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

Solutions of Quasilinear Second-Order Elliptic Boundary Value Problems via C.. Solutions of Quasilinear Second-Order Elliptic Boundary Value Problems via Degree Theory Catherine Bandle a

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This handbook is Volume I in a multi-volume series devoted to stationary partial ential equations It is a collection of self contained, state-of-the-art surveys written bywell-known experts in the field The authors have made an effort to achieve readabilityfor mathematicians and scientists from other fields, and we hope that this series of hand-books will become a new reference for research, learning and teaching

differ-Partial differential equations represent one of the most rapidly developing topics in ematics This is due to their numerous applications in science and engineering on one handand to the challenge and beauty of associated mathematical problems on the other This vol-ume consists of eight chapters covering a variety of elliptic problems and explaining manyuseful ideas, techniques and results Although the central theme is the mathematically rig-orous analysis, many of the contributions are enriched by a plenty of figures originating innumerical simulations

math-We thank all the contributors for their clearly written and elegant articles, and ArjenSevenster 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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1 Solutions of Quasilinear Second-Order Elliptic Boundary Value Problems via

C Bandle and W Reichel

2 Stationary Navier–Stokes Problem in a Two-Dimensional Exterior Domain 71

G.P Galdi

3 Qualitative Properties of Solutions to Elliptic Problems 157

W.-M Ni

4 On Some Basic Aspects of the Relationship between the Calculus of Variations

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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 3

1.1 Introduction 3

1.2 Brouwer degree in finite dimensions 4

1.3 Leray–Schauder degree in Banach spaces 7

1.4 The index of an isolated solution 10

1.5 Asymptotically linear equations 12

1.6 Fixed point alternatives 13

1.7 Degree theory in unbounded domains 14

1.8 Degree theory in cones 14

1.9 Notes 16

2 Existence of solutions 17

2.1 Function spaces 17

2.2 Uniformly elliptic linear operators 18

2.3 Schauder estimates 19

2.4 L p-estimates 20

2.5 Applications to boundary value problems 21

2.6 Comparison principles 23

2.7 Degree between sub- and supersolutions 24

2.8 Emden–Fowler type equations 26

2.9 Multiplicity results 28

2.10 Notes 31

3 Global continuation of solutions 34

3.1 A global implicit function theorem 34

3.2 Applications – continuation of solutions 38

3.3 Further applications 42

3.4 Notes 48

4 Bifurcation theory and related problems 50

HANDBOOK OF DIFFERENTIAL EQUATIONS

Stationary Partial Differential Equations, volume 1

Edited by M Chipot and P Quittner

© 2004 Elsevier B.V All rights reserved

1

2 C Bandle and W Reichel

4.1 Bifurcation from the trivial solution 50

4.2 Bifurcation from infinity 58

4.3 Perturbations at resonance 65

Acknowledgments 68

References 68

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 rior 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).

inte-Such an invariant can easily be found for continuously differentiable functions

F:[0, 1] → R with isolated solutions {x i}k

i=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

i=1sign F(x i ) is invariant under deformations of F which keep the endpoints fixed,

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(x

i )= 0 then

a natural approach is to approximate F by functions F n with isolated solutions, cf ure 4 Heuristically, the quantity described above seems to be stable if we pass to the limit

Fig-F n → F

For analytic functions F : Ω⊂ C → C the argument principle can be employed to

deter-mine 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

given by the boundary integral 2π i1 

γ

F(z)

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 quantitiesk

F (z) −w dz will be replaced by a more general concept, namely the topological

de-gree In many cases it will be impossible to compute it directly For the applications twoproperties 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 andSchauder (1934) [49] Since we are mainly interested in the degree theory as a tool forproving 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 thedesired properties, then in proving that there is at most one quantity satisfying all theseconditions, 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 provingexistence of solutions of linear and nonlinear second-order elliptic boundary value prob-lems and to recount some of the most interesting properties and applications Rather thandescribing more recent topological developments of the notion of degree and its properties

4 C Bandle and W Reichel

avail-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→ RNdenote the identity map

DEFINITION1.1 Suppose that y / ∈ G(∂Ω) The degree is a mapping deg: (G, Ω, y) → Z

with the following properties:

(d1) Normalization: deg(Id, Ω, y) = 1 if y ∈ Ω and deg(Id, Ω, y) = 0 if y /∈  Ω

(d2) Excision: if  Ω= Ω1∪ Ω2with Ω1, Ω2open, 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] → R Nis

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 isfying (d1)–(d3) (cf., e.g., Deimling [26]) One can show the following extension of thehomotopy invariance (d3), cf Amann [3] and Leray and Schauder [49]:

sat-(d3)g General homotopy invariance: let Θ ⊂ [0, 1] × R N be bounded and open in

[0, 1] × R N and denote by Θ t the slice at t , that is,

Θ t=x∈ RN : (t, x) ∈ Θ.

If h :  Θ → RN is continuous and y : [0, 1] → R N is continuous with y(t) /∈

h(t, ∂Θ t ) for all t ∈ [0, 1] then

deg

h(t, ·), Θ t , y(t)

is independent of t.

For the construction of the degree we proceed in several steps

(I) Degree for regular values of C1-maps Let G ∈ C1(  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

6 C Bandle and W Reichel

(II) Degree for singular values of C2-maps Let G ∈ C2(  For y / ∈ G(∂Ω) let y1be

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 y1the followingdefinition

deg(G, Ω, y) := deg(G, Ω, y1)

makes sense The proof is done through the integral representation

EXAMPLE1.1 Consider Ω = (−1, 1) and G(x) = x3 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

G(x) = x2then similarly deg(G, Ω, y1) = sign G(−√δ) + sign G(√

pre-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 ∈ C2(  be such thatG−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 toarbitrary spaces of finite dimensions which are different fromRN 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 → R Nbe a linear homeomorphism Then

deg(G, Ω, y):= degL ◦ 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

THEOREM 1.2 (Brouwer’s fixed point theorem) Every continuous map F :  B1( 0)→



B1( 0), where B1( 0) is the open unit ball {x ∈ R N: x < 1} has a fixed point.

PROOF 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 ex-cluded 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 

1.3 Leray–Schauder degree in Banach spaces

We wish to extend the previous results to infinite-dimensional spaces However, oneneeds to be careful: although Brouwer’s fixed point theorem follows immediately fromthe 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 con-tinuous compact maps And likewise the topological degree can be defined for continuouscompact 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 continuousmap, the Leray–Schauder degree is defined only for compact perturbations of the identity,

namely G = Id −F

THEOREM 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):

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 applicationsdiscussed 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 F ε: Ω → X ε ⊂ X with finite-dimensional range X εsuch thatF (x) − F ε (x) < ε for all x ∈  Ω In general the approximation F εisnot unique However, it turns out that the degree for Id−F ε on Ω ∩ X ε is well defined,

provided 0 < ε  ε0= dist(y, (Id−F )(∂Ω)) We then define

deg(Id−F, Ω, y) := deg(Id −F ε , Ω ∩ X ε , 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

DEFINITION1.4 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| = Id The map r is called a retraction.

8 C Bandle and W Reichel

EXAMPLES (1) The closed unit ball is a retract Consider the map r(x) = x/x2 if

x > 1 and r(x) = x elsewhere.

(2) Dugundji [27] proved that closed convex sets are retracts

THEOREM1.5 (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.

PROOF (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 B ρ ( 0) be

a large ball containing C The map F ◦ r has no fixed point on ∂B ρ ( 0) Consider the homotopy k(t, x) := tF (r(x)) for t ∈ [0, 1] There is no fixed point of k(t, ·) on ∂B ρ ( 0), because for t = 1 this has already been excluded, and for t < 1 we have k(t, x) < ρ

ifx  ρ By the homotopy invariance of the degree we get deg(Id −F ◦ r, B ρ ( 0), 0)=

deg(Id, B ρ ( 0), 0) = 1, i.e., F ◦ r has a fixed point in B ρ ( 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→ C0has a fixed point x ∈ C0,

1.3.2 Tools for calculating the degree

THEOREM 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 proximation Finally the dimension reduction is proved for all compact perturbations of theidentity The basis for the general dimension reduction formula is illustrated next

ap-EXAMPLE1.3 Consider a linear map F :Rn→ Rk Rn given by F (x) = Ax with an

n × n matrix A Since F maps into R k with k < n the matrix A can be written as follows:

LEMMA 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).

PROOF 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)

1.3.3 Degree for linear maps

LEMMA 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

= degId−K| V , B1( 0) ∩ V, 0· degId−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→ Rnis givenby

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 N n and its range R n There exists an integer

n0= n0(λ) 1 such that

N1 N2 · · ·  N n0 = N n0+1= N n0+2= · · · ,

R1 R2 · · ·  R n0= R n0+1= R n0+2= · · ·

The set N n0(λ) is called the generalized nullspace of K − λ Id and m(λ) = dim N n0(λ)is

called the algebraic multiplicity of the eigenvalue λ The set R n0(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 = N n0(λ) ⊕ R n0(λ) Moreover, K maps N n0(λ) to N n0(λ) , R n0(λ) to R n0(λ)

and K − λ Id has a bounded inverse on R n0(λ)

LEMMA 1.9 Let K : X → X be linear and compact with Id −K injective and suppose

multiplicity

REMARK Observe that the degree formula in Lemma 1.9 remains valid for deg(Id−

K − x0, Ω , 0) provided (Id−K)−1x

0∈ 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 x0be such a solution Then x0is the only solution in some ball B ε0(x0) Therefore

deg(Id −F, B ε (x0), y) is independent of ε for 0 < ε < ε0 We define the index of an isolated solution x0by means of the degree as follows:

ind(Id −F, x0, y)= degId−F, B ε (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

THEOREM 1.10 (Leray–Schauder) Under the preceding assumptions and if Id −F(x

0)

is injective we have that ind(Id −F, x0, y) = ±1 More precisely,

ind(Id −F, x0, y)= indId−F(x0), x0, y

PROOF Without loss of generality we may assume that y = 0 and that x0= 0 is the

iso-lated solution Then for x near the origin we have (Id −F )(x) = (Id −F( 0))x − ω(x),

whereω(x)/x → 0 as x → 0 Hence deg(Id −F − tω, B ε ( 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), B

ε ( 0), 0) Lemma 1.9 applies

The condition that Id−F(x0)is injective in the previous theorem is necessary, as the

following examples shows:

EXAMPLE Let F (x) = −x2+ x for x ∈ R The only solution of x − F (x) = 0 is x0= 0.Then Id−F(x0) = 0 The index of x0vanishes, cf Example 1.1 in Section 1.2

In the next theorems we consider potential operator on a Hilbert space H Let

g : B ε (x0) → R be a C1-functional and let ∇g(x) be its gradient, i.e., the Riesz sentation of its Fréchet derivative g(x).

repre-THEOREM1.11 (Rabinowitz [61]) Suppose that ∇g(x) = x − F (x) where F is compact.

If x0is 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) +1

2(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 cal maxima nor minima The index will depend on the type of saddle point as it is

lo-seen in the following example Consider the function g :RN → R given by g(x) =

non-The next definition is due to Hofer [39]

DEFINITION1.12 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 ∩ M cis nonemptyand not path connected

This definition of a critical point of mountain pass type is satisfied by a mountain passpoint 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 tocritical points of mountain pass type

THEOREM1.13 (Hofer [39]) Let g be as in Theorem 1.11 Suppose in addition that it is

in C2(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(x

0) 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

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( ∞).

THEOREM1.14 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.

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 = B R ( 0) we want to calculate deg(Id−tF, Ω, y) for t ∈ [0, 1] For z ∈ ∂B R ( 0) we have

COROLLARY1.15 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 herethe version given by Sattinger [64]

THEOREM 1.16 Let F be compact and asymptotically linear Suppose that Id −F( ∞)

is invertible Assume that F has two different fixed points x1, x2such that (Id −F(x i ))−1

exists for i = 1, 2 Then there exists a third fixed point x3.

PROOF Since Id−F( ∞) is invertible there exists a > 0 such that x − F( ∞)x  ax for all x ∈ X Since F is asymptotically linear we can find a positive number R such that

Hence τ F + (1 − τ)F( ∞) has no fixed point outside of B R0 and as a consequence

deg(Id−τF − (1 − τ)F( ∞), B R0, 0) is well defined and independent of τ Thus setting

i=1ind(Id −F, x i , 0)= 0 or ±2 This contradicts (1.1)

Therefore at least one more fixed point of F must exist. 

1.6 Fixed point alternatives

THEOREM1.17 (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.

PROOF 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 cluded and since (ii) does not hold it is impossible that 0 < t < 1 And since p ∈ Ω also

ex-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 assump-

THEOREM 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.

PROOF Standard arguments show that the hypotheses imply that F : [0, 1] × X → X is compact If B R ( 0) is such that all solutions of x = F (t, x) for t ∈ [0, 1] are a priori known to lie inside B R ( 0) then deg(Id −F (t, ·), B R ( 0), 0) is homotopy invariant Hence deg(Id−F (1, ·), B R ( 0), 0) = deg(Id, B R ( 0), 0) = 1 This shows that F (1, ·) has a fixed

14 C Bandle and W Reichel

By taking F (t, x) = tF (x) we get the following result.

COROLLARY1.19 (Schäfer’s theorem [65]) Let F : X → X be compact Then the ing alternative holds:

follow-(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

LEMMA 1.20 Let F :  Ω → X be compact and assume that F (  Ω ) is bounded Then (Id−F )−1(y) is compact.

PROOF Let{x n}n1 be a sequence of solutions to the equation x − F (x) = y The quence is bounded because we have assumed that F (  Ω) is bounded Since F is compact

se-there exists a subsequence{x n}n 1 such that {F (x n) } converges Hence x n 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

F:C → C because many important cones such as L p

+(D) = {x ∈ L p (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

DEFINITION 1.21 LetC ⊂ X be a cone, r : X → C be a retraction and let U ⊂ C be a

bounded open set with respect to the relative topology ofC If F : C → C is compact and

y ∈ C is such that (Id −F )x = y has no solution on the boundary of U (with respect to the

relative topology ofC) then we define

deg(Id−F, U, y) := degId−F ◦ r, r−1U, y

.

The definition makes sense because it is independent of the particular choice of the

retraction Moreover, the solution set A := {x ∈ U: x − F (x) = y} is the same as the solution set B := {x ∈ r−1U : x − (F ◦ r)(x) = y} since for the latter, the fact that y, (F ◦ r)(x) ∈ C implies x ∈ C In particular, the solution set A (= B) is compact since U is bounded Note that even if r−1U is unbounded the degree is nevertheless defined by the

arguments of the previous section

Many applications of degree theory in cones are due to Amann [3] The next result is

due to Krasnosel’skii, cf [44] It is also found in [9], Appendix 1 For 0 < r < R consider the sets S(r, R) := {x ∈ C: r < x < R} and S(R) = {x ∈ C: x < R} Both sets are

open in the relative topology ofC.

THEOREM 1.22 Let C ⊂ X be a cone and F : C → C be compact Assume there exist numbers 0 < r < R and a point 0 = v ∈ C such that

(i) x = tF (x) for all 0  t  1 and x = r,

(ii) x = F (x) + tv for t  0 and x = R.

In particular, F has a fixed point in S(r, R).

PROOF It follows from (i) that x − tF (x) = 0 on x = r for all t ∈ [0, 1] By the topy invariance (d3) and by the normalization (d1) it follows that deg(Id −tF, S(r), 0) = deg(Id, S(r), 0)= 1 which establishes the first assertion By (ii) and the homotopy invari-

homo-ance we have deg(Id −F − tv, S(R), 0) = deg(Id −F, S(R), 0) for all t > 0 Suppose that deg(Id −F − tv, S(R), 0) = 0 Then by the existence property (d4) the equa- tion F (x) + tv = x has always a solution x t in S(R) For large t we have the esti- mate R  tv − F (x t ) This leads to a contradiction if t is too large and shows that deg(Id −F, S(R), 0) = 0 The last statement now follows from the excision prop-

For later uses let us describe a result concerning the spectrum of compact linear operators

in cones For an elementary proof we refer to Takáˇc [69] A proof using degree theory isgiven in Theorem 3.4 in Section 3.1

THEOREM 1.23 (Krein–Rutman) Let X be a Banach space ordered with respect to a cone C Suppose that Int(C) = ∅ and let T : X → X be a compact linear operator which is strongly positive in the sense that T (C \ {0}) ⊂ Int(C) Then:

16 C Bandle and W Reichel

(1) the spectral radius r(T ) is a positive simple eigenvalue of T ,

(2) the eigenvector u ∈ X \ {0} associated with the eigenvalue r(T ) can be taken in Int( C),

(3) if μ is in the spectrum of T , 0 = μ = r(T ) then μ is an eigenvalue of T satisfying

of the most complete texts is the celebrated book by Krasnosel’skii and Zabreiko [44] whohave made important contributions to the theory and its applications For an introduction

to degree theory see also Deimling [26]

2 Brezis [11,12] reviews degree theory for harmonic maps fromSN toSN which arecritical points of the Dirichlet energy SN |∇u|2dx Similar to the Brouwer degree one can

define for continuous mapsSN→ SN the degree deg(u,SN , y) for y∈ SN In contrast to

degree theory on sets with boundary, the degree for functions u :SN→ SN is independent

of y∈ SN Hence, we set deg(u) = deg(u, S N , y) for every continuous map u :SN→ SN

By Hopf’s result, if two continuous maps u, v :SN→ SNhave the same degree, then there

exists a homotopy connecting u and v Thus, the space C(SN ,SN )is decomposed into itsconnected components characterized by their degree One can try to use this decomposi-

tion for finding harmonic maps in each connected component of C(S N ,SN ) In order toapply direct methods of the calculus of variations one has to define the degree for maps in

the Sobolev space H 1,2 (SN ,SN ), and the question arises if the connected components

re-main the same when passing from C(S N ,SN ) to H 1,2 (SN ,SN ) This is true in dimension

N= 2, cf Schoen and Uhlenbeck [67], but the problem of closedness of the components in

the H 1,2-topology still remains These harmonic map problems are considered in [11,12]

and Struwe [68]; see also the references given there For dimensions N 3 however, the

Schoen–Uhlenbeck approach only works for maps in H 1,N (SN ,SN ), which poses againproblems when minimizing the Dirichlet energy by the direct methods of the calculus of

variations Brezis and Nirenberg [13,14] consider the degree for maps in VMO(SN ,SN )

(vanishing mean oscillation)

3 Counterexample to Brouwer’s fixed point theorem in infinite dimensions Let X be the Banach space of real sequences x = (x n )tending to zero with normx = max n |x n|

Let F : X → X be defined by

(F x)1= (1 + x)/2 and (F x) n+1= x n

F is continuous and maps B ( 0) into itself, but F has no fixed point.

2 Existence of solutions

In the theory of elliptic partial differential equations the Hölder and the Sobolev spaces play

an outstanding role Both are Banach spaces in which a complete theory for linear ellipticdifferential equations is available For convenience we shall summarize the main results.More details and proofs are found in the reference texts of Miranda [50] and Gilbarg andTrudinger [32]

2.1 Function spaces

Let D be an open set inRN and α ∈ [0, 1] be an arbitrary number A function f : D → R N

is said to be Hölder continuous with exponent α if

modu-D k f = ∂ |k| f

∂x k1

1 ∂x k2

2 · · · ∂x k N N

Let C m (  denote the space of functions with continuous mth order derivatives in  D With

this notation we can now define the Hölder spaces

C m,αD

:=f ∈ C mD

: f m +α <∞ The subspaces of C m (  and C m,α (  consisting of functions with compact support in D will be denoted by C m (  and C m,α ( 

18 C Bandle and W Reichel

For bounded domains the Sobolev spaces H m,p (D) , [H0m,p (D)] are obtained as the

completion of C m (  , [C0m (  ] with respect to the norm

For m  1 every function f in H m,p or H0m,p has generalized derivatives up to order m.

2.2 Uniformly elliptic linear operators

For the rest of this chapter let D⊂ RN be a bounded domain From now on we shall use

the summation convention and the abbreviations ∂ i:= ∂

∂x i and ∂ ij2 := ∂2

∂x i ∂x j The operator

L := a ij (x)∂ ij2+ b i (x)∂ i + c(x)

is uniformly elliptic in D provided there exists a positive constant Λ such that

a ij (x)ξ i ξ j  Λξ i ξ i for all x ∈ D and ξ ∈ R N

and a ij , b i , c ∈ L∞(D) Associated to L is the operator

L0:= a ij (x)∂ ij2 + b i (x)∂ i

We say that L satisfies the maximum principle for u ∈ C2(D) ∩ C( D)if

Lu  0 in D, u  0 on ∂D ⇒ max

D u  0. (MP)

Sufficient for the maximum principle is c 0 Moreover, under this condition and if

Lu  0 in D the following strong maximum principle holds:

(i) if u attains its nonnegative maximum in D then u≡ const;

(ii) if u attains its nonnegative maximum at a point x0∈ ∂D which lies on the boundary

of a ball B ⊂ D and if u is continuous in D ∪ {x0} and an outward directionalderivative∂u ∂ν (x0)exists then ∂u ∂ν (x0) > 0 unless u≡ const

If the maximum principle (MP) holds then simple pointwise estimates for the classicalsolutions of the boundary value problem

can be derived For instance, if c  0 and if u ∈ C( D) ∩ C2(D)then

sup|u|  C sup |f |

where C depends only on the diameter of D and b i /Λ∞, i = 1, , N; cf Gilbarg and

Trudinger [32]

The maximum principle holds for the class of operators defined next We assume that at

every point x ∈ ∂D an outward normal ν(x) exists.

DEFINITION2.1 The operator−L is called strictly positive provided there exists a tion φ ∈ C1(  ∩ C2(D) with φ > 0 in D and ∂φ ∂ν (x0) < 0 at points x0∈ ∂D where φ(x0) = 0 such that −Lφ  0 but ≡ 0 in D.

func-LEMMA 2.2 Suppose −L is strictly positive Then (MP) holds in the class of C1(  ∩

C2(D)-functions and (2.2) holds for every C1(  ∩ C2(D)-solution of (2.1), where the constant C depends also on max{c, 0}∞.

PROOF Suppose u ∈ C1(  ∩ C2(D) is such that Lu  0 in D, u  0 on ∂D but

maxD u > 0 Let t∗> 0 be so large that t∗φ > u in D Consider the smallest ¯t ∈ (0,t∗)

such that tφ  u in D for all t ∈ (¯t, t∗) Then we have v = ¯tφ − u  0 in D and there exists

a point x0in D with v(x0) = ∇v(x0) = 0 Let c−= min{c, 0} The function v satisfies

L0v + c−v  Lv  0 in D Since v attains its zero-infimum at x0 with∇v(x0)= 0 the

strong form of the maximum principle implies v ≡ 0, i.e., u ≡ ¯tφ This is impossible, and proves (MP) The L∞-estimate (2.2) carries over like in [32], Section 3.3. There are essentially two classical approaches concerning existence and estimates forsolutions of (2.1): the Schauder theory for classical solutions in the Hölder spaces and the

L p-theory for strong solutions in the Sobolev spaces Both are described in the next twosections

2.3 Schauder estimates

Consider the Dirichlet problem

Assume ∂D ∈ C 2,α , a ij , b i , c, f ∈ C α (  Moreover, let a ij α,b i α,c α  M Let

u ∈ C 2,α (  be a solution of (2.3) Then it satisfies the following Schauder boundary estimate

The Schauder interior estimates come into play if no information of the solutions of

Lu = f in D on the boundary is available They involve norms f m +αdefined as follows:

20 C Bandle and W Reichel

where d x = dist(x, ∂D) and d x,y = min(d x , d y ) Then we define

where C depends only on Λ, α, M and D.

We consider the Dirichlet problem (2.3) where all data are α-Hölder continuous and

∂D ∈ C 2,α Then Schauder’s famous result, obtained by means of a continuity argument(see, e.g., [32]) states:

THEOREM Under the above conditions and if −L is strictly positive the Dirichlet lem (2.3) has a unique solution in C 2,α (  and there exists a constant C such that

Here the regularity assumptions on the data and the solutions are weaker Assume

∂D ∈ C 1,1 , a ij ∈ C( D) , b i , c ∈ L∞(D) , f ∈ L p (D) and 1 < p <∞ A strong solution

For the Dirichlet problem (2.3) we have the following existence theorem (see, for stance, [32]):

in-THEOREM Let L satisfy the assumptions above and assume c(x)  0 If f ∈ L p (D) then the Dirichlet problem (2.3) has a unique strong solution u ∈ H 2,p (D) ∩ H 1,p

(D).

More generally, let us suppose instead of c(x) 0 that whenever (2.3) has a solution

then this solution is unique Then the linear operator L−1: L p (D) → H 2,p ∩ H 1,p

0 (D)iswell defined and bounded, i.e.,

u H 2,p (D)  Cf L p (D)

with u = L−1f Moreover, L−1: L p (D) → H 1,p

0 (D)is compact

For later purposes we shall need the fact that L−1 has a unique compact restriction,

denoted again by L−1, from C(  D) into C 1,α (  Indeed if f is continuous then u = L−1f

belongs to L p (D) for all p > 1 The L p-estimates imply thatu H 2,p (D)  Cf ∞and by

the compact embedding H 2,p (D) → C 1,α (  for p > N and a suitable α ∈ (0, 1 − N/p)

it follows thatu1+α  Cf ∞and L−1: C(  D) → C 1,α (  is compact

2.5 Applications to boundary value problems

2.5.1 Asymptotically linear equations. Consider the boundary value problem

where g(x, s) = o(s) as |s| → ∞ uniformly for x ∈  D This problem can be written in theform

u + L−1λu + g(x, u)= 0,

where L−1is either defined on X = C α (  or on X = C( D)depending on the regularity of

the data The operator Gu = u + L−1(λu + g(x, u)) is asymptotically linear Its derivative

at infinity is given by G( ∞) = Id +λL−1 From Theorem 1.14 we deduce:

THEOREM2.3 Depending on the underlying space X assume that either

(i) g(x, s) is α-Hölder continuous in x∈ D uniformly w.r.t s in bounded intervals and locally Lipschitz continuous in s uniformly w.r.t x∈ D

or

(ii) g(x, s) continuous in x∈ D and s ∈ R.

If λ is not an eigenvalue of −L then (2.4) possesses a solution.

REMARK If λ is an eigenvalue of −L then (2.4) is discussed in Section 4.3.1.

2.5.2 Semilinear boundary value problems. Consider the problem

Lu = g(x, u, ∇u) in D, u = 0 on ∂D,

where L satisfies the conditions of Section 2.4 and L−1: L p (D) → H 1,p

0 (D)exists and iscompact The nonlinearity is continuous in D× R × RNand subject to the condition

g(x, u, ∇u)  M

1+ |u| + |∇u|γ

22 C Bandle and W Reichel

for some positive γ < 1.

THEOREM 2.4 Let p > 1 Under the hypothesis above there exists a solution in

where C is independent of t Using subsequently the inequality (1 + s) γp  c(ε) + εs p,

s  0, together with Minkowski’s inequality we conclude for all t ∈ [0, 1] and any solution u of (2.5) that u H 2,p (D)  C0 for some positive constant C0 The same

holds for the H01,p -norm The operator L−1: L p (D) → H 1,p

0 (D)is compact Likewise,

L−1G [u] : H 1,p

0 (D) → H 1,p

0 (D) is compact where G[u] := g(x, u, ∇u) Consequently

Schäfer’s theorem, cf Corollary 1.19, applies and shows the existence of a solution of (2.5)

in H01,p (D) for every t ∈ [0, 1] and in particular for t = 1 By a regularity step the solutions

2.5.3 Quasilinear boundary value problems. In this section we describe the Leray–Schauder method for solving the boundary value problem

C 2,αβ (  mapping z → U(z) It is not difficult to see that F is a compact operator from

C 1,β (  into itself The solutions of (2.6) can be interpreted as the fixed points of F

in C 1,β ( 

For σ ∈ [0, 1] the equation u = σF (u) is equivalent to the quasilinear problem

a ij (x, u, ∇u) ∂2

ij u = σf (x, u, ∇u) in D, u = 0 on ∂D. (2.7)The next theorem goes back to Leray and Schauder [49] We present it in the version ofGilbarg and Trudinger [32] It is an immediate consequence of Schäfer’s theorem (Corol-lary 1.19)

THEOREM2.5 Under the above assumptions and if for some β > 0 there exists a constant

M > 0 such that every solution of (2.7) satisfies u1+β  M for any σ ∈ [0, 1] then (2.6) has a solution u ∈ C 2,α ( 

This theorem reduces the solvability of quasilinear problems to finding a priori mates This step is by far the most difficult problem in the application of the Leray–Schauder technique

esti-Terminology: In the following L is always a uniformly elliptic operator

with bounded coefficients such that−L is strictly positive.

2.5.4 Eigenvalue problems. In this section we use the Krein–Rutman theorem, cf orem 1.23, to show the existence of an eigenvalue and an eigenfunction for the problem

where m ∈ C α (  is a nonnegative weight m  0, m ≡ 0.

THEOREM 2.6 Let the data be Hölder continuous Then (2.8) has a smallest value λ1 which is positive The corresponding eigenspace is one-dimensional and the eigenfunction φ1(x) may be taken positive in D.

eigen-PROOF By the considerations at the end of Section 2.4 the differential operator has a

compact inverse L−1: C(  D) → C 1,α (  The application of the abstract Theorem 1.23

requires a careful choice of the cone The standard positive cone C+

0(  is not suitablebecause it has empty interior

We follow the proof given by Amann [3] Let e be the solution of the boundary value problem Le + 1 = 0 in D, e = 0 on ∂D By the maximum principle it follows that e > 0

in D and ∂e ∂ν < 0 on ∂D Consider the linear space C e (D) = {v ∈ C0(  : ∃λ > 0 such that

−λe  v  λe} It is complete with respect to the norm v e := inf{λ  0: −λe  v  λe}.

The cone C = {v ∈ C+0(  : ∃λ > 0 such that 0  v  λe} of nonnegative functions in

C e (  has nonempty interior with respect to the · e-topology

Due to the compactness of L−1: C(  D) → C 1,α (  it follows that the operator L−1

maps C(  D) compactly into C e (  Moreover, the operator T : C e (  → C e (  defined

as T u = −L−1(mu)is strongly positive with respect toC because of the strong version of the maximum principle The theorem of Krein–Rutman applies to T and establishes the

2.6 Comparison principles

Consider the boundary value problem

24 C Bandle and W Reichel

We assume that f (x, s) as a function D× R → R is

(H1) Hölder continuous in x∈ D , locally uniformly w.r.t s∈ R,

(H2) locally Lipschitz continuous in s, uniformly w.r.t x∈ D

DEFINITION2.7 Suppose a pair of functions (v, w) in C2(D) ∩ C1(  satisfies

Lv + f (x, v)  0, Lw + f (x, w)  0 in D

with v  0  w on ∂D Then (v, w) is called a pair of sub- and supersolutions for (2.9).

LEMMA2.8 Let (v, w) be a pair of sub- and supersolutions for (2.9) Then the following holds:

(i) Strong comparison principle: Assume v  w Then either v ≡ w or v < w in D.

In the second case suppose x ∈ ∂D is a point where v(x) = w(x) Then ∂v

2.7 Degree between sub- and supersolutions

A pair (v, w) of sub- and supersolutions is called strict if neither v nor w is a solution.

LEMMA2.9 (Monotone iterations) Let (v, w) be a pair of strict sub- and supersolutions

of (2.9) and assume v < w Then there exist a minimal solution u and a maximal solution

u of (2.9) in V = {u ∈ C1(  : v < u < w in D}.

The idea is a follows: let σ > 0 be so large that g(x, s) = f (x, s) + σs is increasing in

s ∈ [−R, R] for all x ∈  D , where R  max(v∞, w∞) If M = L − σ Id then (2.9) is

equivalent to

Mu + g(x, u) = 0 in D, u = 0 on ∂D.

The operator ( −M)−1◦ g(x, ·) : C1(  → C1(  is compact and monotone increasing,

i.e., if u1, u2∈ C1(  satisfy u1 u2then ( −M)−1(g(x, u1))  (−M)−1(g(x, u2)

More-over, it mapsV into itself The sequence v n+1= (−M)−1(g(x, v n )) , v0= v is monotone increasing with u= limn→∞v n Likewise w n+1 = (−M)−1(g(x, w

n )) , w0 = w is a

monotone decreasing sequence with limn→∞w n = ¯u.

THEOREM2.10 Let (v, w) be a pair of strict sub- and supersolutions for (2.9) with v < w

Then u ∈ U is a solution of Lu + f (x, u) = 0 if and only if u solves Lu + ˜ f (x, u)= 0.

Thus, by replacing f by ˜ f we may suppose that f is bounded Moreover, by choosing

σ > 0 sufficiently large, we may suppose that f (x, s) + σs is increasing in s ∈ R for all

x∈ D Let u be the minimal solution of (2.9) in U Consider for t ∈ [0, 1] the following

one-parameter family of problems

Lu − σu + tf (x, u) + σu+ (1 − t)f (x, u ) + σu= 0 in D (2.10)

with u = 0 on ∂D The pair (v, w) remains a pair of strict sub- and supersolutions for (2.10) By the strong comparison principle no solution of (2.10) lies on ∂ U More- over, since we assumed boundedness of f , for all t ∈ [0, 1] every solution lies in the open norm ball B R (h) ⊂ C1(  , where h ∈ U is arbitrary and R is sufficiently large – in par- ticular large enough that v, w ∈ B R (h) Therefore the following topological degree is welldefined

26 C Bandle and W Reichel

Since the unique solution of the linear problem,

for which u= 0 is a solution In addition to hypotheses (H1) and (H2) we introduce:

(H3) f (x, s) > λ1s for large s > 0 where λ1is the smallest eigenvalue of−L,

(H4) lims→0f (x, s)/s = 0 uniformly for x ∈  D

We also use an assumption on the solution set of (2.12) with f (x, s) replaced by

f (x, s) + κ:

(H5) for κ in bounded intervals there exists an upper bound M such that u C1 < M

for every solution of (2.12)

Sufficient conditions for (H5) are given, e.g., in the a priori bound principle of Gidas andSpruck [31] stated next; see also the notes for further results This result is fundamental for

a large number of applications

LEMMA 2.11 (Gidas and Spruck) Suppose f :  D × R → R is continuous in x ∈  D and there exists p ∈ (1, 2∗− 1) and h ∈ C( D), h > 0, in  D with lim s→∞f (x, s)/s p = h(x) uniformly for x∈ D Then there exists a constant M > 0 such that every positive solution u

of (2.12) satisfies u C1< M.

REMARK In order to derive (H5) from Lemma 2.11 it is important to have a version which

applies for positive solutions of the parameter-dependent problem Lu + f (x, u, λ) = 0

in D, u = 0 on ∂D, where λ ∈ I = [λ a , λ b ] Suppose f :  D ×R×I → R is continuous both

in x∈ D and λ ∈ I and there exists p as above and a continuous, positive function h :  D×

I→ R with lims→∞f (x, s, λ)/s p = h(x, λ) uniformly for x ∈  D and λ ∈ I Then there exists a constant M > 0 such that every positive solution u for λ ∈ I satisfies u C1< M.Our result for the generalized Emden–Fowler problem (2.12) is as follows

THEOREM2.12 Suppose f :  D × R → R satisfies (H1)–(H5) Then (2.12) has a positive solution.

Before we can prove Theorem 2.12 by degree theory, we need the following result:

LEMMA 2.13 Suppose f :  D × R → R satisfies (H1)–(H3) Then there exists a value

K > 0 such that

has no nonnegative solution if κ  K.

PROOF Let φ1be the positive first Dirichlet eigenfunction of−L with the positive value λ1 By (H3) there exists K > 0 such that f (x, s) > λ1s − K for all s  0 Suppose

eigen-κ  K and assume u is a nonnegative solution of (2.13) Then Lu + λ1u < 0 in D, and by the strong comparison principle of Lemma 2.8 we know u > 0 in D and ∂ ν u < 0 on ∂D.

The latter implies that{t > 0: u  tφ1in D} is nonempty Hence we may define

τ = sup{t > 0: u  tφ1in D}.

Again by the strong comparison principle of Lemma 2.8 applied to u and τ φ1 we find

that either τ φ1≡ u or τφ1> u in D and τ ∂ ν φ1< ∂ ν u on ∂D The first alternative can

be excluded immediately and the second contradicts the definition of τ Hence there is no

PROOF OF THEOREM 2.12 By (H4) we have f (x, 0)= 0 Since we are interested in

positive solutions only we may assume that f (x, s) = 0 for all s < 0 Then every solution

of (2.12) is positive or identically zero By (H4) the function tφ1is a strict supersolution

to (2.12) for t > 0 small Likewise, −tφ1is a subsolution to (2.12) for t > 0 small After rewriting (2.12) as u + L−1f (x, u)= 0 we find by Theorem 2.10 that

deg

Id+L−1◦ f (x, ·), U ∩ B R ( 0), 0

= 1,

whereU is the open set as in Theorem 2.10 spanned by (−tφ1, tφ1) Let K > 0 be the

constant from Lemma 2.13 By assumption (H5) we know that all solutions of

28 C Bandle and W Reichel

Fig 5 Excision property of the degree.

since no solution of (2.14) exists for κ = K By the excision property (d2) of the Leray–

Schauder degree (see Figure 5), we know that next to the zero-solution a second solution

2.9 Multiplicity results

In this section two examples for the existence of multiple solutions are given The firstresult of Amann [3] shows the existence of an additional solution between a pair of strict

sub- and supersolutions (v, w) in case the maximal and the minimal solutions u, u are

different The arguments are based on the abstract result in Theorem 1.16

THEOREM 2.14 (Amann [3]) Let (v, w) be a pair of strict sub- and supersolutions

to (2.12) Assume that the maximal and the minimal solutions ¯u and u of (2.12) tween v and w satisfy v < u < ¯u < w in D Suppose that the operators L + f

f (x, ¯u ) + ¯u − t if t  ¯u,

and consider the problem

We claim that every solution u of (2.15) also solves (2.12) Indeed if u  ¯u in some main D⊂ D then

subdo-L(u − ¯u) − (u − ¯u) = 0 in D, u = ¯u on ∂D.

By the maximum principle u = ¯u in D In the same way we can exclude that u < u.

Problem (2.15) is equivalent to u + L−1g(x, u) = 0 in C 1,α (  The operator F =

−L−1◦g(x, ·) is asymptotically linear with F( ∞) = L−1◦Id In view of our assumptions

The second multiplicity result is due to P Hess [37] It combines topological degree

methods with variational principles Consider for λ > 0 the problem

where f is a sign-changing function.

THEOREM2.15 Let f :R+→ R be a continuously differentiable function with f (0) > 0 Suppose

(1) there exist numbers 0 < a1< a2< · · · < a m such that f (a k ) = 0 for k = 1, 2, , m,

(2) max{F (s): 0  s  ak−1} < F (a k ), k = 2, , m, where F (s) := s

0f (t) dt Then there exists a number ¯λ such that for all λ > ¯λ there are at least 2m − 1 posi- tive solutions ˆu1, u2, ˆu2, , u m , ˆu m of (2.16) with 0 < ˆu1∞< a1, ˆu k∞< a k and

u k∞> a k−1for k = 2, , m Moreover, ˆu1< ˆu2< · · · < ˆu m and u k < ˆu k

REMARK The hypotheses imply that the graph of f has m positive humps and (m − 1)

negative humps, each positive hump having greater area than the previous negative hump;

a k is the right end point of the kth positive hump (Figure 6).

Fig 6.