Differential and Integral Equations Volume xx, Number xxx, , Pages xx–xx

NONLINEAR EIGENVALUE PROBLEMS FORDEGENERATE ELLIPTIC SYSTEMS

Mabel CuestaDepartement de Mathematique, Universite du Littoral (ULCO)

50, rue F. Buisson, F–62228 Calais, France

Peter TakacInstitut fur Mathematik, Universitat Rostock

Universitatsplatz 1 D–18055 Rostock, Germany

(Submitted by: Klaus Schmitt)

Abstract. The following nonlinear eigenvalue problem for a pair of realparameters (λ, µ) is studied:8><>:

−∆pu = λa(x) |u|α1 |v|β1−1v in Ω;

−∆qv = µ b(x) |v|α2 |u|β2−1u in Ω;

u = v = 0 on ∂Ω.

Here, p, q ∈ (1,∞) are given numbers, Ω is a bounded domain in RNwith a C2-boundary, a, b ∈ L∞(Ω) are given functions, both assumedto be strictly positive on compact subsets of Ω, and the coefficientsαi, βi are nonnegative numbers satisfying either the conditions α1+β1 =p− 1 and α2 + β2 = q − 1, or the condition

(p− 1− α1)(q − 1− α2) = β1β2.

A smooth curve of pairs (λ, µ) in (0,∞)× (0,∞) is found for which thequasilinear elliptic system possesses a solution pair (u, v) consisting of

nontrivial, nonnegative functions u ∈ W 1,p0 (Ω) and v ∈ W 1,q

0 (Ω). Keyroles in the proof are played by the strong comparison principle anda nonlinear Kreın-Rutman theorem obtained by the authors in earlierworks. The main result is applied to some quasilinear elliptic systemsrelated to the above system.

Accepted for publication: March 2010.AMS Subject Classifications: 35J70, 35P30; 47H07, 47H12.

1

2 Mabel Cuesta and Peter Takac

1. Introduction

We consider the following system of quasilinear elliptic boundary-valueproblems:

−∆pu = λ a(x) |u|α1 |v|β1−1v in Ω;

−∆qv = µ b(x) |v|α2 |u|β2−1u in Ω;u = v = 0 on ∂Ω,

(1.1)

where Ω is a bounded domain in RN whose boundary ∂Ω is a C2-manifold∂Ω (which is not assumed to be connected), x = (x1, . . . , xN ) is a genericpoint in Ω, p, q ∈ (1,∞) are given numbers, a, b ∈ L∞(Ω) are given functionssatisfying

a0def= ess inf

x∈Ωa(x) > 0 and b0

def= ess infx∈Ω

b(x) > 0,

and αi, βi are constants with αi ≥ 0 and βi > 0 for i = 1, 2. The quasili-near elliptic operator u 7→ ∆pu

def= div(|∇u|p−2∇u), called the p-Laplacian,is defined for u ∈ W 1,p

0 (Ω) with values ∆pu ∈ W−1,p′(Ω), the dual spaceof W 1,p

0 (Ω), where 1p + 1

p′ = 1. We view System (1.1) as a homogeneousnonlinear eigenvalue problem for the unknown pair of parameters (λ, µ) ∈R∗+ ×R∗+ = (0,∞)2 associated with the unknown pair of nonnegative eigen-functions u ∈ W 1,p

0 (Ω) and v ∈ W 1,q0 (Ω). We will refer to such a couple

(λ, µ) as a “principal eigenvalue” of system (1.1).Notice that System (1.1) is neither variational nor of Hamiltonian type,

in general, except for the cases when either ∂f∂v ≡

∂g∂u or ∂f

∂u ≡∂g∂v where we

have denoted by f(x, u, v) and g(x, u, v), respectively, the right-hand side ofthe first and second equations in (1.1). Computing the partial derivatives

∂f∂v = λβ1 a(x) |u|α1 |v|β1−1, ∂g

∂u = µβ2 b(x) |v|α2 |u|β2−1,

∂f∂u = λα1 a(x) |u|α1−2u|v|β1−1v, ∂g

∂v = µα2 a(x) |v|α1−2v|u|β1−1u,

we observe that the former case occurs if and only if

α1 = β2 − 1, α2 = β1 − 1, and λβ1a(x) = µβ2b(x) for a.e. x ∈ Ω,

whereas the latter case occurs if and only if either

α1 − 1 = β2, α2 − 1 = β1, and λα1a(x) = µα2b(x) for x ∈ Ω,

or α1 = α2 = 0. The special “superlinear” case when α1 = α2 = 0 andβ1β2 > (p − 1)(q − 1) was treated in Ph. Clement, R. F. Manasevich, andE. Mitidieri [2]. Systems with variational and Hamiltonian structures have

Nonlinear eigenvalue problems for degenerate elliptic systems 3

been studied in D. G. de Figueiredo [6], for instance. Our method does notrequire any variational or Hamiltonian structure for System (1.1).

We wish to apply a simplified version of a Kreın-Rutman theorem forhomogeneous nonlinear mappings due to P. Takac [20, Theorem 3.5, page1763]. Given any f ∈ L∞(Ω), we denote by Tp(f) ≡ u ∈W 1,p

0 (Ω) the uniqueweak solution of the boundary-value problem

−∆pu = f(x) in Ω; u = 0 on ∂Ω. (1.2)

It is well known [7, 15, 24] that u ∈ C1,β(Ω) for some β ∈ (0, 1). We denote

X = [C10 (Ω)]2, X+ = (f, g) ∈ X : f ≥ 0 and g ≥ 0 in Ω, and

X+ is

the topological interior of X+ in X. Finally define the map S : X → X byS(u, v) def= (u, v) with

u = Tp(a|u|α1 |v|β1−1v) and v = Tq(b|v|α2 |u|β2−1u)

for (u, v) ∈ X.In Section 2, we treat the case when S is homogeneous whereas in Sec-

tion 3, we deal with a slightly more general case. We prove in both casesthe existence of a curve C1 of principal eigenvalues (λ, µ) for system (1.1).In Section 3 we turn to the nonhomogeneous problem

−∆pu = λa(x)|u|α1 |v|β1−1v + f(x) in Ω;

−∆qv = µb(x)|v|α2 |u|β2−1u+ g(x) in Ω;

u = v = 0 on ∂Ω.

(1.3)

We discuss the solvability of (1.3) when either (a) λ, µ > 0, (λ, µ) lies belowor to the left of C1, and f, g ≥ 0, or (b) (λ, µ) lies on C1 and f, g ≥ 0,or (c) (λ, µ) lies above or to the right of C1, and f, g ≤ 0 (“antimaximumprinciple”). We also briefly discuss the uniqueness of solutions in these cases.

Finally in the appendix we present first a new theorem on the strongcomparison principle for the p-Laplacian and a simplified version of theKreın-Rutman theorem for homogeneous nonlinear mappings of P. Takac[20, Theorem 3.5, page 1763].

2. A curve of principal eigenvalues

2.1. The case when S is homogeneous. We consider C10 (Ω) def= f ∈

C1(Ω) : f = 0 on ∂Ω and its positive cone (C10 (Ω))+ = f ∈ C1

0 (Ω) : f ≥0 in Ω which is normal and has nonempty topological interior (C1

0 (Ω))+

4 Mabel Cuesta and Peter Takac

characterized by v ∈ (C10 (Ω))+ if and only if v ∈ C1

0 (Ω) satisfies the strongmaximum principle ([23, 25]):

v > 0 in Ω and∂v

∂ν< 0 on ∂Ω. (2.1)

We consider also the Cartesian product X = [C10 (Ω)]2, which is a strongly

ordered Banach space endowed with the natural norm and ordering for pairsof functions (f, g) ∈ X. Its positive cone X+ = (f, g) ∈ X : f ≥ 0 and g ≥0 in Ω is normal and has nonempty topological interior

X+, where

X+ =

[(C10 (Ω))+]2.

We define the map S : X → X by S(u, v) def= (u, v) with

u = Tp(a|u|α1 |v|β1−1v) and v = Tq(b|v|α2 |u|β2−1u)

for (u, v) ∈ X, where Tp and Tq have been defined in (1.2).The following lemma describes the interaction between positive eigenval-

ues and the different homogeneities of S1 and S2, the two components of S.

Lemma 2.1. (i) A couple (u, v) ∈ X+ \ 0 is a weak solution of (1.1) for

some (λ, µ) ∈ (R∗+)2 if and only if (u, v) ∈X+ and S(u, v) = (λ−1/(p−1)u,

µ−1/(q−1)v).(ii) For all ρ, σ ∈ R+ and for all (u, v) ∈ X+ we have

S(ρu, σv) =(

(ρα1σβ1)1/(p−1)S1(u, v), (ρβ2σα2)1/(q−1)S2(u, v)).

(iii) If (u, v) ∈ X+ solves (1.1) with (λ′, µ′) in place of (λ, µ), then for anyρ, σ > 0, the pair (ρu, σv) solves (1.1) with (λ, µ) satisfying

λ = ρp−1−α1σ−β1λ′, µ = σq−1−α2ρ−β2µ′. (2.2)

The proof is left to the reader.Let us now look for principal eigenvalues of the map S via the Kreın-

Rutman theorem, cf. Theorem A.2. We have the following.

Theorem 2.2. Assume

α1 + β1 = p− 1 and α2 + β2 = q − 1. (2.3)

Then there exists Λ > 0 and a couple (u1, v1) ∈X+ such that system (1.1)

possesses a positive weak solution (u, v) ∈ X+ associated to some (λ, µ) ∈(R∗+)2 if and only if

λ1β1 µ

1β2 = Λ. (2.4)

Moreover, (u, v) = c(u1, v1) for some constant c > 0.

Nonlinear eigenvalue problems for degenerate elliptic systems 5

Proof. It is easy to see from Lemma 2.1(ii) that S is homogeneous; i.e.,S(tu, tv) = tS(u, v) for every t ∈ R+

def= [0,∞) if and only if (2.3) is satisfied.Furthermore S : X+ → X+ is strongly monotone; that is, if (ui, vi) ∈ X,i = 1, 2, satisfy 0 ≤ u1 ≤ u2 and 0 ≤ v1 ≤ v2 in Ω with u1 6≡ u2 or v1 6≡ v2

in Ω, then S(u2, v2) − S(u1, v1) ∈X+. This result follows from the strong

comparison principle established in [4] and [16]. Finally by a regularity resultof de Thelin [22, Theoreme 1, page 376] or Ladyzhenskaya and Ural’tseva[14, Theoreme 7.1] (for the L∞(Ω) bound of u and v) and the results ofLieberman [15, Theorem 1, page 1203] and DiBenedetto [7] or Tolksdorf [24]for interior regularity it follows that the mapping Tp : L∞(Ω) → C1,β(Ω)is continuous and bounded; that is, it maps bounded sets into boundedsets. It follows from Arzela-Ascoli’s theorem that Tp : L∞(Ω)→ C1,β′(Ω) iscompact whenever 0 < β′ < β; that is, it maps bounded sets into sets withcompact closure. By Theorem A.1 there exists a unique number Λ1 ∈ R ande = (u1, v1) ∈

X+ such that S(u1, v1) = Λ1(u1, v1). Hence, by Lemma 2.1(i),

the couple (λ′, µ′) def= (Λ−(p−1)1 ,Λ−(q−1)

1 ) is a principal eigenvalue for system(1.1). Using this eigenvalue in Lemma 2.1(iii) and condition (2.3) it followsthat, for any (λ, µ) satisfying( λ

Λ−(p−1)1

)1/β1( µ

Λ−(q−1)1

)1/β2

= 1, (2.5)

the couple ρ =(

λ

Λ−(p−1)1

)1/β1 , σ = 1 solves (2.2). Therefore, (λ, µ) is a prin-

cipal eigenvalue of (1.1). Conversely, assume that system (1.1) possesses a

positive weak solution (u, v) ∈X+ associated to some (λ, µ) ∈ (R∗+)2 and

choose ρ, σ > 0 such that(( ρσ )β1λ

) −1p−1 =

((σρ )β2µ

) −1q−1

def= Λ0. Then, byLemma 2.1, parts (ii) and (iii), S(ρu, σv) = Λ0(ρu, σv) and, therefore, bythe uniqueness results of Theorem A.2, Λ0 = Λ1. It follows from the defini-tion of Λ0 above that (λ, µ) satisfies (2.5). The conclusion of the theorem isnow obtained with

Λ def= Λ−(p−1)/β1−(q−1)/β2

1 . (2.6)Finally, the fact that (u, v) = c(u1, v1) with some positive constant c followsfrom the uniqueness in Theorem A.2.

Let us denote the set of all principal eigenvalues (λ, µ) of (1.1) by

C1def=

(λ, µ) ∈ (R∗+)2 : λ1β1 µ

1β2 = Λ

.

We will call it the principal eigenvalue curve of system (1.1).

6 Mabel Cuesta and Peter Takac

2.2. The general case. Let us now consider λ > 0, µ > 0, the mappingS : X+ → X+ defined in Section 2 above, and its square iterate S2 ≡ S S.We wish to apply the Kreın-Rutman theorem to S2. Notice that S2 : X+ →X+ is homogeneous if and only if the following equations are satisfied:

α1 = α2 = 0, β1β2 = (p− 1)(q − 1). (2.7)

Indeed, given (u, v) ∈ X+, set (u, v) = S(u, v) and (˜u, ˜v) = S(u, v). Fort ∈ R+ we have

(tγ1 u, tγ2 v) = S(tu, tv), (2.8)

and, therefore,(tδ1 ˜u, tδ2 ˜v) = S2(tu, tv),

where δ1 = α1γ1+β1γ2p−1 , δ2 = α2γ2+β2γ1

q−1 . Thus, S2 is homogeneous if and onlyif δ1 = δ2 = 1. These equations are equivalent to

α1(γ1 − γ2) = (p− 1)(1− γ1γ2), β1(γ1 − γ2) = (p− 1)(γ21 − 1),

α2(γ1 − γ2) = (q − 1)(γ1γ2 − 1), β2(γ1 − γ2) = (q − 1)(1− γ22).

The case γ1 = γ2 forces γ1γ2 = 1 which yields γ1 = γ2 = 1. Consequently,αi and βi satisfy (2.3). Therefore, without loss of generality, from now onwe may assume γ1 < γ2. Using α1 ≥ 0 we obtain 1 − γ1γ2 ≤ 0, whereasα2 ≥ 0 yields γ1γ2 − 1 ≤ 0. It follows that γ1γ2 = 1 again. This forces alsoα1 = α2 = 0 and β1β2 = (p− 1)(q − 1).

Since these conditions on αi, βi are very restrictive, one would prefer usinga better method than the one we have just described. The idea is the follow-ing. Let us assume throughout this section that α1 < p−1 and α2 < q−1. Weintroduce a new mapping T : X+ → X+ defined by T (u, v) def= (J1(v), J2(u))where, for (u, v) ∈ X, J1(v) is the unique (weak) solution u of

−∆pu = a(x)|u|α1vβ1 in Ω;u = 0 on ∂Ω, (2.9)

and J2(u) is the unique (weak) solution v of−∆qv = b(x)|v|α2uβ2 in Ω;v = 0 on ∂Ω. (2.10)

The existence is obtained by classical minimization while uniqueness followsfrom a convexity argument ([11, Theorem 3, page 151]). The regularity re-sults combined with the strong maximum principle already mentioned implythat the pair (u, v) belongs to

X+.

Nonlinear eigenvalue problems for degenerate elliptic systems 7

Let us to consider the mapping T 2. Notice that

T 2(u, v) = (J1 J2(u), J2 J1(v))

for any (u, v) ∈ X, so we can decouple T 2 and look for eigenvalues of eachcomponent. We have now that the condition (2.14) below for the homogene-ity of J1 J2 and J2 J1 is less restrictive than the one found before for S2.We will need the following result.

Lemma 2.3. Let us denote V = C10 (Ω). Then the mapping Ji : V+ → V+

is nondecreasing for i = 1, 2.

Proof. We prove the result only for J1. Let v1, v2 ∈ V+, 0 ≤ v1 ≤ v2, v1 6≡ 0,and denote mi(x) = a(x)vβ1

i and ui = J1(vi) for i = 1, 2. Since m1 ≤ m2, itfollows that u2 is an upper solution for the following problem:

−∆pu = m1(x)|u|α1 in Ω; u = 0 on ∂Ω. (2.11)

Let us denote by ϕ the positive eigenfunction of the Dirichlet p-Laplacianwith weight m1; that is, there exists λ1 > 0 such that

−∆pϕ = λ1m1(x)ϕp−1 in Ω; ϕ = 0 on ∂Ω. (2.12)

We can assume that 0 ≤ ϕ ≤ 1 on Ω. It follows that, for any constant c > 0sufficiently small, we have λ1(cϕ)p−1 ≤ (cϕ)α1 in Ω, whence cϕ is a lowersolution for problem (2.11). By choosing c even smaller if necessary we canassume cϕ < u2 in Ω. Let us define the sequence zn∞n=0 recursively byz0 = u2 and zn is the unique solution of

−∆pzn = m1(x)zα1n−1 in Ω; zn = 0 on ∂Ω (2.13)

inW 1,p0 (Ω), which is positive. Since obviously this sequence is bounded below

and above by cϕ ≤ zn ≤ u2 in Ω, using regularity and compactness results,we can prove that it converges in the norm of V to some function u ∈ Vwhich is a solution of problem (2.11) and satisfies cϕ ≤ u ≤ u2 in Ω. Werefer the reader to D. H. Sattinger [18] for details on this monotone iterationmethod. Thus, by uniqueness, u = u1 and the conclusion follows.

Now we give the analogue of Lemma 2.1 for the mappings J1 J2 andJ2 J1.

Lemma 2.4. (i) A couple u, v ∈ V+ \ 0 is a weak solution of (1.1) for

some (λ, µ) ∈ (R∗+)2 if and only if u, v ∈V + and J1(v) = λ

−1p−1−α1 u, J2(u) =

µ−1

q−1−α2 v.

8 Mabel Cuesta and Peter Takac

(ii) For any ρ, σ > 0 we have

(J1 J2)(ρu) = ρβ1β2

(p−1−α1)(q−1−α2) (J1 J2)(u),

(J2 J1)(σv) = σβ1β2

(p−1−α1)(q−1−α2) (J2 J1)(v).

The proof is left to the reader.The following theorem holds.

Theorem 2.5. Assume α1 < p− 1, α2 < q − 1 and

β1β2 = (p− 1− α1)(q − 1− α2). (2.14)

Then there exists Λ′ > 0 and a couple (u′, v′) ∈X+ such that system (1.1)

possesses a positive weak solution (u, v) ∈ X+ associated to some (λ, µ) ∈(R∗+)2 if and only if

λ1√

β1(p−1−α1)µ1√

β2(q−1−α2) = Λ′. (2.15)

Moreover, (u, v) = (ρu′, ρµ1/β2v′) with some positive constant ρ.

Proof. It follows from Lemma 2.4(ii) and condition (2.14) that the map-pings Ji Jj : V+ → V+, i, j ∈ 1, 2, i 6= j, are homogeneous. Moreover,both mappings are strongly monotone by the strong comparison principlein [4]. The regularity results quoted before imply that J1 J2 and J2 J1

maps bounded sets into sets with compact closure. By Theorem A.2, thereexists a unique number Λ1 ∈ R∗+ such that (J1J2)(u′) = Λ1u

′ holds for some

u′ ∈V + which is unique up to a positive constant multiple. Similarly, there

exists a unique number Θ1 ∈ R∗+ such that (J2 J1)(v′) = Θ1v′ holds for

some v′ ∈V +. The β2

q−1−α2-homogeneity of J2 applied to (J1J2)(u′) = Λ1u

′

yields(J2 J1)(J2(u′)) = Λβ2/(q−1−α2)

1 J2(u′).

Similarly, the β1

p−1−α1-homogeneity of J1 applied to (J2J1)(v′) = Θ1v

′ yields

(J1 J2)(J1(v′)) = Θβ1/(p−1−α1)1 J1(v′).

The uniqueness of Θ1 and v′ yields

Θ1 = Λβ2/(q−1−α2)1 and v′ = θJ2(u′) for some θ ∈ (0,∞).

Hence, we have also

Λ1 = Θβ1/(p−1−α1)1 and u′ = θ−β1/(p−1−α1)Λ−1

1 J1(v′).

Nonlinear eigenvalue problems for degenerate elliptic systems 9

By Lemma 2.4(i), the pair (u′, J2(u′)) solves (1.1) with λ = Λ1−p+α11 and

µ = 1. Similarly, (J1(v′), v′) solves (1.1) with λ = 1 and µ = Θ1−q+α21 =

Λ−β21 . Thus, if λ, µ are positive real numbers satisfying

µ−1/β2λ−1/(p−1−α1) = Λ1, (2.16)

it is easy to see that the pair

ρ = λ1/(p−1−α1)Λ1, σ = 1 (2.17)

satisfies (2.2). Hence, (λ, µ) is a principal eigenvalue of (1.1). Conversely, ifu, v ∈ V+ solves (1.1) for some (λ, µ) then, by Lemma 2.4(i), it follows that

(J1 J2)(u) =(µ−1/(q−1−α2)λ−1/β1

)β1/(p−1−α1)u = µ−1/β2λ−1/(p−1−α1) u.

By the uniqueness results of Theorem A.2, we have equation (2.16) andu = ρu′ for some ρ > 0. Raising both sides of equation (2.16) to thepower −

√β2/(q − 1− α2), we find that (λ, µ) satisfies (2.15) with Λ′ def=

Λ−√β2/(q−1−α2)

1 .Finally, the result v = µ1/β2ρv′ follows from (2.2) and (2.17).

Remark 2.6. One can easily prove that if we restrict ourselves to the case(2.3) then Λ = Λ′.

We will denote also in this case

C1def=

(λ, µ) ∈ (R∗+)2 : λ1/√β1(p−1−α1)µ1/

√β2(q−1−α2) = Λ′

. (2.18)

Let us also denote by (ϕλ, ϕµ) the positive eigenfunction associated to (λ, µ)∈ C1 with ‖ϕλ‖L∞(Ω) = 1.

The following proposition gives some properties of the principal eigenval-ues from C1.

Proposition 2.7. Assume that α1 < p− 1, α2 < q − 1, and (2.14) holds.(i) Uniqueness. (λ, µ) ∈ R+ × R+ is a principal eigenvalue of (1.1) if andonly if (λ, µ) ∈ C1.

(ii) Simplicity inX+. Let (λ, µ) ∈ C1 and (u, v), (u′, v′) ∈

X+ be a couple

of eigenfunctions associated to (λ, µ). Then there exists ρ > 0 such thatu = ρu′ and v = ρµ1/β2v′.(iii) Simplicity in X. Assume that α1 = α2 = 0 and let (u, v) ∈ X be

an eigenfunction associated to (λ, µ) ∈ C1. Then either (u, v) ∈X+ or

(−u,−v) ∈X+.

10 Mabel Cuesta and Peter Takac

Proof. The uniqueness and simplicity inX+ of principal eigenvalues follow

from the previous theorem and Theorem A.2, so we just prove (iii). It isclear that there exists some γ ∈ R+ such that

−u ≤ γϕλ and − v ≤ γωϕµ,

where ω = β2

q−1 ; let γ be the minimum of such γ’s. We assume by contradic-tion that γ > 0. Then if we have also −u ≡ γϕλ in Ω, it follows from the sec-ond equation of system (1.1) that µb(x)|−u|β2−1(−u) = µb(x)|γϕλ|β2−1γϕλand, consequently, −v = γωϕµ and we are done. Thus, it remains to treatthe case −u 6≡ γϕλ. Hence, we have

−∆p(−u) = λa(x)| − v|β1−1(−v) ≤ ( 6≡)−∆p(γϕλ) in Ω,

−∆q(−v) = µb(x)| − u|β2−1(−u) ≤ ( 6≡)−∆q(γωϕµ) in Ω,

together with −u = γϕλ = −v = γωϕµ = 0 on ∂Ω. It follows from thestrong comparison principle (SCP) of Theorem A.1 that −u γϕλ, −v γωϕµ (see (A.3) in the Appendix for the definition of the strong ordering“” in C1(Ω)). Thus, we can find 0 < ε < 1 such that −u ≤ εγϕλ and−v ≤ (εγ)ωϕµ, a contradiction with our definition of γ.

3. Systems of non-homogeneous equations

We turn to the following system of two non-homogeneous equations:−∆pu = λa(x)|u|α1 |v|β1−1v + f(x) in Ω;

−∆qv = µb(x)|v|α2 |u|β2−1u+ g(x) in Ω;

u = v = 0 on ∂Ω,

(3.1)

where 0 ≤ f, g ∈ L∞(Ω) are given functions. Our aim is to study thesolvability of system (3.1) in the following three cases: (a) λ, µ > 0 and(λ, µ) below or to the left of the eigenvalue curve C1 (in Section (3.1)), (b)(λ, µ) ∈ C1 (in Section (3.2)), and (c) λ, µ > 0 and (λ, µ) above or to the rightof, but close to, the eigenvalue curve C1 (in Section (3.3)). We recall that the

eigenvalue curve C1 has been defined in (2.18) with Λ′ def= Λ−√β2/(q−1−α2)

1 .

3.1. The case when (λ, µ) lies below or to the left of C1. Let us nowconsider system (3.1) when (λ, µ) are in the first quadrant of R2 and below orto the left of the principal eigenvalue curve C1. Of course, in order to assurethe existence of C1 we assume the more general condition (2.14) jointly withβ1, β2 > 0, 0 ≤ α1 < p− 1 and 0 ≤ α2 < q − 1.

Nonlinear eigenvalue problems for degenerate elliptic systems 11

Theorem 3.1. Let f, g ∈ L∞(Ω), f ≥ 0, g ≥ 0, and λ > 0, µ > 0 be suchthat

λ1√

β1(p−1−α1) µ1√

β2(q−1−α2) < Λ′.Then system (3.1) has a unique weak solution (u, v) ∈ X+. If, moreover,α1 = α2 = 0 and f + g 6≡ 0 then there exists a unique solution in X.

Proof. First observe that we can rule out the case when either f ≡ 0 andα1 6= 0, or g ≡ 0 and α2 6= 0, because in these cases either (0, Tq(g)) or(Tp(f), 0) are solutions of (3.1). Otherwise consider (u0, v0) = (0, 0) anddefine recursively for n ∈ N: un = Tp

(a|un−1|α1 |vn−1|β1−1v + f

)and vn =

Tq(b|vn−1|α2 |un−1|β2−1v + g

). It is enough to prove that both sequences,

un∞n=1 and vn∞n=1, are uniformly bounded in L∞(Ω). By the regularityresults already quoted, these sequences will also be uniformly bounded inC1,α

0 (Ω). First observe that

0 ≤ ( 6≡)u2 ≤ · · · ≤ un ≤ un+1 ≤ . . . and

0 ≤ ( 6≡)v2 ≤ · · · ≤ vn ≤ vn+1 ≤ . . . pointwise a.e. in Ω.(3.2)

Hence, the functions undef= un/‖un+1‖∞ and vn

def= vn/‖vn+1‖∞ satisfy 0 ≤un, vn ≤ 1 almost everywhere in Ω, n ∈ N. Consequently, the right-handsides of the following equations:−∆p

(un

‖un‖α1p−1∞ ‖vn‖

β1p−1∞

)= λa(x)uα1

n−1vβ1n−1 + f(x)

‖un‖α1∞ ‖vn‖

β1∞in Ω,

−∆q

(vn

‖vn‖α2q−1∞ ‖un‖

β2q−1∞

)= µb(x)vα2

n−1uβ2n−1 + g(x)

‖vn‖α2∞ ‖un‖

β2∞in Ω,

un = vn = 0 on ∂Ω

(3.3)

are uniformly bounded in L∞(Ω). Employing the regularity result in C1,α(Ω),we obtain that both sequences on the left-hand sides above,

un

‖un‖α1p−1∞ ‖vn‖

β1p−1∞

andvn

‖vn‖α2q−1∞ ‖un‖

β2q−1∞

, (3.4)

are bounded in C1,α(Ω) and, in particular, also in L∞(Ω); that is, there is aconstant C > 0 such that

‖un‖∞

‖un‖α1p−1∞ ‖vn‖

β1p−1∞

≤ C and‖vn‖∞

‖vn‖α2q−1∞ ‖un‖

β2q−1∞

≤ C, (3.5)

n ∈ N. Thus, ‖un‖∞ is uniformly bounded if and only if ‖vn‖∞ is uni-formly bounded. Now assume that, by contradiction, ‖un‖∞ → +∞ and

12 Mabel Cuesta and Peter Takac

‖vn‖∞ → +∞. We combine (3.4) with (3.5) to conclude that also the se-quences un/‖un‖∞ and vn/‖vn‖∞ are uniformly bounded in C1,α(Ω). Hence,for a subsequence denoted again by (un, vn) with n ∈ N, there exist somefunctions u∗, v∗ ∈ C1

0 (Ω), u∗, v∗ ≥ 0 in Ω, ‖u∗‖∞ = ‖v∗‖∞ = 1, such thatun‖un‖∞

→ u∗ andvn‖vn‖∞

→ v∗ in C10 (Ω) as n→∞.

Furthermore, there exist λ∗, µ∗ ∈ R such that also

sndef=

‖vn‖β1∞

‖un||p−1−α1∞

→ λ∗ and tndef=

‖un‖β2∞

‖vn||q−1−α2∞

→ µ∗ as n→∞.

Observe that sβ2n t

p−1−α1n = 1 forces λβ2

∗ µp−1−α1∗ = 1. Passing to the limit in

system (3.3) above, as n→∞, we find−∆pu∗ ≤ λλ∗a(x)uα1

∗ vβ1∗ in Ω;

−∆qv∗ ≤ µµ∗b(x)vα2∗ u

β2∗ in Ω;

u∗ = v∗ = 0 on ∂Ω.

The inequalities are obtained with a help from inequalities (3.2). From thefirst inequality in the system above we deduce that (λλ∗)−1/(p−1−α1)u∗ is alower solution of problem (2.9) and (µµ∗)−1/(q−1−α2)v∗ is a lower solution ofproblem (2.10). Using the uniqueness of positive solutions of these problemswe deduce that (λλ∗)−1/p−1−α1u∗ ≤ J1(v∗) and (µµ∗)−1/q−1−α2v∗ ≤ J2(u∗)and, consequently, using also λβ2

∗ µp−1−α1∗ = 1, we get

µ−1/β2 λ−1/(p−1−α1)u∗ ≤ (J1 J2)(u∗).

From the uniqueness results of Theorem A.2 we infer µ−1/β2 λ−1/(p−1−α1) ≤Λ1. We recall that Λ1 is the unique principal eigenvalue of J1 J2 associatedto some u′ ∈

V +. Thus, using the definition of Λ′ in (2.6), we obtain a

contradiction with the hypothesis λ1√

β1(p−1−α1) µ1√

β2(q−1−α2) < Λ′. We havejust proved that both sequences un∞n=1 and vn∞n=1 are uniformly boundedin C1,α

0 (Ω). Hence, there exists (u, v) ∈ X+ such that un → u and vn → vin C1

0 (Ω) and u, v solves (3.1).Assume now by contradiction that (u, v) ∈ X+ is a second solution of

(3.1); we distinguish between two cases: (a) u 6≡ 0, v 6≡ 0 and (b) eitheru ≡ 0 or v ≡ 0.

In case (a) let us assume that for instance f 6≡ 0 (a similar proof worksif g 6≡ 0). Since −∆pu ≥ f = −∆pu1, we have u ≥ u1 and, after an

Nonlinear eigenvalue problems for degenerate elliptic systems 13

iteration process, u ≥ un for all n ∈ N. Thus, u ≤ u and v ≤ v. Observethat if for instance u = u then it follows from the first equation of system(3.1) that λa(x)uα1vβ1 + f = λa(x)uα1 vβ1 + f and consequently v = v.Thus, let us assume by contradiction that u ≤ ( 6≡)u and v ≤ ( 6≡)v. Denoteω

def= (p− 1−α1)/β1, and consider the smallest k ∈ R such that u ≤ ku andv ≤ kωv. We have ω = β2/(q − 1 − α2). Let us assume, by contradiction,that k > 1. From the first equation of (3.1) we have

−∆p(ku) = λa(x)(ku)α1(kωv)β1 + kp−1f ≥ (6≡)−∆pu in Ω,

and ku = u = 0 on ∂Ω. From the strong comparison principle of [4] weinfer that u ku; see (A.3) for the definition of “”. Taking advantage ofthe strict inequality in the second equation of system (3.1) we get, again byusing the same SCP, that v kωv, in contradiction with our choice of k.

In case (b) if both u ≡ 0 and v ≡ 0 then f = g ≡ 0 and we can provedirectly from the system that µ−1/β2 λ−1/(p−1−α1)u = (J1J2)(u). Therefore,from Theorem A.2, it follows that µ−1/β2 λ−1/(p−1−α1) = Λ1, a contradictionwith our hypothesis. In the case u ≡ 0 it is trivial that the couple (0, Tq(g))is the unique solution of (3.1). A similar result holds if v ≡ 0.

Finally, to prove the uniqueness in X when α1 = α2 = 0, we argue as inPart (iii) of Proposition 2.7 to get the conclusion. We leave the details tothe reader.

As a corollary of this theorem we have the following more general existenceresult.

Corollary 3.2. Let us consider f, g ∈ L∞(Ω) and let λ > 0, µ > 0 be suchthat

λ1√

β1(p−1−α1) µ1√

β2(q−1−α2) < Λ′.

Then system (3.1) possesses at least one solution.

Proof. Assume |f |+ |g| 6≡ 0, otherwise the conclusion is trivial. Let (u1, v1)

∈X+ be the (unique) solution of (3.1) for the functions |f | and |g| instead of

f and g. Then trivially (u1, v1) is an upper solution of our problem. Similarly(−u1,−v1) is a lower solution of our problem. We then apply degree theoryto the mapping Tf,g : X 7→ X defined by

Tf,g(u, v) def=(Tp(a|u|α1 |v|β1−1v + f), Tq(b|v|α2 |u|β2−1u+ g)

)to get the conclusion (cf., for instance, [5]).

14 Mabel Cuesta and Peter Takac

3.2. The case when (λ, µ) ∈ C1. In this section we prove a nonexistenceresult for system (3.1) in the case when (λ, µ) ∈ C1, f ≥ 0, g ≥ 0, andf + g 6≡ 0.

Proposition 3.3. Let f, g ∈ L∞(Ω), f ≥ 0, g ≥ 0, f + g 6≡ 0, and λ > 0,µ > 0 be such that

λ1√

β1(p−1−α1) µ1√

β2(q−1−α2) = Λ′.

Then system (3.1) has no solution in X+. If, moreover, α1 = α2 = 0 thenthere is no solution in X.

Proof. Assume by contradiction that (u, v) ∈ X+ is a solution of (3.1).Then,

−∆pu ≥ λa(x)uα1vβ1 in Ω;

−∆qv ≥ µb(x)vα2uβ2 in Ω;

u = v = 0 on ∂Ω,and arguing as in the first part of the proof of Theorem 3.1 we conclude thatJ1 J2(u) ≤ Λ1u. By Theorem A.2, it follows that u = ρϕλ for some ρ > 0and similarly v = ρµ1β2ϕµ. But this is impossible because of f + g 6≡ 0.

In the case α1 = α2 = 0 let us assume, by contradiction, that there existsa solution (u, v) ∈ X and let us consider the smallest γ ∈ R+ such that−u ≤ γϕλ, −v ≤ γω ϕµ, where ω = β2

q−1 . If γ = 0 then (u, v) ∈ X+ and weargue as previously to arrive at a contradiction with f + g 6≡ 0. Hence, wemay assume γ > 0. If, for instance, f 6≡ 0, then, from the first equation ofsystem (3.1), we have

−∆p(−u) = λa(x)| − v|β1−1(−v)− f ≤ (6≡)−∆p(γϕλ) in Ω.

Applying Theorem A.1, we conclude that−u γϕλ. Then, using the secondequation of (3.1), we get also that −v γωϕµ and thus a contradiction withthe minimality of γ.

3.3. An antimaximum principle for systems. Here we treat the casewhen (λ, µ) lies above or to the right of, but close to, C1.

Let us recall the so-called “antimaximum principle” for a single equation(cf. for instance [9, 10]): Assume that Ω is a bounded domain in RN with aC2-boundary ∂Ω. If f ∈ L∞(Ω), f ≥ 0, f 6≡ 0 in Ω, then there exists δ > 0such that, for each λ ∈ (λ1, λ1 + δ), every weak solution u ∈W 1,p

0 (Ω) of

−∆pu = λ|u|p−2u+ f in Ω u = 0 on ∂Ω

Nonlinear eigenvalue problems for degenerate elliptic systems 15

satisfies −u ∈V +; i.e., u is of class C1 and satisfies u 0 (cf. (A.3) for the

definition of “”).In this section we consider again system (3.1) with α1 = α2 = 0, f ≥ 0

and g ≥ 0. More precisely, we have the following result.

Theorem 3.4. Let α1 = α2 = 0, β1β2 = (p − 1)(q − 1), and (λ1, µ1) ∈ C1.Consider two functions 0 ≤ f, g ∈ L∞(Ω) with f + g 6≡ 0. Then thereexists δ > 0 such that, for all pairs (λ, µ) ∈ R2 with λ1 < λ < λ1 + δ andµ1 < µ < µ1 + δ, every (weak) solution (u, v) of system (3.1) satisfies u 0and v 0 in Ω.

Proof. Assume that, by contradiction, there exist a sequence (λ′n, µ′n)∞n=1

⊂ R2, such that λ′n > λ1, µ′n > µ1, and λ′n → λ1, µ′n → µ1 as n → ∞, andanother sequence of solutions (un, vn) ∈ X of

−∆pun = λ′na(x)|vn|β1−1vn + f(x) in Ω;

−∆qvn = µ′nb(x)|un|β2−1un + g(x) in Ω;

un = vn = 0 on ∂Ω,

(3.6)

with at least one of −un or −vn not in (C10 (Ω))+. We distinguish between

two cases: (a) both ‖un‖∞ and ‖vn‖∞ are bounded and (b) either ‖un‖∞ or‖vn‖∞ is unbounded.

In case (a) it follows that the sequences are bounded in C1,α0 (Ω), by reg-

ularity. Employing Arzela-Ascoli’s theorem, we may pass to the limit inC1

0 (Ω). Then the limit functions u, v ∈ C10 (Ω) satisfy

−∆pu = λa(x)|v|β1−1v + f(x) in Ω;

−∆qv = µb(x)|u|β2−1u+ g(x) in Ω;

u = v = 0 on ∂Ω.

But this contradicts Proposition 3.3.In case (b) we argue as in the proof of Theorem 3.1; see system (3.3).

We observe that both sequences in (3.4) are bounded in C1,α(Ω) and, inparticular, also in L∞(Ω); that is, there is a constant C > 0 such that bothinequalities in (3.5) hold for every n ∈ N. Thus, ‖un‖∞ →∞ if and only if‖vn‖∞ → ∞ as n → ∞. For a subsequence denoted again by (un, vn) withn ∈ N, there exist some functions u∗, v∗ ∈ C1

0 (Ω), ‖u∗‖∞ = ‖v∗‖∞ = 1, suchthat

un‖un‖∞

→ u∗ andvn‖vn‖∞

→ v∗ in C10 (Ω) as n→∞. (3.7)

16 Mabel Cuesta and Peter Takac

Furthermore, there exist λ∗, µ∗ ∈ R such that also

sndef=‖vn‖β1

∞

‖un||p−1∞→ λ∗ and tn

def=‖un‖β2

∞

‖vn||q−1∞→ µ∗ as n→∞,

thanks to α1 = α2 = 0. We have λβ2∗ µ

p−1∗ = 1. Passing to the limit in

system (3.6) above, as n→∞, for un/‖un‖∞ and vn/‖vn‖∞, we find−∆pu∗ = λ1λ∗a(x)|v|β1−1

∗ v∗ in Ω;

−∆qv∗ = µ1µ∗b(x)|u|β2−1∗ u∗ in Ω;

u∗ = v∗ = 0 on ∂Ω.

This shows that (λ1λ∗, µ1µ∗) ∈ C1. By Proposition 2.7(iii), we must have

either (u∗, v∗) ∈X+ or (−u∗,−v∗) ∈

X+.

If (u∗, v∗) ∈X+, we rewrite system (3.6) as follows:

−∆pun = λ1a(x)|vn|β1−1vn + (λ′n − λ1)a(x)|vn|β1−1vn + f(x) in Ω;

−∆qvn = µ1b(x)|un|β2−1un + (µ′n − µ1)b(x)|un|β2−1un + g(x) in Ω;

un = vn = 0 on ∂Ω.(3.8)

From the convergence in (3.7) we thus conclude that also (un, vn) ∈X+

for all n ≥ n0 with n0 ∈ N large enough. But then (3.8) contradicts thenonexistence result in Proposition 3.3. Thus, we must have (−u∗,−v∗) ∈X+. Using the convergence in (3.7) again, we now have (−un,−vn) ∈

X+

for all n ≥ n0 with n0 ∈ N large enough. But this contradicts our hypothesisthat for every n ∈ N we have −un 6∈ (C1

0 (Ω))+ or −vn 6∈ (C10 (Ω))+.

The proposition is proved.

Appendix A.. Appendix

A.1.. Strong comparison principle. In this paragraph we establish aversion of the strong comparison principle (SCP, for brevity) for the p-Laplacian. The present version of the SCP is a modification of those inD. Arcoya and D. Ruiz [1, Proposition 2.6, page 853], M. Cuesta and P. Takac[3, Theorem 1, page 81] and [4, Theorem 2.1, page 725], and M. Lucia andS. Prashanth [16, Theorem 1.3, pages 1006–1007]. Let f and g be two func-tions from L∞(Ω) satisfying f ≤ g almost everywhere in Ω. Assume that

Nonlinear eigenvalue problems for degenerate elliptic systems 17

u, v ∈W 1,p0 (Ω) are any weak solutions to the following boundary value prob-

lems, respectively:

−∆pu = f(x) in Ω; u = 0 on ∂Ω, (A.1)−∆pv = g(x) in Ω; v = 0 on ∂Ω. (A.2)

Then we have u ≤ v in Ω, by the weak comparison principle (WCP, forbrevity) due to P. Tolksdorf [23, Lemma 3.1, page 800]. A classical versionof the SCP would claim that if f and g satisfy also f 6≡ g in Ω then

u < v in Ω and∂u

∂ν>∂v

∂νon ∂Ω. (A.3)

We abbreviate u v if (A.3) holds. It is still an open question if this claimholds without additional hypotheses. For f ≡ 0 and u ≡ 0 in Ω, this is thestrong maximum principle due to P. Tolksdorf [23, Proposition 3.2.1 and3.2.2, page 801] and J. L. Vazquez [25, Theorem 5, page 200].

As usual, ν ≡ ν(x0) denotes the exterior unit normal to ∂Ω at x0 ∈ ∂Ω.We recall that u, v ∈ C1,β(Ω) by a regularity result due to E. DiBenedetto [7,Theorem 2, page 829] and P. Tolksdorf [24, Theorem 1, page 127] (interiorregularity, shown independently), and to G. Lieberman [15, Theorem 1, page1203] (regularity near the boundary).

Here, we verify the SCP (A.3) under the following additional hypotheses.

Theorem A.1. Let 1 < p <∞. Assume that Ω is either a bounded domainin RN whose boundary ∂Ω is a C2-manifold if N ≥ 2, or a bounded openinterval in R1 if N = 1. Let f, g ∈ L∞(Ω) be such that f ≤ g and f 6≡ g inΩ and also g ≥ 0 and g 6≡ 0 in Ω. Then the SCP (A.3) is valid for any weaksolutions u, v ∈W 1,p

0 (Ω) of equations (A.1) and (A.2).

Proof. The case N = 1 is proved in M. Cuesta and P. Takac [4, pages731–732]. In the sequel we therefore assume N ≥ 2.

First, the WCP with 0 ≡ f ≤ g and u ≡ 0 in Ω guarantees v ≥ 0 in Ω.Now we may apply the strong maximum principle ([23, 25]) to obtain (2.1).Take γ > 0 and δ > 0 small enough, such that |∇v(x)| ≥ γ holds for everyx ∈ Ωδ, where

Ωδ = x ∈ Ω : d(x) < δ (A.4)

is the open δ-neighborhood in Ω of the boundary ∂Ω. As usual, d(x) def=dist(x, ∂Ω) denotes the distance from a point x ∈ Ω to the boundary ∂Ω.Set Ω′δ = Ω \ Ωδ. Since ∂Ω is assumed to be a compact manifold of classC2, so is ∂Ω′δ provided δ > 0 is small enough. Indeed, the last claim is aconsequence of d ∈ C2(Ωδ), by [12, Lemma 14.16, page 355] and its proof,

18 Mabel Cuesta and Peter Takac

where it is shown that Ωδ is C1-diffeomorphic to ∂Ω×[0, δ] with x 7→ (x, 0) forall x ∈ ∂Ω, and Ω′δ = Ω\Ωδ is C1-diffeomorphic to Ω. Both diffeomorphismsare considered between manifolds with boundary of class C2. Of course, theycan be replaced by C2-diffeomorphisms, by M. W. Hirsch [13, Theorem 3.5,page 57].

Second, set w = v − u; hence 0 ≤ w ∈ C1,β(Ω) with w = 0 on ∂Ω.Subtracting Equation (A.1) from (A.2), we find out that w satisfies thefollowing linear elliptic inequality in the sense of distributions in Ωδ:

− div(A(x)∇w) def= −∑N

i,j=1∂∂xi

(aij(x) ∂w∂xj

)= g − f ≥ 0 for x ∈ Ωδ.

(A.5)

Here, the coefficients

aij(x) =∫ 1

0 aij ((1− s)∇u(x) + s∇v(x)) ds (A.6)

of the matrix A(x) belong to Cβ(Ωδ) and form a uniformly elliptic operatorin Ωδ, where

aij(z) = |z|p−2(δij + (p− 2) |z|−2 zizj

)is the Jacobian matrix of the mapping z 7→ |z|p−2z for z = (z1, . . . , zN ) ∈RN \ 0, with the Kronecker symbol δij . The uniform ellipticity is verifiedas follows, cf. P. Takac [21, Appendix A, pages 233–235]:

The symmetric matrix with the entries ˆaij(z) = δij + (p − 2) |z|−2 zizjhas only two eigenvalues, equal to 1 and p − 1. The expression |z|p−2 forz = (1 − s)a + sb with a = ∇u(x) and b = ∇v(x) is estimated by theinequalities

cp ·(

max0≤s≤1

|a + sb|)p−2

≤∫ 1

0|a + sb|p−2 ds ≤ Cp ·

(max

0≤s≤1|a + sb|

)p−2

for all a,b ∈ RN with |a|+ |b| > 0, (A.7)

where 0 < cp ≤ Cp < ∞ are some constants and we have substituted bfor the difference b − a in z = a + s(b − a) to simplify our notation. Thefirst inequality is trivial for 1 < p ≤ 2 (take cp = 1), the second one for2 ≤ p < ∞ (take Cp = 1); the remaining inequalities are proved in [21,Lemma A.1, page 233]. The desired uniform ellipticity now follows from ourchoice of Ωδ; we have |∇v| ≥ γ = const > 0 in Ωδ.

Third, in every subdomain σ of Ωδ with σ ∩ ∂Ω = ∅, we apply the strongmaximum principle from D. Gilbarg and N. S. Trudinger [12, Theorem 8.19,page 198] to the linear elliptic inequality (A.5) considered in σ. If ∂Ω is con-nected as in M. Cuesta and P. Takac [3, 4], then so is Ωδ and, consequently,

Nonlinear eigenvalue problems for degenerate elliptic systems 19

we obtain either w = v − u > 0 in Ωδ, or else w = v − u ≡ 0 in Ωδ. If ∂Ω isnot connected as in M. Lucia and S. Prashanth [16], Proposition 4.1 in [16,page 1009] guarantees that either w ≡ 0 in some connected component ofΩδ, or else w > 0 in Ωδ (cf. Cases A and B on page 1009).

If u < v in Ωδ then, for any fixed number η ∈ (0, δ), we can find a constantc > 0 such that v ≥ u + c holds on the boundary ∂Ω′η = ∂Ωη \ ∂Ω ⊂ Ωδ ofthe domain Ω′η = Ω \ Ωη. We combine this boundary inequality with

−∆p(u+ c) = −∆pu = f(x) ≤ g(x) = −∆pv in Ω′η

to conclude that v ≥ u+ c holds throughout Ω′η, by the WCP (P. Tolksdorf[23, Lemma 3.1, page 800]). We have thus obtained u < v in Ω = Ωδ ∪Ω′η asdesired. Furthermore, we can make use of the boundary point principle asshown in R. Finn and D. Gilbarg [8, Lemma 7, page 31] (for N ≥ 3, see also[8, Remarks, page 35]) in order to deduce that −∂u

∂ν (x0) < −∂v∂ν (x0) holds at

an arbitrary boundary point x0 ∈ ∂Ω.Finally, assume u ≡ v in Ωδ. Next, we employ a version of the divergence

theorem proved in M. Cuesta and P. Takac [3, Lemma A.1, page 742]. Forη ∈ (0, δ) small enough we apply the divergence theorem to equations (A.1)and (A.2) over the domain Ω′η = Ω \ Ωη. We thus obtain

−∫∂Ω′η

|∇u(x)|p−2∇u(x) · ν(x) dσ(x) =∫

Ω′η

f(x) dx, (A.8)

−∫∂Ω′η

|∇v(x)|p−2∇v(x) · ν(x) dσ(x) =∫

Ω′η

g(x) dx. (A.9)

Since u ≡ v in Ωδ and ∂Ω′η ⊂ Ωδ, the two surface integrals on the left-handside in equations (A.8) and (A.9) are equal. Therefore, we have∫

Ω′η

f(x) dx =∫

Ω′η

g(x) dx.

Combined with f ≤ g in Ω, this equality forces f ≡ g in Ω′η. From u ≡ vin Ωδ we obtain also f ≡ g in Ωδ. Thus, we arrive at f ≡ g throughoutΩ = Ωδ ∪ Ω′η, a contradiction to our hypothesis f 6≡ g in Ω.

The proposition is proved.

A.2.. Kreın-Rutman theorem for nonlinear mappings. We need thefollowing version of the Kreın-Rutman theorem for nonlinear homogeneousmappings which is essentially due to P. Takac [20, Theorem 3.5, page 1763];

20 Mabel Cuesta and Peter Takac

cf. also R. Nussbaum [17, Proposition 3.1, page 100]. As stated below, thisversion includes also some results from the proof of Theorem 3.5 in [20].

We assume that E = (E,≤) is a strongly ordered Banach space; i.e., E is areal Banach space endowed with an ordering “≤ ” that is compatible with thenorm topology on E and such that the positive cone E+

def= x ∈ E : x ≥ 0has nonempty interior (in E) denoted by

E+. For x, y ∈ E we write x ≤ y

(or equivalently y ≥ x) if and only if y − x ∈ E+. Similarly, we write x < y(or equivalently y > x) in E if and only if y − x ∈ E+ \ 0, whereas x y

(or equivalently y x) in E if and only if y − x ∈E+. In particular,

E+ =

x ∈ E : x 0. For a ≤ b in E, the set [a, b] def= x ∈ E : a ≤ x ≤ b iscalled a closed order interval in E. (The reader is referred to the monographby H. H. Schaefer [19] for details on ordered Banach spaces.)

A (nonlinear) self-mapping T : E+ → E+ is called homogeneous if T (sx) =sTx holds for all x ∈ E+ and s ∈ [0, 1]. We say that T : X ⊂ E → E ismonotone if x ≤ y in X implies Tx ≤ Ty, and strongly monotone if x < yin X implies Tx Ty. Finally, a monotone mapping T : X ⊂ E → Eis called order-compact if the set T ([a, b] ∩ X) = Tx : x ∈ [a, b] ∩ X hascompact closure in E for each pair a ≤ b in E.

Theorem A.2. ([20, Theorem 3.5, page 1763]) Let E be a strongly orderedBanach space and T : E+ → E+ a continuous homogeneous mapping. As-sume that T is strongly monotone and order-compact. Then there exist anumber Λ1 > 0 and e ∈

E+ such that Te = Λ1e. Furthermore, if λ ∈ [0,Λ1]

and u ∈ E+\0 satisfy Tu ≤ λu, then λ = Λ1 and u = ce for some constantc > 0, and thus Tu = Λ1u. Finally, if u ∈ E+ \0 satisfies Tu ≥ Λ1u, thenagain u = ce with a constant c > 0.

We wish to apply this theorem in the Banach space E = C ′0(Ω) (or inE2 = E ×E) of all continuous functions u : Ω→ R with u = 0 on ∂Ω, suchthat u possesses a continuous normal derivative ∂u/∂ν : ∂Ω → R. Recallthat ν ≡ ν(x0) denotes the exterior unit normal to ∂Ω at x0 ∈ ∂Ω. Moreprecisely, C ′0(Ω) is defined to be the completion of the vector space

C10 (Ω) def=

u ∈ C1(Ω) : u = 0 on ∂Ω

under the order norm

‖u‖′ def= maxΩ|u(x)|+ max

∂Ω

∣∣∣∂u∂ν

∣∣∣ for u ∈ C10 (Ω). (A.10)

Nonlinear eigenvalue problems for degenerate elliptic systems 21

The ordering “≤ ” on C ′0(Ω) is induced by the natural pointwise ordering offunctions; that is, u ≤ v in E is defined by u(x) ≤ v(x) for all x ∈ Ω. Thus,

C ′0(Ω)+ =u ∈ C ′0(Ω) : u ≥ 0 in Ω

is the positive cone in C ′0(Ω); the interior

C ′0(Ω) consists of all functions

u ∈ C ′0(Ω) that satisfy both inequalities of the strong maximum principle,

u > 0 in Ω and∂u

∂ν< 0 on ∂Ω. (A.11)

The self-mapping T : C ′0(Ω)→ C ′0(Ω) may typically be of the form Tudef=

v, where u ∈ C ′0(Ω) is arbitrary and v ∈W 1,p0 (Ω) is the unique weak solution

to the Dirichlet boundary-value problem

−∆pv = m(x) |u|p−2u in Ω; v = 0 on ∂Ω. (A.12)

Hence, v ∈ C1,β(Ω) ∩ C ′0(Ω), by regularity. The function m ∈ L∞(Ω) isa positive “weight;” i.e., it satisfies m > 0 almost everywhere in Ω. It iseasy to see that this mapping satisfies all hypotheses of Theorem A.2 above,thanks to Theorem A.1 combined with the uniqueness and regularity of v;cf. [20], Example 5.2 (pages 1771–1773) and Corollary 5.3 (page 1773).

References

[1] D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the p-Laplace operator,Comm. P.D.E., 31 (2006), 849–865, doi: 10.1080/03605300500394447

[2] P. Clement, R.F. Manasevich, and E. Mitidieri, Positive solutions for a quasilinearsystem via blow up, Comm. P.D.E., 18 (1993), 2071–2106.

[3] M. Cuesta and P. Takac, A strong comparison principle for the Dirichlet p-Laplacian,in “Reaction-Diffusion Systems,” G. Caristi and E. Mitidieri, eds., pp. 79–87. InLecture Notes in Pure and Applied Mathematics, Vol. 194. Marcel Dekker, Inc., NewYork–Basel, 1998.

[4] M. Cuesta and P. Takac, A strong comparison principle for positive solutions of de-generate elliptic equations, Differential and Integral Equations, 13 (2000), 721–746.

[5] C. De Coster and M. Henrard, Existence and localization of solution for second orderelliptic BVP in presence of lower and upper solutions without any order, J. DifferentialEquations, 145 (1998), 420–452.

[6] D. G. de Figueiredo, Semilinear elliptic systems: existence, multiplicity, symmetryof solutions, in M. Chipot; eds., “Handbook of Differential Equations: StationaryPartial Differential Equations,” Vol. 5, pp. 1–48. Elsevier Science B.V., Amsterdam,The Netherlands, 2008.

[7] E. DiBenedetto, C1+α local regularity of weak solutions of degenerate elliptic equa-tions, Nonlinear Anal., 7 (1983), 827–850.

[8] R. Finn and D. Gilbarg, Asymptotic behavior and uniqueness of plane subsonic flows,Comm. Pure Appl. Math., 10 (1957), 23–63.

22 Mabel Cuesta and Peter Takac

[9] J. Fleckinger, J.-P. Gossez, P. Takac, and F. de Thelin, Existence, nonexistence etprincipe de l’antimaximum pour le p-laplacien, Comptes Rendus Acad. Sc. Paris, SerieI, 321 (1995), 731–734.

[10] J. Fleckinger, J.-P. Gossez, P. Takac, and F. de Thelin, Nonexistence of solutionsand an anti-maximum principle for cooperative systems with the p-Laplacian, Math.Nachrichten, 194 (1998), 49–78.

[11] J. Fleckinger, J. Hernandez, P. Takac, and F. de Thelin, Uniqueness and positiv-ity for solutions of equations with the p-Laplacian, in “Reaction-Diffusion Systems,”G. Caristi and E. Mitidieri, eds., pp. 141–155. In Lecture Notes in Pure and AppliedMathematics, Vol. 194. Marcel Dekker, Inc., New York–Basel, 1998.

[12] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of SecondOrder,” Springer-Verlag, New York–Berlin–Heidelberg, 1977.

[13] M. W. Hirsch, “Differential Topology,” Springer-Verlag, New York–Berlin–Heidelberg,1976.

[14] O. Ladyzhenskaya and N. Uraltseva, “Equations aux Derivees Partielles du TypeElliptique du Second Ordre,” Ed. Masson, France, 1975.

[15] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Non-linear Anal., 12(11) (1988), 1203–1219.

[16] M. Lucia and S. Prashanth, Strong comparison principle for solutions of quasilinearequations, Proc. Amer. Math. Soc., 132(4) (2003), 1005–1011.

[17] R. D. Nussbaum, Hilbert’s projective metric and iterated nonlinear maps, Mem. Amer.Math. Soc., 391, Providence, R.I., 1988.

[18] D. H. Sattinger, Monotone metods in nonlinear elliptic and parabolic boundary valueproblems, Indiana Univ. Math. J., 21(11) (1972), 979–1000.

[19] H. H. Schaefer, “Topological Vector Spaces,” Springer-Verlag, New York–Berlin–Heidelberg–Tokyo, 1971 (5-th print., 1986).

[20] P. Takac, Convergence in the part metric for discrete dynamical systems in orderedtopological cones, Nonlinear Anal., 26(11) (1996), 1753–1777.

[21] P. Takac, On the Fredholm alternative for the p-Laplacian at the first eigenvalue,Indiana Univ. Math. J., 51(1) (2002), 187–237.

[22] F. de Thelin, Resultats d’existence et de non-existence pour la solution positive etbornee d’une e.d.p. elliptique non lineaire, Annales Faculte des Sciences de Toulouse,VIII(3) (1986–1987), 375–389.

[23] P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with con-ical boundary points, Comm. P.D.E., 8(7) (1983), 773–817.

[24] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J.Differential Equations, 51 (1984), 126–150.

[25] J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations,Appl. Math. Optim., 12 (1984), 191–202.