Multiple Solutions for a Nonlinear Dirichlet Problem via - Mathematics

MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET
PROBLEM VIA MORSE INDEX
Jorge Cossio ∗
Universidad Nacional de Colombia
Alfonso Castro
Harvey Mudd College, USA
Carlos Velez
Universidad Nacional de Colombia
Hawaii, April 2008

In this lecture we study the nonlinear Dirichlet problem
� Δu + f (u) = 0 in Ω,
u = 0 on ∂Ω,
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 2➲
▲ ▲
(0.1)
➲ ❏ ✘

In this lecture we study the nonlinear Dirichlet problem
� Δu + f (u) = 0 in Ω,
u = 0 on ∂Ω,
where Δ is the Laplacian operator, Ω ⊂ R N is a bounded region with
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 3➲
▲ ▲
(0.1)
smooth boundary, and f : R → R is a function of class C 1 asymptotically
linear, i.e.
.
f ′ (∞) := lím
|t|→∞ f ′ (t) ∈ R.
➲ ❏ ✘

Let λ1,λ2,λ3,... be the sequence of eigenvalues of −Δ in Ω with zero
Dirichlet boundary condition.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 4➲
▲ ▲
➲ ❏ ✘

Let λ1,λ2,λ3,... be the sequence of eigenvalues of −Δ in Ω with zero
Dirichlet boundary condition.
As it is well known, to solve (0.1) is equivalent to find critical points of the
functional J : H 1 0
where F(ξ) = � ξ
0
(Ω) → R defined by
� �
1
J(u) =
2 |∇u|2 �
− F(u)
f (s) ds.
Ω
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 5➲
dx, (0.2)
▲ ▲
➲ ❏ ✘

Throughout this paper we assume f satisfies the following hypotheses:
(f1) f (0) = 0 and f ′ (0) < λ1,
(f2) f ′ (∞) ∈ (λk,λk+1),
(f3) There is a constant γ such that f ′ (t) ≤ γ < λk+1 for every t ∈ R.
Under these conditions that we are assuming on f , J ∈ C 2 .
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 6➲
▲ ▲
➲ ❏ ✘

Throughout this paper we assume f satisfies the following hypotheses:
(f1) f (0) = 0 and f ′ (0) < λ1,
(f2) f ′ (∞) ∈ (λk,λk+1),
(f3) There is a constant γ such that f ′ (t) ≤ γ < λk+1 for every t ∈ R.
Under these conditions that we are assuming on f , J ∈ C 2 .
We recall that if u is a critical point of J and there exists a maximal
integer q such that D 2 J(u) is negative definite on some q-dimensional
subspace then the Morse index m(u,J) of u is q.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 7➲
▲ ▲
➲ ❏ ✘

Theorem A. If f satisfies (f1), (f2), (f3), k ≥ 4 is an even number, and all
the solutions of (0.1) are nondegenerate then problem (0.1)
has at least five solutions: u1 = 0, u2 > 0 in Ω, u3 < 0 in Ω, u4, and u5.
Moreover,
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 8➲
▲ ▲
➲ ❏ ✘

Theorem A. If f satisfies (f1), (f2), (f3), k ≥ 4 is an even number, and all
the solutions of (0.1) are nondegenerate then problem (0.1)
has at least five solutions: u1 = 0, u2 > 0 in Ω, u3 < 0 in Ω, u4, and u5.
Moreover,
(a) m(u4,J) = k.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 9➲
▲ ▲
➲ ❏ ✘

Theorem A. If f satisfies (f1), (f2), (f3), k ≥ 4 is an even number, and all
the solutions of (0.1) are nondegenerate then problem (0.1)
has at least five solutions: u1 = 0, u2 > 0 in Ω, u3 < 0 in Ω, u4, and u5.
Moreover,
(a) m(u4,J) = k.
(b) u5 changes sign, m(u5,J) is even and m(u5,J) ≤ k.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 10➲
▲ ▲
➲ ❏ ✘

Theorem A. If f satisfies (f1), (f2), (f3), k ≥ 4 is an even number, and all
the solutions of (0.1) are nondegenerate then problem (0.1)
has at least five solutions: u1 = 0, u2 > 0 in Ω, u3 < 0 in Ω, u4, and u5.
Moreover,
(a) m(u4,J) = k.
(b) u5 changes sign, m(u5,J) is even and m(u5,J) ≤ k.
(c) If m(u5,J) = k, there are two additional solutions u6 and u7 of (0.1).
Moreover m(u6,J) = k − 1.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 11➲
▲ ▲
➲ ❏ ✘

Theorem A. If f satisfies (f1), (f2), (f3), k ≥ 4 is an even number, and all
the solutions of (0.1) are nondegenerate then problem (0.1)
has at least five solutions: u1 = 0, u2 > 0 in Ω, u3 < 0 in Ω, u4, and u5.
Moreover,
(a) m(u4,J) = k.
(b) u5 changes sign, m(u5,J) is even and m(u5,J) ≤ k.
(c) If m(u5,J) = k, there are two additional solutions u6 and u7 of (0.1).
Moreover m(u6,J) = k − 1.
(d) If u4 is of one sign, there are also two additional solutions u6 and u7
of (0.1).
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 12➲
▲ ▲
➲ ❏ ✘

REMARK:
We prove the existence of u2 and u3 using mountain pass arguments of
the Ambrosetti-Rabinowitz type.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 13➲
▲ ▲
➲ ❏ ✘

REMARK:
We prove the existence of u2 and u3 using mountain pass arguments of
the Ambrosetti-Rabinowitz type.
We show the existence of u4 by using Lyapunov-Schmidt arguments to
reduce the solvability of (0.1) to a finite dimensional problem.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 14➲
▲ ▲
➲ ❏ ✘

REMARK:
We prove the existence of u2 and u3 using mountain pass arguments of
the Ambrosetti-Rabinowitz type.
We show the existence of u4 by using Lyapunov-Schmidt arguments to
reduce the solvability of (0.1) to a finite dimensional problem.
We use degree theory to show the existence of u5.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 15➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Because of (f1), u1 ≡ 0 is a strict local minimum of J, so
dloc(∇J,u1) = 1.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 16➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Because of (f1), u1 ≡ 0 is a strict local minimum of J, so
dloc(∇J,u1) = 1.
The existence of the two signed-solutions u2 > 0 in Ω and u3 < 0 in Ω is
proved by using the Mountain Pass Theorem. Moreover,
dloc(∇J,u2) = dloc(∇J,u3) = −1, (0.3)
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 17➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Also, if S2 (resp. S3) is a region that contains all the positive (resp.
negative) solutions of (0.1) and no one else, Castro and Cossio proved
that
d(∇J,S2,0) = −1 = d(∇J,S3,0). (0.4)
Since f ′ (∞) is not an eigenvalue of −Δ, it can be shown that there exists
an R > 0 such that BR(0) contains all the critical points of J, and
d(∇J,BR(0),0) = (−1) k . (0.5)
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 18➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Lemma 1. (the Lyapunov-Schmidt reduction method)
Let H be a real separable Hilbert space. Let X and Y be closed
subspaces of H such that H = X ⊕Y . Let J : H → R be a functional of
class C 2 . If there are m > 0 and α > 1 such that
〈∇J(x + y) − ∇J(x + y1),y − y1〉 ≥ m�y − y1� α
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 19➲
for all x ∈ X, y, y1 ∈ Y
▲ ▲
(0.6)
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Lemma 1. (the Lyapunov-Schmidt reduction method)
Let H be a real separable Hilbert space. Let X and Y be closed
subspaces of H such that H = X ⊕Y . Let J : H → R be a functional of
class C 2 . If there are m > 0 and α > 1 such that
〈∇J(x + y) − ∇J(x + y1),y − y1〉 ≥ m�y − y1� α
then:
There exists a C 1 function ψ : X → Y such that
J(x + ψ(x)) = mín
y∈Y
J(x + y).
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 20➲
for all x ∈ X, y, y1 ∈ Y
▲ ▲
(0.6)
➲ ❏ ✘

Sketch of the Proof of Theorem A:
The function ˆ
J : X → R defined by ˆ
J(x) = J(x + ψ(x)) is of class C 2 .
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 21➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
The function ˆ
J : X → R defined by ˆ
J(x) = J(x + ψ(x)) is of class C 2 .
An element x0 ∈ X is a critical point of ˆ
J if and only if x0 + ψ(x0) ∈ H is a
critical point of J.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 22➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
The function ˆ
J : X → R defined by ˆ
J(x) = J(x + ψ(x)) is of class C 2 .
An element x0 ∈ X is a critical point of ˆ
J if and only if x0 + ψ(x0) ∈ H is a
critical point of J.
Moreover,
dloc(∇J,x0 + ψ(x0)) = dloc(∇ ˆ
J,x0). (0.7)
m(u0,J) = m(x0, ˆ
J). (0.8)
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 23➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Let X be the subspace of H spanned by {ϕ1,ϕ2,...,ϕk } and let
Y := X ⊥ , thus H = X ⊕Y. Since J ∈ C 2 (H,R), there is a constant γ
such that f ′ (t) ≤ γ < λk+1 for every t ∈ R, and
〈∇J(u),v〉 =
it follows that
�
(∇u · ∇v − f (u)v) dx, ∀u,v ∈ H
Ω
1 0 (Ω), (0.9)
〈∇J(x + y) − ∇J(x + y1),y − y1〉 ≥ m�y − y1� 2
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 24➲
for all x ∈ X, y, y1 ∈ Y
▲ ▲
(0.10)
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Let X be the subspace of H spanned by {ϕ1,ϕ2,...,ϕk } and let
Y := X ⊥ , thus H = X ⊕Y. Since J ∈ C 2 (H,R), there is a constant γ
such that f ′ (t) ≤ γ < λk+1 for every t ∈ R, and
〈∇J(u),v〉 =
it follows that
�
(∇u · ∇v − f (u)v) dx, ∀u,v ∈ H
Ω
1 0 (Ω), (0.9)
〈∇J(x + y) − ∇J(x + y1),y − y1〉 ≥ m�y − y1� 2
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 25➲
for all x ∈ X, y, y1 ∈ Y
then, by Lemma 1 There exists a C 1 function ψ : X → Y such that
J(x + ψ(x)) = mín
y∈Y
J(x + y).
▲ ▲
(0.10)
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Let X be the subspace of H spanned by {ϕ1,ϕ2,...,ϕk } and let
Y := X ⊥ , thus H = X ⊕Y. Since J ∈ C 2 (H,R), there is a constant γ
such that f ′ (t) ≤ γ < λk+1 for every t ∈ R, and
〈∇J(u),v〉 =
it follows that
�
(∇u · ∇v − f (u)v) dx, ∀u,v ∈ H
Ω
1 0 (Ω), (0.9)
〈∇J(x + y) − ∇J(x + y1),y − y1〉 ≥ m�y − y1� 2
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 26➲
for all x ∈ X, y, y1 ∈ Y
then, by Lemma 1 There exists a C 1 function ψ : X → Y such that
J(x + ψ(x)) = mín
y∈Y
J(x + y).
▲ ▲
(0.10)
➲ ❏ ✘

Sketch of the Proof of Theorem A:
The function ˆ
J : X → R defined by ˆ
J(x) = J(x + ψ(x)) is of class C 2 , and
x0 ∈ X is a critical point of ˆ
J if and only if x0 + ψ(x0) ∈ H is a critical
point of J.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 27➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
The function ˆ
J : X → R defined by ˆ
J(x) = J(x + ψ(x)) is of class C 2 , and
x0 ∈ X is a critical point of ˆ
J if and only if x0 + ψ(x0) ∈ H is a critical
point of J.
Because f ′ (∞) ∈ (λk,λk+1) it follows that
ˆ
J(x) → −∞ as �x� → ∞ for x ∈ X. (0.11)
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 28➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
The function ˆ
J : X → R defined by ˆ
J(x) = J(x + ψ(x)) is of class C 2 , and
x0 ∈ X is a critical point of ˆ
J if and only if x0 + ψ(x0) ∈ H is a critical
point of J.
Because f ′ (∞) ∈ (λk,λk+1) it follows that
ˆ
J(x) → −∞ as �x� → ∞ for x ∈ X. (0.11)
Because dimX < ∞, there exists x4 ∈ X such that
J(x4) ˆ = máxJ.
ˆ
X
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 29➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
The function ˆ
J : X → R defined by ˆ
J(x) = J(x + ψ(x)) is of class C 2 , and
x0 ∈ X is a critical point of ˆ
J if and only if x0 + ψ(x0) ∈ H is a critical
point of J.
Because f ′ (∞) ∈ (λk,λk+1) it follows that
ˆ
J(x) → −∞ as �x� → ∞ for x ∈ X. (0.11)
Because dimX < ∞, there exists x4 ∈ X such that
J(x4) ˆ = máxJ.
ˆ
X
Hence, u4 = x4 + ψ(x4) is a critical point of J.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 30➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
The function ˆ
J : X → R defined by ˆ
J(x) = J(x + ψ(x)) is of class C 2 , and
x0 ∈ X is a critical point of ˆ
J if and only if x0 + ψ(x0) ∈ H is a critical
point of J.
Because f ′ (∞) ∈ (λk,λk+1) it follows that
ˆ
J(x) → −∞ as �x� → ∞ for x ∈ X. (0.11)
Because dimX < ∞, there exists x4 ∈ X such that
J(x4) ˆ = máxJ.
ˆ
X
Hence, u4 = x4 + ψ(x4) is a critical point of J.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 31➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Observe that x4 is a point of minimum of − ˆ
J. Thus, dloc(∇(− ˆ
J),x4) = 1
and dloc(∇ ˆ
J,x4) = (−1) k .
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 32➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Observe that x4 is a point of minimum of − ˆ
J. Thus, dloc(∇(− ˆ
J),x4) = 1
and dloc(∇ ˆ
J,x4) = (−1) k .
In virtue of (0.7),
dloc(∇J,u4) = (−1) k = 1. (0.12)
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 33➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Observe that x4 is a point of minimum of − ˆ
J. Thus, dloc(∇(− ˆ
J),x4) = 1
and dloc(∇ ˆ
J,x4) = (−1) k .
In virtue of (0.7),
dloc(∇J,u4) = (−1) k = 1. (0.12)
Now, because of our hypothesis of nondegeneracy of critical points of J
and the characterization of x4, m(x4, ˆ
J) = k. Lemma 1 implies that
m(u4,J) = k and we have proved part (a) of Theorem A.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 34➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Let K be the set of critical points of J. Let S1, S2, S3 be disjoint open
bounded regions in H such that S1 ∩ K = {u1}, S2 ∩ K is the set of
positive solutions of (1), and S3 ∩ K is the set of negative solutions of
(0.1). We consider two cases, which will lead us to prove (b) and (d) of
Theorem A.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 35➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Suppose first that u4 changes sign. Let S4 be an open bounded region
disjoint from S1 ∪ S2 ∪ S3 such that S4 ∩ K = {u4}. Then, because of
(0.4), (0.5), (0.12) and the excision property of Leray-Schauder degree,
(−1) k = d(∇J,BR(0),0)
= d(∇J,S1,0) + d(∇J,S2,0) + d(∇J,S3,0) + d(∇J,S4,0)
+ d(∇J,BR(0) \ S1 ∪ S2 ∪ S3 ∪ S4,0)
= 1 − 1 − 1 + 1 + d(∇J,BR(0) \ S1 ∪ S2 ∪ S3 ∪ S4,0).
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 36➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Suppose first that u4 changes sign. Let S4 be an open bounded region
disjoint from S1 ∪ S2 ∪ S3 such that S4 ∩ K = {u4}. Then, because of
(0.4), (0.5), (0.12) and the excision property of Leray-Schauder degree,
(−1) k = d(∇J,BR(0),0)
= d(∇J,S1,0) + d(∇J,S2,0) + d(∇J,S3,0) + d(∇J,S4,0)
+ d(∇J,BR(0) \ S1 ∪ S2 ∪ S3 ∪ S4,0)
= 1 − 1 − 1 + 1 + d(∇J,BR(0) \ S1 ∪ S2 ∪ S3 ∪ S4,0).
Therefore, d(∇J,BR(0) \ S1 ∪ S2 ∪ S3 ∪ S4,0) = 1.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 37➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Consequently, there must exist another critical point u5 of J whose local
degree is
1 = (−1) m(u5,J) .
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 38➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Consequently, there must exist another critical point u5 of J whose local
degree is
Thus, m(u5,J) is even.
1 = (−1) m(u5,J) .
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 39➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Consequently, there must exist another critical point u5 of J whose local
degree is
Thus, m(u5,J) is even.
Since
〈D 2 J(u)y,y〉 ≥ (1 − γ
1 = (−1) m(u5,J) .
λk+1
)�y� 2 ; ∀u ∈ H ∀y ∈ Y. (0.13)
it follows that m(u5,J) ≤ k. Observe that u5 is a sign-changing solution of
(0.1). This prove Theorem A part (b).
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 40➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Now suppose u4 is of one sign. Without loss of generality we can
assume u4 ∈ S2. Let P be a subregion of S2 such that P ∩ K = {u2,u4}.
From (0.3), (0.4), (0.12) and the excision property of Leray-Schauder
degree it follows that
−1 = d(∇J,S2,0)
= dloc(∇J,u2) + dloc(∇J,u4) + d(∇J,S2 \ P,0)
= −1 + (−1) k + d(∇J,S2 \ P,0).
Hence, the existence property of the Leray-Schauder degree implies
there must be another signed solution of (0.1).
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 41➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
Let us denote by u5 this solution. Arguing as in the case in which u4
changes sign, one can get the existence of u6 and u7 which are
sign-changing solutions of (0.1). We have completed the proof of (b) and
(d) in Theorem A.
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 42➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
We finally prove (c). Let us suppose m(u5,J) = k. Writing u5 as
u5 = x5 + ψ(x5), Lemma 1 asserts that m(x5, J) ˆ = k and so x5 is a local
maximum of J.
ˆ
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 43➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
We finally prove (c). Let us suppose m(u5,J) = k. Writing u5 as
u5 = x5 + ψ(x5), Lemma 1 asserts that m(x5, J) ˆ = k and so x5 is a local
maximum of J. ˆ
Thus, x4 and x5 are points of minima of − ˆ
J. The Mountain Pass
Theorem implies that there exists a critical point x6 of the mountain pass
type for −J. ˆ
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 44➲
▲ ▲
➲ ❏ ✘

Sketch of the Proof of Theorem A:
We finally prove (c). Let us suppose m(u5,J) = k. Writing u5 as
u5 = x5 + ψ(x5), Lemma 1 asserts that m(x5, J) ˆ = k and so x5 is a local
maximum of J. ˆ
Thus, x4 and x5 are points of minima of − ˆ
J. The Mountain Pass
Theorem implies that there exists a critical point x6 of the mountain pass
type for −J. ˆ
Because of nondegeneracy, m(x6,− ˆ
J) = 1 (see [Hofer]). Hence,
m(x6, ˆ
J) = k − 1. Again, by Lemma 1 u6 = x6 + ψ(x6) is a critical point of
J whose Morse index is k − 1. Finally, using a degree counting, as
before, we get u7. In this way we got (c), and our proof is complete. �
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 45➲
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References
Referencias
[1] A. Castro and J. Cossio, Multiple Solutions for a Nonlinear Dirichlet
Problem, SIAM J. Math. Anal., Vol. 25 (1994), 1554-1561.
[2] A. Castro, J. Cossio, and J. M. Neuberger, A Sign-Changing
Solution for a Superlinear Dirichlet Problem, Rocky Mountain J.
Math., Vol. 27, 4 (1997), 1041-1053.
[3] A. Castro, J. Cossio, and J. M. Neuberger, On Multiple Solutions of
a Nonlinear Dirichlet Problem, Nonlinear Analysis, Vol. 30, 6 (1997),
3657-3662.
[4] A. Castro and A. C. Lazer, Critical Point Theory and the Number of
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MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 46➲
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References
Solutions of a Nonlinear Dirichlet Problem, Ann. Mat. Pura Appl.,
Vol. 70, No. 4 (1979), pp. 113-137.
[5] J. Cossio and S. Herrón, Nontrivial Solutions for a Semilinear
Dirichlet Problem with Nonlinearity Crossing Multiple Eigenvalues,
Journal of Dynamics and Differential Equations, Vol. 16, No. 3, July
2004, 795-803.
[6] J. Cossio and C. Vélez, Soluciones no Triviales para un Problema
de Dirichlet Asintóticamente Lineal, Rev. Colombiana Mat., Vol. 37,
No. 1, (2003), 25-36.
[7] H. Hofer, The Topological Degree at a Critical Point of Mountain
Pass Type, Proc. Sympos. Pure Math., Vol. 45, (1986), 501-509.
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MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 47➲
▲ ▲

References
[8] P. Rabinowitz, Minimax Methods in Critical Point Theory with
Applications to Differential Equations, Regional Conference Series
in Mathematics, 65, Providence, R.I., AMS (1986).
MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM VIA MORSE INDEX – 48➲
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Fin
Copyright c○ 2008
Jorge Cossio ∗
Universidad Nacional de Colombia
Alfonso Castro
Harvey Mudd College, USA
Carlos Velez
Universidad Nacional de Colombia
Realizada con ujislides c○ 2002-3 Sergio Barrachina (barrachi@icc.uji.es)