Bounded Components of Positive Solutions of Nonlinear Abstract Equations
In this work a general class of nonlinear abstract equations satisfying a generalized strong maximum principle is considered in order to show that any bounded component of positive solutions bifurcating from the curve of trivial states (λ, u) = (λ, 0) at a nonlinear eigenvalue λ = λ₀ must meet the c...
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nasplib_isofts_kiev_ua-123456789-1245802025-02-10T00:29:05Z Bounded Components of Positive Solutions of Nonlinear Abstract Equations Cano-Casanova, S. Lopez-Gomez, J. Molina-Meyer, M. In this work a general class of nonlinear abstract equations satisfying a generalized strong maximum principle is considered in order to show that any bounded component of positive solutions bifurcating from the curve of trivial states (λ, u) = (λ, 0) at a nonlinear eigenvalue λ = λ₀ must meet the curve of trivial states (λ, 0) at another singular value λ₁ ≠ λ₀. Since the unilateral theorems of P. H. Rabinowitz [13, Theorems 1.27 and 1.40] are not true as originally stated (c.f. the counterexample of E. N. Dancer [6]), in order to get our main result the unilateral theorem of J. Lopez-Gomez [11, Theorem 6.4.3] is required. 2005 Article Bounded Components of Positive Solutions of Nonlinear Abstract Equations / S. Cano-Casanova, J. Lopez-Gomez, M. Molina-Meyer // Український математичний вісник. — 2005. — Т. 2, № 1. — С. 38-51. — Бібліогр.: 13 назв. — англ. 1810-3200 2000 MSC. 34A34, 34C23, 35B32, 35B50, 35J25. https://nasplib.isofts.kiev.ua/handle/123456789/124580 en Український математичний вісник application/pdf Інститут прикладної математики і механіки НАН України |
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In this work a general class of nonlinear abstract equations satisfying a generalized strong maximum principle is considered in order to show that any bounded component of positive solutions bifurcating from the curve of trivial states (λ, u) = (λ, 0) at a nonlinear eigenvalue λ = λ₀ must meet the curve of trivial states (λ, 0) at another singular value λ₁ ≠ λ₀. Since the unilateral theorems of P. H. Rabinowitz [13, Theorems 1.27 and 1.40] are not true as originally stated (c.f. the counterexample of E. N. Dancer [6]), in order to get our main result the unilateral theorem of J. Lopez-Gomez [11, Theorem 6.4.3] is required. |
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Cano-Casanova, S. Lopez-Gomez, J. Molina-Meyer, M. Bounded Components of Positive Solutions of Nonlinear Abstract Equations Український математичний вісник |
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Cano-Casanova, S. Lopez-Gomez, J. Molina-Meyer, M. |
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Bounded Components of Positive Solutions of Nonlinear Abstract Equations |
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Bounded Components of Positive Solutions of Nonlinear Abstract Equations |
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Bounded Components of Positive Solutions of Nonlinear Abstract Equations |
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Bounded Components of Positive Solutions of Nonlinear Abstract Equations |
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Bounded Components of Positive Solutions of Nonlinear Abstract Equations |
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bounded components of positive solutions of nonlinear abstract equations |
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Інститут прикладної математики і механіки НАН України |
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Bounded Components of Positive Solutions of Nonlinear Abstract Equations / S. Cano-Casanova, J. Lopez-Gomez, M. Molina-Meyer // Український математичний вісник. — 2005. — Т. 2, № 1. — С. 38-51. — Бібліогр.: 13 назв. — англ. |
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Український математичний вісник |
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AT canocasanovas boundedcomponentsofpositivesolutionsofnonlinearabstractequations AT lopezgomezj boundedcomponentsofpositivesolutionsofnonlinearabstractequations AT molinameyerm boundedcomponentsofpositivesolutionsofnonlinearabstractequations |
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2025-12-02T04:17:41Z |
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1850368654387970048 |
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Український математичний вiсник
Том 2 (2005), № 1, 38 – 51
Bounded Components of Positive Solutions of
Nonlinear Abstract Equations
Santiago Cano-Casanova, Julián López-Gómez
and Marcela Molina-Meyer
(Presented by I. V. Skrypnik)
Abstract. In this work a general class of nonlinear abstract equations
satisfying a generalized strong maximum principle is considered in order
to show that any bounded component of positive solutions bifurcating
from the curve of trivial states (λ, u) = (λ, 0) at a nonlinear eigenvalue
λ = λ0 must meet the curve of trivial states (λ, 0) at another singu-
lar value λ1 6= λ0. Since the unilateral theorems of P. H. Rabinowitz
[13, Theorems 1.27 and 1.40] are not true as originally stated (c.f. the
counterexample of E. N. Dancer [6]), in order to get our main result the
unilateral theorem of J. López-Gómez [11, Theorem 6.4.3] is required.
2000 MSC. 34A34, 34C23, 35B32, 35B50, 35J25.
Key words and phrases. Positive Solutions, Compact solution com-
ponents, Nonlinear abstract equations, Bifurcation theory.
1. Introduction
Throughout this work U stands for an ordered real Banach space
whose positive cone, P , is normal and it has nonempty interior, and we
consider the nonlinear abstract equation
F(λ, u) := L(λ)u+ R(λ, u) = 0 , (λ, u) ∈ X := R × U , (1.1)
where
(HL) The family K(λ) := IU − L(λ) ∈ L(U), λ ∈ R, is compact and
real analytic, and L(λ̂) is a linear topological isomorphism for some
λ̂ ∈ R, where IU stands for the identity map of U and L(U) is the
space of linear continuous endomorphisms of U .
Received 13.01.2004
The work has been partially supported by grants BFM2000-0797 and REN2003-00707
of the Spanish Ministry of Science and Technology.
ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України
S. Cano-Casanova, J. López-Gómez, M. Molina-Meyer 39
(HR) R ∈ C(R×U ;U) is compact on bounded sets and limu→0
R(λ,u)
‖u‖ = 0
uniformly on compact intervals of R .
(HP) The solutions of (1.1) satisfy the strong maximum principle in the
sense that
(λ, u) ∈ R × (P \ {0}) and F(λ, u) = 0 imply u ∈ IntP ,
where IntP stands for the interior of the cone P .
Subsequently, given u1, u2 ∈ U , we write u1 > u2 if u1−u2 ∈ P \{0}, and
u1 ≫ u2 if u1 − u2 ∈ IntP . Also, it will be said that (λ, u) is a positive
solution of (1.1), if (λ, u) is a solution of (1.1) with u > 0. Thanks to
Assumption (HP), any positive solution (λ, u) of (1.1) must be strongly
positive, in the sense that u≫ 0.
Under Assumptions (HL) and (HR), F(λ, 0) = 0 for each λ ∈ R . The
main result of this paper concerns the bounded components of positive
solutions of (1.1) emanating from (λ, u) = (λ, 0) at a nonlinear eigenvalue
λ0 ∈ R with geometric multiplicity one. By a component of positive
solutions of (1.1) it is meant a maximal (for the inclusion) relatively
closed and connected subset of the set of positive solutions of (1.1) (in
R × IntP ). A value σ ∈ R is said to be an eigenvalue of the family L(λ)
if dimN [L(σ)] ≥ 1. The set of eigenvalues of L(λ) will be denoted by S.
Thanks to (HL), L(λ) is Fredholm of index zero for any λ ∈ R and, hence,
S provides us with the set of singular values of the family L(λ). In other
words, L(λ) is a linear topological isomorphism if λ ∈ R \ S. Moreover,
it follows from [11, Theorem 4.4.4] that S is discrete and that it consists
of algebraic eigenvalues of L, i.e., for any σ ∈ S there exist C > 0,
ε > 0 and ν ≥ 1 such that for any λ ∈ (σ − ε, σ + ε) \ {σ} the operator
L−1(λ) is well defined and ‖L−1(λ)‖L(U) ≤ C
|λ−σ|ν if 0 < |λ − σ| < ε.
Thus, thanks to the abstract spectral theory developed in [11, Chapter
4], the algebraic multiplicity χ[L; ·] : S → N introduced by J. Esquinas
and J. López-Gómez in [8] and [7] is well defined. Actually, χ[L;λ0] = 1
if λ0 ∈ S is a simple eigenvalue of L(λ) as discussed by M. G. Crandall
and P. H. Rabinowitz [4], i.e., if
dimN [L0] = 1 and L1(N [L0]) ⊕R[L0] = U , (1.2)
where L0 := L(λ0), L1 := dL
dλ (λ0) = −dK
dλ (λ0), and, for any T ∈ L(U),
N [T ] and R[T ] stand for the null space and the range of T . A value σ ∈ S
is said to be a nonlinear eigenvalue of L(λ) if (σ, 0) is a bifurcation point
of (1.1) from (λ, 0), λ ∈ R, for any R(λ, u) satisfying (HR). According to
[11, Theorem 4.3.4], χ[L;σ] ∈ 2N+1 if σ is a nonlinear eigenvalue of L(λ),
40 Bounded Components of Positive Solutions...
and, thanks to [11, Theorem 6.6.2], for any σ ∈ S there is η ∈ {−1, 1}
such that
Ind (0,K(λ)) = η sign (λ− σ)χ[L;σ] , λ ∼ σ , λ 6= σ ,
where Ind (0,K(λ)) is the local index of K(λ) at zero (the topological de-
gree of L(λ) in any open bounded set containing zero). Thus, Ind (0,K(λ))
changes as λ crosses σ if, and only if, χ[L;σ] ∈ 2N + 1. Consequently,
thanks to [11, Theorem 6.2.1], σ ∈ S is a nonlinear eigenvalue of L(λ) if
χ[L;σ] ∈ 2N + 1, and, therefore, σ ∈ S is a nonlinear eigenvalue of L(λ)
if, and only if, χ[L;σ] ∈ 2N + 1. The main result of this paper can be
stated as follows.
Theorem 1.1. Suppose λ0 ∈ S is a nonlinear eigenvalue of L(λ) such
that
N [L(λ0)] = span [ϕ0] , ϕ0 ∈ P \ {0} , (1.3)
and K(λ0) is strongly positive, i.e.,
K(λ0)(P \ {0}) ⊂ IntP . (1.4)
Then, there exists a component CP
λ0
of the set of positive solutions of
F(λ, u) = 0 emanating from (λ, u) = (λ, 0) at λ = λ0. Moreover, there
exists λ1 ∈ S \ {λ0} such that (λ1, 0) ∈ C̄P
λ0
if CP
λ0
is bounded in X :=
R × U .
Note that, thanks to (HL) and (1.4),
K(λ0)ϕ0 = ϕ0 ≫ 0 . (1.5)
Thus, the existence of CP
λ0
follows by adapting some of the unilateral
results of P. H. Rabinowitz [13]; in Section 2 complete details will be
provided.
The distribution of this paper is as follows. In Section 2 we show
the existence of CP
λ0
, in Section 3 we complete the proof of Theorem 1.1,
and in Section 4 we derive from Theorem 1.1 a celebrated unilateral the-
orem attributable to E. N. Dancer [5]. Finally, in Section 5 we give an
application of Theorem 1.1, and in Section 6 we construct an example
showing the necessity of condition χ[L;λ0] ∈ 2N + 1 for the validity of
Theorem 1.1. Throughout the remaining of this paper it will be assumed
that λ0 ∈ S satisfies all the requirements of Theorem 1.1.
S. Cano-Casanova, J. López-Gómez, M. Molina-Meyer 41
2. Unilateral Bifurcation. The Existence of CP
λ0
The set of non-trivial solutions of (1.1) is defined through
S := F−1(0) \ [(R \ S) × {0}] .
Note that (λ, u) ∈ S if either u 6= 0, or else u = 0 and λ ∈ S. Since
χ[L;λ0] ∈ 2N+1, thanks to [11, Corollary 6.3.2] there exists a component
of S, subsequently denoted by Cλ0 , such that (λ0, 0) ∈ Cλ0 . Subsequently,
we suppose that ϕ0 has been normalized so that
N [L(λ0)] = span [ϕ0] , ‖ϕ0‖ = 1 , ϕ0 > 0 . (2.1)
Thanks to (1.5), ϕ0 ≫ 0. Now, let Y be a closed subspace of U such
that U = N [L(λ0)]⊕Y . Thanks to Hahn-Banach’s theorem, there exists
ϕ∗
0 ∈ U ′ such that
Y = {u ∈ U : 〈ϕ∗
0, u〉 = 0 } , 〈ϕ∗
0, ϕ0〉 = 1 ,
where 〈·, ·〉 stands for the duality between U and U ′. Now, for each
η ∈ (0, 1) and sufficiently small ε > 0 we set
Qε,η := { (λ, u) ∈ X : |λ− λ0| < ε , |〈ϕ∗
0, u〉| > η ‖u‖ } .
Since the mapping u 7→ |〈ϕ∗
0, u〉| − η ‖u‖ is continuous, Qε,η is an open
subset of X consisting of the two disjoint components Q+
ε,η and Q−
ε,η
defined though
Q+
ε,η := { (λ, u) ∈ X : |λ− λ0| < ε , 〈ϕ∗
0, u〉 > η ‖u‖ } ,
Q−
ε,η := { (λ, u) ∈ X : |λ− λ0| < ε , 〈ϕ∗
0, u〉 < −η ‖u‖ } .
(2.2)
The following result collects the main consequences from [11, Theorem
6.2.1, Proposition 6.4.2], which are consequences from the reflection ar-
gument of P. H. Rabinowitz [13]. Subsequently, we denote by BR(x) the
open ball of radius R > 0 centered at x ∈ X.
Theorem 2.1. For each sufficiently small δ > 0,
Cλ0 ∩Bδ(λ0, 0) ⊂ Qε,η ∪ {(λ0, 0)}
and each of the sets S \
[
Q−
ε,η ∩Bδ(λ0, 0)
]
and S \
[
Q+
ε,η ∩Bδ(λ0, 0)
]
contains a component, denoted by C+
λ0
and C−
λ0
, respectively, such that
(λ0, 0) ∈ C+
λ0
∩ C−
λ0
and
Cλ0 ∩Bδ(λ0, 0) =
(
C+
λ0
∪ C−
λ0
)
∩Bδ(λ0, 0) . (2.3)
Moreover, for each (λ, u) ∈ (Cλ0 \ {(λ0, 0)}) ∩ Bδ(λ0, 0), there exists a
unique pair (s, y) ∈ R × Y such that u = sϕ0 + y and |s| > η ‖u‖.
Furthermore, λ = λ0 + o(1) and y = o(s) as s→ 0.
42 Bounded Components of Positive Solutions...
It should be noted that if (λ, u) ∈ C+
λ0
∩ Bδ(λ0, 0), u 6= 0, then u =
sϕ0 + y with s > η ‖u‖ > 0, and, hence, u
s = ϕ0 + y
s . Thus, since
lims→0
y
s = 0, for sufficiently small s > 0, u
s ∈ IntP and, consequently,
u ∈ IntP . Therefore, for any sufficiently small δ > 0, we have that
[
C+
λ0
\ {(λ0, 0)}
]
∩Bδ(λ0, 0) ⊂ R × IntP . (2.4)
This shows the existence of the component CP
λ0
of R × IntP containing
(λ0, 0) (cf. the statement of Theorem 1.1). Actually, CP
λ0
is the maximal
sub-continuum of C+
λ0
in R × IntP .
The following result, which is [11, Theorem 6.4.3], provides us with
an updated version of the unilateral theorem of P.H. Rabinowitz [13,
Theorem 1.27], which is not true as originally stated (cf. E. N. Dancer
[6]).
Theorem 2.2. For each ∗ ∈ {−,+}, the component C∗
λ0
satisfies some
of the following alternatives:
1. C∗
λ0
is unbounded in X.
2. There exists λ1 ∈ S \ {λ0} such that (λ1, 0) ∈ C∗
λ0
.
3. C∗
λ0
contains a point (λ, y) ∈ R × (Y \ {0}).
Thanks to (HL), (1.4) and (1.5), the theorem of M. G. Krein and M.
A. Rutman [10] (cf. H. Amann [1, Theorem 3.2]) as well), shows the
validity of the following result, which is needed to conclude the proof of
Theorem 1.1.
Theorem 2.3. Let Spr (K(λ0)) denote the spectral radius of K(λ0).
Then,
(a) Spr (K(λ0)) = 1 is an algebraically simple eigenvalue of K(λ0) and,
hence,
N [L0] = N [L2
0] = span [ϕ0] . (2.5)
Thus, 0 is an algebraically simple eigenvalue of L0, i.e.,
U = N [L0] ⊕R[L0] . (2.6)
Moreover, no other eigenvalue of K(λ0) admits a positive eigenvec-
tor.
(b) For every y ∈ IntP , the equation u − K(λ0)u = y cannot admit a
positive solution.
S. Cano-Casanova, J. López-Gómez, M. Molina-Meyer 43
Proof. As Spr (K(λ0)) is the unique eigenvalue of K(λ0) associated with it
there is a positive eigenvector, and 1 is an eigenvalue to the eigenfunction
ϕ0, we have that Spr (K(λ0)) = 1. Moreover, 1 is algebraically simple,
i.e.,
N [IU − K(λ0)] = N [(IU − K(λ0))
2] ,
and, hence, due to (HL), (2.5) holds. Now, since L0 is Fredholm of index
zero, to prove (2.6) it suffices to show that ϕ0 6∈ R[L0]. On the contrary,
suppose that ϕ0 ∈ R[L0]. Then, there exists u ∈ U \ N [L0] such that
L0u = ϕ0, and, hence, L2
0u = L0ϕ0 = 0. Thus, u ∈ N [L2
0] \N [L0], which
contradicts (2.5) and concludes the proof of (2.6). This completes the
proof of Part (a). Part (b) is an straightforward consequence from H.
Amann [1, Theorem 3.2].
As a consequence from (2.6) we can make the choice
Y = R[L0] , (2.7)
which will be maintained throughout the remaining of the proof of The-
orem 1.1.
3. Completion of the Proof of Theorem 1.1
Assume CP
λ0
is bounded. Since CP
λ0
⊂ C+
λ0
, some of the following
alternatives occurs. Either
CP
λ0
= C+
λ0
\ {(λ0, 0)} , (3.1)
or else
CP
λ0
is a proper subset of C+
λ0
\ {(λ0, 0)} . (3.2)
Suppose (3.1). Then, C̄P
λ0
= C+
λ0
satisfies some of the alternatives of
Theorem 2.2. Alternative 1 cannot be satisfied, since C̄P
λ0
is compact.
Suppose Alternative 3 occurs. Then, thanks to the choice (2.7), there
exists (λ, y) ∈ R × R[L0], y 6= 0, such that (λ, y) ∈ C̄P
λ0
⊂ R × P . Since
y 6= 0, necessarily y ∈ IntP , by (HP). Thus, there exists u ∈ U such that
L0u = u− K(λ0)u = y .
Since ϕ0 ∈ IntP , for each sufficiently large α > 0 we have that uα :=
u+ αϕ0 ≫ 0. Moreover,
uα − K(λ0)uα = y ,
44 Bounded Components of Positive Solutions...
because K(λ0)ϕ0 = ϕ0, which contradicts Theorem 2.3(b). Therefore,
Alternative 2 of Theorem 2.2 must be satisfied. This concludes the proof
of Theorem 1.1 in case (3.1).
Now, suppose (3.2). Then, since C+
λ0
∩Bδ(λ0, 0) =
[
CP
λ0
∩Bδ(λ0, 0)
]
∪
{(λ0, 0)} for each sufficiently small δ > 0, fixing one of these δ’s, there
exists (λ1, u) 6∈ Bδ(λ0, 0) such that
(λ1, u) ∈ C+
λ0
∩ (R × ∂P ) ∩ ∂CP
λ0
.
Let {(µn, un)}n≥1 be any subsequence of CP
λ0
such that limn→∞(µn, un) =
(λ1, u). Then,
F(λ1, u) = 0 and u ∈ P .
If u > 0, then, thanks to (HP), u ∈ IntP , which contradicts u ∈ ∂P .
Thus, u = 0, and, hence, (λ1, 0) ∈ C̄P
λ0
. Moreover, λ1 6= λ0, since
(λ1, u) = (λ1, 0) 6∈ Bδ(λ0, 0), which concludes the proof of Theorem 1.1.
Note that, thanks to [11, Lemma 6.1.2], λ1 ∈ S, since these are the
unique possible bifurcation values from (λ, 0) for (1.1).
4. Improving Dancer’s Unilateral Theorem
As an immediate consequence from Theorem 1.1, the following uni-
lateral result holds.
Theorem 4.1. Suppose λ0 ∈ S is a nonlinear eigenvalue of L(λ) satis-
fying (1.3) and (1.4), and, in addition, no other σ ∈ S \ {λ0} admits an
eigenfunction in P \ {0}. Then, the component CP
λ0
is unbounded in X.
Proof. On the contrary, suppose that CP
λ0
is bounded. Then, thanks
to Theorem 1.1, there exists λ1 ∈ S \ {λ0} such that (λ1, 0) ∈ C̄P
λ0
. Let
{(µn, un)}n≥1 be any sequence of CP
λ0
such that limn→∞(µn, un) = (λ1, 0)
and set vn := un
‖un‖ , n ≥ 1. Then,
vn = K(µn)vn − R(µn, un)
‖un‖
, n ≥ 1 ,
and, hence, by (HL) and (HR), there exists a subsequence of {vn}n≥1,
again labeled by n, such that limn→∞ vn = ψ. Necessarily ψ > 0. More-
over, passing to the limit as n → ∞ gives ψ = K(λ1)ψ, or, equivalently,
L(λ1)ψ = 0, which is impossible, since we are assuming that λ0 is the
unique element of S to a positive eigenvector. This contradiction shows
that CP
λ0
is unbounded and concludes the proof.
S. Cano-Casanova, J. López-Gómez, M. Molina-Meyer 45
In the special case when K(λ) = λK, λ ∈ R, for some linear strongly
positive compact operator K ∈ L(U), necessarily λ0 := 1
Spr K is the
unique element of S associated with it there is a positive eigenvector, and,
therefore, thanks to Theorem 4.1, CP
λ0
must be unbounded. Consequently,
Theorem 4.1 is a substantial improvement of [11, Theorem 6.5.5], which
is a well known result attributable to E. N. Dancer [5].
5. An Application of Theorem 1.1
In this section we consider the semi-linear weighted boundary value
problem {
Eu = λW (x)u− a(x)ur in Ω ,
u = 0 on ∂Ω ,
(5.1)
where Ω is a bounded domain of RN with boundary ∂Ω of class C2+ν for
some ν ∈ (0, 1), r ∈ (1,∞), λ ∈ R is regarded as a bifurcation parameter,
E is a second order uniformly elliptic operator of the form
E := −
N∑
i,j=1
αij
∂2
∂xi∂xj
+
N∑
i=1
αi
∂
∂xi
+ α0
with αij = αji ∈ Cν(Ω̄), αi, α0 ∈ Cν(Ω̄), 1 ≤ i, j ≤ N , and
(Ha) a ∈ Cν(Ω̄) and, setting
a+ := max{a, 0}, a− := a+ − a,
Ω0
a+ := Ω \ supp a+, Ω0
a− := Ω \ supp a−,
Ω0
a+ and Ω0
a− are two proper open subsets of Ω of class C2+ν with
a finite number of well separated components. Moreover, either
N ∈ {1, 2}, or else N ≥ 3 and for some constant γ > 0 the following
is satisfied
[dist (·, ∂Ω0
a−)]−γa− ∈ C
(
supp a−, (0,∞)
)
,
r < max
{
N + 2
N − 2
,
N + 1 + γ
N − 1
}
.
(Hw) W ∈ Cν(Ω̄) changes of sign in Ω0
a+ and
max
λ∈R
σ[E − λW ; Ω] > 0 , (5.2)
where, for any elliptic operator L in a bounded domain D, σ[L;D]
stands for the principal eigenvalue of L in D under homogeneous
Dirichlet boundary conditions.
46 Bounded Components of Positive Solutions...
Thanks to (5.2), for each D ∈
{
Ω0
a+ ,Ω
}
the weighted boundary value
problem {
Eϕ = λWϕ in D ,
ϕ = 0 on ∂D ,
(5.3)
possesses two principal eigenvalues, λD
1 < λD
2 . By a principal eigenvalue
of (5.3) it is meant a value of λ for which (5.3) possesses an eigenfunction
ϕ > 0. It should be noted that, necessarily, σ[E − λD
j W ;D] = 0 for each
(D, j) ∈
{
Ω0
a+ ,Ω
}
× {1, 2}. Moreover, by the monotonicity of σ[·;D]
with respect to D, λ
Ω0
a+
1 < λΩ
1 < λΩ
2 < λ
Ω0
a+
2 , and, thanks to a celebrated
result by P. Hess and T. Kato [9], setting Σ(λ) := σ[E − λW ; Ω], λ ∈ R,
one has that Σ′(λΩ
1 ) > 0 and Σ′(λΩ
2 ) < 0, where ′ = d
dλ . Set λ0 := λΩ
1
and denote by ϕ0 the principal eigenfunction associated to Σ(λ0) = 0,
normalized so that ‖ϕ0‖ = 1. Then, since Σ(λ) is a simple eigenvalue,
there is an analytic mapping λ 7→ ϕ(λ) ∈ C2+ν
0 (Ω̄) such that ϕ(λ0) = ϕ0
and
(E − λW )ϕ(λ) = Σ(λ)ϕ(λ) .
Now, differentiating with respect to λ and particularizing at λ = λ0 gives
(E − λ0W )ϕ′(λ0) = Wϕ0 + Σ′(λ0)ϕ0
and, hence, Σ′(λ0) = −〈ϕ∗
0,Wϕ0〉, where N [E∗ − λ0W ] = span[ϕ∗
0] with
〈ϕ∗
0, ϕ0〉 = 1. Therefore, since Σ′(λ0) > 0, we find that
Wϕ0 6∈ R[E − λ0W ] . (5.4)
Now, suppose u is a positive solution of (5.1). Then, u|∂Ω0
a+
> 0 and
(E − λW )u = −aur = a−ur ≥ 0 in Ω0
a+ .
Thus, thanks to characterization of the maximum principle of J. López-
Gómez and M. Molina-Meyer [12], it is apparent that σ[E−λW ; Ω0
a+ ] > 0,
and, therefore, λ ∈ (λ
Ω0
a+
1 , λ
Ω0
a+
2 ), by the strict concavity of λ 7→ Σ(λ).
Thus, by the a priori bounds found by S. Cano-Casanova [2], there exists
a constant C > 0 such that for any positive solution (λ, uλ) of (5.1),
‖uλ‖Cν(Ω̄) ≤ C. Now, let M > 0 be sufficiently large so that σ[E +
M ; Ω] > 0 and λΩ
1W (x) +M > 0 for each x ∈ Ω̄. By elliptic regularity,
the positive solutions of (5.1) are given by the zeroes in U := Cν
0 (Ω̄) of
the equation
L(λ)u+ R(λ, u) = 0 , (5.5)
where, for each (λ, u) ∈ R × U , we have denoted
L(λ)u := u− (E +M)−1[(λW +M)u] ,
S. Cano-Casanova, J. López-Gómez, M. Molina-Meyer 47
and
R(λ, u) := (E +M)−1(aur) .
It should be noted that the inverse operator (E +M)−1 ∈ L(U) is com-
pact, by elliptic regularity and Ascoli-Arzela’s theorem, and strongly or-
der preserving, by the strong maximum principle; the space U being
ordered by the cone of point-wise non-negative functions. Thus, (5.5) fits
into the abstract setting of Section 1 with
K(λ)u := (E +M)−1[(λW +M)u] , u ∈ U .
By the choice of M , K(λ0) is strongly positive. Moreover, setting
L0 := L(λ0) , L1 :=
dL
dλ
(λ0) = −(E +M)−1(W ·) ,
one has that N [L0] = span[ϕ0] and L1ϕ0 6∈ R[L0]. Indeed, if
L0u = −(E +M)−1(Wϕ0)
for some u ∈ U , then
u− (E +M)−1[(λ0W +M)u] = −(E +M)−1(Wϕ0)
and, by elliptic regularity, u ∈ C2+ν
0 (Ω̄). Thus, (E − λ0W )u = −Wϕ0,
which contradicts (5.4). Hence, the transversality condition of
M. G. Crandall and P. H. Rabinowitz [4] is satisfied and, consequently,
χ[L;λ0] = 1. Therefore, since λΩ
1 and λΩ
2 are the unique values of λ where
positive solutions of (5.1) can bifurcate from (λ, 0), as an immediate con-
sequence from Theorem 1.1 the following result is obtained.
Theorem 5.1. There is a bounded component of the set of positive
solutions of (5.1), say CP , such that (λΩ
1 , 0), (λΩ
2 , 0) ∈ C̄P . Moreover,
PλCP ⊂ (λ
Ω0
a+
1 , λ
Ω0
a+
2 ), where Pλ stands for the λ-projection operator.
It should be noted that, thanks to Theorem 4.1 and Theorem 5.1, (5.1)
cannot admit an abstract representation as a fixed point equation of the
form (1.1) with K(λ) = λK for some fixed compact strongly positive
operator K, and, consequently, even if the unilateral results of P. H.
Rabinowitz [13] would be correct as originally stated, Theorem 5.1 could
not be a consequence from them.
48 Bounded Components of Positive Solutions...
6. Three Different Types of Bounded Components
Under the assumptions of Theorem 1.1, L(λ) must possess two dif-
ferent eigenvalues λ ∈ S, at least λ0 and λ1, associated with each the
operator has a positive eigenfunction. This is far from being true if, in-
stead of assuming that λ0 is a nonlinear eigenvalue, one assumes that
χ[L;λ0] ∈ 2N, since, in this case, the component CP
λ0
might emanate
from the curve (λ, 0) exclusively at λ0. Therefore, the oddity of the mul-
tiplicity is crucial for the validity of Theorem 1.1. Actually, there are
boundary value problems of the form (5.1) that possess bounded com-
ponents exhibiting each of these behaviors. For example, consider the
one-dimensional prototype model in Ω = (0, 1)
{
−u′′ + µu = λ sin(2πx)u− a(x)u2 ,
u(0) = u(1) = 0 ,
(6.1)
where
a(x) =
−0.2 sin
(
π
0.2(0.2 − x)
)
if 0 ≤ x ≤ 0.2 ,
sin
(
π
0.6(x− 0.2)
)
if 0.2 < x ≤ 0.8 ,
−0.2 sin
(
π
0.2(x− 0.8)
)
if 0.8 < x ≤ 1 ,
(6.2)
and (λ, µ) ∈ R2 are regarded as two real parameters. Note that a > 0 in
(0.2, 0.8), a < 0 in (0, 0.2)∪ (0.8, 1), and a(0) = a(0.2) = a(0.8) = a(1) =
0. For an adequate choice of the parameter µ, this problem fits into the
abstract setting of Section 5 by choosing Eµ := − d2
dx2 +µ, W := sin(2π · )
and r = 2. Indeed, since N = 1, Ω0
a+ = (0, 0.2) ∪ (0.8, 1) and
max
λ∈R
σ[Eµ − λW ; Ω] = σ[− d2
dx2
+ µ; Ω] = π2 + µ ,
because of the symmetry of the problem around 0 (cf. [3] for further
details), it turns out that condition (5.2) holds as soon as µ > −π2.
Actually, for each µ > −π2, there exist λΩ
1 (µ) < 0 < λΩ
2 (µ) = −λΩ
1 (µ)
such that
σ[Eµ − λΩ
1 (µ)W ; Ω] = σ[Eµ − λΩ
2 (µ)W ; Ω] = 0 .
Moreover, As µ decreases approaching −π2, λΩ
1 (µ) increases, and, hence,
λΩ
2 (µ) decreases, approaching 0, i.e., limµ↓−π2 λΩ
1 (µ)=0=limµ↓−π2 λΩ
2 (µ).
As a result, Theorem 5.1 applies when µ > −π2 while it cannot be applied
if µ ≤ −π2. Actually, the mapping λ 7→ Σµ(λ) := σ[Eµ−λW ; Ω] satisfies
Σ−π2(0) = 0, Σ′
−π2(0) = 0, and Σ−π2(λ) < 0 for each λ ∈ R \ {0}.
Therefore, χ[E−π2 −λW ; 0] = 2 and Theorem 5.1 cannot applied to cover
S. Cano-Casanova, J. López-Gómez, M. Molina-Meyer 49
this transition situation, though the problem still possesses a bounded
component of positive solutions emanating from (λ, 0) at λ = 0 if µ =
−π2 (cf. the second plot of Figure 6.1). When µ < −π2 there are
no bifurcation points to positive solutions from (λ, 0) and, actually, if
µ is sufficiently close to −π2, then (3.1) exhibits an isola of positive
solutions (cf. the third plot of Figure 6.1). The bifurcation diagrams
of Figure 6.1 were computed by coupling a pseudo-spectral method with
collocation and a path-continuation solver (cf. [3]). The left plot of Figure
6.1 shows the component for µ = 0. In this case, λΩ
1 ∼ −28.0233 and
λΩ
2 ∼ 28.0233. The central plot of Figure 6.1 show the perturbations of
the positive solutions of the left plot as µ decreases from zero up to reach
the value µ = −9.8693 > π2 = −9.86960... . Now, λΩ
1 ∼ −0.13861 and
λΩ
2 ∼ 0.13861; as these values are very close, the central plot of Figure
6.1 shows them super-imposed. As the computational model is discrete
and π2 is irrational there is no way to get the bifurcation diagram for
µ = −π2, though it must be very similar to the central diagram. The
right plot shows the isola of solutions obtained for µ = −40, for which
(λ, 0) always is linearly unstable.
−50 0 50
−200
0
200
400
600
800
1000
1200
1400
−50 0 50
−200
0
200
400
600
800
1000
1200
1400
−50 0 50
−200
0
200
400
600
800
1000
1200
1400
Figure 6.1 Three components of positive solutions for
µ = 0,−9.8693,−40, respectively.
50 Bounded Components of Positive Solutions...
In Figure 6.1 we are plotting the value of λ against the L∞-norm of the
corresponding positive solution. Stable solutions are indicated by solid
lines, unstable by dotted lines. As there are some ranges of values of λ
where the model possesses at least two solutions with very similar L∞-
norms, the plot did not allow us distinguishing them, but rather plotted
twice these pieces. This is why the bifurcation diagrams exhibit a darker
arc of curve.
It should be clear that in case µ = −π2 the component of positive
solutions of (3.1) bifurcating from (λ, 0) at λ = 0 must be bounded and
that it emanates from the curve (λ, 0) exclusively at λ = 0.
References
[1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered
Banach spaces, SIAM Rev. 18 (1976), 620–709.
[2] S. Cano-Casanova, Compact Components of positive solutions for Superlinear In-
definite Elliptic Problems of Mixed Type // Top. Meth. Non. Anal. 23 (2004),
45–72.
[3] S. Cano-Casanova, J. López-Gómez, and M. Molina-Meyer, Isolas: compact solu-
tion components separated away from a given equilibrium curve, Hiroshima Math.
J. 34 (2004), 177-199.
[4] M. G. Crandall, and P. H. Rabinowitz, Bifurcation from simple eigenvalues //
J. Funct. Anal. 8 (1971), 321–340.
[5] E. N. Dancer, Global solution branches for positive mappings // Arch. Rat. Mech.
Anal. 52 (1973), 181–192.
[6] E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric
multiplicity one // Bull. London Math. Soc. 34 (2002), 533–538.
[7] J. Esquinas, Optimal multiplicity in local bifurcation theory, II: General case //
J. Diff. Eqns. 75 (1988), 206–215.
[8] J. Esquinas, and J. López-Gómez, Optimal multiplicity in local bifurcation theory,
I: Generalized generic eigenvalues // J. Diff. Eqns. 71 (1988), 72–92.
[9] P. Hess, and T. Kato, On some linear and nonlinear eigenvalue problems with an
indefinite weight function // Comm. Part. Diff. Eqns. 5 (1980), 99–1030.
[10] M. G. Krein, and M. A. Rutman, Linear operators leaving invariant a cone in a
Banach space // Amer. Math. Soc. Transl. 10 (1962), 199–325.
[11] J. López-Gómez, [2001] Spectral Theory and Nonlinear Functional Analysis, Re-
search Notes in Mathematics 426, CRC Press, Boca Raton 2001.
[12] J. López-Gómez, and M. Molina-Meyer, The maximum principle for cooperative
weakly elliptic systems and some applications // Diff. Int. Eqns. 7 (1994), 383–
398.
[13] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems //
7 (1971), 487–513.
S. Cano-Casanova, J. López-Gómez, M. Molina-Meyer 51
Contact information
Santiago
Cano-Casanova
Departamento de Matemática Aplicada y
Computación
Universidad Pontificia Comillas de Madrid
28015-Madrid,
Spain
E-Mail: scano@dmc.icai.upco.es
Julián
López-Gómez
Departamento de Matemática Aplicada
Universidad Complutense de Madrid
28040-Madrid,
Spain
E-Mail: Lopez_Gomez@mat.ucm.es
Marcela
Molina-Meyer
Departamento de Matemáticas
Universidad Carlos III de Madrid
28911-Leganés, Madrid,
Spain
E-Mail: mmolinam@math.uc3m.es
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