THE FIRST EIGENVALUE OF p-LAPLACIAN SYSTEMS WITH NONLINEAR BOUNDARY CONDITIONS D. A. KANDILAKIS, M.

THE FIRST EIGENVALUE OF p-LAPLACIAN SYSTEMS WITH NONLINEAR BOUNDARY CONDITIONS

D. A. KANDILAKIS, M. MAGIROPOULOS, AND N. B. ZOGRAPHOPOULOS

Received 12 October 2004 and in revised form 21 January 2005

We study the properties of the positive principal eigenvalue and the corresponding eigenspaces of two quasilinear elliptic systems under nonlinear boundary conditions. We prove that this eigenvalue is simple, unique up to positive eigenfunctions for both sys- tems, and isolated for one of them.

1. Introduction Let Ω be an unbounded domain in RN , N ≥ 2, with a noncompact and smooth boundary ∂Ω. In this paper we prove certain properties of the principal eigenvalue of the following quasilinear elliptic systems

(1.1)

in Ω, in Ω,

(1.2)

−∆pu = λa(x)|u|p−2u + λb(x)|u|α−1|v|β+1u, −∆qv = λd(x)|v|q−2v + λb(x)|u|α+1|v|β−1v, −∆pu = λa(x)|u|p−2u + λb(x)|u|α|v|βv in Ω, −∆qv = λd(x)|v|q−2v + λb(x)|u|α|v|βu in Ω

satisfying the nonlinear boundary conditions

(1.3)

|∇u|p−2∇u · η + c1(x)|u|p−2u = 0 on ∂Ω, |∇v|q−2∇v · η + c2(x)|v|q−2v = 0 on ∂Ω,

where η is the unit outward normal vector on ∂Ω. As it will be clear later, under condition (H1), 1 < p, q < N, α,β ≥ 0 and

α + 1

= 1,

α + 1 <

(1.4)

p +

β + 1 q

pq∗ N , β + 1 <

p∗q N ,

systems (1.1), (1.2) are in fact nonlinear eigenvalue problems. Our procedure here will be based on the proper space setting provided in [14], (see Section 2). In this section, we also state the assumptions on the coeﬃcient functions.

308 The ﬁrst eigenvalue of p-Laplacian systems

Problems of such a type arise in a variety of applications, for example, non-Newtonian ﬂuids, reaction-diﬀusion problems, theory of superconductors, biology, and so forth, (see [2, 15] and the references therein). As a consequence, there are many works treating non- linear systems from diﬀerent points of view, for example, [4, 7, 9, 11, 13].

Properties of the principal eigenvalue are of prime interest since for example they are closely associated with the dynamics of the associated evolution equations (e.g., global bi- furcation, stability) or with the description of the solution set of corresponding perturbed problems (e.g., [17]). These properties are: existence, positivity, simplicity, uniqueness up to eigenfunctions which do not change sign and isolation, which hold in the case of the Lapla- cian operator in a bounded domain. It is well known that these properties also hold for the p-Laplacian scalar eigenvalue problem (in both bounded and unbounded domains) and were recently obtained in [12] under nonlinear boundary conditions while the case of some (p, q)-Laplacian systems with Dirichlet boundary conditions was also successfully treated in [1, 10, 16, 18].

Note that we discuss the case of a potential (or gradient) system, which is a restriction. However, this is in some sense natural because the aforementioned properties of the prin- cipal eigenvalue are stronger than in the scalar equation case; for example the principal eigenvalue of the system is the only eigenvalue which admits a nonnegative eigenfunc- tion in the sense that both components do not change sign. It is also remarkable that the associated “eigenspaces” are generally not linear subspaces.

Starting with the system (1.1)–(1.3), we proceed as follows: in Section 2, we give the space setting and the assumptions on the coeﬃcient functions. In Section 3, using the compactness of the corresponding operators we prove the existence and positivity of λ1 and we state a regularity result based on the iterative procedure of [5]. In Section 4, we prove the simplicity and the uniqueness up to positive (componentwise) eigenfunc- tions. This is done by using the Picone’s identity (see [1]). Finally, in Section 5, we prove Theorem 2.3 by establishing the connection between the two systems with respect to ex- istence and simplicity of the common principal eigenvalue λ1 as well as the regularity of the eigenfunctions. In addition, we show that λ1 is isolated for the system (1.2)-(1.3).

2. Preliminaries and statement of the results Let Ω be an unbounded domain in RN , N ≥ 2, with a noncompact and smooth bound- ary ∂Ω. For m > 0 and r ∈ (1,+∞) let wm(x) = 1/(1 + |x|)m and assume that the space (cid:1) Ω(1/(1 + |x|)m)|u|r < +∞} is supplied with the norm Lr(wm,Ω) := {u :

(cid:2) (cid:3) (cid:6)1/r

.

(cid:5)m |u|r

(2.1)

Ω

(cid:6)u(cid:6)wm,r =

1 (cid:4) 1 + |x|

We require the following hypotheses:

(H1) 1 < p, q < N, α,β ≥ 0 with (α + 1)/ p + (β + 1)/q = 1, α + 1 < pq∗/N and β + 1 <

p∗q/N.

Here p∗ and q∗ are the critical Sobolev exponents deﬁned by

p∗ =

q∗ =

.

(2.2)

pN N − p ,

qN N − q

D. A. Kandilakis et al.

309 (H2)

(i) There exists positive constants α1, A1 with α1 ∈ (p + ((β + 1)(N − p)/q∗),N) and

a.e. in Ω,

(2.3)

0 < a(x) ≤ A1wα1(x)

(ii) there exists positive constants α2, D1 with α2 ∈ (q + ((α + 1)(N − q)/ p∗),N) and

a.e. in Ω,

(2.4)

0 < d(x) ≤ D1wα2(x)

(iii) m{x ∈ Ω : b(x) > 0} > 0 and

a.e. in Ω,

(2.5)

0 ≤ b(x) ≤ B1ws(x)

where B1 > 0 and s ∈ (max{p, q},N).

(H3) c1(·) and c2(·) are positive and continuous functions deﬁned on RN with

(2.6)

k1wp−1(x) ≤ c1(x) ≤ K1wp−1(x), l1wq−1(x) ≤ c2(x) ≤ L1wq−1(x),

Let C∞

for some positive constants k1, K1, l1, L1. δ (Ω) be the space of C∞

0 (RN )-functions restricted to Ω. For m ∈ (1,+∞), the

weighted Sobolev space Em is the completion of C∞

δ (Ω) in the norm

(cid:7) (cid:3) (cid:3) (cid:8)1/m |u|m

.

|||u|||m = |∇u|m +

(2.7)

Ω

1 (1 + |x|)m

By [14, Lemma 2] we see that if c(·) is a positive continuous function deﬁned on Rn then the norm

(cid:7) (cid:3) (cid:3) (cid:8)1/m |∇u|m +

c(x)|u|m

(2.8)

Ω

∂Ω

(cid:6)u(cid:6)1,m =

is equivalent to ||| · |||m. The proof of the following lemma is also provided in [14].

Lemma 2.1. (i) If

p ≤ r ≤

,

(2.9)

pN N − p , N > α ≥ N − r

N − p p

310 The ﬁrst eigenvalue of p-Laplacian systems

then the embedding E ⊆ Lr(wα,Ω) is continuous. If the upper bound for r in the ﬁrst in- equality and the lower bound in the second is strict, then the embedding is compact.

(ii) If

p ≤ m ≤

,

(2.10)

p(N − 1) N − p , N > β ≥ N − 1 − m

N − p p

then the embedding E ⊆ Lm(wβ,∂Ω) is continuous. If the upper bounds for m are strict, then the embedding is compact.

It is natural to consider our systems on the space E = Ep × Eq supplied with the norm

(cid:9) (cid:9) (cid:9) (cid:9)(u,v)

(2.11)

pq = (cid:6)u(cid:6)1,p + (cid:6)v(cid:6)1,q.

We now deﬁne the functionals Φ, I, J : E → R as follows:

(cid:3) (cid:3) (cid:3) (cid:3)

Φ(u,v) =

|∇v|p + |∇u|p +

c1(x)|u|p +

c2|v|q

α + 1 p

β + 1 q

α + 1 p

∂Ω

Ω

Ω (cid:3)

(cid:3) (cid:3) − λ

a(x)|u|p − λ

d(x)|v|q − λ

b(x)|u|α+1|v|β+1,

β + 1 q

Ω

(cid:3) (cid:3) (cid:3)

α + 1 p (cid:3)

I(u,v) =

|∇u|p + |∇v|p +

c1(x)|u|p +

c2|v|q,

β + 1 q

α + 1 p

∂Ω

Ω

Ω (cid:3)

(cid:3) (cid:3)

J(u,v) =

a(x)|u|p +

d(x)|v|q +

b(x)|u|α+1|v|β+1.

β + 1 q

α + 1 p

Ω

(2.12)

In view of (H1)–(H3), the functionals Φ, I, J are well deﬁned and continuously diﬀeren- tiable on E. By a weak solution of (1.1) we mean an element (u0,v0) of E which is a critical point of the functional Φ.

The main results of this work are the following theorems.

Theorem 2.2. Let Ω be an unbounded domain in RN , N ≥ 2, with a noncompact and smooth boundary ∂Ω. Assume that the hypotheses (H1), (H2), and (H3) hold. Then

(i) System (1.1)–(1.3) admits a positive principal eigenvalue λ1 given by (cid:10)

(cid:11) .

I(u,v) : J(u,v) = 1

(2.13)

λ1 = inf

Each component of the associated normalized eigenfunction (u1,v1) is positive in Ω and of class C1,δ

loc (Ω) for some δ ∈ (0,1).

(ii) The set of eigenfunctions corresponding to λ1 forms a one dimensional manifold E1 ⊆

E deﬁned by

(cid:10)(cid:4) (cid:5) (cid:11) .

: c ∈ R\{0}

(2.14)

E1 =

cu1, ±|c|p/qv1

Furthermore, a componentwise positive eigenfunction always corresponds to λ1.

D. A. Kandilakis et al.

311 Theorem 2.3. Assume that the hypotheses of Theorem 2.2 hold.

(a) System (1.2)-(1.3) shares the same positive principal eigenvalue λ1 and the same prop-

erties of the associated eigenfunctions with (1.1)–(1.3).

(b) The set of eigenfunctions corresponding to λ1 forms a one dimensional manifold E2 ⊆

E deﬁned by

(cid:10) (cid:5) ± (cid:11) .

: c > 0

(2.15)

E2 =

(cid:4) c u1,c p/qv1

(c) λ1 is isolated for the system (1.2)-(1.3), in the sense that there exists η > 0 such that

the interval (0,λ1 + η) does not contain any other eigenvalue than λ1.

3. Existence and regularity

In this section, we prove the existence of a positive principal eigenvalue and the regularity of the corresponding eigenfunctions for the system (1.1)–(1.3). Existence. The operators I, J are continuously Fr´echet diﬀerentiable, I is coercive on E ∩ {J(u,v) ≤ const}, J is compact and J (cid:11)(u,v) = 0 only at (u,v) = 0. So the assumptions of Theorem 6.3.2 in [3] are fulﬁlled implying the existence of a principal eigenvalue λ1, satisfying

I(u,v).

(3.1)

λ1 = inf

J(u,v)=1

Moreover, if (u1,v1) is a minimizer of (2.13) then (|u1|, |v1|) should be also a minimizer. Hence, we may assume that there exists an eigenfunction (u1,v1) corresponding to λ1, such that u1 ≥ 0 and v1 ≥ 0, a.e. in Ω. Regularity. We show ﬁrst that wpu1 and wqv1 are essentially bounded in Ω. To that pur- pose deﬁne uM(x) := min{u1(x),M}. It is clear that uk p+1 M ∈ Ep, for k ≥ 0. Multiplying the ﬁrst equation of (1.1) by uk p+1

(cid:3) (cid:3) (cid:13)

uk p+1 M

uk p+1 M dx

(cid:12) (cid:12)∇u1

c1(x)up−1

Ω

∂Ω

M and integrating over Ω, we get (cid:14) (cid:12) (cid:12)p−2∇u1 · ∇ (cid:3)

dx + (cid:3)

(3.2)

k p+α+1 dx.

a(x)u(k+1)p

b(x)vβ+1

≤ λ1

dx + λ1

u1

Ω

(cid:3) (cid:3) (cid:13) (cid:14) (cid:12) (cid:12)puk p (cid:12) (cid:12)∇uM

uk p+1 M

M dx =

(cid:12) (cid:12)∇uk+1 M

dx = (k p + 1)

Note that (cid:3) (cid:12) (cid:12) (cid:12)p−2∇u1 · ∇ (cid:12)∇u1

Ω

k p + 1 (k + 1)p

(cid:12) (cid:12)pdx, (3.3)

so since (k p + 1)/(k + 1)p ≤ 1, then (cid:3)

(cid:3) (cid:13) (cid:14)

uk p+1 M

uk p+1 M dx

dx +

Ω

∂Ω

(cid:12) (cid:12)∇u1

c1(x)up−1 1 (cid:8)p/ p∗

(cid:12) (cid:12)p−2∇u1 · ∇ (cid:7) (cid:3)

(3.4)

dx

u(k+1)p∗ M

Ω

≥ c3

k p + 1 (k + 1)p

1 (1 + |x|)p

312 The ﬁrst eigenvalue of p-Laplacian systems

due to Lemma 2.1(i) and (2.8). Let t = p(1 − (β + 1/q∗))−1, which is less than p∗ because of H(1). Then H(2)(i) and H¨older inequality imply that

(cid:3) (cid:3)

dx

a(x)u(k+1)p

dx ≤ A1

u(k+1)p 1

Ω

(cid:5)α1

1 (cid:4) 1 + |x|

(cid:3)

= A1 (cid:5)p2/t dx

1 (cid:5)α1−p2/t (cid:4) 1 + |x|

u(k+1)p 1 (cid:4) 1 + |x|

Ω (cid:7) (cid:3)

(cid:8)p/t (cid:8)(t−p)/t(cid:7) (cid:3)

dx

≤ A1

u(k+1)t 1

Ω

1 (1 + |x|)(tα1−p2)/(t−p)

1 (1 + |x|)p

(3.5)

(observe that (tα1 − p2)/(t − p) > N by H(2)(i)). Also, because of (H1), we may assume that

(cid:3) (cid:3)

dx ≤

b(x)vβ+1

dx,

(3.6)

uk p+α+1 1

u(k+1)p 1

Ω

otherwise we could consider

(cid:10)  

u1(x),M

uM(x) =

(3.7)



min 0,

(cid:11) , u1(x) ≥ 1, u1(x) < 1

as a test function. So

(cid:3) (cid:3)

dx

(cid:5)s vβ+1

b(x)vβ+1

dx ≤ B1

u(k+1)p 1

Ω

1 (cid:4) 1 + |x|

(cid:3)

dx

= B1 (cid:5)s(p/t)

Ω (cid:7) (cid:3)

(cid:8)p/t

u(k+1)p 1 (cid:4) 1 + |x| (cid:8)(t−p)/t(cid:7) (cid:3)

dx

Ω

≤ B1

vβ+1 1 (cid:5)s(1−(p/t)) (cid:4) 1 + |x| v(β+1)(t/t−p) 1 (1 + |x|)s

u(k+1)t 1 (1 + |x|)s

Ω (cid:8)(t−p)/t(cid:7) (cid:3)

(cid:8)p/t (cid:7) (cid:3)

dx

,

≤ B1

vq∗ 1

u(k+1)t 1

Ω

1 (1 + |x|)q

1 (1 + |x|)p

(3.8)

by H(2)(iii). On combining (3.2)–(3.8), we conclude that

(cid:18) (cid:19)1/(k+1)(cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9)uM

(3.9)

wp,(k+1)p∗ ≤ C1/(k+1)

wp,(k+1)t,

(cid:9)u1

k + 1 (k p + 1)1/ p

where C is independent of M and k. We now follow the same steps as in the proof of [8, Theorem 2] or [5, Lemma 3.2]. Let k1 = (p∗/t) − 1. Since (k1 p + 1)/(k1 + 1)p ≤ 1, we can

D. A. Kandilakis et al.

313 choose k = k1 in (3.9) to get

(cid:18) (cid:19)1/(k1+1)(cid:9) (cid:9) (cid:9)uM

(3.10)

(cid:9)u1 (cid:9) (cid:9) wp, p∗, (cid:5)1/ p (cid:9) (cid:9) wp,(k1+1)p∗ ≤ C1/(k1+1)

k1 + 1 (cid:4) k1 p + 1

while by letting M → ∞ we obtain that

(cid:18) (cid:19)1/(k1+1)(cid:9) (cid:9) (cid:9)

(3.11)

wp,(k1+1)p∗ ≤ C1/(k1+1)

(cid:9) (cid:9)u1 (cid:9)u1 (cid:9) (cid:9) wp, p∗.

k1 + 1 (cid:5) (cid:4) 1/ p k1 p + 1

(wp,Ω). Note that if k ≥ k1 then (k p + 1)/(k + 1)p ≤ 1. Choosing in

Hence, u1 ∈ L(k1+1)p∗ (1.1) k = k2 with (k2 + 1)t = (k1 + 1)p∗, that is, k2 = (p∗/t)2 − 1, we have

(cid:18) (cid:19)1/(k2+1)(cid:9) (cid:9) (cid:9)

(3.12)

wp,(k2+1)p∗ ≤ C1/(k1+1)

(cid:9) (cid:9)u1 (cid:9)u1 (cid:5)1/ p (cid:9) (cid:9) wp,(k1+1)p∗.

k2 + 1 (cid:4) k2 p + 1

(wp,Ω). Proceeding by induction we arrive at

Hence, u1 ∈ L(k2+1)p∗

(cid:18) (cid:19)1/(kn+1)(cid:9) (cid:9) (cid:9) (cid:9) (cid:9)

(3.13)

wp,(kn+1)p∗ ≤ C1/(kn+1)

wp,(kn−1+1)p∗.

(cid:9) (cid:9)u1 (cid:9)u1 (cid:5)1/ p

kn + 1 (cid:4) kn p + 1

From (3.10) and (3.13) we conclude that

n(cid:21)

(cid:20)n

i=1 1/(ki+1)

wp,(kn+1)p∗ ≤ C

i=1

wp, p∗ √

(cid:18) (cid:19)1/(ki+1)(cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9) (cid:9)u1 (cid:9)u1 (cid:5)1/ p

  √

(3.14)

ki+1

ki + 1 (cid:4) ki p + 1 (cid:7)

ki+1

n(cid:21)

(cid:20)n

i=1 1/(ki+1)

(cid:8)1/ = C     (cid:9) (cid:9)u1 (cid:9) (cid:9) wp, p∗. (cid:5)1/ p

ki + 1 (cid:4) ki p + 1

i=1

y+1 > 1 for y > 0, and limy→∞(y + 1/(y p + 1)1/ p)1/

y+1 = 1,

√ √

(cid:20)n

ki+1

i=1 1/

i=1 1/(ki+1)K

√ (cid:9) (cid:9)

(3.15)

(cid:9) (cid:9)u1 (cid:9) (cid:9)u1

Since (y + 1/(y p + 1)1/ p)1/ there exists K > 1 independent of kn such that (cid:9) (cid:9) wp,(kn+1)p∗ ≤ C

wp, p∗,

(cid:29) (cid:28)

where 1/(ki + 1) = (t/ p∗)i and 1/

ki + 1 = (

(cid:9) (cid:9) (cid:9) (cid:9)

(3.16)

t/ p∗)i. Letting now n → ∞ we conclude that (cid:9) (cid:9)u1

wp, p∗,

wp, ∞ ≤ c

(cid:9) (cid:9)u1

for some positive constant c. By [8], u1 ∈ C1,δ

loc (Ω). Similarly v1 ∈ C1,δ

loc (Ω).

Finally, we notice that for the principal eigenvalue, each component of an eigenfunc- tion is either positive or negative in Ω due to the Harnack inequality [8] and if we assume that u1(x0) = 0 for some x0 ∈ ∂Ω then by [19, Theorem 5] we have |∇u1(x0)|p−2∇u1(x0) · η(x0) < 0, contradicting (1.3). Thus u1 > 0 (or u1 < 0) on Ω. Similarly v1 > 0 (or v1 < 0) on Ω.

314 The ﬁrst eigenvalue of p-Laplacian systems

4. The eigenfunctions corresponding to λ1 In this section, we complete the proof of Theorem 2.2 establishing the simplicity of λ1. More precisely, we show that if (u2,v2) is another pair of eigenfunctions corresponding to λ1, then there exists c ∈ R\{0} such that (u2,v2) = (cu1, ±|c|p/qv1). To that end, we employ a technique similar to the one described in [1]. Namely, we will prove that if (w1,w2) is a positive on ¯Ω solution of the problem

in Ω, in Ω,

(4.1)

on ∂Ω, on ∂Ω,

−∆pu ≤ λa(x)|u|p−2u + λb(x)|u|α−1|v|β+1u, −∆qv ≤ λd(x)|v|q−2v + λb(x)|u|α+1|v|β−1v, |∇u|p−2∇u · η + c1(x)|u|p−2u = 0, |∇v|q−2∇v · η + c2(x)|v|q−2v = 0,

for some λ > 0, and (w(cid:11)

1,w(cid:11)

2) is a positive on ¯Ω solution of

−∆pu ≥ λa(x)|u|p−2u + λb(x)|u|α−1|v|β+1u in Ω, −∆qv ≥ λd(x)|v|q−2v + λb(x)|u|α+1|v|β−1v in Ω,

(4.2)

|∇u|p−2∇u · η + c1(x)|u|p−2u = 0 on ∂Ω, |∇v|q−2∇v · η + c2(x)|v|q−2v = 0 on ∂Ω

then (w(cid:11)

1,w(cid:11) Let ϕ ∈ C∞

2) = (cw1,c p/qw2) for a constant c > 0. δ (Ω), ϕ > 0, then ϕp/(w(cid:11)

1)p−1 ∈ Ep. By Picone’s identity [1], we get 

(cid:3) (cid:3) (cid:3) (cid:5) (cid:12) (cid:12)p

R

|∇ϕ|p −  · ∇ =

0 ≤

Ω

(cid:4) ϕ,w(cid:11) 1 (cid:12) (cid:12)∇w(cid:11) 1 ∇w(cid:11) 1 (cid:5)p−1   ϕp (cid:4) w(cid:11) 1 (cid:3) (cid:3) (cid:3) (cid:12) (cid:12)p = − · η |∇ϕ|p +

∆pw(cid:11) 1

(cid:12) (cid:12)∇w(cid:11) 1 ∇w(cid:11) 1

ϕp (cid:5)p−1

Ω

∂Ω

Ω (cid:3)

(cid:4) w(cid:11) 1 (cid:3) (cid:5)α(cid:4) (cid:5)β+1(cid:5) ≤ |∇ϕ|p − λ (cid:4) a(x)

ϕp (cid:4) (cid:5)p−1 w(cid:11) 1 (cid:5)p−1 + b(x)

(cid:4) w(cid:11) 1 (cid:4) w(cid:11) 1

w(cid:11) 2

ϕp (cid:5)p−1

Ω (cid:3)

(4.3)

∂Ω

(cid:12) (cid:12)p − · η (cid:4) w(cid:11) Ω 1 (cid:12) (cid:12)∇w(cid:11) 1 (cid:4) w(cid:11) 1 (cid:3)

ϕp (cid:5)p−1 (cid:3)

(cid:3) (cid:5)β+1 = |∇ϕ|p − λ − λ

a(x)ϕp

b(x)ϕp

Ω

Ω (cid:3)

Ω (cid:12) (cid:12)∇w(cid:11) 1

(cid:4) w(cid:11) 2 ∇w(cid:11) 1 (cid:5) p−1 (cid:5)p−1 (cid:4) w(cid:11) 1 (cid:4) w(cid:11) 1 (cid:4) (cid:5)α w(cid:11) 1 (cid:5)p−1 (cid:4) w(cid:11) 1 (cid:12) (cid:12)p − · η, ∇w(cid:11) 1

ϕp (cid:5)p−1

∂Ω

(cid:4) w(cid:11) 1

while the boundary conditions imply that

(cid:3) (cid:3) (cid:3) (cid:5)β+1 |∇ϕ|p − λ − λ

0 ≤

a(x)ϕp

b(x)ϕp

Ω

Ω (cid:3)

(cid:4) w(cid:11) 2 (cid:5)p−1 (cid:5)p−1 (cid:5)α (cid:4) w(cid:11) 1 (cid:5)p−1 (cid:4) w(cid:11) 1

(4.4)

(cid:4) w(cid:11) 1 (cid:4) w(cid:11) 1 (cid:5)p−1.

+

c1(x)

Ω ϕp (cid:5)p−1

∂Ω

(cid:4) w(cid:11) 1 (cid:4) w(cid:11) 1

D. A. Kandilakis et al.

315 Letting ϕ → w1 in Ep we obtain

(cid:3) (cid:3) (cid:3) (cid:3) (cid:5)α−p+1(cid:4) (cid:12) (cid:12)p − λ − λ

.

0 ≤

a(x)w p

b(x)w p

(cid:5)β+1 +

(4.5)

(cid:12) (cid:12)∇w1

c1(x)w p

(cid:4) w(cid:11) 1

w(cid:11) 2

Ω

∂Ω

Note also that (cid:3)

(cid:3) (cid:3) (cid:3) (cid:12) (cid:12)p ≤ λ

.

+

a(x)w p

b(x)wα+1

(4.6)

(cid:12) (cid:12)∇w1

c1(x)w p

1 + λ

wβ+1 2

Ω

∂Ω

Ω

On combining (4.5) and (4.6) we get

(cid:3) (cid:13) (cid:5)α−p+1(cid:4) (cid:5)β+1 (cid:14) .

0 ≤

b(x)

(4.7)

wβ+1 2

− w p 1

wα+1 1

(cid:4) w(cid:11) 1

w(cid:11) 2

Ω

Similarly,

(cid:3) (cid:13) (cid:5)β+1−q(cid:4) (cid:5)α+1 (cid:14) .

0 ≤

b(x)

(4.8)

wβ+1 2

− wq 2

wα+1 1

(cid:4) w(cid:11) 2

w(cid:11) 1

Ω

We can now work as in Theorem 2.7 in [1] to get the desired result.

Returning to our problem, we obtain E1 as the set of eigenfunctions corresponding to λ1, simply by applying the previous result to the case of our system with λ = λ1, and taking (u1,v1) instead of (w1,w2). One has now to combine the fact that the nonnegative solutions are given by (cu1,c p/qv1), c > 0, with the trivial observation that if (u,v) is an eigenfunction then (−u,v), (u, −v), (−u, −v) are also eigenfunctions.

The same technique can be used for proving that nonnegative solutions in Ω cor- respond only to the ﬁrst eigenvalue. Assume, for instance, that there exists an eigen- pair (λ∗,u2,v2) for the problem (1.1) such that λ∗ > λ1, u2 ≥ 0 and v2 ≥ 0, a.e. in Ω. Then (u1,v1) is a solution of (1.2) with λ = λ∗ and (u2,v2) is a solution of (1.3). Then (u2,v2) = (cu1,c p/qv1), for some c > 0, which is a contradiction.

5. The second system

In this section, we present the proof of Theorem 2.3.

(a) Since for positive solutions systems (1.1) and (1.2) coincide, we deduce that (λ1,u1, v1) is also an eigenpair for the system (1.2). Assume that there exists another nontrivial eigenpair (λ∗,u∗,v∗) of (1.2), such that 0 < λ∗ < λ1. Then the following equality must be satisﬁed

λ∗ =

(cid:5) (cid:5) ,

(5.1)

(cid:4) u∗,v∗ I (cid:4) u∗,v∗ ˜J

with ˜J(u∗,v∗) > 0, where ˜J(·, ·) is deﬁned by

(cid:3) (cid:3) (cid:3)

a(x)|u|p +

d(x)|v|q +

b(x)|u|α|v|βuv.

(5.2)

˜J(u,v) =

α + 1 p

β + 1 q

Ω

316 The ﬁrst eigenvalue of p-Laplacian systems

Note that ˜J(·, ·) is also compact. From (5.1) we also have that

(cid:4) (cid:4) (cid:5) (cid:5) (cid:5) (cid:5) ≥

λ∗ =

(cid:5) (cid:5) ,

(5.3)

I J

u∗,v∗ u∗,v∗

(cid:4) u∗,v∗ (cid:4) u∗,v∗ (cid:4) u∗,v∗ J (cid:4) u∗,v∗ ˜J

since

(cid:5) (cid:5) ≥ 1.

(5.4)

(cid:4) u∗,v∗ J (cid:4) u∗,v∗ ˜J

Normalizing (u∗,v∗) by setting

" (cid:4)

u∗ =:

v∗ =:

(5.5)

(cid:12) (cid:12) (cid:5)#1/ p , (cid:12) (cid:12) (cid:5)#1/q ,

J

" J (cid:12) (cid:12)u∗ (cid:4) u∗,v∗ (cid:12) (cid:12)v∗ u∗,v∗

we get that

(cid:5)

I

= (cid:5) (cid:5) ,

(5.6)

J

I (cid:4) u∗,v∗ J (cid:5) (cid:4) u∗,v∗

(cid:4) u∗,v∗ (cid:4) u∗,v∗ = 1.

(5.7)

From relations (5.3)–(5.7) we conclude that

(cid:5) (cid:4) (cid:4) (cid:5) (cid:5) = I

λ∗ ≥

(cid:4) u∗,v∗

(5.8)

≥ λ1,

I J

u∗,v∗ u∗,v∗

a contradiction.

(b) Let (u,v) be an eigenfunction of (1.2) corresponding to λ1. If uv ≥ 0 a.e., then the right-hand sides of (1.1) and (1.2) are equal, so (u,v) is an eigenfunction of (1.1), and we are done. On the other hand we cannot have uv < 0 on a set of positive measure, because then

>

(5.9)

λ1 =

= λ1,

I(u,v) J(u,v)

I(u,v) ˜J(u,v)

a contradiction.

(c) Suppose that there exists a sequence of eigenpairs (λn,un,vn) of (1.2) with λn → λ1. By the variational characterization of λ1 we know that λn ≥ λ1. So we may assume that λn ∈ (λ1,λ1 + η) for each n ∈ N. Furthermore, without loss of generality, we may assume that (cid:6)(un,vn)(cid:6) = 1, for all n ∈ N. Hence, there exists (˜u, ˜v) ∈ E such that (un,vn) (cid:1) (˜u, ˜v). The simplicity of λ1 implies that (˜u, ˜v) = (u1,v1) or (˜u, ˜v) = (−u1, −v1). Let us suppose

D. A. Kandilakis et al.

317 that (un,vn) (cid:1) (u1,v1) in E. For any two pairs of eigenfunctions (un,vn), (um,vm), multi- plying the ﬁrst equation by un − um and integrating by parts we derive

(cid:3) (cid:14) (cid:5)(cid:4)

dx

Ω

∇un − ∇um (cid:14)(cid:4) (cid:5) dx

un − um

+

∂Ω (cid:3)

(cid:5) dx

un − um

= λn

a(x) Ω (cid:3)

Ω

(cid:14)(cid:4) (cid:5) dx (cid:12)vn (cid:12)vm (cid:12) (cid:12)βvm

un − um (cid:3)

c1(x) (cid:13)(cid:12) (cid:12)un (cid:13)(cid:12) (cid:12)un b(x) $ (cid:3) (cid:5)

(cid:12)α(cid:12) (cid:12)

.

(cid:13)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)p−2∇um (cid:12)p−2∇un − (cid:12)∇um (cid:12)∇un (cid:3) (cid:13)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)p−2um (cid:12)p−2un − (cid:12)um (cid:12)un (cid:14)(cid:4) (cid:12) (cid:12) (cid:12) (cid:12)p−2um (cid:12)p−2un − (cid:12)um (cid:12)α(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)α(cid:12) (cid:12) (cid:12)βvn − (cid:12)um (cid:12) (cid:12)p−2um

+ λn (cid:4) λn − λm

(cid:12) (cid:12)um (cid:4) un − um (cid:12)vm (cid:12) (cid:12)βvmdx

a(x)

+

Ω

(cid:5) dx + (cid:12) (cid:12)um b(x)

(5.10)

From the second equation we similarly derive

(cid:3) (cid:5) (cid:14)(cid:4)

dx

Ω

∇vn − ∇vm (cid:14)(cid:4) (cid:5) dx

vn − vm

+

c2(x)

∂Ω (cid:3)

= λn

vn − vm

d(x) Ω (cid:3)

(cid:5) dx (cid:14)(cid:4) (cid:5) dx (cid:12)vn (cid:12)vm (cid:12) (cid:12)βum

vn − vm

Ω

(cid:3) (cid:13)(cid:12) (cid:12)vn (cid:13)(cid:12) (cid:12)un b(x) $ (cid:3) (cid:5) (cid:12)α(cid:12) (cid:12) % . (cid:13)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)q−2∇vm (cid:12)q−2∇vn − (cid:12)∇vm (cid:12)∇vn (cid:3) (cid:13)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)q−2vm (cid:12)q−2vn − (cid:12)vm (cid:12)vn (cid:14)(cid:4) (cid:12) (cid:12) (cid:12) (cid:12)q−2vm (cid:12)q−2vn − (cid:12)vm (cid:12)α(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)α(cid:12) (cid:12) (cid:12)βun − (cid:12)um (cid:12) (cid:12)q−2vm

+ λn (cid:4) λn − λm

(cid:12) (cid:12)vm (cid:4) vn − vm (cid:12) (cid:12)um (cid:12)vm (cid:12) (cid:12)βumdx

a(x)

+

(cid:5) dx +

b(x)

Ω

(5.11)

From (5.10) and (5.11), by using the compactness of the operator ˜J and the monotonicity of the p-Laplacian operator [6], we obtain

Ω

Ω (cid:3)

(cid:3) (cid:3) (cid:12) (cid:12)pdx −→ (cid:3)

(5.12)

Ω

(cid:12) (cid:12)qdx −→ (cid:12) (cid:12)∇un (cid:12) (cid:12)∇vn (cid:12) (cid:12) (cid:12)pdx, (cid:12)∇u1 (cid:12) (cid:12) (cid:12)qdx. (cid:12)∇v1

Exploiting the strict convexity of Ep and Eq we get that (un,vn) → (u1,v1) in E. For a ﬁxed n ∈ N and for every (φ,ψ) ∈ E we have (cid:3)

(cid:3) (cid:12) (cid:12)∇un

c1(x)

Ω

∂Ω

(cid:12) (cid:12)un (cid:3) (cid:12) (cid:12)p−2∇un∇φ dx + (cid:3) (cid:12) (cid:12)un = λn (cid:12) (cid:12)p−2unφ dx (cid:12)α(cid:12) (cid:12) (cid:12) (cid:12)un (cid:12)vn

a(x)

b(x)

Ω

(cid:12) (cid:12)βvnφ dx, (cid:3) (cid:3)

(5.13)

(cid:12) (cid:12)∇vn

c1(x)

Ω

∂Ω

Ω (cid:12) (cid:12)p−2vnψ dx (cid:12)α(cid:12) (cid:12) (cid:12) (cid:12)un

(cid:12) (cid:12)p−2unφ dx + λn (cid:12) (cid:12)vn (cid:3) (cid:12) (cid:12)q−2∇vn∇ψ dx + (cid:3) (cid:12) (cid:12)vn = λn (cid:12)vn

d(x)

(cid:12) (cid:12)q−2vnψ dx + λn

b(x)

Ω

(cid:12) (cid:12)βunψ dx,

318 The ﬁrst eigenvalue of p-Laplacian systems

n =: {x ∈ Ω : un(x)<0} and (cid:2)−

n =: {x ∈Ω : vn(x) < 0}. By (c) we must have m(Ω−

n ∪ (cid:2)−

n . Denoting by u−

n ) > n = min{0,vn} and choosing

n = min{0,un} and v−

Let (cid:1)− 0, with Ω− φ ≡ u−

n = (cid:1)− n and ψ ≡ v−

n , it follows that

(cid:3) (cid:3)

(cid:1)− n

n |pdx c1(x)|u− (cid:3)

n vndx,

(cid:12) (cid:12)∇u− n (cid:12) (cid:12)pdx + (cid:3) (cid:12)α(cid:12) (cid:12) (cid:12) (cid:12)βu− = λn (cid:12)vn (cid:12) (cid:12)u− n

b(x)

(cid:1)− n

(cid:3)

a(x) (cid:3)

(5.14)

(cid:1)− n (cid:12) (cid:12)qdx

(cid:12) (cid:12)∇v− n

c1(x)

(cid:2)− n

∂Ω∩(cid:1)− n (cid:12) (cid:12) (cid:12)pdx + λn (cid:12)u− n (cid:12) (cid:12)v− n (cid:3)

n dx.

(cid:12) (cid:12)qdx + (cid:3) (cid:12)α(cid:12) (cid:12) = λn (cid:12) (cid:12)un (cid:12) (cid:12)βunv− (cid:12)v− n

d(x)

∂Ω∩(cid:2)− n (cid:12) (cid:12) (cid:12)qdx + λn (cid:12)v− n

b(x)

(cid:2)− n

n v+

n v−

n and u+

n are negative, from the above system of equations we

Since the products u− obtain

(cid:3) (cid:3) (cid:12) (cid:12)pdx (cid:12) (cid:12)∇u− n

c1(x)

(cid:1)− n

n v−

n dx,

(cid:12) (cid:12)pdx + (cid:3) (cid:12) (cid:12)u− n (cid:3) (cid:12)α(cid:12) (cid:12) (cid:12) (cid:12)βu− ≤ λn (cid:12) (cid:12)u− n (cid:12)v− n

b(x)

(cid:1)− n

(cid:3)

a(x) (cid:3)

(5.15)

(cid:1)− n (cid:12) (cid:12)qdx

(cid:12) (cid:12)∇v− n

c2(x)

(cid:2)− n

∂Ω∩(cid:1)− n (cid:12) (cid:12) (cid:12)pdx + λn (cid:12)u− n (cid:12) (cid:12)v− n (cid:3)

n v−

n dx.

(cid:12) (cid:12)qdx + (cid:3) (cid:12)α(cid:12) (cid:12) (cid:12) (cid:12)βu− ≤ λn (cid:12) (cid:12)u− n (cid:12)v− n

d(x)

∂Ω∩(cid:2)− n (cid:12) (cid:12) (cid:12)qdx + λn (cid:12)v− n

b(x)

(cid:2)− n

From H¨older and Young inequalities we derive that

n v−

n dx

(cid:3) (cid:12)α(cid:12) (cid:12) (cid:12) (cid:12)βu− (cid:12)v− n

b(x)

(cid:1)− n

n v−

n dx

(cid:1)− n

(cid:12) (cid:12)u− n (cid:3) (cid:12)α(cid:12) (cid:12) (cid:12) (cid:12)βu− (cid:5)s (cid:12) (cid:12)u− n (cid:12)v− n ≤ B1 (cid:3)

(5.16)

(cid:1)− n (cid:7) (cid:3)

(cid:12) (cid:12)α+1 (cid:12) (cid:12)β+1dx (cid:5)s (cid:12) (cid:12)u− n (cid:12) (cid:12)v− n = B1 (cid:8) (cid:3) (cid:12) (cid:12)qdx

.

(cid:12) (cid:12)u− n (cid:12) (cid:12)v− n (cid:12) (cid:12)pdx + ≤ c3

1 (cid:4) 1 + |x| 1 (cid:4) 1 + |x| 1 (1 + |x|)s

1 (1 + |x|)s

(cid:1)− n

Thus

(cid:5)&(cid:9) (cid:9) (cid:9)p (cid:9) (cid:9)q ’ . (cid:9) (cid:9)u− n (cid:9)u− n (cid:9) (cid:9)v− n

(5.17)

Lp(ws,(cid:1)−

n ) +

Lq(ws,(cid:1)− n )

(cid:4) λ1 + η (cid:9) (cid:9)p 1, p ≤ c4

Similarly,

(cid:5)&(cid:9) (cid:9) (cid:9)q (cid:9) (cid:9)p ’ . (cid:9) (cid:9)v− n (cid:9)v− n (cid:9) (cid:9)u− n

(5.18)

Lq(ws,(cid:2)−

n ) +

Lp(ws,(cid:2)− n )

(cid:4) λ1 + η (cid:9) (cid:9)q 1, p ≤ c5

D. A. Kandilakis et al.

319 For r > 0 let Br denote the open ball with radius r centered at 0 ∈ Rn. For ε > 0 let rε > 0 be such that

Lp(ws,(cid:1)−

n ∩Brε ) +

Lq(ws,(cid:1)−

(cid:5)(cid:13)(cid:9) (cid:9) (cid:9)p (cid:9) (cid:9)q

(5.19)

Lq(ws,(cid:2)−

n ∩Brε ) +

Lp(ws,(cid:2)−

n ∩Brε ) + ε n ∩Brε ) + ε

(cid:5)(cid:13)(cid:9) (cid:9) (cid:9)q (cid:9) (cid:9)p (cid:14) , (cid:14) . (cid:9) (cid:9)u− n (cid:9) (cid:9)v− n (cid:9)u− n (cid:9)v− n (cid:9) (cid:9)v− n (cid:9) (cid:9)u− n (cid:4) λ1 + η (cid:4) λ1 + η (cid:9) (cid:9)p 1, p ≤ c4 (cid:9) (cid:9)q 1, q ≤ c5

Let 0 < δ < min{p∗ − p, q∗ − q} and suppose that γ1 ∈ (N(p∗ − p − δ)/ p∗,s − (N − p)(δ/ p)) and γ2 ∈ (N(q∗ − q − δ)/q∗,s − (N − q)(δ/q)). Lemma 2.1 implies that Ep ⊆ Lpp∗/(p+δ)(wζ1,Ω) and Eq ⊆ Lqq∗/(q+δ)(wζ2,Ω), where ζ1 = (s − γ1)p∗/(p + δ) and ζ2 = (s − γ2)q∗/(q + δ). Applying once again the H¨older inequality we derive that

(cid:7) (cid:3) (cid:8)(p∗−p−δ)/ p∗ (cid:9) (cid:9)p ≤

dx

Lp(ws,(cid:1)−

n ∩Brε )

(cid:9) (cid:9)u− n

1 (1 + |x|)γ1 p∗/(p∗−p−δ)

n ∩Brε

(cid:1)− (cid:7) (cid:3)

(cid:8)(p+δ)/ p∗ × (cid:12) (cid:12)pp∗/(p+δ)dx

(5.20)

(cid:1)−

(cid:12) (cid:12)u− n

1 (1 + |x|)(s−γ1)p∗/(p+δ)

n ∩Brε

(cid:7) (cid:3) (cid:8)(p∗−p−δ)/ p∗

dx

(cid:1)−

(cid:9) (cid:9)u− n ≤ c6 (cid:9) (cid:9)p 1,p,

1 (1 + |x|)γ1 p∗/(p∗−p−δ)

n ∩Brε

(note that γ1 p∗/(p∗ − p − δ) > N). A similar inequality also holds for v− n :

(cid:2) (cid:3) (cid:6)(q∗−q−δ)/q∗ (cid:9) (cid:9) (cid:9)q

dx

(cid:9) (cid:9)v− n (cid:9)v− n

(5.21)

Lq(ws,(cid:1)−

n ∩Brε )

(cid:1)−

≤ c7 (cid:9) (cid:9)q 1, q.

1 (1 + |x|)γ2q∗/(q∗−q−δ)

n ∩Brε

Combining (5.19), (5.20), and (5.21) we get

(cid:9) (cid:9)u− n (cid:9) (cid:9)p 1,p − c8ε (cid:7) (cid:3) (cid:8)(p∗−p−δ)/ p∗ (cid:9) (cid:9)p

dx

1, p

(cid:9) (cid:9)u− n ≤ c9

1 (1 + |x|)γ1 p∗/(p∗−p−δ)

n ∩Brε

(cid:1)− (cid:7) (cid:3)

(cid:6)(q∗−q−δ)/q∗ (cid:9) (cid:9) (cid:9)q

dx

(cid:9)v− n

+ c10

1,q

(cid:1)−

n ∩Brε

(cid:7) (cid:3) (cid:8)(p∗−p−δ)/ p∗

1 (1 + |x|)γ2q∗/(q∗−q−δ)  

’ (cid:9) (cid:9)q

dx

1, q

&(cid:9) (cid:9)u− n (cid:9) (cid:9)v− n ≤ c11 (cid:9) (cid:9)p 1, p + 

1 (1 + |x|)γ1 p∗/(p∗−p−δ)

n ∩Brε

(cid:1)− (cid:7) (cid:3)

(cid:8)(q∗−q−δ)/q∗  

dx

.

+

(cid:1)−



1 (1 + |x|)γ2q∗/(q∗−q−δ)

n ∩Brε

(5.22)

320 The ﬁrst eigenvalue of p-Laplacian systems

(cid:7) (cid:3) (cid:8)(p∗−p−δ)/ p∗   ’ (cid:9) (cid:9)q

dx

Similarly, (cid:9) (cid:9) (cid:9)q (cid:9)v− 1,q − c12ε n &(cid:9) (cid:9)u− n

1,q

(cid:9) (cid:9)v− n ≤ c13 (cid:9) (cid:9)p 1,p + 

1 (1 + |x|)γ1q∗/(p∗−p−δ)

n ∩Brε

(cid:2)− (cid:7) (cid:3)

(cid:8)(q∗−q−δ)/q∗  

dx

.

+

(cid:2)−



1 (1 + |x|)γ2q∗/(q∗−q−δ)

n ∩Brε

(5.23)

We can now add inequalities (5.22), (5.23) to get

(cid:7) (cid:3) (cid:8)(p∗−p−δ)/ p∗

dx

1 − ε(cid:11) ≤ c14

1 (1 + |x|)γ1 p∗/(p∗−p−δ)

n ∩Brε

(5.24)

(cid:1)− (cid:7) (cid:3)

(cid:8)(q∗−q−δ)/q∗

dx

.

+ c15

(cid:2)−

1 (1 + |x|)γ2q∗/(q∗−q−δ)

n ∩Brε

By taking ε suﬃciently small we see that

(cid:5) (cid:4) Ω−

m

n ∩ Brε

(5.25)

> c16 > 0,

(w1,BK (0)) and vn → v1 in Lq∗

where the constant c16 is independent of λn and un. Since un → u1 in Ep and vn → v1 in Eq, we have that un → u1 in Lp∗ (w1,Ω) and vn → v1 in Lq∗ (w2,Ω). Consequently, un → u1 in Lp∗ (w2,BK (0)). By Egorov’s theorem we conclude that un(x) (vn(x)) converges uniformly to u1(x) (resp., v1(x)) on Brε (0) with the exception of a set with arbitrarily small measure. But this contradicts (5.25) and the conclusion follows. The proof is complete.

Acknowledgments

The ﬁrst author is supported by the Greek Ministry of Education at the University of the Aegean under the Project EPEAEK II-PYTHAGORAS with title “Theoretical and Numerical Study of Evolutionary and Stationary PDEs Arising as Mathematical Mod- els in Physics and Industry.” The third author acknowledges support by the Operational Program for Educational and Vocational Training II (EPEAEK II) and particularly by the PYTHAGORAS Program no. 68/831 of the Ministry of Education of the Hellenic Republic.

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D. A. Kandilakis: Department of Sciences, Technical University of Crete, 73100 Chania, Greece E-mail address: dkan@science.tuc.gr

M. Magiropoulos: Science Department, Technological and Educational Institute of Crete, 71500

Heraclion, Greece

E-mail address: mageir@stef.teiher.gr

N. B. Zographopoulos: Department of Applied Mathematics, University of Crete, 71409 Heraklion,

Greece

E-mail address: nzogr@tem.uoc.gr