0. Moreover, there exists a sequence (xm ) such that u(xm ) = 0
and either xm ↓ x0 or xm ↑ x0 . Assume for instance that xm ↓ x0 and choose m
sufficiently large so that xm < x0 + ε and
(x0 ,xm )
σ1
[L1 ; Dx0 , Dxm ] > 0 .
Note that in fact the principal eigenvalue goes to infinity as the length of the interval
goes to zero. It is clear that α(x) v(x) does not change of sign in (x0 , xm ). Therefore,
it follows from the first equation of (3.1a) and the maximum principle that u does
not vanish on (x0 , xm ). This is impossible. Similarly, we get also a contradiction if
xm ↑ x0 . The same argument shows that the zeros of v do not accumulate. Lemma 3.4. Let (u, v) be a non-trivial solution of (3.1a) such that u(z 1 ) = u(z2 ) =
0, for some z1 , z2 ∈ [a, b], z1 < z2 , u < 0 in (z1 , z2 ) and v(z1 ) ≤ 0. Then,
v(z2 ) > 0.
Remark 3.5. Changing the signs of u and v it is clear that if (u, v) is a non-trivial
solution of (3.1a) such that u(z1 ) = u(z2 ) = 0, u > 0 in (z1 , z2 ) and v(z1 ) ≥ 0,
then v(z2 ) < 0.
Proof of Lemma 3.4. By (A1) and Theorem 2.1, σ1I [Lj ; Bpj ,a , Bqj ,b ] > 0. Thus, we
find from Theorem 2.3(b) that
(z1 ,z2 )
σ1
(a,b)
[L1 ; Dz1 , Dz2 ] ≥ σ1
[L1 ; Da , Db ] > 0 .
So, if we assume that v ≤ 0 in (z1 , z2 ), then
L1 u = −α v ≥ 0
in (z1 , z2 ) ,
u(z1 ) = u(z2 ) = 0 ,
and hence we find from Theorem 2.1 that u ≥ 0, which is a contradiction. Therefore,
v changes of sign in (z1 , z2 ). Note that in particular v 6= 0. On the other hand,
L2 v = β u ≤ 0
in (z1 , z2 ) ,
v(z1 ) ≤ 0 ,
INVERTIBILITY OF LINEAR NON-COOPERATIVE ELLIPTIC SYSTEMS
9
(z ,z )
and since σ1 1 2 [L2 ; Dz1 , Dz2 ] > 0, if we assume that v(z2 ) ≤ 0, then we get from
Theorem 2.1 that v ≤ 0 in (z1 , z2 ), which is not possible. Therefore, v(z2 ) > 0.
This completes the proof. Proof of Theorem 3.1. We will argue by contradiction. Assume that (3.1) possesses
a solution (u, v) 6= (0, 0). Then, u 6= 0 and v 6= 0. Indeed, if we assume that v = 0,
then
L1 u = 0 in (a, b) ,
D a u = Db u = 0 ,
and since σ1I [L1 ; Da , Db ] > 0, we obtain u = 0. Similarly, if u = 0 then we find
from the second equation of (3.1a) that v = 0. Therefore, u 6= 0 and v 6= 0. We
claim that u and v change of sign. On the contrary, assume that u > 0. Then,
L2 v = βu ≥ 0 in (a, b) and so we find from the maximum principle that v ≥ 0.
Thus, L1 u = −αv ≤ 0 and hence u ≤ 0, which contradicts u > 0. Therefore, u
changes of sign. By symmetry, v changes of sign as well.
By Lemmas 3.3, 3.4 the set of zeros of u where it changes of sign is discrete in [a, b].
Let
a = z 0 < z1 < · · · < z N = b
be such set. By changing the signs of u and v, if necessary, we can assume that
u>0
in
(z2j , z2j+1 )
0≤j≤N.
u<0
in (z2j−1 , z2j )
In particular,
u>0
in (a, z1 )
Due to Theorem 2.3(c) we have
(a,z1 )
σ1
(a,b)
[L2 ; Bp2 ,a , Dz1 ] > σ1
[L2 ; Bp2 ,a , Bq2 ,b ] > 0 .
(3.2)
Moreover, Bp2 ,a v = 0. We claim that v(z1 ) < 0. On the contrary, assume that
Dz1 v ≥ 0. Then, since
L2 v = β u ≥ 0
Bp2 ,a v = 0 ,
in (a, z1 ) ,
D z1 v ≥ 0 ,
it follows from (3.2) and Theorem 2.1 that v > 0 in (a, z1 ). Thus,
L1 u = −α v < 0 in (a, z1 ) ,
Bp1 ,a u = 0 ,
D z1 u = 0
(3.3)
On the other hand, it follows from Theorem 2.3(c) that
(a,z1 )
σ1
(a,b)
[L1 ; Bp1 ,a , Dz1 ] > σ1
[L1 ; Bp1 ,a , Bq1 ,b ] > 0 ,
and so we can apply Theorem 2.1 to (3.3) to get u < 0 in (a, z1 ). This contradiction
shows that v(z1 ) < 0. Now, thanks to Lemma 3.3 and 3.4, some of the following
situations occurs. Either
v(zN ) < 0
and
u<0
in
(zN , b) ,
10
JULIÁN LÓPEZ-GÓMEZ
AND
ROSA M. PARDO
or
v(zN ) > 0
and
u>0
in
(zN , b) .
(3.4)
Without loss of generality we can assume that (3.4) occurs. If necessary, we can
change the signs of u and v. Then,
L2 v = β u > 0 in (zN , b) ,
(zN ,b)
and since σ1
v(zN ) > 0 ,
Bq2 ,b v = 0 ,
[L2 ; DzN , Bq2 ,b ] > 0, we obtain that v > 0 in (zN , b). Hence,
L1 u = −α v ≤ 0 in (zN , b) ,
u(zN ) = 0 ,
Bq1 ,b u = 0 ,
(z ,b)
and since σ1 N [L1 ; DzN , Bq1 ,b ] > 0, we find that u < 0 in (zN , b), which is impossible. This contradiction completes the proof of the theorem. 4. Uniqueness of coexistence states for predator-prey elliptic systems
In this section we consider the following nonlinear elliptic boundary value problem
E1 u = `(x) u − A(x) u2 − B(x) u v
in I := (a, b) ,
E2 v = m(x) v + C(x) u v − D(x) v
(4.1)
2
Bp1 ,a u = Bq1 ,b u = Bp2 ,a v = Bq2 ,b v = 0 ,
where Ej , j = 1 , 2, are two second order differential operators of the form
Ej = −αj (x)
d
d2
+ βj (x)
,
2
dx
dx
j = 1, 2,
with αj , βj ∈ C[a, b], αj (x) > 0 for all x ∈ [a, b], and we assume ` , m , A , B , C , D ∈
C[a, b] to satisfy the following hypothesis:
(H1) A > 0, D > 0, i.e. there exist x1 , x2 ∈ (a, b) such that A(x1 ) > 0, D(x2 ) > 0.
(H2) B > 0, C > 0, and { x ∈ I : B(x) = 0 }, { x ∈ I : C(x) = 0 } have empty
interior.
The boundary operators Bpj ,a and Bqj ,b are of the same type as those of (1.3). Note
that (H1) does not exclude the possibility that A(x), or D(x), vanishes on some
subinterval of I. Our main result shows that (4.1) possesses at most a coexistence
state. By a coexistence state we mean a solution couple (u0 , v0 ) with both component positive. Due to the strong maximum principle, if (u0 , v0 ) is a coexistence
state, then u0 (x) > 0 and v0 (x) > 0 for all x ∈ I. Moreover, u00 (a) > 0 if p1 = 0
and u00 (b) < 0 if q1 = 0. Similarly, v00 (a) > 0 if p2 = 0 and v00 (b) < 0 if q2 = 0.
Theorem 4.1. Suppose (H1) and (H2). Then, (4.1) admits at most a coexistence
state.
Proof. Assume that (4.1) admits a coexistence state, say (u1 , v1 ). Let (u2 , v2 ) be
any other coexistence state of (4.1). Set
U := u1 − u2 ,
V := v1 − v2 ,
INVERTIBILITY OF LINEAR NON-COOPERATIVE ELLIPTIC SYSTEMS
L1 := E1 + A (u1 + u2 ) + B v1 − ` ,
11
L2 := E2 + D (v1 + v2 ) − C u1 − m .
After some straightforward manipulations it can be easily seen that
L1 U = −B(x) u2 (x) V , L2 V = C(x) v2 (x) U
in I := (a, b) ,
Bp1 ,a U = Bq1 ,b U = Bp2 ,a V = Bq2 ,b V = 0 ,
(4.2a)
(4.2b)
The fact that (u1 , v1 ) is a solution of (4.1) gives
(E1 + A u1 + B v1 − `) u1 = 0 ,
(E2 + D v1 − C u1 − m) v1 = 0 .
Thus, since u1 > 0, v1 > 0, it follows from the uniqueness of the principal eigenvalue
that
0 = σ1I [E1 + A u1 + B v1 − `; Bp1 ,a , Bq1 ,b ] = σ1I [E2 + D v1 − C u1 − m; Bp2 ,a , Bq2 ,b ] .
Therefore, Theorem 2.3(a) implies
σ1I [L1 ; Bp1 ,a , Bq1 ,b ] > 0 ,
σ1I [L2 ; Bp2 ,a , Bq2 ,b ] > 0 .
Finally, Theorem 3.1 shows that (U, V ) = (0, 0) is the unique solution of (4.3). This
completes the proof. Now, we use the theory developed in [4] and [6] to characterize the existence of
a coexistence state for (4.1) in a case not previously considered in the literature.
We shall assume that there exists I1 := (aA , bA ) such that I 1 ⊂ (a, b), A(x) = 0
for x ∈ [aA , bA ] and A(x) > 0 otherwise. Similarly, we assume that there exists
I2 := (aD , bD ) such that I 2 ⊂ (a, b), D(x) = 0 for x ∈ [aD , bD ] and D(x) > 0 in
its complement. Under these assumptions, thanks to the results of [4] and [6] the
positive solutions of (4.1) do not admit a priori bounds. To precise, given a second
order differential operator E of the same type as Ej , p, q ∈ [0, ∞], and n , h ∈ C[a, b],
h > 0, we consider the following nonlinear boundary value problem
E w = n(x) w − h(x) w 2
in I ,
Bp,a w = Bq,b w = 0 .
(4.3)
For this problem, the following result is known, [4].
Theorem 4.2. Suppose that there exist ah < bh such that [ah , bh ] ⊂ (a, b) and that
h(x) = 0 if and only if x ∈ [ah , bh ]. Then, (4.3) possesses a positive solution if, and
only if,
(a,b)
(a ,b )
σ1 [E − n; Bp,a , Bq,b ] < 0 < σ1 h h [E − n; Dah , Dbh ] .
Let θ[E−n;h;p;q] denote the maximal non-negative solution of (4.3). Due to Theorem
(a ,b )
(a,b)
4.2, θ[E−n;h;p;q] = 0 if σ1 [E −n; Bp,a , Bq,b ] ≥ 0 or σ1 h h [E −n; Dah , Dbh ] ≤ 0, and
θ[E−n;h;p,q] 0 in any other case. By we mean that it lies in the interior of the
cone of positive functions of C01 [a, b]. If h(x) satisfies the assumptions of Theorem
4.2 and θ[E−n;h;p;q] 0 then θ[E−n;h;p;q] is a global attractor for the positive solutions
of the parabolic problem associated with (4.3), [4]. Moreover,
(ah ,bh )
σ1
lim
kθ[E−n;h;p;q]k∞ = ∞ ,
[E−n;Dah ,Dbh ]↓0
and therefore, the positive solutions of (4.1) do not admit a priori bounds. As all
the results found for this type of systems dealt with the case where a priori bounds
are available, the following characterization of the existence is new.
12
JULIÁN LÓPEZ-GÓMEZ
AND
ROSA M. PARDO
Theorem 4.3. Suppose that
(aA ,bA )
[E1 − `; DaA , DbA ] > 0 ,
(4.4a)
[E2 − C θ[E1 −`;A;p1 ;q1 ] − m; DaD , DbD ] > 0 .
(4.4b)
σ1
(aD ,bD )
σ1
Then, (4.1) possesses a coexistence state if, and only if,
(a,b)
[E1 + B θ[E2 −m;D;p2 ;q2 ] − `; Bp1 ,a , Bq1 ,b ] < 0 ,
(4.5a)
(a,b)
[E2 − C θ[E1 −`;A;p1 ;q1 ] − m; Bp2 ,a , Bq2 ,b ] < 0 ,
(4.5b)
σ1
σ1
i.e. if any of the semi-trivial positive solutions is linearly unstable.
Remark 4.4. Since θ[E2 −m;D;p2 ,q2 ] ≥ 0, (4.5a) implies
(a,b)
σ1
[E1 − `; Bp1 ,a , Bq1 ,b ] < 0 .
(4.6)
If (4.6) holds, then (4.4a) and Theorem 4.2 give θ[E1 −`;A;p1 ;q1 ] 0.
Remark 4.5. It can be easily seen that (4.4) holds provided bA − aA and aD − bD
are sufficiently small. The fact that (4.5) is equivalent to the instability of any of
the semi-trivial states (those with one component vanishing) follows by a direct
calculation from the strong maximum principle. Note that the linearization of
(4.1) at any semi-trivial state possesses a triangular structure and so estimating its
spectrum is reduced to estimating the spectrum of a boundary value problem for a
single equation.
Proof. Suppose (4.4). Let (u0 , v0 ) be a coexistence state of (4.1). Then,
E1 u0 = `(x) u0 − A(x) u20 − B(x) u0 v0 < `(x) u0 − A(x) u20 ,
and hence, u0 > 0 is a strict subsolution of
E1 w = `(x) w − A(x) w 2
in I ,
Bp1 ,a w = Bq1 ,b w = 0 .
By (4.4a) we can use the method of sub and supersolutions to show that this implies
0 < u0 < θ[E1 −`;A;p1 ;q1 ] .
(4.7)
Moreover, due to Theorem 4.2 we get (4.6). Thus, (4.6) is necessary for the existence
of a solution with the first component positive. Now, going back to the second
equation of (4.1) yields
E2 v0 = m(x) v0 + C(x) u0 v0 − D(x) v02 < m(x) v0 + C(x) θ[E1 −`;A;p1 ;q1 ] v0 − D(x) v02 ,
and hence, v0 > 0 is a strict subsolution of
(E2 − C(x) θ[E1 −`;A;p1 ;q1 ] ) w = m(x) w − D(x) w 2 in I , Bp2 ,a w = Bq2 ,b w = 0 .
INVERTIBILITY OF LINEAR NON-COOPERATIVE ELLIPTIC SYSTEMS
13
Therefore, the method of sub and supersolutions implies
0 < v0 < θ[E2 −C θ[E1 −`;A;p1 ;q1 ] −m;D;p2 ;q2 ] .
(4.8)
In particular, it follows from Theorem 4.2 that
(a,b)
σ1
[E2 − C θ[E1 −`;A;p1 ;q1 ] − m; Bp2 ,a , Bq2 ,b ] < 0 .
(aD ,bD )
Similarly, since σ1
[E2 − m; DaD , DbD ] > 0, the second equation of (4.1) gives
v0 > θ[E2 −m;D;p2 ;q2 ] .
Going back to the first equation and substituting the previous estimate gives
E1 u0 = `(x) u0 − A(x) u20 − B(x) u0 v0 < `(x) u0 − A(x) u20 − B(x) θ[E2 −m;D;p2 ;q2 ] u0 .
So, u0 > 0 is a strict subsolution of
(E1 + B(x) θ[E2 −m;D;p2 ;q2 ] ) w = `(x) w − A(x) w 2 in I , Bp1 ,a w = Bq1 ,b w = 0 ,
and therefore, it follows from the method of sub and supersolutions that
0 < u0 < θ[E1 +B θ[E2 −m;D;p2 ;q2 ] −`;A;p1 ;q1 ] .
Due to Theorem 4.2, this relation gives
(a,b)
σ1
[E1 + B θ[E2 −m;D;p2 ;q2 ] − `; Bp1 ,a , Bq1 ,b ] < 0 .
This shows that (4.5) is necessary for the existence of a coexistence state.
We now show that under condition (4.4), (4.5) is sufficient for the existence of a
coexistence state. To show the sufficiency of this condition we can apply Theorem
4.1 of [6]. Thought this result was obtained for Dirichlet boundary conditions, it can
be easily adapted to deal with general boundary conditions. We will omit the details
here. To apply this theorem is useful regarding to (4.1) as the particularization at
(λ, µ) = (0, 0) of
(E1 − `) u = λ u − A(x) u2 − B(x) u v
in I := (a, b) ,
(E2 − m) v = µ v + C(x) u v − D(x) v
2
Bp1 ,a u = Bq1 ,b u = Bp2 ,a v = Bq2 ,b v = 0 ,
where (λ, µ) is a two-dimensional continuation parameter. Observe that any coexistence state of (4.1) must satisfy (4.7) and (4.8). Moreover, due to (4.4), it
follows from the theory developed in [4] that the functions on the right hand sides
14
JULIÁN LÓPEZ-GÓMEZ
AND
ROSA M. PARDO
of these inequalities are bounded above uniformly on compact subsets of the parameter space. Therefore, under condition (4.4) we have a priori bounds for the
coexistence states. This completes the proof. Finally, a remark about the stability of the coexistence states of (4.1). Let (u 0 , v0 )
be a coexistence state of (4.1). Then, the stability of (u0 , v0 ) as a solution of the
parabolic system associated with (4.1) is given by the real parts of the eigenvalues
of
(E1 + 2 A u0 + B v0 ) u = −B(x) u0 (x) v + τ u
in I := (a, b) ,
(E2 + 2 D v0 − C u0 ) v = C(x) v0 (x) u + τ v
Bp1 ,a u = Bq1 ,b u = Bp2 ,a v = Bq2 ,b v = 0 .
If Re τ > 0 for any eigenvalue τ , then (u0 , v0 ) is asymptotically stable. If it
possesses some eigenvalue τ with Re τ < 0, then (u0 , v0 ) is unstable. Since (u0 , v0 )
is a coexistence state of (4.1) we have
σ1I [E1 + A u0 + B v0 − `; Bp1 ,a , Bq1 ,b ] = 0 , σ1I [E2 + D v0 − C u0 − m; Bp2 ,a , Bq2 ,b ] = 0 .
Thus, it follow from Theorem 2.3(a) that
σ1I [E1 + 2 A u0 + B v0 − `; Bp1 ,a , Bq1 ,b ] > 0 ,
σ1I [E2 + 2 D v0 − C u0 − m; Bp2 ,a , Bq2 ,b ] > 0 ,
and hence we find from Theorem 3.1 that τ = 0 can not be an eigenvalue of the
linearization of (4.1) at (u0 , v0 ). Therefore, if (u0 , v0 ) loses its stability is because
some sort of Hopf bifurcation occurs.
Acknowledgements: This work was partially supported by the European Network Reaction Diffusion Equations under contract ERBCHRX CT93-0409, and by
the Spanish Gobernment under grant DGICYT PB93-0465. We sincerely thank to
the referee for a useful comment related to Theorem 4.3 and to Prof. H Amann for
suggesting us the use of different boundary conditions for each of the componentes
of the system.
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