Chipot m quittner p (eds ) handbook of differential equations stationary partial differential equations vol 1

H ANDBOOK
OF D IFFERENTIAL E QUATIONS
S TATIONARY PARTIAL
D IFFERENTIAL E QUATIONS
VOLUME I


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H ANDBOOK
OF D IFFERENTIAL E QUATIONS
S TATIONARY PARTIAL
D IFFERENTIAL E QUATIONS
Volume I

Edited by

M. CHIPOT
Institute of Mathematics, University of Zurich, Zurich, Switzerland

P. QUITTNER
Institute of Applied Mathematics, Comenius University,
Bratislava, Slovak Republic

2004
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Preface
This handbook is Volume I in a multi-volume series devoted to stationary partial differential equations. It is a collection of self contained, state-of-the-art surveys written by
well-known experts in the field. The authors have made an effort to achieve readability
for mathematicians and scientists from other fields, and we hope that this series of handbooks will become a new reference for research, learning and teaching.
Partial differential equations represent one of the most rapidly developing topics in mathematics. This is due to their numerous applications in science and engineering on one hand
and to the challenge and beauty of associated mathematical problems on the other. This volume consists of eight chapters covering a variety of elliptic problems and explaining many
useful ideas, techniques and results. Although the central theme is the mathematically rigorous analysis, many of the contributions are enriched by a plenty of figures originating in
numerical simulations.
We thank all the contributors for their clearly written and elegant articles, and Arjen
Sevenster at Elsevier for efficient collaboration.
M. Chipot and P. Quittner

v


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List of Contributors
Bandle, C., Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland (Ch. 1)
Galdi, G.P., University of Pittsburgh, 15261 Pittsburgh, USA (Ch. 2)
Ni, W.-M., University of Minnesota, Minneapolis, MN 55455, USA (Ch. 3)
Pedregal, P., Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain (Ch. 4)
Reichel, W., Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland (Ch. 1)
Shafrir, I., Technion, Israel Institute of Technology, 32000 Haifa, Israel (Ch. 5)
Takáˇc, P., Universität Rostock, D-18055 Rostock, Germany (Ch. 6)

Tarantello, G., Università di Roma ‘Tor Vergata’, Dipartimento di Matematica, Via della
Ricerca Scientifica, 1, 00133 Rome, Italy (Ch. 7)
Véron, L., Université de Tours, Parc de Grandmont, 37200 Tours, France (Ch. 8)

vii


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Contents
Preface
List of Contributors

v
vii

1. Solutions of Quasilinear Second-Order Elliptic Boundary Value Problems via
Degree Theory
C. Bandle and W. Reichel
2. Stationary Navier–Stokes Problem in a Two-Dimensional Exterior Domain
G.P. Galdi
3. Qualitative Properties of Solutions to Elliptic Problems
W.-M. Ni
4. On Some Basic Aspects of the Relationship between the Calculus of Variations
and Differential Equations
P. Pedregal
5. On a Class of Singular Perturbation Problems
I. Shafrir
6. Nonlinear Spectral Problems for Degenerate Elliptic Operators

P. Takáˇc
7. Analytical Aspects of Liouville-Type Equations with Singular Sources
G. Tarantello
8. Elliptic Equations Involving Measures
L. Véron
Author Index
Subject Index

1
71
157

235
297
385
491
593

713
721

ix


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CHAPTER 1

Solutions of Quasilinear Second-Order Elliptic

Boundary Value Problems via Degree Theory

Catherine Bandle and Wolfgang Reichel
Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland
E-mail: {catherine.bandle;wolfgang.reichel}@unibas.ch

Contents
1. Degree theory . . . . . . . . . . . . . . . . . . .
1.1. Introduction . . . . . . . . . . . . . . . .
1.2. Brouwer degree in finite dimensions . . .
1.3. Leray–Schauder degree in Banach spaces
1.4. The index of an isolated solution . . . . .
1.5. Asymptotically linear equations . . . . .
1.6. Fixed point alternatives . . . . . . . . . .
1.7. Degree theory in unbounded domains . .
1.8. Degree theory in cones . . . . . . . . . .
1.9. Notes . . . . . . . . . . . . . . . . . . . .
2. Existence of solutions . . . . . . . . . . . . . .
2.1. Function spaces . . . . . . . . . . . . . .
2.2. Uniformly elliptic linear operators . . . .
2.3. Schauder estimates . . . . . . . . . . . . .
2.4. Lp -estimates . . . . . . . . . . . . . . . .
2.5. Applications to boundary value problems
2.6. Comparison principles . . . . . . . . . . .
2.7. Degree between sub- and supersolutions .
2.8. Emden–Fowler type equations . . . . . .
2.9. Multiplicity results . . . . . . . . . . . . .
2.10. Notes . . . . . . . . . . . . . . . . . . . .
3. Global continuation of solutions . . . . . . . . .
3.1. A global implicit function theorem . . . .

3.2. Applications – continuation of solutions .
3.3. Further applications . . . . . . . . . . . .
3.4. Notes . . . . . . . . . . . . . . . . . . . .
4. Bifurcation theory and related problems . . . .

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HANDBOOK OF DIFFERENTIAL EQUATIONS

Stationary Partial Differential Equations, volume 1
Edited by M. Chipot and P. Quittner
© 2004 Elsevier B.V. All rights reserved
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3
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31

34
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48
50


2

C. Bandle and W. Reichel

4.1. Bifurcation from the trivial solution
4.2. Bifurcation from infinity . . . . . .
4.3. Perturbations at resonance . . . . . .
Acknowledgments . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . .

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50
58
65
68
68


Solutions of quasilinear second-order elliptic boundary value problems via Degree Theory

3

1. Degree theory
1.1. Introduction
In this chapter we shall develop a tool for proving the existence of solutions of nonlinear
equations in a Banach space X of the form
F (x) = y,

x ∈ Ω ⊂ X,

where F : Ω ⊂ X → X is a continuous map. We want to study the solutions in the interior of Ω knowing the restriction of F onto the boundary ∂Ω. This will be achieved by
considering a topological invariant defined on the triple (F, Ω, y).
Such an invariant can easily be found for continuously differentiable functions
F : [0, 1] → R with isolated solutions {xi }ki=1 of F (x) = y. Let us fix F (0) and F (1).
It is clear that for given y ∈
/ {F (0), F (1)} the number of solutions varies with F but
k
sign
F
(x
)
is
invariant
under deformations of F which keep the endpoints fixed,
i
i=1
cf. Figures 1 and 2. More generally, F (0) and F (1) can also be deformed as long as they
do not cross y. As soon as one of the endpoints coincides with y, the invariance under
deformations is lost, cf. Figure 3. If the solutions are not isolated or if F (xi ) = 0 then
a natural approach is to approximate F by functions Fn with isolated solutions, cf. Figure 4. Heuristically, the quantity described above seems to be stable if we pass to the limit
Fn → F .
For analytic functions F : Ω ⊂ C → C the argument principle can be employed to determine the number of solutions F (z) = w in a given domain. More precisely, if γ is a simple
closed curve in Ω on which F is different from w then the number of solutions inside γ is
F (z)
1
given by the boundary integral 2πi
γ F (z)−w dz. Obviously this integral is invariant under
“small” deformations of F on γ .
In the subsequent sections these simple observations will be generalized to large classes

of equations in finite and infinite-dimensional spaces. The quantities ki=1 sign F (xi ) and
F (z)
1
2πi γ F (z)−w dz will be replaced by a more general concept, namely the topological degree. In many cases it will be impossible to compute it directly. For the applications two
properties will be crucial:
1. If the degree is different from zero then a solution of F (x) = y exists.
2. The degree is invariant under certain deformations.
The definition and use of the degree goes back to Brouwer (1912) [16] and Leray and
Schauder (1934) [49]. Since we are mainly interested in the degree theory as a tool for
proving the existence of solutions to certain equations and less in its geometrical meaning
we shall adopt an axiomatic approach common in analysis. It consists first in listing the
desired properties, then in proving that there is at most one quantity satisfying all these
conditions, and finally in discussing one of several possible constructions of the degree.
This text is intended for nonspecialists. The goal is to present a powerful tool for proving
existence of solutions of linear and nonlinear second-order elliptic boundary value problems and to recount some of the most interesting properties and applications. Rather than
describing more recent topological developments of the notion of degree and its properties


4

C. Bandle and W. Reichel

Fig. 1.

Fig. 2.

Fig. 3.

Fig. 4.

we discuss in some detail different classes of boundary value problems for which variational methods do not apply.
The completeness of the proofs varies. Full details are given if the proofs are not available in the literature or if they contribute to a better understanding. The more difficult
technical proofs are only sketched and references are suggested.

1.2. Brouwer degree in finite dimensions
In finite dimensions the notion of degree goes back to Brouwer [16]. The proofs of the next
two sections can be found in [26]. Let Ω ⊂ RN be a bounded open set and G : Ω → RN
be a continuous map. Let Id : RN → RN denote the identity map.
/ G(∂Ω). The degree is a mapping deg : (G, Ω, y) → Z
D EFINITION 1.1. Suppose that y ∈
with the following properties:
(d1) Normalization: deg(Id, Ω, y) = 1 if y ∈ Ω and deg(Id, Ω, y) = 0 if y ∈
/ Ω.


Solutions of quasilinear second-order elliptic boundary value problems via Degree Theory

5

(d2) Excision: if Ω = Ω1 ∪ Ω2 with Ω1 , Ω2 open, disjoint and y ∈
/ G(∂Ω1 ∪ ∂Ω2 ) then
deg(G, Ω, y) = deg(G, Ω1 , y) + deg(G, Ω2 , y).
(d3) Homotopy invariance: if h : [0, 1] × Ω → RN is continuous and y : [0, 1] → RN is
continuous with y(t) ∈
/ h(t, ∂Ω) for all t ∈ [0, 1] then
deg h(t, ·), Ω, y(t) is independent of t.
(d4) Existence: if deg(G, Ω, y) = 0 then G(x) = y has a solution x ∈ Ω.
It can be shown that (d1)–(d3) imply (d4). Moreover, there is at most one function satisfying (d1)–(d3) (cf., e.g., Deimling [26]). One can show the following extension of the
homotopy invariance (d3), cf. Amann [3] and Leray and Schauder [49]:
(d3)g General homotopy invariance: let Θ ⊂ [0, 1] × RN be bounded and open in

[0, 1] × RN and denote by Θt the slice at t, that is,
Θt = x ∈ RN : (t, x) ∈ Θ .
/
If h : Θ → RN is continuous and y : [0, 1] → RN is continuous with y(t) ∈
h(t, ∂Θt ) for all t ∈ [0, 1] then
deg h(t, ·), Θt , y(t) is independent of t.
For the construction of the degree we proceed in several steps.
(I) Degree for regular values of C 1 -maps. Let G ∈ C 1 (Ω) and denote by G (x) its
Jacobian and by det G (x) the determinant of the Jacobian. Furthermore y ∈ RN is called
a regular value of G if det G (x) = 0 for all x ∈ G−1 (y). Otherwise y is called a singular
value. If y ∈
/ G(∂Ω) is a regular value then we define
deg(G, Ω, y) :=

sign det G (x).
x∈G−1 (y)

It can be shown that for small ε > 0 the following integral representation holds
deg(G, Ω, y) =

φε G(x) − y det G (x) dx,
Ω

where φε (x) = ε−N φ1 (x/ε) and φ1 ∈ C0∞ (RN ) with φ1 (0) > 0, RN φ1 (x) dx = 1. This
integral representation plays a key role in the analytic approach to degree theory.


6

C. Bandle and W. Reichel

(II) Degree for singular values of C 2 -maps. Let G ∈ C 2 (Ω). For y ∈
/ G(∂Ω) let y1 be
a regular value with |y1 − y| < dist(y, G(∂Ω)). By Sard’s lemma, which states that the
set of singular values has N -dimensional Lebesgue measure 0, such a value always exists.
Since it can be shown that deg(G, Ω, y1 ) is independent of the choice of y1 the following
definition
deg(G, Ω, y) := deg(G, Ω, y1)
makes sense. The proof is done through the integral representation.
E XAMPLE 1.1. Consider Ω = (−1, 1) and G(x) = x 3 . The value y = 0 is a singular value,
but any neighboring value y1 = δ is regular. Then deg(G,
Ω, y1 ) =√
sign G (δ 1/3 ) = 1. If
√
2
G(x) = x then similarly deg(G, Ω, y1 ) = sign G (− δ) + sign G ( δ) = 0.
E XAMPLE 1.2. Consider Ω = {x12 + x22 < 1} and G(x1 , x2 ) = (x13 − x1 x22 , x23 ). The value
y = (0, 0) is singular, and the neighboring value y1 = (0, δ 3 ) with δ > 0 is regular. The preimage G−1 (y1 ) consists of the three points (0, δ), (δ, δ) and (−δ, δ). In the first point G
has a negative and in the last two points a positive determinant. Hence deg(G, Ω, y) = 1.
(III) Degree for continuous maps. An important fact of the degree is that it can be
extended to maps which are merely continuous. Let G ∈ C(Ω) and y ∈
/ G(∂Ω). Let
H ∈ C 2 (Ω) be such that G−H ∞ < dist(y, G(∂Ω)). Then it turns out that deg(H, Ω, y)
is independent of the choice of H . Therefore we can set
deg(G, Ω, y) := deg(H, Ω, y).
(IV) Degree in finite-dimensional spaces. The concept of degree is easily extended to
arbitrary spaces of finite dimensions which are different from RN . Let (X, · ) be an
N -dimensional normed space. Suppose Ω ⊂ X is an open, bounded set, G ∈ C(Ω) and let
y∈
/ G(∂Ω). Let L : X → RN be a linear homeomorphism. Then

deg(G, Ω, y) := deg L ◦ G ◦ L−1 , LΩ, Ly
is independent of the choice of L.
A consequence of the elementary properties of degree theory is the following theorem.
T HEOREM 1.2 (Brouwer’s fixed point theorem). Every continuous map F : B1 (0) →
B1 (0), where B1 (0) is the open unit ball {x ∈ RN : x < 1} has a fixed point.
P ROOF. If there is no fixed point on the boundary of B1 (0) we consider the homotopy
h(t, x) = Id −tF (x). There is no zero of h(t, ·) on ∂B1 (0), because for t = 1 this is excluded by assumption and for 0 t < 1 we have x − tF (x)
1 − t > 0 if x = 1. Thus
deg(h(t, ·), B1 (0), 0) is well defined. From the homotopy invariance (d3) we conclude that
deg(h(t, ·), B1 (0), 0) = deg(Id, B1 (0), 0) = 1 which by (d4) establishes the assertion.


Solutions of quasilinear second-order elliptic boundary value problems via Degree Theory

7

1.3. Leray–Schauder degree in Banach spaces
We wish to extend the previous results to infinite-dimensional spaces. However, one
needs to be careful: although Brouwer’s fixed point theorem follows immediately from
the elementary properties of the degree, its generalization to infinite dimensions is false
(cf. notes). A large class of nonlinear maps for which it is still valid is the class of continuous compact maps. And likewise the topological degree can be defined for continuous
compact perturbations of the identity.
Suppose (X, · ) is a real Banach space. Let Ω = ∅ be an open, bounded set in X and let
F : Ω → X be compact which means that F is continuous and maps bounded closed sets
into compact sets. In contrast to the Brouwer degree, which is defined for any continuous
map, the Leray–Schauder degree is defined only for compact perturbations of the identity,
namely G = Id −F .
T HEOREM 1.3. Let the above assumptions hold. If y ∈
/ (Id −F )(∂Ω) then there exists a
unique mapping deg : (Id −F, Ω, y) → Z for which the properties (d1), (d2) and (d4) of

Definition 1.1 hold with G replaced by Id −F and for which (d3) holds in the following
form:
(d3) Homotopy invariance: if k : [0, 1] × Ω → X is compact in R × X and y :
[0, 1] → X is continuous with y(t) ∈
/ (Id −k(t, ·))(∂Ω) for all t ∈ [0, 1] then
deg(Id −k(t, ·), Ω, y(t)) is independent on t.
As for the Brouwer degree one can generalize (d3) :
(d3)g General homotopy invariance: let Θ ⊂ [0, 1] × X be bounded and open in
[0, 1] × X with Θt = {x ∈ X: (t, x) ∈ Θ}. If k : Θ → X is compact and
y : [0, 1] → X is continuous with y(t) ∈
/ (Id −k(t, ·))(∂Θt ) for all t ∈ [0, 1] then
deg(Id −k(t, ·), Θt , y(t)) is independent of t.
The class of maps Id −F , F compact is by no means the most general class for which
the degree can be defined. It is, however, sufficiently broad to include the applications
discussed here.
The fundamental idea in infinite-dimensional degree theory goes back to Schauder. It
consists of the following approximation of compact maps F defined on bounded sets Ω:
for every ε > 0 there exists a continuous map Fε : Ω → Xε ⊂ X with finite-dimensional
range Xε such that F (x) − Fε (x) < ε for all x ∈ Ω . In general the approximation Fε is
not unique. However, it turns out that the degree for Id −Fε on Ω ∩ Xε is well defined,
provided 0 < ε ε0 = dist(y, (Id −F )(∂Ω)). We then define
deg(Id −F, Ω, y) := deg(Id −Fε , Ω ∩ Xε , y).
This definition makes sense since the latter is independent of the choice of the Schauder
approximation and independent of ε ∈ (0, ε0 ).
1.3.1. Retracts and Schauder’s fixed point theorem
D EFINITION 1.4. A subset R of a Banach space X is called a retract of X if there exists
a continuous map r : X → R such that r|R = Id. The map r is called a retraction.


8

C. Bandle and W. Reichel

E XAMPLES . (1). The closed unit ball is a retract. Consider the map r(x) = x/ x
x > 1 and r(x) = x elsewhere.
(2). Dugundji [27] proved that closed convex sets are retracts.

2

if

T HEOREM 1.5 (Schauder). (i) Let X be a Banach space, C ⊂ X nonempty closed bounded
and convex. If F : C → C is compact then F has a fixed point.
(ii) The same is true if C is homeomorphic to a closed bounded and convex set.
P ROOF. (i) By Dugundji’s theorem C is a retract. Let r : X → C be the retraction. Consider
the map F ◦ r : X → C. Any fixed point of F ◦ r is a fixed point of F . Let Bρ (0) be
a large ball containing C. The map F ◦ r has no fixed point on ∂Bρ (0). Consider the
homotopy k(t, x) := tF (r(x)) for t ∈ [0, 1]. There is no fixed point of k(t, ·) on ∂Bρ (0),
because for t = 1 this has already been excluded, and for t < 1 we have k(t, x) < ρ
if x
ρ. By the homotopy invariance of the degree we get deg(Id −F ◦ r, Bρ (0), 0) =
deg(Id, Bρ (0), 0) = 1, i.e., F ◦ r has a fixed point in Bρ (0). This proves the theorem if C is
closed bounded and convex.
(ii) Suppose now that C = g(C0 ) where C0 is closed bounded and convex and
g : C0 → C is a homeomorphism. Then g −1 ◦ F ◦ g : C0 → C0 has a fixed point x ∈ C0 ,
i.e., g(x) ∈ C is a fixed point of F .
1.3.2. Tools for calculating the degree
T HEOREM 1.6 (Dimension reduction). Let (X, · ) be a Banach space and (X0 , · )
be a closed subspace. Suppose F : Ω ⊂ X → X0 is compact. Let y ∈ X0 be such that
y∈

/ (Id −F )(∂Ω). Then
deg(Id −F, Ω, y) = deg(Id −F |X0 ∩Ω , X0 ∩ Ω, y).
The property is first established for maps with finite-dimensional range. Then it is used
to show that the Leray–Schauder degree does not depend on the particular Schauder approximation. Finally the dimension reduction is proved for all compact perturbations of the
identity. The basis for the general dimension reduction formula is illustrated next.
E XAMPLE 1.3. Consider a linear map F : Rn → Rk Rn given by F (x) = Ax with an
n × n matrix A. Since F maps into Rk with k < n the matrix A can be written as follows:
A=

B
0

C
0

,

where B, C are k × k and k × (n − k) matrices. The derivative of Id −F at x is Id −A given
by
Id −A =

Idk×k −B
0

−C
.
Id(n−k)×(n−k)

Therefore det(Id −A) = det(Id −B).

Solutions of quasilinear second-order elliptic boundary value problems via Degree Theory

9

L EMMA 1.7. Suppose F1 , F2 : Ω → X are compact and y ∈
/ (Id −F1 )(∂Ω). If F1 = F2
on ∂Ω then deg(Id −F1 , Ω, y) = deg(Id −F2 , Ω, y).
P ROOF. We define the homotopy k(t, x) := tF1 (x)+(1−t)F2 (x) for t ∈ [0, 1]. On ∂Ω we
have k(t, x) = F1 (x) = F2 (x). Therefore y ∈
/ (Id −k(t, ·)(∂Ω)) and deg(Id −k(t, ·), Ω, y)
is invariant for t ∈ [0, 1].
1.3.3. Degree for linear maps
L EMMA 1.8 (Product formula). (a) Let K, L : X → X be linear and compact with
Id −K, Id −L injective and suppose 0 ∈ Ω. Then
deg (Id −K) ◦ (Id −L), Ω, 0 = deg(Id −K, Ω, 0) · deg(Id −L, Ω, 0).
(b) Let K : X → X be linear and compact with Id −K injective. Let also X = V ⊕ W
with closed subspaces V , W such that K : V → V and K : W → W . Then
deg Id −K, B1 (0), 0
= deg Id −K|V , B1 (0) ∩ V , 0 · deg Id −K|W , B1 (0) ∩ W, 0 .
Part (a) reflects the multiplication rule for the determinant of products of matrices.
Part (b) is best understood by an example: suppose the block-matrix A : Rn → Rn is given
by
A=

B
0

0
C

,

with a k × k-matrix B and an (n − k) × (n − k)-matrix C. Thus A maps the k-dimensional
subspace V and the (n − k)-dimensional subspace W into itself. It is immediate that
det(Id −A) = det(Id −B) · det(Id −C), and therefore Part (b) holds for this example.
In order to state a degree formula for Id −K, where K is a compact linear operator, we
recall the main facts from the classical Fredholm–Riesz–Schauder theory. Let 0 = λ ∈ R be
an eigenvalue of a compact linear operator K. Its eigenspace is finite-dimensional, and the
dimension of the eigenspace is called the geometric multiplicity of λ. For each n = 1, 2, . . .
consider the operator (K − λ Id)n , its nullspace Nn and its range Rn . There exists an integer
n0 = n0 (λ) 1 such that
N1

N2

···

Nn0 = Nn0 +1 = Nn0 +2 = · · · ,

R1

R2

···

Rn0 = Rn0 +1 = Rn0 +2 = · · · .

The set Nn0 (λ) is called the generalized nullspace of K − λ Id and m(λ) = dim Nn0 (λ) is
called the algebraic multiplicity of the eigenvalue λ. The set Rn0 (λ) is called the generalized

range.


10

C. Bandle and W. Reichel

If λ is simple we have the well-known Fredholm alternative X = N1 ⊕ R1 . In the general
case one has X = Nn0 (λ) ⊕ Rn0 (λ) . Moreover, K maps Nn0 (λ) to Nn0 (λ) , Rn0 (λ) to Rn0 (λ)
and K − λ Id has a bounded inverse on Rn0 (λ) .
L EMMA 1.9. Let K : X → X be linear and compact with Id −K injective and suppose
0 ∈ Ω. Then
deg(Id −K, Ω, 0) = (−1)β ,

where β =

m(λ).
λ>1

The sum is taken over all eigenvalues λ > 1 of K and m(λ) is the algebraic multiplicity
of λ.
To understand the formula take a real matrix A in Jordan normal form. Calculating
sign det(Id −A) amounts to counting the number of negative entries in the diagonal. Thus
the contribution comes only from the eigenvalues of A larger than 1, each with its algebraic
multiplicity.
R EMARK . Observe that the degree formula in Lemma 1.9 remains valid for deg(Id −
K − x0 , Ω, 0) provided (Id −K)−1 x0 ∈ Ω.

1.4. The index of an isolated solution
Suppose the solution set of (Id −F )(x) = y with F compact consists of isolated points,

and let x0 be such a solution. Then x0 is the only solution in some ball Bε0 (x0 ). Therefore
deg(Id −F, Bε (x0 ), y) is independent of ε for 0 < ε < ε0 . We define the index of an isolated
solution x0 by means of the degree as follows:
ind(Id −F, x0 , y) = deg Id −F, Bε (x0 ), y

for small ε.

In general, it is difficult to determine the index. We shall list some cases where this can be
done. Recall that if F is compact and differentiable then its Fréchet derivative F (x0 ) is a
compact linear operator.
T HEOREM 1.10 (Leray–Schauder). Under the preceding assumptions and if Id −F (x0 )
is injective we have that ind(Id −F, x0 , y) = ±1. More precisely,
ind(Id −F, x0 , y) = ind Id −F (x0 ), x0 , y
= (−1)β ,

β=

m(λ).
λ>1

The sum is taken over all eigenvalues λ > 1 of F (x0 ) and m(λ) is the algebraic multiplicity
of λ.


Solutions of quasilinear second-order elliptic boundary value problems via Degree Theory

11

P ROOF. Without loss of generality we may assume that y = 0 and that x0 = 0 is the isolated solution. Then for x near the origin we have (Id −F )(x) = (Id −F (0))x − ω(x),
where ω(x) / x → 0 as x → 0. Hence deg(Id −F − tω, Bε (0), 0) is well defined for

all t ∈ [0, 1] and ε > 0 sufficiently small. Moreover it is independent of t. Consequently,
we get that ind(Id −F, 0, 0) coincides with deg(Id −F (0), Bε (0), 0). Lemma 1.9 applies
and proves the assertion.
The condition that Id −F (x0 ) is injective in the previous theorem is necessary, as the
following examples shows:
E XAMPLE . Let F (x) = −x 2 + x for x ∈ R. The only solution of x − F (x) = 0 is x0 = 0.
Then Id −F (x0 ) = 0. The index of x0 vanishes, cf. Example 1.1 in Section 1.2.
In the next theorems we consider potential operator on a Hilbert space H. Let
g : Bε (x0 ) → R be a C 1 -functional and let ∇g(x) be its gradient, i.e., the Riesz representation of its Fréchet derivative g (x).
T HEOREM 1.11 (Rabinowitz [61]). Suppose that ∇g(x) = x − F (x) where F is compact.
If x0 is an isolated local minimum of g then ind(∇g, x0 , 0) = 1.
Rather than giving the proof we illustrate this result in the finite-dimensional case. Let
g : RN → R. If 0 is a critical point of g then under suitable regularity assumptions we have
for small |x| that g(x) = g(0) + 12 (g (0)x, x) + o(|x|2), where g (0) is the Hessian of g
at 0. If 0 is a nondegenerate minimum all eigenvalues are positive and thus its index is 1.
Notice that the index of a nondegenerate isolated maximum is (−1)N . It depends on the
dimension N of the underlying space.
The next example deals with saddle points, i.e., critical points which are neither local maxima nor minima. The index will depend on the type of saddle point as it is
seen in the following example. Consider the function g : RN → R given by g(x) =
2
− si=1 ai xi2 + N
/ {0, N} then 0 is a saddle point
i=s+1 bi xi where ai > 0 and bi > 0. If s ∈
s
and ind(∇g, 0, 0) = (−1) . The case s = 1 has received special attention. Its topological
properties can be described in a more general setting as follows: Suppose U ⊂ X is a nonempty open set. For a C 1 -functional g : U → R and c ∈ R we define Mc := g −1 ((−∞, c)).
The next definition is due to Hofer [39].
D EFINITION 1.12. Let 0 be a critical point of g with g(0) = c. The point 0 is said to be
of mountain pass type if for all open neighborhoods W of 0 the set W ∩ Mc is nonempty
and not path connected.

This definition of a critical point of mountain pass type is satisfied by a mountain pass
point in the sense of Ambrosetti and Rabinowitz. Notice that in the previous example 0
is of mountain pass type if and only if s = 1. Hofer [39] has extended Theorem 1.11 to
critical points of mountain pass type.
T HEOREM 1.13 (Hofer [39]). Let g be as in Theorem 1.11. Suppose in addition that it is
in C 2 (U, R) for some open subset U ⊂ H. Suppose that 0 is an isolated critical point of


12

C. Bandle and W. Reichel

mountain pass type. Assume also that if the smallest eigenvalue of g (x0 ) is zero, then it is
simple. Then ind(∇g, x0 , 0) = −1.

1.5. Asymptotically linear equations
A map G : X → X is called asymptotically linear if there exists a bounded linear operator
A : X → X such that
lim

x →∞

G(x) − Ax
= 0.
x

The linear operator A is uniquely determined and is therefore called the derivative of G
at infinity, written as G (∞). It can be shown that if G is compact then the same is true
for G (∞).
T HEOREM 1.14. Let G : X → X be asymptotically linear such that G (∞) is invertible.

Assume also that G − G (∞) is compact. Then the nonlinear problem G(x) = y has a
solution for every y ∈ X.
P ROOF. We have to show that the equation G(x) + G (∞)x = y with G = G − G (∞) has
a solution. Set F = −G ◦ [G (∞)]−1 . Then the problem reduces to (Id −F )(z) = y, where
F is compact and z = G (∞)x. By definition of the derivative at infinity it follows that
F (z) / z → 0 as z → ∞. For Ω = BR (0) we want to calculate deg(Id −tF, Ω, y)
for t ∈ [0, 1]. For z ∈ ∂BR (0) we have
z − tF (z) − y

z

1−t

F (z)
z

− y

R
− y ,
2

provided R is sufficiently large. If R is even bigger than 2 y then we have that y ∈
/
(Id −tF )(∂Ω) and by homotopy invariance of the degree we get deg(Id −F, Ω, y) =
deg(Id, Ω, y) = 1. This completes the proof.
C OROLLARY 1.15. It is sufficient for Theorem 1.14 to have an invertible linear operator A
such that lim sup x →∞ G(x) − Ax / x < 1/ A−1 .
The following multiplicity result goes back to Amann [2], see also [3]. We present here
the version given by Sattinger [64].

T HEOREM 1.16. Let F be compact and asymptotically linear. Suppose that Id −F (∞)
is invertible. Assume that F has two different fixed points x1 , x2 such that (Id −F (xi ))−1
exists for i = 1, 2. Then there exists a third fixed point x3 .
P ROOF. Since Id −F (∞) is invertible there exists a > 0 such that x − F (∞)x
a x
for all x ∈ X. Since F is asymptotically linear we can find a positive number R0 such that


Solutions of quasilinear second-order elliptic boundary value problems via Degree Theory

F (x) − F (∞)x

a
2

x for all x

x − τ F (x) − (1 − τ )F (∞)x

R0 . Hence, for all x

13

R0 and for all τ ∈ [0, 1],

x − F (∞)x − τ F (x) − F (∞)x
a
R0 .
2

Hence τ F + (1 − τ )F (∞) has no fixed point outside of BR0 and as a consequence
deg(Id −τ F − (1 − τ )F (∞), BR0 , 0) is well defined and independent of τ . Thus setting
τ = 0 and τ = 1 we get
deg Id −F (∞), BR0 , 0 = deg(Id −F, BR0 , 0).

(1.1)

By Lemma 1.9 the left-hand side of (1.1) equals (−1)β = ±1 where β is related to the
multiplicity of the eigenvalues of F (∞) larger than one. On the other hand if we assume
that xi , i = 1, 2, are the only fixed-points of F in BR0 then by the excision property (d2)
the right-hand side of (1.1) is 2i=1 ind(Id −F, xi , 0) = 0 or ±2. This contradicts (1.1).
Therefore at least one more fixed point of F must exist.

1.6. Fixed point alternatives
T HEOREM 1.17 (Leray–Schauder alternative). Let Ω ⊂ X be bounded, open and assume
p ∈ Ω. Let furthermore F : Ω → X be compact. Then the following alternative holds:
(i) F has a fixed point in Ω
or
(ii) there exists λ ∈ (0, 1) and x ∈ ∂Ω such that x = λF (x) + (1 − λ)p.
P ROOF. Suppose for contradiction that neither (i) nor (ii) holds. We want to show that
deg(Id −tF, Ω, (1 − t)p) is well defined. So suppose that for some t ∈ [0, 1] there is
x ∈ ∂Ω with x − tF (x) = (1 − t)p. Since (i) does not hold the possibility t = 1 is excluded and since (ii) does not hold it is impossible that 0 < t < 1. And since p ∈ Ω also
t = 0 is excluded. Hence, homotopy invariance applies and yields deg(Id −F, Ω, 0) =
deg(Id, Ω, p) = 1 which shows that F has a fixed point in Ω. This contradicts the assumption that (i) does not hold.
T HEOREM 1.18 (Principle of a priori bounds). For t ∈ [0, 1] let F (t, ·) : X → X be a
family of compact operators with F (0, ·) ≡ 0. Assume, moreover, that F (t, x) is continuous
in t uniformly w.r.t. x in balls in X. Suppose that the set S = {x: ∃t ∈ [0, 1]: x = F (t, x)}
is bounded. Then F (1, ·) has a fixed point.
P ROOF. Standard arguments show that the hypotheses imply that F : [0, 1] × X → X is
compact. If BR (0) is such that all solutions of x = F (t, x) for t ∈ [0, 1] are a priori

known to lie inside BR (0) then deg(Id −F (t, ·), BR (0), 0) is homotopy invariant. Hence
deg(Id −F (1, ·), BR (0), 0) = deg(Id, BR (0), 0) = 1. This shows that F (1, ·) has a fixed
point.


14

C. Bandle and W. Reichel

By taking F (t, x) = tF (x) we get the following result.
C OROLLARY 1.19 (Schäfer’s theorem [65]). Let F : X → X be compact. Then the following alternative holds:
(i) x − tF (x) = 0 has a solution for every t ∈ [0, 1]
or
(ii) S = {x : ∃t ∈ [0, 1]: x − tF (x) = 0} is unbounded.

1.7. Degree theory in unbounded domains
Up to now the degree was defined only in bounded domains. We indicate a generalization
to unbounded domains which will be needed in the next section.
Assume Ω ⊂ X is open and possibly unbounded. Let us consider the class of maps
F : Ω → X where (Id −F )−1 (y) is compact for every y ∈
/ (Id −F )(∂Ω). In order to define
deg(Id −F, Ω, y) take any bounded open neighborhood V ⊂ Ω of (Id −F )−1 (y) and set
deg(Id −F, Ω, y) =: deg(Id −F, V, y).
This definition makes sense because the excision property (d2) implies that deg(Id −F,
V, y) is the same for every bounded open neighborhood V of (Id −F )−1 (y).
The following lemma is useful for practical purposes.
L EMMA 1.20. Let F : Ω → X be compact and assume that F (Ω) is bounded. Then
(Id −F )−1 (y) is compact.
P ROOF. Let {xn }n 1 be a sequence of solutions to the equation x − F (x) = y. The sequence is bounded because we have assumed that F (Ω) is bounded. Since F is compact
there exists a subsequence {xn }n 1 such that {F (xn )} converges. Hence xn converges

to x, and from the continuity of F we conclude that the limit solves x − F (x) = y.

1.8. Degree theory in cones
Krasnosel’skii derived a theorem to find nontrivial fixed points of cone preserving maps,
cf. [44]. A cone C is a closed, convex subset of the Banach space X with the following
properties:
(i) if x, y ∈ C and α, β 0 then αx + βy ∈ C,
(ii) if x ∈ C and x = 0 then −x ∈
/ C.
A cone induces a partial ordering x y in X whenever y − x ∈ C.
The Leray–Schauder degree theory cannot be applied immediately to functions
p
F : C → C because many important cones such as L+ (D) = {x ∈ Lp (D): x 0 a.e.} for
p 1 have empty interior. By Dugundji’s theorem [27] one knows that C is a retract.
Hence it is possible to extend the degree to arbitrary cones in a natural way.