nonlinear eigenvalue problems for degenerate elliptic ... - LMPA - ULCO

Differential and Integral Equations

Volume xx, Number xxx, , Pages xx–xx

NONLINEAR EIGENVALUE PROBLEMS FOR DEGENERATE ELLIPTIC SYSTEMS Mabel Cuesta Département de Mathématique, Université du Littoral (ULCO) 50, rue F. Buisson, F–62228 Calais, France ć ˇ Peter Taka Institut f¨ ur Mathematik, Universität Rostock Universit¨ atsplatz 1 D–18055 Rostock, Germany (Submitted by: Klaus Schmitt) Abstract. The following nonlinear eigenvalue problem for a pair of real parameters (λ, µ) is studied: 8 β1 −1 α1 > v : u=v=0

in Ω; in Ω; on ∂Ω.

Here, p, q ∈ (1, ∞) are given numbers, Ω is a bounded domain in RN with a C 2 -boundary, a, b ∈ L∞ (Ω) are given functions, both assumed to be strictly positive on compact subsets of Ω, and the coefficients αi , 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 ) = β1 β2 . A smooth curve of pairs (λ, µ) in (0, ∞) × (0, ∞) is found for which the quasilinear elliptic system possesses a solution pair (u, v) consisting of nontrivial, nonnegative functions u ∈ W01,p (Ω) and v ∈ W01,q (Ω). Key roles in the proof are played by the strong comparison principle and a nonlinear Kre˘ın-Rutman theorem obtained by the authors in earlier works. The main result is applied to some quasilinear elliptic systems related to the above system.

Accepted for publication: March 2010. AMS Subject Classifications: 35J70, 35P30; 47H07, 47H12. 1

2

ć ˇ Mabel Cuesta and Peter Taka

1. Introduction We consider the following system of quasilinear elliptic boundary-value problems:  α1 β1 −1 v in Ω;   −∆p u = λ a(x) |u| |v| α2 β2 −1 (1.1) −∆q v = µ b(x) |v| |u| u in Ω;   u=v=0 on ∂Ω, where Ω is a bounded domain in RN whose boundary ∂Ω is a C 2 -manifold ∂Ω (which is not assumed to be connected), x = (x1 , . . . , xN ) is a generic point in Ω, p, q ∈ (1, ∞) are given numbers, a, b ∈ L∞ (Ω) are given functions satisfying def

a0 = ess inf a(x) > 0 x∈Ω

and

def

b0 = ess inf b(x) > 0, x∈Ω

and αi , βi are constants with αi ≥ 0 and βi > 0 for i = 1, 2. The quasilidef near elliptic operator u 7→ ∆p u = div(|∇u|p−2 ∇u), called the p-Laplacian, 0 is defined for u ∈ W01,p (Ω) with values ∆p u ∈ W −1,p (Ω), the dual space of W01,p (Ω), where p1 + p10 = 1. We view System (1.1) as a homogeneous nonlinear eigenvalue problem for the unknown pair of parameters (λ, µ) ∈ R∗+ × R∗+ = (0, ∞)2 associated with the unknown pair of nonnegative eigenfunctions u ∈ W01,p (Ω) and v ∈ W01,q (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, ∂g ∂f ∂g in general, except for the cases when either ∂f ∂v ≡ ∂u or ∂u ≡ ∂v where we have denoted by f (x, u, v) and g(x, u, v), respectively, the right-hand side of the first and second equations in (1.1). Computing the partial derivatives ( ∂f ∂g α1 β1 −1 , α2 β2 −1 , ∂v = λβ1 a(x) |u| |v| ∂u = µβ2 b(x) |v| |u| ∂f ∂u

= λα1 a(x) |u|α1 −2 u|v|β1 −1 v,

∂g ∂v

= µα2 a(x) |v|α1 −2 v|u|β1 −1 u,

we observe that the former case occurs if and only if α1 = β2 − 1, α2 = β1 − 1, and λβ1 a(x) = µβ2 b(x) for a.e. x ∈ Ω, whereas the latter case occurs if and only if either α1 − 1 = β2 , α2 − 1 = β1 , and λα1 a(x) = µα2 b(x) for x ∈ Ω, or α1 = α2 = 0. The special “superlinear” case when α1 = α2 = 0 and β1 β2 > (p − 1)(q − 1) was treated in Ph. Clément, R. F. Manásevich, and E. 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 not require any variational or Hamiltonian structure for System (1.1). We wish to apply a simplified version of a Kre˘ın-Rutman theorem for homogeneous nonlinear mappings due to P. Takáˇc [20, Theorem 3.5, page 1763]. Given any f ∈ L∞ (Ω), we denote by Tp (f ) ≡ u ∈ W01,p (Ω) the unique weak solution of the boundary-value problem − ∆p u = f (x) in Ω;

u = 0 on ∂Ω.

(1.2)

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

X = [C01 (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 by def S(u, v) = (˜ u, v˜) with u ˜ = Tp (a|u|α1 |v|β1 −1 v)

and

v˜ = Tq (b|v|α2 |u|β2 −1 u)

for (u, v) ∈ X. In Section 2, we treat the case when S is homogeneous whereas in Section 3, we deal with a slightly more general case. We prove in both cases the existence of a curve C1 of principal eigenvalues (λ, µ) for system (1.1). In Section 3 we turn to the nonhomogeneous problem   −∆p u = λa(x)|u|α1 |v|β1 −1 v + f (x) in Ω;   (1.3) −∆q v = µb(x)|v|α2 |u|β2 −1 u + g(x) in Ω;    u = v = 0 on ∂Ω. We discuss the solvability of (1.3) when either (a) λ, µ > 0, (λ, µ) lies below or 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 (“antimaximum principle”). We also briefly discuss the uniqueness of solutions in these cases. Finally in the appendix we present first a new theorem on the strong comparison principle for the p-Laplacian and a simplified version of the Kre˘ın-Rutman theorem for homogeneous nonlinear mappings of P. Takáˇc [20, Theorem 3.5, page 1763]. 2. A curve of principal eigenvalues def

2.1. The case when S is homogeneous. We consider C01 (Ω) = {f ∈ C 1 (Ω) : f = 0 on ∂Ω} and its positive cone (C01 (Ω))+ = {f ∈ C01 (Ω) : f ≥ 0 in Ω} which is normal and has nonempty topological interior (C01 (Ω))◦+

4


characterized by v ∈ (C01 (Ω))◦+ if and only if v ∈ C01 (Ω) satisfies the strong maximum principle ([23, 25]): ∂v v > 0 in Ω and < 0 on ∂Ω. (2.1) ∂ν We consider also the Cartesian product X = [C01 (Ω)]2 , which is a strongly ordered Banach space endowed with the natural norm and ordering for pairs of 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 + = [(C01 (Ω))◦+ ]2 . def

We define the map S : X → X by S(u, v) = (˜ u, v˜) with u ˜ = Tp (a|u|α1 |v|β1 −1 v)

and

v˜ = Tq (b|v|α2 |u|β2 −1 u)

for (u, v) ∈ X, where Tp and Tq have been defined in (1.2). The following lemma describes the interaction between positive eigenvalues 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) = (ρα1 σ β1 )1/(p−1) S1 (u, v), (ρβ2 σ α2 )1/(q−1) S2 (u, v) . (iii) If (u, v) ∈ X+ solves (1.1) with (λ0 , µ0 ) in place of (λ, µ), then for any ρ, σ > 0, the pair (ρu, σv) solves (1.1) with (λ, µ) satisfying λ = ρp−1−α1 σ −β1 λ0 ,

µ = σ q−1−α2 ρ−β2 µ0 .

(2.2)

The proof is left to the reader. Let us now look for principal eigenvalues of the map S via the Kre˘ınRutman 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 1

1

λ β1 µ β2 = Λ. Moreover, (u, v) = c(u1 , v1 ) for some constant c > 0.

(2.4)


5

Proof. It is easy to see from Lemma 2.1(ii) that S is homogeneous; i.e., def

S(tu, tv) = tS(u, v) for every t ∈ R+ = [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 result of de Thélin [22, Théorème 1, page 376] or Ladyzhenskaya and Ural’tseva [14, Théorème 7.1] (for the L∞ (Ω) bound of u and v) and the results of Lieberman [15, Theorem 1, page 1203] and DiBenedetto [7] or Tolksdorf [24] for interior regularity it follows that the mapping Tp : L∞ (Ω) → C 1,β (Ω) is continuous and bounded; that is, it maps bounded sets into bounded 0 sets. It follows from Arzel´ a-Ascoli’s theorem that Tp : L∞ (Ω) → C 1,β (Ω) is compact whenever 0 < β 0 < β; that is, it maps bounded sets into sets with compact closure. By Theorem A.1 there exists a unique number Λ1 ∈ R and ◦

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

−(p−1)

−(q−1)

the couple (λ0 , µ0 ) = (Λ1 , Λ1 ) is a principal eigenvalue for system (1.1). Using this eigenvalue in Lemma 2.1(iii) and condition (2.3) it follows that, for any (λ, µ) satisfying λ 1/β1 µ 1/β2 = 1, (2.5) −(p−1) −(q−1) Λ1 Λ1 1/β1 λ the couple ρ = −(p−1) , σ = 1 solves (2.2). Therefore, (λ, µ) is a prinΛ1

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 −1 −1 def choose ρ, σ > 0 such that ( σρ )β1 λ p−1 = ( σρ )β2 µ q−1 = Λ0 . Then, by Lemma 2.1, parts (ii) and (iii), S(ρu, σv) = Λ0 (ρu, σv) and, therefore, by the uniqueness results of Theorem A.2, Λ0 = Λ1 . It follows from the definition of Λ0 above that (λ, µ) satisfies (2.5). The conclusion of the theorem is now obtained with def −(p−1)/β1 −(q−1)/β2 Λ = Λ1 . (2.6) Finally, the fact that (u, v) = c(u1 , v1 ) with some positive constant c follows from the uniqueness in Theorem A.2. Let us denote the set of all principal eigenvalues (λ, µ) of (1.1) by 1 1 def C1 = (λ, µ) ∈ (R∗+ )2 : λ β1 µ β2 = Λ . We will call it the principal eigenvalue curve of system (1.1).

6


2.2. The general case. Let us now consider λ > 0, µ > 0, the mapping S : X+ → X+ defined in Section 2 above, and its square iterate S 2 ≡ S ◦ S. We wish to apply the Kre˘ın-Rutman theorem to S 2 . Notice that S 2 : X+ → X+ is homogeneous if and only if the following equations are satisfied: α1 = α2 = 0,

β1 β2 = (p − 1)(q − 1).

(2.7)

˜˜, v˜˜) = S(˜ Indeed, given (u, v) ∈ X+ , set (˜ u, v˜) = S(u, v) and (u u, v˜). For t ∈ R+ we have (tγ1 u ˜, tγ2 v˜) = S(tu, tv), (2.8) and, therefore, ˜˜, tδ2 v˜˜) = S 2 (tu, tv), (tδ1 u 1 +β1 γ2 2 +β2 γ1 where δ1 = α1 γp−1 , δ2 = α2 γq−1 . Thus, S 2 is homogeneous if and only if δ1 = δ2 = 1. These equations are equivalent to

α1 (γ1 − γ2 ) = (p − 1)(1 − γ1 γ2 ), β1 (γ1 − γ2 ) = (p − 1)(γ12 − 1), α2 (γ1 − γ2 ) = (q − 1)(γ1 γ2 − 1), β2 (γ1 − γ2 ) = (q − 1)(1 − γ22 ). The case γ1 = γ2 forces γ1 γ2 = 1 which yields γ1 = γ2 = 1. Consequently, αi and βi satisfy (2.3). Therefore, without loss of generality, from now on we may assume γ1 < γ2 . Using α1 ≥ 0 we obtain 1 − γ1 γ2 ≤ 0, whereas α2 ≥ 0 yields γ1 γ2 − 1 ≤ 0. It follows that γ1 γ2 = 1 again. This forces also α1 = α2 = 0 and β1 β2 = (p − 1)(q − 1). Since these conditions on αi , βi are very restrictive, one would prefer using a better method than the one we have just described. The idea is the following. Let us assume throughout this section that α1 < p−1 and α2 < q−1. We def introduce a new mapping T : X+ → X+ defined by T (u, v) = (J1 (v), J2 (u)) where, for (u, v) ∈ X, J1 (v) is the unique (weak) solution u ˜ of −∆p u ˜ = a(x)|˜ u|α1 v β1 in Ω; (2.9) u ˜=0 on ∂Ω, and J2 (u) is the unique (weak) solution v˜ of −∆q v˜ = b(x)|˜ v |α2 uβ2 v˜ = 0

in Ω; on ∂Ω.

(2.10)

The existence is obtained by classical minimization while uniqueness follows from a convexity argument ([11, Theorem 3, page 151]). The regularity results combined with the strong maximum principle already mentioned imply ◦

that the pair (˜ u, v˜) belongs to X + .


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 each component. We have now that the condition (2.14) below for the homogeneity of J1 ◦ J2 and J2 ◦ J1 is less restrictive than the one found before for S 2 . We will need the following result. Lemma 2.3. Let us denote V = C01 (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)viβ1 and ui = J1 (vi ) for i = 1, 2. Since m1 ≤ m2 , it follows that u2 is an upper solution for the following problem: − ∆p u = m1 (x)|u|α1 in Ω;

u = 0 on ∂Ω.

(2.11)

Let us denote by ϕ the positive eigenfunction of the Dirichlet p-Laplacian with weight m1 ; that is, there exists λ1 > 0 such that − ∆p ϕ = λ1 m1 (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 > 0 sufficiently small, we have λ1 (cϕ)p−1 ≤ (cϕ)α1 in Ω, whence cϕ is a lower solution for problem (2.11). By choosing c even smaller if necessary we can assume cϕ < u2 in Ω. Let us define the sequence {zn }∞ n=0 recursively by z0 = u2 and zn is the unique solution of α1 − ∆p zn = m1 (x)zn−1 in Ω;

zn = 0 on ∂Ω

(2.13)

in W01,p (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 ∈ V which is a solution of problem (2.11) and satisfies cϕ ≤ u ≤ u2 in Ω. We refer the reader to D. H. Sattinger [18] for details on this monotone iteration method. Thus, by uniqueness, u = u1 and the conclusion follows. Now we give the analogue of Lemma 2.1 for the mappings J1 ◦ J2 and J2 ◦ J1 . Lemma 2.4. (i) A couple u, v ∈ V+ \ {0} is a weak solution of (1.1) for ◦

−1

some (λ, µ) ∈ (R∗+ )2 if and only if u, v ∈ V + and J1 (v) = λ p−1−α1 u, J2 (u) = −1

µ q−1−α2 v.

8


(ii) For any ρ, σ > 0 we have β1 β2

(J1 ◦ J2 )(ρu) = ρ (p−1−α1 )(q−1−α2 ) (J1 ◦ J2 )(u), β1 β2

(J2 ◦ J1 )(σv) = σ (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 β1 β2 = (p − 1 − α1 )(q − 1 − α2 ).

(2.14)

◦

Then there exists Λ0 > 0 and a couple (u0 , v 0 ) ∈ 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 λ

√

1 β1 (p−1−α1 )

µ

√

1 β2 (q−1−α2 )

= Λ0 .

(2.15)

Moreover, (u, v) = (ρu0 , ρµ1/β2 v 0 ) with some positive constant ρ. Proof. It follows from Lemma 2.4(ii) and condition (2.14) that the mappings Ji ◦ Jj : V+ → V+ , i, j ∈ {1, 2}, i 6= j, are homogeneous. Moreover, both mappings are strongly monotone by the strong comparison principle in [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, there exists a unique number Λ1 ∈ R∗+ such that (J1 ◦J2 )(u0 ) = Λ1 u0 holds for some ◦

u0 ∈ V + which is unique up to a positive constant multiple. Similarly, there exists a unique number Θ1 ∈ R∗+ such that (J2 ◦ J1 )(v 0 ) = Θ1 v 0 holds for ◦

β2 -homogeneity of J2 applied to (J1 ◦J2 )(u0 ) = Λ1 u0 some v 0 ∈ V + . The q−1−α 2 yields β /(q−1−α2 ) (J2 ◦ J1 )(J2 (u0 )) = Λ1 2 J2 (u0 ).

Similarly, the

β1 p−1−α1 -homogeneity

of J1 applied to (J2 ◦J1 )(v 0 ) = Θ1 v 0 yields β /(p−1−α1 )

(J1 ◦ J2 )(J1 (v 0 )) = Θ1 1

J1 (v 0 ).

The uniqueness of Θ1 and v 0 yields β /(q−1−α2 )

Θ1 = Λ1 2

and

v 0 = θJ2 (u0 )

for some θ ∈ (0, ∞).

Hence, we have also β /(p−1−α1 )

Λ1 = Θ1 1

and

0 u0 = θ−β1 /(p−1−α1 ) Λ−1 1 J1 (v ).


9

1 and By Lemma 2.4(i), the pair (u0 , J2 (u0 )) solves (1.1) with λ = Λ1−p+α 1 1−q+α2 0 0 = µ = 1. Similarly, (J1 (v ), v ) solves (1.1) with λ = 1 and µ = Θ1 −β2 Λ1 . Thus, if λ, µ are positive real numbers satisfying

µ−1/β2 λ−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, if u, v ∈ V+ solves (1.1) for some (λ, µ) then, by Lemma 2.4(i), it follows that β /(p−1−α1 ) u = µ−1/β2 λ−1/(p−1−α1 ) u. (J1 ◦ J2 )(u) = µ−1/(q−1−α2 ) λ−1/β1 1 By the uniqueness results of Theorem A.2, we have equation (2.16) and u = ρu0 for some ρ > 0. Raising both sides of equation (2.16) to the p def power − β2 /(q − 1 − α2 ), we find that (λ, µ) satisfies (2.15) with Λ0 = √ − β2 /(q−1−α2 ) . Λ1 Finally, the result v = µ1/β2 ρv 0 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 = Λ0 . We will denote also in this case o n √ √ def C1 = (λ, µ) ∈ (R∗+ )2 : λ1/ β1 (p−1−α1 ) µ1/ β2 (q−1−α2 ) = Λ0 .

(2.18)

Let us also denote by (ϕλ , ϕµ ) the positive eigenfunction associated to (λ, µ) ∈ C1 with kϕλ kL∞ (Ω) = 1. The following proposition gives some properties of the principal eigenvalues 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 and only if (λ, µ) ∈ C1 . ◦

◦

(ii) Simplicity in X + . Let (λ, µ) ∈ C1 and (u, v), (u0 , v 0 ) ∈ X + be a couple of eigenfunctions associated to (λ, µ). Then there exists ρ > 0 such that u = ρu0 and v = ρµ1/β2 v 0 . (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

ć ˇ Mabel Cuesta and Peter Taka ◦

Proof. The uniqueness and simplicity in X + of principal eigenvalues follow from the previous theorem and Theorem A.2, so we just prove (iii). It is clear that there exists some γ ∈ R+ such that −u ≤ γϕλ and − v ≤ γ ω ϕµ , β2 where ω = q−1 ; let γ be the minimum of such γ’s. We assume by contradiction that γ > 0. Then if we have also −u ≡ γϕλ in Ω, it follows from the second equation of system (1.1) that µb(x)| − u|β2 −1 (−u) = µb(x)|γϕλ |β2 −1 γϕλ and, consequently, −v = γ ω ϕµ and we are done. Thus, it remains to treat the 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 (γ ω ϕµ ) in Ω, together with −u = γϕλ = −v = γ ω ϕµ = 0 on ∂Ω. It follows from the strong comparison principle (SCP) of Theorem A.1 that −u γϕλ , −v γ ω ϕµ (see (A.3) in the Appendix for the definition of the strong ordering “” in C 1 (Ω)). Thus, we can find 0 < ε < 1 such that −u ≤ εγϕλ and −v ≤ (εγ)ω ϕµ , a contradiction with our definition of γ. 3. Systems of non-homogeneous equations We turn to the following system of two non-homogeneous equations:   −∆p u = λa(x)|u|α1 |v|β1 −1 v + f (x) in Ω;   (3.1) −∆q v = µb(x)|v|α2 |u|β2 −1 u + g(x) in Ω;    u = v = 0 on ∂Ω, where 0 ≤ f, g ∈ L∞ (Ω) are given functions. Our aim is to study the solvability 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 right of, but close to, the eigenvalue curve C1 (in Section (3.3)). We√recall that the def

−

eigenvalue curve C1 has been defined in (2.18) with Λ0 = Λ1

β2 /(q−1−α2 )

.

3.1. The case when (λ, µ) lies below or to the left of C1 . Let us now consider system (3.1) when (λ, µ) are in the first quadrant of R2 and below or to the left of the principal eigenvalue curve C1 . Of course, in order to assure the existence of C1 we assume the more general condition (2.14) jointly with β1 , β2 > 0, 0 ≤ α1 < p − 1 and 0 ≤ α2 < q − 1.


11

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

1

√

1

λ β1 (p−1−α1 ) µ β2 (q−1−α2 ) < Λ0 . 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 and α1 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) and define recursively for n ∈ N: un = Tp a|un−1 |α1 |vn−1 |β1 −1 v + f and vn = Tq b|vn−1 |α2 |un−1 |β2 −1 v + g . It is enough to prove that both sequences, ∞ ∞ {un }∞ n=1 and {vn }n=1 , are uniformly bounded in L (Ω). By the regularity results already quoted, these sequences will also be uniformly bounded in C01,α (Ω). First observe that 0 ≤ (6≡)u2 ≤ · · · ≤ un ≤ un+1 ≤ . . .

and

0 ≤ (6≡)v2 ≤ · · · ≤ vn ≤ vn+1 ≤ . . .

pointwise a.e. in Ω.

def

(3.2)

def

Hence, the functions u ˆn = un /kun+1 k∞ and vˆn = vn /kvn+1 k∞ satisfy 0 ≤ u ˆn , vˆn ≤ 1 almost everywhere in Ω, n ∈ N. Consequently, the right-hand sides of the following equations:  f (x) β1 un 1  = λa(x)ˆ uαn−1 in Ω, vˆn−1 + −∆  p α β α1 β1  kun k∞1 kvn k∞1  p−1 p−1  ku k kv k n ∞ n ∞ g(x) α2 vn 2 −∆ = µb(x)ˆ vn−1 u ˆβn−1 + in Ω, (3.3) q α2 α2 β2 β2  kv k ku k  n n ∞ ∞ q−1 q−1  kvn k∞ kun k∞   un = vn = 0 on ∂Ω are uniformly bounded in L∞ (Ω). Employing the regularity result in C 1,α (Ω), we obtain that both sequences on the left-hand sides above, un vn and (3.4) α1 β1 α2 β2 , p−1 p−1 q−1 q−1 kun k∞ kvn k∞ kvn k∞ kun k∞ are bounded in C 1,α (Ω) and, in particular, also in L∞ (Ω); that is, there is a constant C > 0 such that kun k∞ kvn k∞ and (3.5) α1 β1 ≤ C α2 β2 ≤ C, p−1 p−1 q−1 q−1 kun k∞ kvn k∞ kvn k∞ kun k∞ n ∈ N. Thus, kun k∞ is uniformly bounded if and only if kvn k∞ is uniformly bounded. Now assume that, by contradiction, kun k∞ → +∞ and

12


kvn k∞ → +∞. We combine (3.4) with (3.5) to conclude that also the sequences un /kun k∞ and vn /kvn k∞ are uniformly bounded in C 1,α (Ω). Hence, for a subsequence denoted again by (un , vn ) with n ∈ N, there exist some functions u∗ , v∗ ∈ C01 (Ω), u∗ , v∗ ≥ 0 in Ω, ku∗ k∞ = kv∗ k∞ = 1, such that un vn → u∗ and → v∗ in C01 (Ω) as n → ∞. kun k∞ kvn k∞ Furthermore, there exist λ∗ , µ∗ ∈ R such that also def

sn =

kvn kβ∞1 1 kun ||p−1−α ∞

→ λ∗

and

kun kβ∞2

def

tn =

2 kvn ||q−1−α ∞

→ µ∗

as n → ∞.

1 1 Observe that sβn2 tp−1−α = 1 forces λβ∗ 2 µp−1−α = 1. Passing to the limit in n ∗ system (3.3) above, as n → ∞, we find   −∆p u∗ ≤ λλ∗ a(x)uα∗ 1 v∗β1 in Ω;  

−∆q v∗ ≤ µµ∗ b(x)v∗α2 uβ∗ 2    u∗ = v∗ = 0 on ∂Ω.

in Ω;

The inequalities are obtained with a help from inequalities (3.2). From the first inequality in the system above we deduce that (λλ∗ )−1/(p−1−α1 ) u∗ is a lower solution of problem (2.9) and (µµ∗ )−1/(q−1−α2 ) v∗ is a lower solution of problem (2.10). Using the uniqueness of positive solutions of these problems we deduce that (λλ∗ )−1/p−1−α1 u∗ ≤ J1 (v∗ ) and (µµ∗ )−1/q−1−α2 v∗ ≤ J2 (u∗ ) 1 and, consequently, using also λβ∗ 2 µp−1−α = 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 associated ◦

to some u0 ∈ V + . Thus, using the definition of Λ0 in (2.6), we obtain a √

1

√

1

contradiction with the hypothesis λ β1 (p−1−α1 ) µ β2 (q−1−α2 ) < Λ0 . We have ∞ just proved that both sequences {un }∞ n=1 and {vn }n=1 are uniformly bounded 1,α in C0 (Ω). Hence, there exists (u, v) ∈ X+ such that un → u and vn → v in C01 (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) either u ≡ 0 or v ≡ 0. In case (a) let us assume that for instance f 6≡ 0 (a similar proof works if g 6≡ 0). Since −∆p u ˜ ≥ f = −∆p u1 , we have u ˜ ≥ u1 and, after an


13

iteration process, u ˜ ≥ un for all n ∈ N. Thus, u ≤ u ˜ and v ≤ v˜. Observe that if for instance u = u ˜ then it follows from the first equation of system (3.1) that λa(x)uα1 v β1 + f = λa(x)uα1 v˜β1 + f and consequently v = v˜. Thus, let us assume by contradiction that u ≤ (6≡)˜ u and v ≤ (6≡)˜ v . Denote def

ω = (p − 1 − α1 )/β1 , and consider the smallest k ∈ R such that u ˜ ≤ ku and v˜ ≤ k ω v. 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 (k ω v)β1 + k p−1 f ≥ (6≡) − ∆p u ˜ in Ω, and ku = u ˜ = 0 on ∂Ω. From the strong comparison principle of [4] we infer that u ˜ ku; see (A.3) for the definition of “”. Taking advantage of the strict inequality in the second equation of system (3.1) we get, again by using the same SCP, that v˜ k ω v, in contradiction with our choice of k. In case (b) if both u ≡ 0 and v ≡ 0 then f = g ≡ 0 and we can prove directly from the system that µ−1/β2 λ−1/(p−1−α1 ) u ˜ = (J1 ◦J2 )(˜ u). Therefore, −1/(p−1−α ) −1/β 1 2 λ = Λ1 , a contradiction from Theorem A.2, it follows that µ with 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 in Part (iii) of Proposition 2.7 to get the conclusion. We leave the details to the reader. As a corollary of this theorem we have the following more general existence result. Corollary 3.2. Let us consider f, g ∈ L∞ (Ω) and let λ > 0, µ > 0 be such that λ

√

1 β1 (p−1−α1 )

µ

√

1 β2 (q−1−α2 )

< Λ0 .

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 theory to the mapping Tf,g : X 7→ X defined by def Tf,g (u, v) = Tp (a|u|α1 |v|β1 −1 v + f ), Tq (b|v|α2 |u|β2 −1 u + g) to get the conclusion (cf., for instance, [5]).

14


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

√

1 β1 (p−1−α1 )

µ

√

1 β2 (q−1−α2 )

= Λ0 .

Then system (3.1) has no solution in X+ . If, moreover, α1 = α2 = 0 then there is no solution in X. Proof. Assume by contradiction that (u, v) ∈ X+ is a solution of (3.1). Then,  α1 β1  in Ω; −∆p u ≥ λa(x)u v  −∆q v ≥ µb(x)v α2 uβ2 in Ω;    u = v = 0 on ∂Ω, and arguing as in the first part of the proof of Theorem 3.1 we conclude that J1 ◦ J2 (u) ≤ Λ1 u. By Theorem A.2, it follows that u = ρϕλ for some ρ > 0 and similarly v = ρµ1β2 ϕµ . But this is impossible because of f + g 6≡ 0. In the case α1 = α2 = 0 let us assume, by contradiction, that there exists a solution (u, v) ∈ X and let us consider the smallest γ ∈ R+ such that β2 −u ≤ γϕλ , −v ≤ γ ω ϕµ , where ω = q−1 . If γ = 0 then (u, v) ∈ X+ and we argue as previously to arrive at a contradiction with f + g 6≡ 0. Hence, we may assume γ > 0. If, for instance, f 6≡ 0, then, from the first equation of system (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 second equation of (3.1), we get also that −v γ ω ϕµ and thus a contradiction with the minimality of γ. 3.3. An antimaximum principle for systems. Here we treat the case when (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 a C 2 -boundary ∂Ω. If f ∈ L∞ (Ω), f ≥ 0, f 6≡ 0 in Ω, then there exists δ > 0 such that, for each λ ∈ (λ1 , λ1 + δ), every weak solution u ∈ W01,p (Ω) of −∆p u = λ|u|p−2 u + f in Ω

u = 0 on ∂Ω


15

◦

satisfies −u ∈ V + ; i.e., u is of class C 1 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, β1 β2 = (p − 1)(q − 1), and (λ1 , µ1 ) ∈ C1 . Consider two functions 0 ≤ f, g ∈ L∞ (Ω) with f + g 6≡ 0. Then there exists δ > 0 such that, for all pairs (λ, µ) ∈ R2 with λ1 < λ < λ1 + δ and µ1 < µ < µ1 + δ, every (weak) solution (u, v) of system (3.1) satisfies u 0 and v 0 in Ω. Proof. Assume that, by contradiction, there exist a sequence {(λ0n , µ0n )}∞ n=1 ⊂ R2 , such that λ0n > λ1 , µ0n > µ1 , and λ0n → λ1 , µ0n → µ1 as n → ∞, and another sequence of solutions (un , vn ) ∈ X of   −∆p un = λ0n a(x)|vn |β1 −1 vn + f (x) in Ω;   (3.6) −∆q vn = µ0n b(x)|un |β2 −1 un + g(x) in Ω;   u = v = 0 on ∂Ω, n

n

with at least one of −un or −vn not in (C01 (Ω))◦+ . We distinguish between two cases: (a) both kun k∞ and kvn k∞ are bounded and (b) either kun k∞ or kvn k∞ is unbounded. In case (a) it follows that the sequences are bounded in C01,α (Ω), by regularity. Employing Arzel` a-Ascoli’s theorem, we may pass to the limit in C01 (Ω). Then the limit functions u, v ∈ C01 (Ω) satisfy  β1 −1 v + f (x)  in Ω; −∆p u = λa(x)|v|  −∆q v = µb(x)|u|β2 −1 u + 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 C 1,α (Ω) and, in particular, also in L∞ (Ω); that is, there is a constant C > 0 such that both inequalities in (3.5) hold for every n ∈ N. Thus, kun k∞ → ∞ if and only if kvn k∞ → ∞ as n → ∞. For a subsequence denoted again by (un , vn ) with n ∈ N, there exist some functions u∗ , v∗ ∈ C01 (Ω), ku∗ k∞ = kv∗ k∞ = 1, such that un vn → u∗ and → v∗ in C01 (Ω) as n → ∞. (3.7) kun k∞ kvn k∞

16


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

sn =

kvn kβ∞1 kun ||p−1 ∞

→ λ∗

and

kun kβ∞2

def

tn =

kvn ||q−1 ∞

→ µ∗

as n → ∞,

thanks to α1 = α2 = 0. We have λβ∗ 2 µ∗p−1 = 1. Passing to the limit in system (3.6) above, as n → ∞, for un /kun k∞ and vn /kvn k∞ , we find   −∆p u∗ = λ1 λ∗ a(x)|v|β∗ 1 −1 v∗ in Ω;   −∆q v∗ = µ1 µ∗ b(x)|u|β∗ 2 −1 u∗ in Ω;    u∗ = v∗ = 0 on ∂Ω. This shows that (λ1 λ∗ , µ1 µ∗ ) ∈ 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:  β1 −1 v + (λ0 − λ )a(x)|v |β1 −1 v + f (x)  1 n n n n  −∆p un = λ1 a(x)|vn |  −∆q vn = µ1 b(x)|un |β2 −1 un + (µ0n − µ1 )b(x)|un |β2 −1 un + g(x)   u = v = 0 on ∂Ω. n

in Ω; in Ω;

n

(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 the nonexistence 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 hypothesis that for every n ∈ N we have −un 6∈ (C01 (Ω))◦+ or −vn 6∈ (C01 (Ω))◦+ . The proposition is proved. Appendix A..

Appendix

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


17

u, v ∈ W01,p (Ω) are any weak solutions to the following boundary value problems, respectively: −∆p u = f (x) in Ω; −∆p v = g(x) in Ω;

u = 0 on ∂Ω, v = 0 on ∂Ω.

(A.1) (A.2)

Then we have u ≤ v in Ω, by the weak comparison principle (WCP, for brevity) due to P. Tolksdorf [23, Lemma 3.1, page 800]. A classical version of the SCP would claim that if f and g satisfy also f 6≡ g in Ω then ∂u ∂v u < v in Ω and > on ∂Ω. (A.3) ∂ν ∂ν We abbreviate u v if (A.3) holds. It is still an open question if this claim holds without additional hypotheses. For f ≡ 0 and u ≡ 0 in Ω, this is the strong maximum principle due to P. Tolksdorf [23, Proposition 3.2.1 and 3.2.2, page 801] and J. L. Vázquez [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 ∈ C 1,β (Ω) by a regularity result due to E. DiBenedetto [7, Theorem 2, page 829] and P. Tolksdorf [24, Theorem 1, page 127] (interior regularity, shown independently), and to G. Lieberman [15, Theorem 1, page 1203] (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 domain in RN whose boundary ∂Ω is a C 2 -manifold if N ≥ 2, or a bounded open interval in R1 if N = 1. Let f, g ∈ L∞ (Ω) be such that f ≤ g and f 6≡ g in Ω and also g ≥ 0 and g 6≡ 0 in Ω. Then the SCP (A.3) is valid for any weak solutions u, v ∈ W01,p (Ω) of equations (A.1) and (A.2). Proof. The case N = 1 is proved in M. Cuesta and P. Takáˇc [4, pages 731–732]. 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 every x ∈ Ωδ , where Ωδ = {x ∈ Ω : d(x) < δ} (A.4) def

is the open δ-neighborhood in Ω of the boundary ∂Ω. As usual, d(x) = dist(x, ∂Ω) denotes the distance from a point x ∈ Ω to the boundary ∂Ω. Set Ω0δ = Ω \ Ωδ . Since ∂Ω is assumed to be a compact manifold of class C 2 , so is ∂Ω0δ provided δ > 0 is small enough. Indeed, the last claim is a consequence of d ∈ C 2 (Ωδ ), by [12, Lemma 14.16, page 355] and its proof,

18


where it is shown that Ωδ is C 1 -diffeomorphic to ∂Ω×[0, δ] with x 7→ (x, 0) for all x ∈ ∂Ω, and Ω0δ = Ω \ Ωδ is C 1 -diffeomorphic to Ω. Both diffeomorphisms are considered between manifolds with boundary of class C 2 . Of course, they can be replaced by C 2 -diffeomorphisms, by M. W. Hirsch [13, Theorem 3.5, page 57]. Second, set w = v − u; hence 0 ≤ w ∈ C 1,β (Ω) with w = 0 on ∂Ω. Subtracting Equation (A.1) from (A.2), we find out that w satisfies the following linear elliptic inequality in the sense of distributions in Ωδ : P def ∂ ∂w − div(A(x)∇w) = − N a (x) ij i,j=1 ∂xi ∂xj (A.5) = g − f ≥ 0 for x ∈ Ωδ . Here, the coefficients aij (x) =

R1 0

a îj ((1 − s)∇u(x) + s∇v(x)) ds

(A.6)

C β (Ω

of the matrix A(x) belong to δ ) and form a uniformly elliptic operator in Ωδ , where a îj (z) = |z|p−2 δij + (p − 2) |z|−2 zi zj is the Jacobian matrix of the mapping z 7→ |z|p−2 z for z = (z1 , . . . , zN ) ∈ RN \ {0}, with the Kronecker symbol δij . The uniform ellipticity is verified as follows, cf. P. Tak´ aˇc [21, Appendix A, pages 233–235]: ˆîj (z) = δij + (p − 2) |z|−2 zi zj The symmetric matrix with the entries a has only two eigenvalues, equal to 1 and p − 1. The expression |z|p−2 for z = (1 − s)a + sb with a = ∇u(x) and b = ∇v(x) is estimated by the inequalities p−2 p−2 Z 1 |a + sb|p−2 ds ≤ Cp · max |a + sb| cp · max |a + sb| ≤ 0≤s≤1

for all a, b ∈ R

0 N

0≤s≤1

with |a| + |b| > 0,

(A.7)

where 0 < cp ≤ Cp < ∞ are some constants and we have substituted b for the difference b − a in z = a + s(b − a) to simplify our notation. The first inequality is trivial for 1 < p ≤ 2 (take cp = 1), the second one for 2 ≤ p < ∞ (take Cp = 1); the remaining inequalities are proved in [21, Lemma A.1, page 233]. The desired uniform ellipticity now follows from our choice of Ωδ ; we have |∇v| ≥ γ = const > 0 in Ωδ . Third, in every subdomain σ of Ωδ with σ ∩ ∂Ω = ∅, we apply the strong maximum 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 connected as in M. Cuesta and P. Takáˇc [3, 4], then so is Ωδ and, consequently,


19

we obtain either w = v − u > 0 in Ωδ , or else w = v − u ≡ 0 in Ωδ . If ∂Ω is not 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 of Ωδ , 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 constant c > 0 such that v ≥ u + c holds on the boundary ∂Ω0η = ∂Ωη \ ∂Ω ⊂ Ωδ of the domain Ω0η = Ω \ Ωη . We combine this boundary inequality with −∆p (u + c) = −∆p u = f (x) ≤ g(x) = −∆p v

in Ω0η

to conclude that v ≥ u + c holds throughout Ω0η , by the WCP (P. Tolksdorf [23, Lemma 3.1, page 800]). We have thus obtained u < v in Ω = Ωδ ∪ Ω0η as desired. Furthermore, we can make use of the boundary point principle as shown in R. Finn and D. Gilbarg [8, Lemma 7, page 31] (for N ≥ 3, see also ∂v [8, Remarks, page 35]) in order to deduce that − ∂u ∂ν (x0 ) < − ∂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. Takáˇc [3, Lemma A.1, page 742]. For η ∈ (0, δ) small enough we apply the divergence theorem to equations (A.1) and (A.2) over the domain Ω0η = Ω \ Ωη . We thus obtain Z Z p−2 − |∇u(x)| ∇u(x) · ν(x) dσ(x) = f (x) dx, (A.8) ∂Ω0η

Z −

Ω0η

|∇v(x)|p−2 ∇v(x) · ν(x) dσ(x) =

∂Ω0η

Z g(x) dx.

(A.9)

Ω0η

Since u ≡ v in Ωδ and ∂Ω0η ⊂ Ωδ , the two surface integrals on the left-hand side in equations (A.8) and (A.9) are equal. Therefore, we have Z Z f (x) dx = g(x) dx. Ω0η

Ω0η

Combined with f ≤ g in Ω, this equality forces f ≡ g in Ω0η . From u ≡ v in Ωδ we obtain also f ≡ g in Ωδ . Thus, we arrive at f ≡ g throughout Ω = Ωδ ∪ Ω0η , a contradiction to our hypothesis f 6≡ g in Ω. The proposition is proved. A.2. . Kre˘ın-Rutman theorem for nonlinear mappings. We need the following version of the Kre˘ın-Rutman theorem for nonlinear homogeneous mappings which is essentially due to P. Takáˇc [20, Theorem 3.5, page 1763];

20


cf. also R. Nussbaum [17, Proposition 3.1, page 100]. As stated below, this version 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 a real Banach space endowed with an ordering “ ≤ ” that is compatible with the def norm topology on E and such that the positive cone E+ = {x ∈ E : x ≥ 0} ◦

has 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 + = def {x ∈ E : x 0}. For a ≤ b in E, the set [a, b] = {x ∈ E : a ≤ x ≤ b} is called a closed order interval in E. (The reader is referred to the monograph by H. H. Schaefer [19] for details on ordered Banach spaces.) A (nonlinear) self-mapping T : E+ → E+ is called homogeneous if T (sx) = sT x holds for all x ∈ E+ and s ∈ [0, 1]. We say that T : X ⊂ E → E is monotone if x ≤ y in X implies T x ≤ T y, and strongly monotone if x < y in X implies T x T y. Finally, a monotone mapping T : X ⊂ E → E is called order-compact if the set T ([a, b] ∩ X) = {T x : x ∈ [a, b] ∩ X} has compact closure in E for each pair a ≤ b in E. Theorem A.2. ([20, Theorem 3.5, page 1763]) Let E be a strongly ordered Banach space and T : E+ → E+ a continuous homogeneous mapping. Assume that T is strongly monotone and order-compact. Then there exist a ◦

number Λ1 > 0 and e ∈ E + such that T e = Λ1 e. Furthermore, if λ ∈ [0, Λ1 ] and u ∈ E+ \{0} satisfy T u ≤ λu, then λ = Λ1 and u = ce for some constant c > 0, and thus T u = Λ1 u. Finally, if u ∈ E+ \ {0} satisfies T u ≥ Λ1 u, then again u = ce with a constant c > 0. We wish to apply this theorem in the Banach space E = C00 (Ω) (or in = E × E) of all continuous functions u : Ω → R with u = 0 on ∂Ω, such that u possesses a continuous normal derivative ∂u/∂ν : ∂Ω → R. Recall that ν ≡ ν(x0 ) denotes the exterior unit normal to ∂Ω at x0 ∈ ∂Ω. More precisely, C00 (Ω) is defined to be the completion of the vector space E2

def

C01 (Ω) =

u ∈ C 1 (Ω) : u = 0 on ∂Ω

under the order norm ∂u def kuk0 = max |u(x)| + max ∂Ω ∂ν Ω

for u ∈ C01 (Ω).

(A.10)


21

The ordering “ ≤ ” on C00 (Ω) is induced by the natural pointwise ordering of functions; that is, u ≤ v in E is defined by u(x) ≤ v(x) for all x ∈ Ω. Thus, C00 (Ω)+ = u ∈ C00 (Ω) : u ≥ 0 in Ω ◦

is the positive cone in C00 (Ω); the interior C00 (Ω) consists of all functions u ∈ C00 (Ω) that satisfy both inequalities of the strong maximum principle, ∂u < 0 on ∂Ω. (A.11) u > 0 in Ω and ∂ν def

The self-mapping T : C00 (Ω) → C00 (Ω) may typically be of the form T u = v, where u ∈ C00 (Ω) is arbitrary and v ∈ W01,p (Ω) is the unique weak solution to the Dirichlet boundary-value problem − ∆p v = m(x) |u|p−2 u in Ω;

v = 0 on ∂Ω.

(A.12)

Hence, v ∈ C 1,β (Ω) ∩ C00 (Ω), by regularity. The function m ∈ L∞ (Ω) is a positive “weight;” i.e., it satisfies m > 0 almost everywhere in Ω. It is easy 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. Clément, R.F. Man´ asevich, and E. Mitidieri, Positive solutions for a quasilinear system via blow up, Comm. P.D.E., 18 (1993), 2071–2106. [3] M. Cuesta and P. Tak´ aˇc, A strong comparison principle for the Dirichlet p-Laplacian, in “Reaction-Diffusion Systems,” G. Caristi and E. Mitidieri, eds., pp. 79–87. In Lecture Notes in Pure and Applied Mathematics, Vol. 194. Marcel Dekker, Inc., New York–Basel, 1998. [4] M. Cuesta and P. Tak´ aˇc, A strong comparison principle for positive solutions of degenerate elliptic equations, Differential and Integral Equations, 13 (2000), 721–746. [5] C. De Coster and M. Henrard, Existence and localization of solution for second order elliptic BVP in presence of lower and upper solutions without any order, J. Differential Equations, 145 (1998), 420–452. [6] D. G. de Figueiredo, Semilinear elliptic systems: existence, multiplicity, symmetry of solutions, in M. Chipot; eds., “Handbook of Differential Equations: Stationary Partial Differential Equations,” Vol. 5, pp. 1–48. Elsevier Science B.V., Amsterdam, The Netherlands, 2008. [7] E. DiBenedetto, C 1+α local regularity of weak solutions of degenerate elliptic equations, 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.

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