Acta Mathematica Sinica, English Series
1999, Oct., Vol.15, No.4, p. 439–454
An Extension of the Hess-Kato Theorem to Elliptic Systems
and Its Applications to Multiple Solution Problems
Kungching Chang
Institute of Mathematics, Peking University, Beijing 100871, P. R. China
E-mail: [email protected]
Abstract The Hess-Kato Theorem on the simplicity of the first eigenvalue of a second order elliptic
operator is extended to elliptic systems. The theorem is applied to figure out the critical groups
of a mountain pass point for functionals which have nonlinear ellipitic systems as Euler Lagrange
Equations. A muliple solution result is obtained via the ordered Banach space method and the critical
group calculations.
Keywords The principal eigenvalue, Maximum principle, Cooperative, Indefinite weight
1991MR Subject Classification 35J50, 35P05, 35B50
0
Introduction
It is well known that the first eigenvalue of the second order elliptic operator on a bounded
domain is simple. In 1980, H. Hess and T. Kato observed that this result also holds for the
eigenvalue problem with respect to indefinite weight functions. The conclusion is very useful
in many problems: bifurcation, linearized stability, and the sub- and super-solutions etc. In
addition, the Hess-Kato Theorem plays an important role in the study of multiple solutions
of semilinear elliptic BVP. According to this theorem, the critical groups of a mountain pass
point are completely figured out, i.e. the critical groups of all orders are trivial except the first
order having rank one. However, this theorem does not hold for elliptic systems, if there are no
additional conditions. In a recent paper [1], the strong Maximum Principle has been extended
to elliptic systems under the cooperative and the fully coupled assumptions. The main purpose
of this paper is to extend Hess-Kato Theorem to this case. This is the first part of this paper.
Theorem 1.1 (see below) is our main result in the first part. Extended versions of the Maximum
Principle and the Anti-Maximum Principle for systems are given in Theorems 1.2 and 1.3.
Received December 29, 1998, Accepted January 14, 1999
Supported by CNSF and MCME
Kungching Chang
440
In the second part, we first extend the critical group characterization of the mountain pass
points to elliptic systems with variational structures under the above assumptions. This is
Theorem 2.1, which is very basic in the study of multiple solutions. As an example, the result
for the scalar case, due to the author [2] and Dancer, Du [3], in which the existence of at least
seven nontrivial solutions under certain conditions of the nonlinear term at the origin and at
infinity, in combination with the assumption that there exists a pair of negative strict sub- and
positive strict super-solutions, is extended to the vector case. (Theorem 2.4). Unlike the scalar
case, the cut off techniques match with the variational structures; in the vector case, this is not
true in general. Some results concerning the connection between the order structures and the
variational structures are developed to replace the cut off techniques (see Lemma 2.4 and the
proof of Theorem 2.3). Furthermore, in the same problem, if the seventh solution is assumed
to be nondegenerate, then one more solution is obtained. This result is new even for the scalar
case (Theorem 2.3).
1
Hess-Kato Theorem for Elliptic Systems
Let Ω be a bounded open domain, with smooth boundary in Rn . Let M ∈ C (Ω, M (p, R)),
where M (p, R) denotes the set of real p × p matrices, and let L =diag{D1 , D2 , . . . , Dq }, where
Di , i = 1, 2, . . . , p, are second order uniformly elliptic operators with continuous coefficients and
the first eigenvalue µi ≥ 0 under the Dirichlet data.
For a vector function U ∈ C = C (Ω, Rp ), the following notations are used: U ≥ 0
means all components of U are nonnegative functions; U > 0 means U ≥ 0 but U is not the
null element of C, U 0 means all the components uj (x) > 0, j = 1, . . . , p, ∀ x ∈ Ω. Let
E = C0 (Ω, Rp ) = {U ∈ C|U |∂Ω = 0}.
We make the following assumptions:
(I) M = (mij (x)) is cooperative, i.e. mij (x) ≥ 0, ∀ i = j, ∀ x ∈ Ω.
(II) M is fully coupled, i.e. the index set {1, . . . , p} cannot be split into two disjoint
nonempty subsets I and J such that mij (x) ≡ 0 in Ω for i ∈ I and j ∈ J.
(III) maxx∈Ω max1≤i≤p mi i (x) > 0.
Our main result for this section is the following
Theorem 1.1
Under assumptions (I), (II) and (III), there exist a unique λ1 > 0 and φ 0, φ ∈ E satisfying
(1)
Moreover we have L φ = λ1 M φ in Ω.
Moreover, we have
(2)
dim ker (L − λ1 M ) = dim coker (L − λ1 M ) = 1.
(3)
The M -algebraic multiplicity of λ1 is odd.
(4)
If λ > 0 is an eigenvalue of L U = λ M U , then λ ≥ λ1 .
An Extension of the Hess-Kato Theorem to Elliptic Systems
441
The M -algebraic multiplicity of λ1 is the integer r satisfying
ker (I − λ1 L−1 M ) ker (I − λ1 L−1 M )2 · · · ker (I − λ1 L−1 M )r = ker (I − λ1 L−1 M )r+1 .
A weaker result for elliptic systems has been obtained in [4], in which without the assumption
(II), only φ > 0 is known, but not (2) and (3). We do not know whether λ1 is M -algebraically
simple; and leave it as an open question, although Theorem 1.1 is sufficient in many applications.
Our result is based on a version of Maximum Principle for elliptic systems due to G. Sweers
[1]. Namely,
Theorem 0
(IV).
Under assumptions (I), (II) and
There exists a positive strict supersolution U of the equation
(L − M ) U = 0 in Ω,
U |∂Ω = 0 ,
there exist a unique positive eigenfunction (normalized) ϕ 0 and µ > 0 satisfying
(L − M ) ϕ = µ ϕ in Ω,
U |∂Ω = 0 ,
Moreover, if F ∈ Lq (Ω, RP ), q > n, F > 0, and
(L − M ) U = F
(1.1)
in Ω,
U |∂Ω = 0,
then U 0.
The argument of Sweers is based on the following deduction: Let L̂ be the closure of the
operator L in C (Ω, Rp ), i.e. D (L̂) = {U ∈ E|LU ∈ C (Ω, Rp )}, where LU is in the distribution
sense. Let X = D (L̂) be with the graph norm: ||U ||X = ||U ||E + ||LU ||C . According to
the Maximum Principle for each component of U there exists a constant C1 > 0 such that
||U ||E ≤ C1 ||LU ||C . Thus, L̂ : X → C is a continuous isomorphism. We choose
β > Max {mii (x) | 1 ≤ i ≤ p, x ∈ Ω} + 1,
(1.2)
and write Lβ = (L + β I)∧ . Obviously. Lβ : X → C is a continuous isomorphism: Define
q
A = L−1
β (M + β I). Then A : C → E is positive and compact, by virtue of the L elliptic
◦
theory and the compact embedding X → Wq2 ∩ W 1q (Ω, Rp ) → E for q > n.
The restriction of A to E is irreducible, according to the assumption (II). Let
T = (I − A)
−1
L−1
β
=
∞
j=0
Aj L−1
β .
It is well defined, due to the fact that the spectral radius r(A) < 1, which is a consequence of
the irreducibility of A and the assumption (IV). Moreover, we have
(i) Both (I − A) ∈ L (X, X), and T ∈ L (C, X) are continuous isomorphisms, and
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442
(ii) The restriction of T to E is positive, compact and irreducible.
From a simple calculation, it follows that T −1 = L̂ − M. Now, ∀ f ∈ C, the system in finding
U ∈ X of (L̂−M ) U = f, is equivalent to U = T f. And the eigenvalue problem (L̂−M ) ψ = µ ψ,
is equivalent to that of ψ = µ T ψ.
Since the restriction of T to E is positive, compact and irreducible, we can apply a theorem
due to de Pagter [5] and the Krein-Rutman Theorem to prove Theorem 0. Moreover, we have
Corollary 1.1
Under the assumptions of Theorem 0, there exists ψ E ∗ , ψ > 0, such that
(L̂∗ − M ∗ ) ψ = µ ψ and ψ, φ > 0 ,
where E ∗ is the dual space of E, and L̂∗ and M ∗ are the dual operators of L̂ and M respectively.
Moreover, dim {U ∈ X|(L̂ − M ) U = µ U } = 1.
Proof Since T is positive, compact and irreducible in E, we conclude that r (T ) > 0, according
to de Pagter [5]. Applying the Krein Rutman Theorem, there exist φ ∈ E, ψ ∈ E ∗ , φ > 0, ψ >
0 and ψ, φ > 0, which satisfy T φ = r (T ) φ, and T ∗ ψ = r (T ) ψ. Moreover, dim {x ∈
E| T x = r (T ) x} = 1.
Let µ = r (T )−1 . Since T ∗−1 = L̂∗ − M ∗ ∈ L (E ∗ , X ∗ ), we have (L̂ − M ) φ = µ φ, and
(L̂∗ − M ∗ ) ψ = µ ψ .
Now, we return to the proof of our main theorem.
Lemma 1.1
If B = (bij ) ∈ C (Ω, M (p, R)) satisfies
max
p
1≤i≤p j=1
bij (x) ψj (x) < µi ψi (x),
∀ x ∈ Ω,
∀ i,
where µj and ψj > 0 are the first eigenvalue and the first eigenfunction of Dj on Ω with
0-Dirichlet boundary condition respectively, j = 1, 2, . . . , p, then there exists a positive strict
super-solution U of the system: (L − B) U = 0 , U |∂Ω = 0.
Proof
We set U = (ψ1 , ψ2 , . . . , ψp ). Obviously, L U > B U .
By the strong Maximum Principle, we can find a constant C > 0 depending on M, L, and
Ω, such that
p
mij (x) ψj (x) < C ψi (x)
∀ x ∈ Ω, i = 1, 2, . . . , p .
j=1
1
∀ λ ∈ R , let β1 (λ) = λ C + 1. We consider the operator
L̂ − (λ M − β1 (λ) I) ∈ L (X , C) .
By Lemma 1.1 and Theorem 0, there exist µ(λ) > 0 and ϕλ ∈ X with ϕλ 0, ||ϕλ || = 1,
satisfying
L̂ − (λ M − β1 (λ) I) ϕλ = µ (λ) ϕλ .
Setting µ (λ) = µ (λ) − β1 (λ), it follows that (L̂ − λ M ) ϕλ = µ (λ) ϕλ . Moreover, we have
ker (µ (λ) I − (L̂ − λ M )) = ker (µ(λ) I − (L̂ − (λ M − β1 (λ) I))) = span{ϕλ }.
An Extension of the Hess-Kato Theorem to Elliptic Systems
443
We are going to show that there exists a unique λ1 > 0, which satisfies the equation µ (λ) = 0.
Lemma 1.2
Assume (I) and (II). If ∃ U > 0 and λ0 > 0 such that
(L̂ − λ0 M ) U ≤ 0,
(1.3)
then µ (λ0 ) ≤ 0.
Proof
Assume µ (λ0 ) > 0. By definition, there exists a unique vector ϕλ0 0 such that
(L̂ − λ0 M ) ϕλ0 = µ (λ0 ) ϕλ0 . Let β be defined as in (1.2), and let Aλ0 = λ0 L−1
λ0 β (M + β I)
−1 −1
−1
and Tλ0 = (I − Aλ0 ) Lλ0 β = (L̂ − λ0 M ) . The latter is well defined, because now ϕλ is a
positive strict supersolution of the system (L̂−λ0 M ) U = 0. And then, Tλ0 is positive, compact
and irreducible. Applying Corollary 1.1, there exists ψλ0 ∈ E ∗ with ψλ0 > 0, such that
Tλ∗0 ψλ0 = r (Tλ0 ) ψλ0 ,
(1.4)
and ψλ0 , ϕλ0 > 0.
Now, (1.3) implies U ≤ Aλ0 U. Setting V = Aλ0 U, we have V ≥ U > 0, and
L̂λ0 β V = λ0 (M + β I) U ≤ λ0 (M + β I) V .
Therefore
S = (L̂ − λ0 M ) V ≤ 0.
(1.5)
Combining (1.4) and (1.5), we obtain
r (Tλ0 )2 ψλ0 , S = ψλ0 , Tλ20 S = ψλ0 , Tλ0 V.
(1.6)
However, M is fully coupled, and there exists an integer m such that Am
λ0 v 0. This implies
Tλ0 v 0. The LHS of (1.6) is nonpositive, while the RHS is positive. This is a contradiction.
Corollary 1.2.
Proof
Assume (I), (II) and (III). ∃ λ0 > 0 such that µ (λ0 ) ≤ 0.
By assumption (III), we may assume m+
11 (x) = m11 (x) ∨ 0 > 0. Applying Hess-Kato
Theorem for the equations, we obtain λ0 > 0, and w C0 (Ω), w > 0 such that (D1 −λ0 m11 ) w =
0. Setting U = (w, 0, . . . .0)T , we have
(L̂ − λ0 M ) U ≤ 0 ,
in view of the cooperativeness. The conclusion follows from Lemma 1.2.
Noticing that µ (λ) is geometrically simple, and µ (λ) < | ν (λ)| for any other eigenvalue
ν (λ) of L̂ − λ M , we have
Lemma 1.3
Proof
The function λ → (µ (λ), ϕλ ) is C 1 .
Consider the map:
Γ : S1 × R1 × R1 −→ C,
Γ (x, µ; λ) = (L̂ − λ M ) x − µ x ,
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444
where S1 is the unit sphere of Wq2 (Ω)
◦
W
1
q
(Ω), q > n. Since Γ (ϕλ , µ(λ); λ) = θ, and
Γx (ϕλ , µ (λ); λ) y + Γµ (ϕλ , µ(λ); λ) µ = (L̂ − λM − µ(λ)) y + µ ϕλ ,
∀ y ∈ Tϕλ (S1 ), ∀ µ ∈ R1 , the linear map (Γx (ϕλ , µ(λ); λ), Γµ (ϕλ , µ(λ); λ)) is surjective. Applying the Implicit Function Theorem, the conclusion follows.
Corollary 1.3
Under assumptions (I), (II) and (1.3), there exists λ1 ∈ (0, λ0 ) which satisfies
the equation µ (λ1 ) = 0.
Proof
Obviously µ (+0) > 0, our conclusion follows from Lemma 1.2 and Lemma 1.3.
Assume (I), (II) and (III), let λ1 be the smallest positive root of µ(λ) = 0.
Thus, we have proved the existence of an eigenvalue λ1 > 0 with ϕλ1 0, satisfying
(L̂ − λ1 M ) ϕλ1 = 0. It remains to show that this is unique among those which are associated
with strongly positive eigenvectors, and that the M -algebraic multiplicity is odd.
/ Im (L̂ − λ1 M ), and this, in
Recall that λ1 is said to be M -algebraically simple, if M ϕλ1 ∈
−1
M.
turn, is equivalent to that λ−1
1 is algebraically simple for the operator A = L̂
Lemma 1.4
Proof
λ1 is M algebraically simple if and only if µ (λ1 ) = 0.
Consider the equation: (L̂ − λ M ) ϕλ = µ (λ) ϕλ . Differentiating it with respect to λ at
λ1 , we obtain
(L̂ − λ1 M ) ϕλ1 = (M + µ (λ1 ) I) ϕλ1 .
On the one hand, if µ (λ1 ) = 0, then M ϕλ1 ∈ Im(L̂ − λ1 M ), i.e. λ1 is not M -algebraically
simple.
On the other hand, if λ1 is not M -algebraically simple, then, by definition, there exists
W ∈ X such that (L̂ − λ1 M ) W = M ϕλ1 .
Setting Z = ϕλ1 − W, we have (L̂ − λ1 M ) Z = µ (λ1 ) ϕλ1 . We choose ψ ∈ ker (L̂∗ −
λ1 M ∗ ), ψ 0, then
µ (λ1 )
ψ, ϕλ1 = ψ, (L̂ − λ1 M ) Z = 0 .
But, ψ, ϕλ1 > 0, we conclude that µ (λ1 ) = 0.
Now, we introduce a family of matrices satisfying the assumptions (I), (II) and (III). Set
M1 = M − diag M,
t̄ = max max mii (x) ,
1≤i≤p x∈Ω̄
and for any diffeomorphism s on [0, t̄) with s (t) > 0, set
M (t) = Ms (t) = (1 − t/t̄) M1 + ( diag M − s(t) I)
t ∈ [0, t̄) .
We have M (0) = M . Now, the first eigenvalue of L̂ − λ M (t) depends on t, and is denoted by
µ (λ, t). According to Lemma 1.2, µ is C 1 in λ and t.
Lemma 1.5
∀ λ > 0, ∂t µ (λ, t) > 0.
An Extension of the Hess-Kato Theorem to Elliptic Systems
Proof
445
∀ t1 , t2 ∈ [0, t̄) with t1 < t2 , let ϕi and ϕ∗i be the first eigenvectors associated with
L̂ − λ M (ti ) and (L̂ − λ M (ti ))∗ , i = 1, 2, respectively, i.e.
(L̂ − λ M (ti )) ϕi = µ (λ, ti ) ϕi ,
(L̂∗ − λ M ∗ (ti )) ϕ∗i = µ (λ, ti ) ϕ∗i ,
i = 1, 2. We have
(µ (λ, t2 ) − µ (λ, t1 ))
ϕ1 , ϕ∗2 = λ
[M (t1 ) − M (t2 )] ϕ1 , ϕ∗2 = λ
[(t2 − t1 ) t̄−1 M1 + (s (t2 ) − s (t1 ) I)] ϕ1 , ϕ∗2 .
It follows that
∂t µ (λ, t1 ) = λ
(t̄−1 M1 + s (t1 )) ϕ1 , ϕ∗1 /
ϕ1 , ϕ∗1 > 0.
In the following, ∀ t ∈ [0, t̄), we denote λ = ξ (t), the smallest positive root of µ(λ, t) = 0.
According to Corollary 1.2, ξ (t) is well defined. In particular, ξ (0) = λ1 .
Lemma 1.6
The map ξ : [0, t̄) → [λ1 , ∞) is a homeomorphism.
Proof 1◦ We prove that ξ → +∞ as t → t̄. If not, ξ (t̄ − 0) = ξ ∗ < ∞. By definition,
∀t ∈ [0, t̄), ∃ ϕt θ with ||ϕt || = 1 satisfying ϕt = ξ (t) L̂−1 M (t) ϕt . Since L̂−1 is compact,
ϕt → ϕ∗ as t → t̄. Hence
ϕ∗ = ξ ∗ L̂−1 (diag M − t̄ I) ϕ∗ ,
with ||ϕ∗ || = 1, and ϕ∗ ≥ θ.
Since the RHS is nonpositive, while the LHS is nonnegative, there must be ϕ∗ = θ. This
contradicts ||ϕ∗ || = 1.
2◦
t −→ ξ (t) is strictly monotone increasing. In fact, ∀ t1 < t2 , we have ϕi θ satisfying
L̂ ϕi = ξ (ti ) M (ti ) ϕi ,
i = 1, 2.
It follows that
(L̂ − ξ (t2 ) M (t1 )) ϕ2 = ξ (t2 )[(t1 − t2 )/t̄ M1 + (s (t1 ) − s (t2 )) I] ϕ2 < 0.
By Corollary 1.3, ξ (t1 ) < ξ (t2 ). This proves the strict monotonicity of ξ.
3◦
From the definition µ (ξ (t), t) ≡ 0, by Lemma 1.3, µ is C 1 in λ and t, and since ξ (t)
exists almost everywhere, we obtain
∂λ µ (ξ (t), t) ξ (t) + ∂t µ (ξ (t), t) = 0
a.e.
By Lemma 1.5, ∂λ µ (ξ (t), t) = 0 a.e. This shows that ξ(t) is M (t) algebraically simple for a.e.
t, according to Lemma 1.4.
4◦
ξ (t) is continuous.
If not, ∃t0 ∈ [0, t̄) such that ξ (t0 − 0) < ξ (t0 + 0).
We denote the real eigenvalues of the problem
(L̂ − λ M (t)) ϕ = 0 ,
(1.7)
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446
according to their ordering ν1 (t) < ν2 (t) < · · · . To solve (1.7) is equivalent to solving
(L̂ − λ M (t)) ϕ = µ (λ, t) ϕ ,
(1.8)
µ (λ, t) = 0 .
One may order the eigenvalues {µ1 (λ, t), µ2 (λ, t), . . . , µk (λ, t), . . .} of (1.8) by their absolute
values. By the Krein Rutman Theorem, µ1 (λ, t) is real and satisfies
µ1 (λ, t) < | µ2 (λ, t) | ≤ | µ3 (λ, t) | ≤ · · · .
First, we claim
ξ (t) = ν1 (t) < νj (t),
∀ j ≥ 2,
∀ t ∈ [0, t̄) .
Indeed, ξ (t) is the smallest positive root of µ1 (λ, t) = 0, i.e.
µ1 (ξ (t), t) = 0,
µ1 (λ, t) > 0,
∀ λ ∈ [0, ξ (t)).
It follows that
| µj (λ, t) | > 0,
∀ λ ∈ [0, ξ (t)),
∀j≥2.
Since νk (t) must be a zero of µj (λ, t) for some j ≥ 1, one proves νk (t) ≥ ξ (t) ∀ t ∈
[0, t̄) ∀ k ≥ 2.
However, according to the analytic perturbation theory (cf. [6] Lemma 8(a)), for each
λ0 = ξ (t0 ), the solution of (1.7) bifurcates as follows:
1
1
λ (t) = λ0 + (a(t − t0 )) r + ◦ (|t − t0 | r ) ,
where a = ϕ∗ , (I + t̄−1 M1 ) ϕ and r is the algebraic multiplicity of λ0 . This excludes the
possibility of intersection of the two branches of the real roots. Therefore νk (t) > ξ(t), ∀ t ∈
[0, t̄), ∀ k ≥ 2.
We study the problem of the complexification of the space X. Let us define
1
(ζ I − A (t))−1 dζ,
P (t) =
2πi Γ
(1.9)
where A (t) = L̂−1 M (t), and Γ is the boundary circle of the disk D (ξ0 , ), ξ0 = ξ (t0 −0)−1 , > 0
such that D (ξ0 , )\ξ0 is in the resolvent set of A (t0 ). There exists δ > 0 such that there are
no eigenvalues of A (t) on Γ for |t − t0 | < δ. Then P (t) is a projection and equals the sum of
eigenprojections for all eigenvalues of A (t) lying inside Γ.
Now we take > 0 so small that νj (t) > ν1 (t) + 2 , ∀ j ≥ 2, ∀ | t − t0 | < δ, and ξ (t0 + 0) >
ξ (t0 − 0) + . There is no real eigenvalue of A (t) in D (ξ0 , ) for |t − t0 | ≤ δ, except ξ (t) for
t ∈ (t0 − δ, t0 ]. However, by 3◦ ,
dim Im P (t) = 1,
dim Im P (t) = even,
∀ t ∈ (t0 − δ, t0 ) ,
and
∀ t ∈ (t0 , t0 + δ) .
An Extension of the Hess-Kato Theorem to Elliptic Systems
447
The latter is due to the fact that A (t) is a real operator, and the complex eigenvalues occur in
pairs. The contradiction shows that ξ (t0 + 0) = ξ (t0 − 0), i.e. ξ is continuous. When t0 = 0,
the proof is the same if we extend M (t) to negative t.
Now we set s (t) = t, and show
Lemma 1.7
Proof
The M (t)-algebraic multiplicity of ξ (t) is odd.
Following Lemma 8(a) in [6], if ξ (t0 ) is not M (t0 )-algebraically simple, then λ0 = ξ (t0 )
bifurcates to r ≥ 2 branches of eigenvalues:
1
1
λ (t) = λ0 + [a (t − t0 )] r + ◦ (|t − t0 | r ) ,
(1.10)
where a = ϕ∗ , (I + t̄−1 M1 ) ϕ, and ϕ, ϕ∗ are the positive eigenvectors w.r.t. λ0 , i.e.
L ϕ = λ0 M (t0 ) ϕ,
and
L∗ ϕ∗ = λ0 M ∗ (t0 ) ϕ∗ .
Obviously, r cannot be even. because among λ (t), there are either no real values or a pair of
real values: one bigger than and the other less than λ0 for |t − t0 | small. This contradicts the
fact that ξ (t) is continuous and ξ (t) > λ0 when t > t0 .
The proof is complete.
Finally, we return to the
Proof of Theorem 1.1
It remains to show that there exists a unique positive eigenvalue λ1 ,
which is associated with a strongly positive eigenvector.
Suppose to the contrary, i.e. ∃ 0 < λ1 < λ2 both corresponding to strongly positive eigenvectors ϕ1 , ϕ2 θ with (L̂ − λi M ) ϕi = θ, i = 1, 2. According to Lemma 1.6, ∃ t0 ∈ (0, t̄)
such that λ2 = ξ (t0 ), i.e. (L̂ − λ2 Ms (t0 )) ϕ = θ, for some ϕ θ, where s (t) = t, i.e.,
−1
−1
λ−1
2 = r (Lλ2 β (M + β I)) = r (Lλ2 β (Ms (t0 ) + β I)).
Now let us set L̃ = Lλ2 β , M̃ = M + β I, t̃ = β + t̄, and λ̃1 = λ2 . Again, define a C 1 strictly
monotone increasing function s̃ on [0, t̃), which graph contains the segment connecting (0, 0)
and (t1 + , t̄ (t1 + )/t̃) and the point (t̃, t̃), where t1 = t̃ t0 /t̄ and 0 < <
1
2
(t̃ − t1 ). In this
case,
Ms (t0 ) + β I = (1 − t0 /t̄) M1 + diag M + (β − t0 ) I
= (1 − t1 /t̃) M̃1 + diag M̃ − (t̄ t1 /t̃) I
= (1 − t1 /t̃) M̃1 + diag M̃ − s̃ (t1 ) I
= M̃s̃ (t1 ) ,
and M + β I = M̃s̃ (0). According to the Krein Rutman Theorem, λ−1
= r (L̃−1 M̃s̃ (t1 )) =
2
r (L̃−1 M̃s̃ (0)), and then λ2 = ξ˜ (t1 ) = ξ˜ (0), where ξ˜ (t) is the smallest positive eigenvalue
of the problem L̃ ϕ = λ M̃s̃ (t) ϕ. However, applying Lemma 1.6 to (L̃, M̃s̃ ), it follows that
ξ̃ (0) < ξ˜ (t1 ). This is a contradiction.
Theorem 1.1 is proved completely.
The following theorem is an improved version of the Maximum Principle for elliptic systems.
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448
Let L = diag {D1 , . . . , Dp }, where Di is a second order uniformly elliptic operator. Let B
be a continuous, cooperative and fully coupled matrix defined on Ω ⊂ Rn .
Let λ1 > 0 be the first positive eigenvalue for the problem
L u = λ B u,
in Ω,
u ∈ H01 (Ω, Rp ).
Theorem 1.2
∀ 0 < λ < λ1 , for any h ∈ Lq (Ω, Rp ), q > n with h ≥ 0 but not identical to
the zero vector, there exists a unique U θ satisfying the system

 L U = λ B U + h,
in Ω,
 U = 0.
∂Ω
Proof
By Sweers [1], we only want to find a positive strict super-solution for the the operator
L − λ B. Indeed, first we choose c > 0 such that B + c I is positive, then the first eigenvalue
µλ > 0 associated with the positive eigenfunction φλ θ of the equation exists:
(L + λ c) φλ = µλ (B + c I) φλ ,
according to Theorem 1.1. If we can show λ < µλ , then L φλ > λ B φλ in Ω, i.e. φλ is a strict
super-solution.
However, if λ ≥ µλ , then (L + µλ c) φλ ≤ µλ (B + c I) φλ , i.e. L φλ ≤ µλ B φλ ; in other
words, φλ is a positive subsolution of the equation (L − µλ B) v = 0. By Lemma 1.2, λ1 ≤ µλ .
Thus λ1 ≤ λ. This is a contradiction. The theorem is proved.
Theorem 1.3 (Anti-Maximum Principle)
Suppose that M satisfies (I), (II) and (III). ∀ h Lq
(Ω, Rp ), q > n, with h > 0, ∃ δ > 0 such that the equation:
L U = λ M U + h in Ω
has a solution U 0 provided that λ ∈ (λ1 , λ1 + δ).
Proof
We consider the operator
A = L − λ1 M : X := W 2,q ∩ W01,q (Ω, Rp ) → Y := Lq (Ω, Rp ) .
By Theorem 1.1, ker (A) = span {φ1 }, ker (A∗ ) = span {φ∗1 } with φ∗1 , φ1 > 0. We decompose
/ R (A) = {φ∗1 }⊥ .
Y = R (A)⊕ span {φ1 }, in view of the facts: 1◦ codim R (A) = 1, and 2◦ φ1 ∈
Thus ∀ h ∈ Lq (Ω, Rp ), we have h = α φ1 + h1 , where h1 ∈ R (A), and
α=
φ∗1 , h
φ∗1 , φ1 > 0,
if
h > 0.
Then the solution of the equation (L − λ M ) U = h can be written as U = β φ1 + U1 , where
α
λ1 −λ
and U1 X1 , which is the complement of span {φ1 }, and is isomorphic to R (A) under
the map A; we have U1 = A−1 h1 . Since
β=
R(A)
||U1 ||2,q ≤ C1 ||h1 ||q ≤ C2 ||h||q ,
An Extension of the Hess-Kato Theorem to Elliptic Systems
449
we have
||U1 ||C 1 (Ω,Rp ) ≤ C3 ||h||q ,
for q > n .
0
◦
One chooses an element e in P, and there exist γ1 , γ2 > 0 such that −γ1 e ≤ U1 ≤ γ1 e, and
γ2 e ≤ φ1 . Then − γγ12 φ1 ≤ U1 ≤ γγ12 φ1 . It follows that
α
γ1
α
φ1 + (λ1 − λ) U1 ≤
1 + (λ − λ1 )
φ1 0,
U=
λ1 − λ
λ1 − λ
γ2
provided that λ ∈ (λ1 , λ1 + δ) for some δ > 0.
2
Applications to Multiple Solution Problems
We study the following nonlinear elliptic system:
(L U) (x) = f (x, U (x)),
U ∈ H01 (Ω, Rp ) ,
(2.1)
where L = − · I, f (x, ξ) = ∂ξ F (x, ξ) and F ∈ C 2 (Ω × Rp , R1 ) satisfies |fξ (x, ξ)| ≤ C (1 +
|ξ|α ), for 0 ≤ α < ∞ if n = 2, and 0 ≤ α <
4
n−2
if n ≥ 3.
(2.1) is the Euler Lagrange equation of the following functional on H01 (Ω, Rp )
1
| ∇ U (x)|2 − F (x, U (x)) dx .
J (U) =
2
Ω
Theorem 1.1 is used in characterizing the Mountain Pass point U of the above functional
via critical groups Cq (J, U ), [cf. 7]. Namely, we obtain
Theorem 2.1
Suppose that f ∈ C 1 (Ω × Rp , Rp ), and that U is a Mountain Pass point. If
M (x) = fξ (x, U (x)) is cooperative and fully coupled, then rank Cq (J, U ) = δq1 .
Proof
In order to apply Theorem 1.6 Chapter II [7], it remains to prove that if d2 J (U ) is
nonnegative and if 0 ∈ σ (d2 J (U )), then dim ker d2 J (U ) = 1.
Now d2 J (U ) = id − L−1 M (x). If V ∈ ker d2 J (U )\{0}, then L V = M V. It implies that
the first eigenvalue λ1 = 1. To see this, we note that d2 J (U ) is nonnegative, i.e.
|∇ W |2 ≥
M W · W,
∀ W ∈ H01 .
On the one hand, we obtain from the variational characterization that
|∇W |2 M
W
·
W
>
0
≥ 1;
λ1 = inf MW · W on the other,
λ1 ≤
Hence λ1 = 1, as claimed.
|∇V |2 /
M V · V = 1.
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450
Notice that LV = M V is equivalent to Lβ V = (M + βI) V . For the latter, the assumption
(III) is satisfied. Applying Theorem 1.1, it follows that dim ker (d2 J(U )) = 1.
Now, we present here an example showing how Theorem 2.1 is applied to the study of the
multiplicity of solutions of elliptic systems.
Let P be a positive cone in C01 (Ω, Rp ). We make the assumption:
(CF) ∂ξ2i ξj F (x, ξ) > 0, ∀i = j, ∀(x, ξ) ∈ Ω × Rp , and p > 1.
On any given bounded set of ξ, one can find c ≥ 0 such that the matrix Mc = M (x, ξ) + c I
is order preserving, where M (x, ξ) = fξ (x, ξ). Let Lc = L + c I and fc = f + c I. We have
Lemma 2.1
For any bounded set B of C (Ω, Rp ), there exists c > 0 such that the operator
1
p
U −→ K U = L−1
c fc (x, U(x)) is strongly order preserving from B to C0 (Ω, R ), ∀ 1 < q < ∞.
U0 ∈ Wq2 (Ω, Rp ) is called a sub- (or super-) solution of (2.1), if
L U0 ≤ (or ≥ resp.) f (x, U0 (x)) .
This implies
U0 ≤ (or ≥ resp.) K U0 .
(2.2)
Let us consider the gradient flow η t of the functional J under the equivalent norm ||U|| =
1
( |∇ U|2 + c|U|2 ) 2 of H01 . We have
Lemma 2.2
If U± is a sub-/super- solution of (2.1), then η t (U± ± υ) ∈ U± ± int (P) ∀ t >
0, ∀υ ∈ Ṗ = P\{θ}. Consequently, if [U, U] is an order interval in the space C01 (Ω, Rp ) of a pair
of sub- and super-solutions of (2.1), then it is positively invariant under the flow η t .
Indeed, U± ± υ − J (U± ± υ) = K(U± ± υ) ∈ U± ± int (P). The last assertion follows
Proof
from the strong Maximum Principle for elliptic systems (cf. Theorem 0) and (2.2).
Let M0 = M (x, θ), and let {0 < µ01 < µ02 ≤ · · ·} be the positive eigenvalues of L w.r.t. M0 .
We introduce an assumption
(H0 ) Assume f (x, θ) = θ and µ01 < µ02 < 1.
Lemma 2.3
Assume (H0 ). If M0 is cooperative and fully coupled, then there exist a positive
strict sub solution and a negative strict super solution of (2.1), all with small norms. Moreover,
ind (J , θ) ≥ 2.
Proof
Let λ1 be the first eigenvalue of Lc with weight Mc = M0 + cI. First, we claim λ1 < 1.
Indeed, let ϕ1 be the eigenvector associated with µ01 ; we have
2
0
|∇ ϕ1 | = µ1
M0 ϕ1 · ϕ1 <
M0 ϕ1 · ϕ1 .
Then
Lc U · U
< 1.
λ1 = Min Mc U · U
Since Mc satisfies the assumptions (I), (II) and (III), then Theorem 1.1 may be applied. There
exists a unique normalized eigenvector φ ∈ int(P) corresponding to λ1 . Hence, Lc (φ) =
An Extension of the Hess-Kato Theorem to Elliptic Systems
451
λ1 Mc φ < Mc φ. Therefore for small ε > 0, we have Lc (φ) < fc (x, ε φ), i.e. L(ε φ) < f (x, ε φ).
Similarly, −εφ is a negative strict supersolution.
Finally, let ϕj be the eigenfunction corresponding to µ0j . We have
1
|∇ ϕj |2 < 0,
|∇ ϕj |2 − M0 ϕj · ϕj = 1 − 0
µj
for j = 1, 2. This shows ind (J , θ) ≥ 2.
Now, we arrive at
Theorem 2.2
Assume (CF), (H0 ) and that there exist a pair of positive strict super- and
negative strict sub-solutions U and U. If 0 is not in the spectrum of L − M0 , then Eq. (2.1)
possesses at least four nontrivial solutions in the order interval [U, U].
Proof
Applying Lemma 2.3, we obtain two pairs of strict sub- and super-solutions [U, −εφ],
[εφ, U]. There exist two local minima of J, U1 and U2 , in the above two order intervals respectively. And then by the critical group characterizations, we have rank Cq (J, Ui ) = δq0 , i = 1, 2.
Since [U, U] is positively invariant under the gradient flow η t , there exists a Mountain Pass
point U3 ∈ [U, U] (cf. [8]). We claim U3 = θ. Indeed, by the assumption, θ is a nondegenerate
critical point, with index j0 =ind (J , θ) ≥ 2 > 1; but C1 (J, U3 ) = 0.
Moreover, we apply Theorem 2.1, to conclude that rank Cq (J, U3 ) = δq1 .
We now turn to the proof of the existence of the fourth nontrivial solution by contradiction.
If there is no other critical point, then we find the Leray Schauder degree of the vector field
id − K on the order interval O = [U, U], (which is finitely bounded i.e. O ∩ Ek is bounded ∀ k,
where {Ek } is a sequence of finitely dimensional linear spaces, with ∪ Ek = C01 (Ω, Rp )):
deg (id − K, O, θ) =
3
iLS (Ui ) + iLS (θ).
(2.3)
i=1
According to the relationship between Leray Schauder index and the critical groups, and the
coincidence of the critical groups in different topologies H 1 and C 1 ([7]), we have
RHS of (2.3) = 2 − 1 + (−1)j0 ≡ 0 (mod 2) .
But LHS of (2.3) = 1. This is a contradiction.
Remark 2.1
In case p = 1, Theorem 2.2 holds. In this case (CF) does not make sense, so it
is to be deleted.
How many solutions are there outside the order interval [U, U]? We need an assumption of
f at infinity.
(H∞ ) Assume that there exists a matrix M∞ (x), denoted by M∞ , satisfying conditions
(I), (II) and (III) in § 1, and that
|f (x, ξ) − M∞ (x) ξ| = ◦ (|ξ|)
as
|ξ| → ∞ .
∞
∞
∞
Let {0 < µ∞
1 < µ2 ≤ · · ·} be the positive eigevalues of L w.r.t. M∞ . We assume µ1 < µ2 <
1.
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452
Lemma 2.4
Under the assumptions of Theorem 2.2, there exists a maximal (minimal) solu-
∗
tion U (or U∗ ) in the interval [U, U], which is a local minimum of the functional J restricted
to the cone U∗ + P (or U∗ − P resp.).
Proof
Define U∗ = limm→∞ Km U. It is easy to see that Km U is decreasing and bounded
from below in the ordered space C01 . Therefore the limit exists, and is maximal in [U, U].
Let us consider the functional J˜ = J|u∗ +P . Since the negative gradient flow points inward
in U ∗ + P, it is sufficient to show that U∗ is the unique minimum of J˜ in the order interval
[U∗ , U]. This is due to the strong Maximum Principle, and the fact that the intersection of a
small neighborhood of U∗ in C01 and P is in [U∗ , U].
Again, the order interval [U∗ , U] is positively invariant, and has no critical points except U∗ .
Applying the deformation property inherited from H01 (Ω, Rp ) of the functional J˜ on [U∗ , U],
which is a consequence of the (PS) condition for J on [U∗ , U], we conclude that J˜ has a critical
point in [U∗ , U]. But since J˜ is bounded from below in [U∗ , U], U∗ must be the unique minimum.
Theorem 2.3
Assume (CF), (H∞ ), and that U < U is a pair of negative strict sub- and
positive strict super-solutions of (2.1). If 0 is not a spectrum of L − M∞ , then there exist at
/ P ∪ (−P). Moreover,
least three distinct solutions of (2.1) outside [U, U] : U± ∈ ±P and U0 ∈
if U0 is nondegenerate, then there exist at least four solutions ontside [U, U].
Proof
By (H∞ ) and the invertibility of L − M∞ , J satisfies the (PS) condition. Let J+ be the
restriction of J to the cone U∗ + P. We claim that J+ is unbounded from below. Indeed, by
definition, |f (x, ξ) − M∞ (x) ξ| = o(|ξ|), as |ξ| → ∞. It follows from a simple calculation that
1
F (x, U (x)) − (M∞ (x) U (x) · U (x)) dx = o (||U||2 ) .
2
Thus, letting ϕ1 be the first eigenvector,
t2 1 |∇ ϕ1 |2 → −∞
J+ (t ϕ1 + U ) = ◦ (t ) +
1− ∞
2
µ1
∗
2
as t → ∞.
We apply the Mountain Pass lemma on the cone U∗ + P to obtain a mountain pass point
U+ of J+ . Since the negative gradient flow points inward into the interior of P, U+ ∈ int (P),
and then it is a critical point of J. Therefore
rank Cq (J, U+ ) = rank Cq (J|C01 , U+ ) = rank Cq (J+ , U+ ) = δq1 .
Similarly, one obtains a Mountain Pass point U− in the cone U∗ − P, with rank Cq (J, U− ) = δq1 .
Applying a theorem in [7, p. 122], rank Hq (H01 , Ja ) = δj∞ q for −a large, where j∞ =ind
/ [U, U] ∪ P ∪ (−P). Denote
(id − L−1 M∞ ). We shall prove that there exists a third solution U0 ∈
1
p
˜
X1 = C0 (Ω, R ), J = J|X1 .
To this end, let W = [U, U] ∪ P ∪ (−P), Z = W ∪ J˜a , and Z1 = (−P) ∪ P ∪ J˜a . If there is no
critical point outside W , then by deformation, we have
H∗ (X, J˜a ) ∼
= H∗ (Z, J˜a ) .
(2.4)
An Extension of the Hess-Kato Theorem to Elliptic Systems
453
By an argument as used in [9], we prove that J˜a is a strong deformation retract of Z1 \{θ}.
Again Z1 is a strong deformation retract of Z, and we have
H∗ (Z, J˜a ) ∼
= H∗ (Z1 , J˜a ) ∼
= H∗ (Z1 , Z1 \{θ}) .
(2.5)
Indeed, let L = id − (−)−1 M∞ . Then by Theorem 1.1, L has the smallest eigenvalue, which
is negative, geometrically simple and is associated with the eigenfunction Φ ∈ int(P). Define a
deformation η1 (U, t) = e−L t U , ∀ t ≥ 0. ∀ U ∈ P ∪ (−P)\{θ}, we have U, Φ = 0, and hence
||e−L t U|| → ∞, as t → +∞. Applying Lemma 2.2, we see that P ∪ (−P) is invariant w.r.t.
η1 . Noticing that
d
J (e−L t U) = −||L e−L t U||2 + o (||e−L t U||2 ) → −∞ ,
dt
and that J (e−L t U) → −∞ as t → +∞, the flow meets J −1 (a) transversally at finite time,
∀ U ∈ P ∪ (−P)\{θ}. Thus η1 (Z1 \{θ}, +∞) ⊂ J˜a . It remains to show that Z1 is a strong
deformation retract of Z. In fact, there exists a constant c < 0 such that
Min {J˜ (U) U ∈ [U, U] } > c .
One chooses a < c, and define

U,
if U ∈ J˜a ,


 1
[c − (1 − t) a − t J (U)] U, if U ∈ J˜c \J˜a ,
η2 (U, t) =
c−a



(1 − t) U,
if U ∈ J˜c .
It follows that η2 (Z, 1) = Z1 . And also
H∗ (Z1 , Z1 \{θ}) ∼
= H∗ (P ∪ (−P); P ∪ (−P) \ {θ}).
(2.6)
Combining (2.4), (2.5) and (2.6), we obtain rank Hq (X, J˜a ) = δq1 . This contradicts rank
Hq (H01 , Ja ) = δqj∞ .
/ W,
The contradiction ensures the existence of the third solution. The third solution U0 ∈
which distinguishes itself from U± ∈ ±P.
Furthermore, we find a big ball BR in X such that
deg (id − K, BR ∪ [U, U], θ) = (−1)j∞ .
If there were no solutions other than U± and U0 , and if U0 is nondegenarate, then by the
additivity of the Leray Schauder degree, we would have
deg (id − K, BR ∪ [U, U], θ) = deg (id − K, [U, U], θ) + 2 + (±1) ≡ 0 (mod 2).
This is a contradiction, and the theorem is proved.
Remark 2.2
The condition that 0 is not a spectrum of L−M∞ may be replaced by some kind
of Landesman-Lazer conditions, i.e. conditions which preserve the (PS) condition at resonance.
Combining Theorems 2.2 and 2.3, we obtain
Kungching Chang
454
Theorem 2.4
Under the assumptions (H0 ), (H∞ ) and (CF), if 0 is not a spectrum of L−M0 ,
then there exist at least seven distinct nontrivial solutions of (2.1).
Remark 2.3.
Theorem 2.4 extends a result obtained in [3] to elliptic systems. In [3], the
scalar case of Theorem 2.4 is proved via the Conley index.
Remark 2.4.
Assume
(H∞ ) ∃ θ > 2 and r > 0 such that θ F (x, ξ) ≤ ξ · f (x, ξ), ∀ x ∈ Ω, |ξ| ≥ r.
In Theorem 2.4, the same conclusion holds if (H∞ ) is replaced by (H∞ ).
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