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 systems, 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
−∆ p u = λa(x)|u| p−2 u + λb(x)|u|α−1 |v |β+1 u,
in Ω,
−∆q v = λd(x)|v |q−2 v + λb(x)|u|α+1 |v |β−1 v,
in Ω,
−∆ p u = λa(x)|u| p−2 u + λb(x)|u|α |v |β v
in Ω,
−∆q v = λd(x)|v |q−2 v + λb(x)|u|α |v |β u
in Ω
(1.1)
(1.2)
satisfying the nonlinear boundary conditions
|∇u| p−2 ∇u · η + c1 (x)|u| p−2 u = 0
|∇v |
q −2
∇v · η + c2 (x)|v |
q −2
on ∂Ω,
v = 0 on ∂Ω,
(1.3)
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 β+1
= 1,
+
p
q
α+1 <
pq∗
p∗ q
, β+1 <
,
N
N
(1.4)
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.
Copyright © 2006 Hindawi Publishing Corporation
Boundary Value Problems 2005:3 (2005) 307–321
DOI: 10.1155/BVP.2005.307
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 nonlinear 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 bifurcation, 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 Laplacian 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 principal 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 eigenfunction 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) eigenfunctions. 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 existence 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 boundary ∂Ω. For m > 0 and r ∈ (1,+∞) let wm (x) = 1/(1 + |x|)m and assume that the space
Lr (wm ,Ω) := {u : Ω (1/(1 + |x|)m )|u|r < +∞} is supplied with the norm
uwm ,r =
Ω
1
m |u|r
1 + |x|
1/r
.
(2.1)
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∗ =
pN
,
N−p
q∗ =
qN
.
N −q
(2.2)
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,
0 < a(x) ≤ A1 wα1 (x)
(2.3)
(ii) there exists positive constants α2 , D1 with α2 ∈ (q + ((α + 1)(N − q)/ p∗ ),N) and
0 < d(x) ≤ D1 wα2 (x) a.e. in Ω,
(2.4)
(iii) m{x ∈ Ω : b(x) > 0} > 0 and
0 ≤ b(x) ≤ B1 ws (x) a.e. in Ω,
(2.5)
where B1 > 0 and s ∈ (max{ p, q},N).
(H3) c1 (·) and c2 (·) are positive and continuous functions defined on RN with
k1 w p−1 (x) ≤ c1 (x) ≤ K1 w p−1 (x),
(2.6)
l1 wq−1 (x) ≤ c2 (x) ≤ L1 wq−1 (x),
for some positive constants k1 , K1 , l1 , L1 .
Let Cδ∞ (Ω) be the space of C0∞ (RN )-functions restricted to Ω. For m ∈ (1,+∞), the
weighted Sobolev space Em is the completion of Cδ∞ (Ω) in the norm
1
|∇u| +
|u|m
m
Ω
Ω (1 + |x |)
1/m
m
|||u|||m =
.
(2.7)
By [14, Lemma 2] we see that if c(·) is a positive continuous function defined on Rn then
the norm
u1,m =
1/m
m
Ω
|∇u| +
m
∂Ω
c(x)|u|
(2.8)
is equivalent to ||| · |||m . The proof of the following lemma is also provided in [14].
Lemma 2.1. (i) If
p≤r ≤
pN
,
N−p
N > α ≥ N −r
N−p
,
p
(2.9)
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 inequality and the lower bound in the second is strict, then the embedding is compact.
(ii) If
p≤m≤
p(N − 1)
,
N−p
N > β ≥ N −1−m
N−p
,
p
(2.10)
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 = E p × Eq supplied with the norm
(u,v)
pq
= u1,p + v 1,q .
(2.11)
We now define the functionals Φ, I, J : E → R as follows:
Φ(u,v) =
α+1
p
−λ
Ω
α+1
p
I(u,v) =
α+1
p
J(u,v) =
α+1
p
|∇u| p +
∂Ω
Ω
a(x)|u| p − λ
|∇u| p +
β+1
q
Ω
Ω
α+1
p
a(x)|u| p +
c1 (x)|u| p +
β+1
q
β+1
q
|∇v | p +
Ω
β+1
q
∂Ω
c2 |v|q
Ω
d(x)|v|q − λ
Ω
β+1
q
|∇v | p +
α+1
p
Ω
b(x)|u|α+1 |v|β+1 ,
∂Ω
c1 (x)|u| p +
β+1
q
∂Ω
c2 |v|q ,
Ω
d(x)|v|q +
Ω
b(x)|u|α+1 |v|β+1 .
(2.12)
In view of (H1)–(H3), the functionals Φ, I, J are well defined and continuously differentiable 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
λ1 = inf I(u,v) : J(u,v) = 1 .
(2.13)
Each component of the associated normalized eigenfunction (u1 ,v1 ) is positive in Ω and of
1,δ
(Ω) for some δ ∈ (0,1).
class Cloc
(ii) The set of eigenfunctions corresponding to λ1 forms a one dimensional manifold E1 ⊆
E defined by
E1 =
cu1 , ±|c| p/q v1 : c ∈ R\{0} .
Furthermore, a componentwise positive eigenfunction always corresponds to λ1 .
(2.14)
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 properties 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
E2 = ± c u1 ,c p/q v1 : c > 0 .
(2.15)
(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 Fréchet differentiable, I is coercive on E ∩
{J(u,v) ≤ const}, J is compact and J (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
λ1 = inf I(u,v).
(3.1)
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 w p u1 and wq v1 are essentially bounded in Ω. To that purk p+1
pose define uM (x) := min{u1 (x),M }. It is clear that uM ∈ E p , for k ≥ 0. Multiplying the
k p+1
first equation of (1.1) by uM and integrating over Ω, we get
Ω
∇u1 p−2 ∇u1 · ∇ uk p+1 dx +
M
≤ λ1
(k+1)p
Ω
a(x)u1
dx + λ1
p−1 k p+1
∂Ω
c1 (x)u1 uM
dx
(3.2)
β+1
Ω
b(x)v1 u1 k p+α+1 dx.
Note that
Ω
∇u1 p−2 ∇u1 · ∇ uk p+1 dx = (k p + 1)
M
Ω
∇uM p uk p dx = k p + 1
M
(k + 1) p
Ω
k+1 p
∇u dx,
M
(3.3)
so since (k p + 1)/(k + 1) p ≤ 1, then
Ω
∇u1 p−2 ∇u1 · ∇ uk p+1 dx +
M
kp + 1
≥ c3
(k + 1) p
p−1 k p+1
∂Ω
c1 (x)u1 uM
1
(k+1)p∗
u
dx
p M
Ω (1 + |x |)
p/ p∗
dx
(3.4)
The first eigenvalue of p-Laplacian systems
312
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 Hölder inequality imply that
Ω
(k+1)p
a(x)u1
dx
≤ A1
1
(k+1)p
α u1
dx
1 + |x| 1
Ω
= A1
≤ A1
α1 − p2 /t Ω
(k+1)p
1
1 + |x|
Ω
u1
p2 /t dx
1 + |x|
1
(1 + |x|)(tα1 − p2 )/(t− p)
(t− p)/t dx
1
u(k+1)t dx
p 1
Ω (1 + |x |)
p/t
(3.5)
(observe that (tα1 − p2 )/(t − p) > N by H(2)(i)). Also, because of (H1), we may assume
that
β+1 k p+α+1
Ω
b(x)v1 u1
dx ≤
β+1 (k+1)p
Ω
b(x)v1 u1
dx,
(3.6)
otherwise we could consider

min u1 (x),M ,
uM (x) = 
0,
u1 (x) ≥ 1,
u1 (x) < 1
(3.7)
as a test function. So
Ω
β+1 (k+1)p
b(x)v1 u1
dx
≤ B1
Ω
1
β+1 (k+1)p
s v1 u1
dx
1 + |x|
= B1
β+1
(k+1)p
v1
u1
s(1−(p/t)) s(p/t) dx
Ω 1 + |x |
1 + |x|
(β+1)(t/t− p) (t− p)/t (k+1)t
v1
dx
s
Ω (1 + |x |)
≤ B1
≤ B1
1
q∗
v1 dx
q
Ω (1 + |x |)
u1
dx
s
Ω (1 + |x |)
p/t
(t− p)/t 1
u(k+1)t dx
p 1
Ω (1 + |x |)
p/t
,
(3.8)
by H(2)(iii). On combining (3.2)–(3.8), we conclude that
uM w p ,(k+1)p∗
≤C
1/(k+1)
k+1
(k p + 1)1/ p
1/(k+1)
u 1 w p ,(k+1)t ,
(3.9)
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
u M w p ,(k1 +1)p∗
≤C
1/(k1 +1)
k1 + 1
1/ p
k1 p + 1
1/(k1 +1)
u 1 w p , p∗ ,
(3.10)
w p , p∗ .
(3.11)
while by letting M → ∞ we obtain that
u 1 w p ,(k1 +1)p∗
k1 + 1
1/ p
k1 p + 1
1/(k1 +1)
≤ C 1/(k1 +1) u 1 ∗
Hence, u1 ∈ L(k1 +1)p (w p ,Ω). Note that if k ≥ k1 then (k p + 1)/(k + 1) p ≤ 1. Choosing in
(1.1) k = k2 with (k2 + 1)t = (k1 + 1)p∗ , that is, k2 = (p∗ /t)2 − 1, we have
u 1 w p ,(k2 +1)p∗
≤C
1/(k1 +1)
k2 + 1
1/ p
k2 p + 1
1/(k2 +1)
u 1 w p ,(k1 +1)p∗ .
(3.12)
∗
Hence, u1 ∈ L(k2 +1)p (w p ,Ω). Proceeding by induction we arrive at
u 1 ≤C
w p ,(kn +1)p∗
1/(kn +1)
kn + 1
1/ p
kn p + 1
1/(kn +1)
u1 w p ,(kn−1 +1)p∗ .
(3.13)
From (3.10) and (3.13) we conclude that
u 1 w p ,(kn +1)p∗
≤C
n
n
i=1 1/(ki +1)
i =1
ki + 1
1/ p
ki p + 1
1/(ki +1)
u 1 w p , p∗

√ 1/√ki +1
1/
ki +1
n
n
ki + 1


1/(ki +1)
u 1 i
=
1
=C
 
1/ p
w p , p∗ .
ki p + 1
i =1
√
Since (y + 1/(y p + 1)1/ p )1/ y+1 > 1 for y > 0, and lim y→∞ (y + 1/(y p + 1)1/ p )1/
there exists K > 1 independent of kn such that
u 1 w p ,(kn +1)p∗
(3.14)
≤C
n
i=1 1/(ki +1)
K
n
i=1 1/
√
ki +1 u 1 w p , p ∗ ,
√
y+1
= 1,
(3.15)
where 1/(ki + 1) = (t/ p∗ )i and 1/ ki + 1 = ( t/ p∗ )i . Letting now n → ∞ we conclude that
u1 wp , ∞
≤ c u 1 w p , p ∗ ,
(3.16)
1,δ
1,δ
for some positive constant c. By [8], u1 ∈ Cloc
(Ω). Similarly v1 ∈ Cloc
(Ω).
Finally, we notice that for the principal eigenvalue, each component of an eigenfunction 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/q v1 ). 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 Ω̄ solution of the problem
−∆ p u ≤ λa(x)|u| p−2 u + λb(x)|u|α−1 |v |β+1 u,
in Ω,
−∆q v ≤ λd(x)|v |q−2 v + λb(x)|u|α+1 |v |β−1 v,
in Ω,
p −2
p −2
u = 0,
on ∂Ω,
|∇v |q−2 ∇v · η + c2 (x)|v |q−2 v = 0,
on ∂Ω,
|∇u|
∇u · η + c1 (x)|u|
(4.1)
for some λ > 0, and (w1 ,w2 ) is a positive on Ω̄ solution of
−∆ p u ≥ λa(x)|u| p−2 u + λb(x)|u|α−1 |v |β+1 u
−∆q v ≥ λd(x)|v |
|∇u|
p −2
q −2
α+1
v + λb(x)|u|
∇u · η + c1 (x)|u|
p −2
β −1
|v |
in Ω,
in Ω,
v
(4.2)
u = 0 on ∂Ω,
|∇v |q−2 ∇v · η + c2 (x)|v |q−2 v = 0
on ∂Ω
then (w1 ,w2 ) = (cw1 ,c p/q w2 ) for a constant c > 0.
Let ϕ ∈ Cδ∞ (Ω), ϕ > 0, then ϕ p /(w1 ) p−1 ∈ E p . By Picone’s identity [1], we get
0≤
Ω
R ϕ,w1 =
Ω
|∇ϕ| p −
Ω

∇
ϕp
p −1
w1

 · ∇w p ∇ w 1
1
ϕp p
p−1 ∆ p w1 −
p−1 ∇w1 ∇w1 · η
Ω
Ω w1
∂Ω w1
p −1
α β+1 ϕp p
≤
|∇ϕ| − λ
+ b(x) w1 w2
p−1 a(x) w1
Ω
Ω w1
ϕp p
−
p−1 ∇w1 ∇w1 · η
∂Ω w1
ϕp
|∇ϕ| p +
=
w p −1
=
|∇ϕ| − λ a(x)ϕ 1 p−1 − λ
Ω
Ω
w1
p
ϕ
p
−
p−1 ∇w1 ∇w1 · η,
∂Ω w1
p
p
α
α
(4.3)
β+1
w
b(x)ϕ 1 p−1 w2
Ω
w1
p
while the boundary conditions imply that
0≤
p −1
w
a(x)ϕ p 1 p−1 − λ
Ω
Ω
w1
p
p −1
ϕ
+
c1 (x) p−1 w1
.
∂Ω
w1
|∇ϕ| p − λ
β+1
w
b(x)ϕ p 1 p−1 w2
Ω
w1
(4.4)
D. A. Kandilakis et al. 315
Letting ϕ → w1 in E p we obtain
0≤
Ω
∇w1 p − λ
p
a(x)w1
Ω
−λ
Ω
α− p+1 β+1
p
b(x)w1 w1
w2
+
p
∂Ω
c1 (x)w1 .
(4.5)
Note also that
Ω
∇w1 p +
p
∂Ω
c1 (x)w1 ≤ λ
p
Ω
a(x)w1 + λ
β+1
b(x)w1α+1 w2 .
Ω
(4.6)
On combining (4.5) and (4.6) we get
0≤
Ω
β+1
α− p+1 β+1 p
− w1 w1
w2
.
(4.7)
β+1
β+1−q α+1 q
− w2 w2
w1
.
(4.8)
b(x) w1α+1 w2
Similarly,
0≤
Ω
b(x) w1α+1 w2
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/q v1 ), 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 correspond only to the first eigenvalue. Assume, for instance, that there exists an eigenpair (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/q v1 ), 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
I u∗ ,v∗
,
λ∗ = ˜
J u∗ ,v∗
(5.1)
˜ ·, ·) is defined by
˜ ∗ ,v∗ ) > 0, where J(
with J(u
α+1
˜
=
J(u,v)
p
Ω
a(x)|u| p +
β+1
q
Ω
d(x)|v|q +
Ω
b(x)|u|α |v|β uv.
(5.2)
316
The first eigenvalue of p-Laplacian systems
˜ ·, ·) is also compact. From (5.1) we also have that
Note that J(
I u∗ ,v∗ J u∗ ,v∗
I u ,v
≥ ∗ ∗ ,
λ∗ = ˜
J u∗ ,v∗ J u∗ ,v∗
J u∗ ,v∗
(5.3)
since
J u∗ ,v∗
≥ 1.
J˜ u∗ ,v∗
(5.4)
Normalizing (u∗ ,v∗ ) by setting
u ∗ u∗ =: " #1/ p ,
J u∗ ,v∗
v∗ v∗ =: " #1/q ,
J u∗ ,v∗
(5.5)
we get that
I u∗ ,v∗
,
I u ,v = J u∗ ,v∗
J u∗ ,v∗ = 1.
∗
∗
(5.6)
(5.7)
From relations (5.3)–(5.7) we conclude that
I u∗ ,v∗
= I u∗ ,v ∗ ≥ λ1 ,
λ∗ ≥ J u∗ ,v∗
(5.8)
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
I(u,v) I(u,v)
= λ1 ,
>
λ1 = ˜
J(u,v) J(u,v)
(5.9)
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 (un ,vn ) = 1, for all n ∈ N. Hence, there exists (ũ, ṽ) ∈ E such that (un ,vn ) (ũ, ṽ).
The simplicity of λ1 implies that (ũ, ṽ) = (u1 ,v1 ) or (ũ, ṽ) = (−u1 , −v1 ). Let us suppose
D. A. Kandilakis et al. 317
that (un ,vn ) (u1 ,v1 ) in E. For any two pairs of eigenfunctions (un ,vn ), (um ,vm ), multiplying the first equation by un − um and integrating by parts we derive
∇un p−2 ∇un − ∇um p−2 ∇um ∇un − ∇um dx
Ω
+
∂Ω
= λn
Ω
p−2
p −2 c1 (x) un un − um um un − um dx
p−2
p−2 a(x) un un − um um un − um dx
+ λn
Ω
α β
α β b(x) un vn vn − um vm vm un − um dx
+ λn − λm
$
Ω
p −2
a(x)um um un − um dx +
%
Ω
α β
b(x)um vm vm dx .
(5.10)
From the second equation we similarly derive
∇vn q−2 ∇vn − ∇vm q−2 ∇vm ∇vn − ∇vm dx
Ω
+
∂Ω
= λn
Ω
q−2
q−2 c2 (x) vn vn − vm vm vn − vm dx
q −2
q −2 d(x) vn vn − vm vm vn − vm dx
+ λn
Ω
α β
α β b(x) un vn un − um vm um vn − vm dx
+ λn − λm
$
Ω
q −2
a(x)vm vm vn − vm dx +
Ω
%
α β
b(x)um vm um dx .
(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
∇un p dx −→
Ω
∇vn q dx −→
Ω
Ω
∇u1 p dx,
∇v1 q dx.
(5.12)
Exploiting the strict convexity of E p and Eq we get that (un ,vn ) → (u1 ,v1 ) in E. For a fixed
n ∈ N and for every (φ,ψ) ∈ E we have
Ω
∇un p−2 ∇un ∇φ dx +
= λn
Ω
Ω
p −2
a(x)un ∇vn q−2 ∇vn ∇ψ dx +
= λn
Ω
∂Ω
p −2
c1 (x)un un φ dx + λn
∂Ω
Ω
un φ dx
α β
b(x)un vn vn φ dx,
p −2
c1 (x)vn vn ψ dx
q −2
d(x)vn vn ψ dx + λn
Ω
α β
b(x)un vn un ψ dx,
(5.13)
318
The first eigenvalue of p-Laplacian systems
Let ᐁn− =: {x ∈ Ω : un (x)<0} and ᐂn− =: {x ∈ Ω : vn (x) < 0}. By (c) we must have m(Ω−n ) >
0, with Ω−n = ᐁn− ∪ ᐂn− . Denoting by u−n = min{0,un } and vn− = min{0,vn } and choosing
φ ≡ u−n and ψ ≡ vn− , it follows that
ᐁn−
−p
∇u dx +
n
= λn
ᐂn−
ᐁn−
= λn
c1 (x)|u−n | p dx
∂Ω∩ᐁn−
p
a(x)u−n dx + λn
− q
∇v dx +
n
∂Ω∩ᐂn−
ᐁn−
α β
b(x)u−n vn u−n vn dx,
(5.14)
q
c1 (x)v− dx
n
q
d(x)vn− dx + λn
ᐂn−
ᐂn−
α β
b(x)un vn− un vn− dx.
Since the products u−n vn+ and u+n vn− are negative, from the above system of equations we
obtain
ᐁn−
−p
∇u dx +
n
≤ λn
ᐁ−
∂Ω∩ᐁn−
p
c1 (x)u−n dx
p
a(x)u− dx + λn
n
ᐁ−
α β
b(x)u−n vn− u−n vn− dx,
n
n
− q
− q
∇v dx +
c2 (x)vn dx
n
ᐂn−
∂Ω∩ᐂn−
q
α β
≤ λn
d(x)vn− dx + λn
b(x)u−n vn− u−n vn− dx.
ᐂn−
(5.15)
ᐂn−
From Hölder and Young inequalities we derive that
ᐁ−
α β
b(x)u−n vn− u−n vn− dx
n
≤ B1
= B1
ᐁn−
ᐁn−
≤ c3
α − β − −
1
v u v dx
s u−
n
n
n n
1 + |x|
α+1 − β+1
1
v dx
s u−
n
n
1 + |x|
ᐁn−
−p
1
u dx +
(1 + |x|)s n
ᐁn−
(5.16)
− q
1
v dx .
(1 + |x|)s n
Thus
'
−p
− q
& u ≤ c4 λ1 + η u− pp
n 1, p
n L (ws ,ᐁn− ) + vn Lq (ws ,ᐁn− ) .
(5.17)
'
− q
−p
& v ≤ c5 λ1 + η v − q q
n 1, p
n L (ws ,ᐂn− ) + un L p (ws ,ᐂn− ) .
(5.18)
Similarly,
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
−p
− q
u ≤ c4 λ1 + η u− pp
n 1, p
n L (ws ,ᐁn− ∩Brε ) + vn Lq (ws ,ᐁn− ∩Brε ) + ε ,
− q
−p
v ≤ c5 λ1 + η v − q q
n 1, q
n L (ws ,ᐂn− ∩Brε ) + un L p (ws ,ᐂn− ∩Brε ) + ε .
(5.19)
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 E p ⊆
∗
∗
L pp /(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 Hölder inequality we derive that
−p
u p
n L (ws ,ᐁn− ∩Brε )
≤
ᐁn− ∩Brε
×
(1 + |x|)γ1 p∗ /(p∗ − p−δ)
dx
1
(1 + |x|)(s−γ1 )p∗ /(p+δ)
ᐁn− ∩Brε
≤ c6
(p∗ − p−δ)/ p∗
1
− pp∗ /(p+δ)
u dx
(5.20)
n
(p∗ − p−δ)/ p∗
1
ᐁn− ∩Brε
(p+δ)/ p∗
dx
(1 + |x|)γ1 p∗ /(p∗ − p−δ)
−p
u ,
n 1,p
(note that γ1 p∗ /(p∗ − p − δ) > N). A similar inequality also holds for vn− :
− q
v q
n
L (ws ,ᐁn− ∩Brε )
≤ c7
(q∗ −q−δ)/q∗
1
ᐁ − ∩B
n
rε
(1 + |x|)γ2 q∗ /(q∗ −q−δ)
dx
− q
v .
n 1, q
(5.21)
Combining (5.19), (5.20), and (5.21) we get
−p
u − c8 ε
n 1,p
≤ c9
ᐁn− ∩Brε
+ c10
(p∗ − p−δ)/ p∗
1
(1 + |x|)γ1 p∗ /(p∗ − p−δ)
dx
n
rε
(1 + |x|)γ2
& p
− q
≤ c11 u−
n 1, p + vn 1, q
dx
q∗ /(q∗ −q−δ)

' 
− q
v n
1,q
1
ᐁn− ∩Brε
(1 + |x|)γ1 p∗ /(p∗ − p−δ)
+
n 1, p
(q∗ −q−δ)/q∗
1
ᐁ − ∩B
−p
u (p∗ − p−δ)/ p∗
dx
1
ᐁn− ∩Brε
(1 + |x|)γ2 q∗ /(q∗ −q−δ)
dx
(q∗ −q−δ)/q∗ 


.
(5.22)
320
The first eigenvalue of p-Laplacian systems
Similarly,
− q
v − c12 ε
n 1,q

(p∗ − p−δ)/ p∗
& p
' 1
q
−
−
≤ c13 un 1,p + vn 1,q
dx
 ᐂn− ∩Brε (1 + |x|)γ1 q∗ /(p∗ − p−δ)
(q∗ −q−δ)/q∗ 

1
+
ᐂn− ∩Brε
(1 + |x|)γ2 q∗ /(q∗ −q−δ)
dx

.
(5.23)
We can now add inequalities (5.22), (5.23) to get
1 − ε ≤ c14
ᐁn− ∩Brε
+ c15
(p∗ − p−δ)/ p∗
1
(1 + |x|)γ1 p∗ /(p∗ − p−δ)
dx
1
ᐂn− ∩Brε
(1 + |x|)γ2 q∗ /(q∗ −q−δ)
(5.24)
(q∗ −q−δ)/q∗
dx
.
By taking ε sufficiently small we see that
m Ω−n ∩ Brε > c16 > 0,
(5.25)
where the constant c16 is independent of λn and un . Since un → u1 in E p and vn → v1 in
∗
∗
Eq , we have that un → u1 in L p (w1 ,Ω) and vn → v1 in Lq (w2 ,Ω). Consequently, un → u1
∗
∗
in L p (w1 ,BK (0)) and vn → v1 in Lq (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 Models 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