Existence and Uniqueness of Positive Eigenvalues for Certain
Eigenvalue Systems∗
Ruying Xue
Department of Mathematics, Zhejiang University,
Hangzhou 310027, P. R. China
e-mail: ryxue@zju.edu.cn
Abstract
In this paper we consider certain eigenvalue systems. Imposing some reasonable hypotheses,
we prove that the eigenvalue systems have a unique eigenvalue with positive eigenfunctions,
and that the eigenfunction is unique up to a scalar multiple.
Key words and phrases: eigenvalue problem; eigenfunction; fixed point index; maximum principle.
Classificition 35J60, 35P30
1
Introduction
Let Ω be a bounded domain in Rn with smooth boundary ∂Ω, ∆p denotes the p-Laplacian defined
by ∆p = div(gradψp (u)), for p ∈ (1, ∞),and ψp (u) = |u|p−1 u.
It is well-known that the eigenvalue problem
−∆p u + aψp (u) = λa(x)ψp (u) in Ω, u = 0
on ∂Ω
(1.1)
has a unique positive eigenvalue λ0 > 0 associated to which the eigenvalue problem (1.1) possesses
positive eigenfunctions. In fact,
Z
Z
1,p
p
p
λ0 = inf { | 5 u| + a|u| dx : u ∈ W0 (Ω) with
a(x)|u|p dx = 1},
Ω
Ω
and the eigenfunctions for λ0 are the minimizers of the functional
Z
Z
J(u) =
| 5 u|p + a|u|p dx − λ0
a(x)|u|p dx, u ∈ W01,p (Ω).
Ω
(1.2)
Ω
Moreover, such a minimizer φ0 R∈ W01,p (Ω) exists and is unique up to a scalar multiple, and
φ0 ∈ L∞ (Ω); we normalize it by Ω a(x)|φ0 |p dx = 1 and φ0 ≥ 0 in Ω. Then also φ0 > 0 in Ω and
φ0 ∈ C 1.β (Ω̄) ,as a consequence of Lemma 2.3 in Section 2.
∗ Projects
supported by the Natural Science Foundation of China
1
The aim of this paper is to generalize the preceding results to eigenvalue problems of the form

Pk
 −∆p u + aψp (u) = λp−1P i=1 ai (x)uαi v p−1−αi in Ω,
N
(1.3)
−∆q v + bψq (v) = λq−1 j=1 bi (x)uq−1−βj v βj in Ω,

u = v = 0 on ∂Ω,
where 1 < p < +∞, 1 < q < +∞, a and b are some positive numbers, 0 ≤ αi ≤ p − 1(i =
1, , ..., k), 0 ≤ βj ≤ q−1(j = 1, 2, ..., N ) , ai (x)(i = 1, 2, ...k) and bj (x)(j = 1, , ...N ) are nonnegative,
never vanishing functions belonging to C α (Ω̄) for some α ∈ (0, 1). Imposing some reasonable
hypotheses we prove that the eigenvalue problem (1.3) has a unique positive eigenvalue for which
the eigenfunctions are positive, and that the eigenfunction is unique up to a scalar multiple.
Under a positive eigenfunction of eigenvalue problem (1.3) we mean a weak solution (u, v) ∈
W01,p (Ω) × W01,q (Ω) with u > 0 and v > 0 in Ω. A nonnegative solution of eigenvalue problem (1.3)
is a pair (u, v) ∈ W01,p (Ω) × W01,q (Ω) which does not vanish identically in Ω and satisfies (1.3), each
component is nonnegative.
This article is organized as follows. In Section 2 we recall some well-known results for the
single equation with ∆p . In Section 3 we prove that (1.3) has a unique eigenvalue with positive
eigenfunctions. The uniqueness of the eigenfunction of (1.3) is considered in Section 4.
In this paper, we shall write (u1 , v1 ) ≥ (u2 , v2 ) if u1 ≥ u2 and v1 ≥ v2 , (u1 , v1 ) > (u2 , v2 ) if u1 >
R
1
u2 and v1 > v2 . We also denote by |u|0,Ω̄ =→ Sup|u| , and denote by kukA(x),p = ( Ω A(x)|u|p dx) p .
Ω
2
Some Known Results for A Single Equation with ∆p
In this section, we give some known results for the following Dirichlet problem
½
−∆p u + aψp (u) = f (x) in Ω,
u = g on ∂Ω,
(2.1)
where a ∈ (0, ∞) is a constant, 0 ≤ f ∈ L∞ (Ω) with the norm ||f ||∞,Ω , and 0 ≤ g ∈ C 1,α (∂Ω)
with Hőlder norm |g|1+α,∂Ω . By variational methods the Dirichlet problem (2.1) has a unique weak
solution u ∈ W01,p (Ω).
The following weak comparison principle and strong comparison principle near the boundary
∂Ω are proved in [1].
Lemma 2.1 Assume that u, u0 ∈ W 1,p (Ω), respectively, are weak solutions of
−∆p u + aψp (u) = f in Ω, u = g on ∂Ω,
−∆p u0 + aψp (u0 ) = f 0 in Ω, u0 = g 0 on ∂Ω,
with f ≤ f 0 and g ≤ g 0 . Then u ≤ u0 almost everywhere in Ω.
Lemma 2.2 Assume that u, u0 ∈ C 1,α (Ω̄), respectively, are weak solutions of
−∆p u + aψp (u) = f in Ω, u = 0 on ∂Ω,
−∆p u0 + aψp (u0 ) = f 0 in Ω, u0 = 0 on ∂Ω
2
with f, f 0 ∈ C α (Ω̄), 0 ≤ f ≤ f 0 , and f does not vanish identically in Ω.Then there exists a positive
constant δ > 0 small enough such that if Σ ⊂ Ωδ is a connected componnent of Ωδ , then either
u ≡ u0 in Σ or else
∂u
∂u0
u < u0 in Σ̄ − ∂Ω
and
>
on Σ̄ ∩ ∂Ω.
∂ν
∂ν
here ν denotes the exterior unit normal to ∂Ω, Ωδ = {x ∈ Ω; dist(x, ∂Ω) < δ} .
The following regularity result is proved in [2].
Lemma 2.3 Assume that u ∈ W 1,p (Ω) is a bounded weak solution of (2.1). Let C 0 > 0 be a
constant such that
kuk∞,Ω ≤ C 0 , kf k∞,Ω ≤ C 0
and
|g|1+α,∂Ω ≤ C 0 .
Then there exists a constant β, 0 < β < 1, depending solely on α, p, a and n, and another constant
C, 0 ≤ C < ∞,depending solely upon α, p, a, n, Ω and C 0 such that u ∈ C 1,β (Ω̄) with the Hőlder
norm
|u|1+β,Ω̄ ≤ C.
Vazquez[3] has proved the following strong maximum and boundary point principle for the
Dirichlet problem (2.1). Lemma 2.4 Assume that u is a function satisfying the following conditions:
(1). u ∈ C 1 (Ω̄) ,and u ≥ 0 in Ω,
(2). ∆p u ∈ L2loc (Ω̄),
(3). −∆p u ≥ aψp (u) a.e. in Ω.
Then, if u does not vanish identically in Ω, it is positive everywhere in Ω and
∂u
(x0 ) < 0 whenever u(x0 ) = 0 for x0 ∈ ∂Ω.
∂ν
Here ν = ν(x0 ) denotes the exterior unit normal of ∂Ω at x0 .
Denote by C0 (Ω̄) = {u ∈ C(Ω̄), u = 0 on ∂Ω} and let V+ be the positive cone in C0 (Ω̄) ×
C0 (Ω̄), V+ = {(u, v) ∈ C0 (Ω̄) × C0 (Ω̄), u ≥ 0, v ≥ 0 in Ω}. Let f (x, u, v), g(x, u, v) be nonnegative
functions defined in Ω̄ × [0, +∞) × [0, +∞) and satisfy f, g ∈ C α (Ω̄ × [0, +∞) × [0, +∞)) for
some α ∈ (0, 1). For any (u, v) ∈ V+ , by variational methods there exists a unique weak solution
(ū, v̄) ∈ W01,p (Ω) × W01,q (Ω) satisfying

 −∆p ū + aψp (ū) = f (x, u, v) in Ω,
−∆q v̄ + aψq (v̄) = g(x, u, v) in Ω,
(2.2)

ū = v̄ = 0 on ∂Ω.
We recall that (ū, v̄) is a bounded, nonnegative weak solution of (2.2) by Lemma refL2.1, and
that (ū, v̄) ∈ C01,β (Ω̄) × C01,β (Ω̄) for some β ∈ (0, 1) by Lemma 2.3. We define the mapping
T = (Tp , Tq ) : (u, v) 7−→ (ū, v̄) = T (u, v). Observe that T is a self-mapping of V+ . Moreover,
T : V+ 7−→ V+ is continuous and relatively compact.
3
3
Existence and Uniqueness of The Eigenvalue
Theorem 3.1 Assume that αi and βj satisfy one of the following conditions:
(1). 0 ≤ αi < p − 1, 0 ≤ βj < q − 1 for all i ∈ {1, 2, ..., k}, j ∈ {1, 2, ..., N },
(2). α1 = 0 and β1 = 0.
Then the eigenvalue problem (1.3) has a unique eigenvalue λ0 associated to which (1.3) possesses
positive eigenfunctions
Proof: First, we prove the existence of the eigenvalue of (1.3). Let T be the mapping from V+
to V+ defined by (2.2) for the special functions
Pk
f (x, u, v) = i=1 ai (x)uαi v p−1−αi ,
PN
g(x, u, v) = j=1 bj (x)uq−1−βj v βj .
It is obvious that the eigenvalue problem (1.3) has an eigenvalue λ0 with nonnegative eigenfunctions
if and only if λ0 T has a fixed point in V +. We denote by λ1 the first eigenvalue of
−∆p u + aψp (u) = λp−1
a1 (x)up−1 , u|∂Ω = 0, u ∈ W01,p (Ω),
1
and denote by φ1 the eigenfunction associated to λ1 . Let λ2 be the first eigenvalue of
−∆q u + bψq (u) = λq−1
b1 (x)uq−1 , u|∂Ω = 0, u ∈ W01,p (Ω),
2
and let φ2 be the eigenfunctions associated to λ2 . It is well known that 0 ≤ φ1 ∈ L∞ (Ω̄), 0 ≤ φ2 ∈
L∞ (Ω̄). Combining Lemma 2.1, Lemma 2.3 with Lemma ?? we arrive at 0 < φ1 ∈ C 1,β (Ω̄), 0 <
φ2 ∈ C 1,β (Ω̄), and
φ1 > 0, φ2 > 0 in Ω and
∂φ1
∂φ2
< 0,
< 0 on ∂Ω.
∂ν
∂ν
(3.1)
Since ∂Ω is bounded and closed, (3.1) implies that
C0 φ2 (x) ≤ φ1 ≤
1
φ2 (x)
C0
(3.2)
for some positive constant C0 .
Denote by T² (u, v) = T (u, v) + ²(φ1 , φ2 ). Since the mapping T is continuous and relatively
compact in V+ , the mapping T² : V+ 7−→ V+ is continuous and relatively compact. Moreover, we
have
inf {|T² (u, v)|0,Ω̄ : |u|0,Ω̄ + |v|0.Ω̄ = 1, (u, v) ∈ V+ }
≥ ²|φ1 |0,Ω̄ + ²|φ2 |0.Ω̄ > 0.
. Then, by Theorem 5.4.33 in [2], there exists λ² > 0 and (u² , v² ) ∈ V+ with |u² |0,Ω̄ + |v² |0,Ω̄ = 1
satisfying
λ² (u² , v² ) = T² (u² , v² ) = T (u² , v² ) + ²(φ1 , φ2 ).
(3.3)
Since T is a compact operator defined on V+ , and |u² |0,Ω̄ + |v² |0,Ω̄ = 1, we may assume that there
exists a pair (u0 , v0 ) in V+ and a nonnegative number λ1 such that
T (u² , v² ) 7−→ (u1 , v1 ) and λ² 7−→ λ1 , as ² 7−→ 0.
4
Clearly, if λ1 > 0, we have
(u² , v² ) =
1
1
def.
T² (u² , v² ) + ²(φ1 , φ2 ) → 1 (u1 , v1 ) → = (u0 , v0 ),
λ²
λ
1
λ1 T (u0 , v0 ),
as ² → 0, and |u0 |0,Ω̄ + |v0 |0,Ω̄ = 1 and (u0 , v0 ) =
λ1 6= 0.
so that it remains only to prove that
To this end,we first note that (3.3) implies
1
(φ1 , φ2 ).
λ²
(3.4)
(u² , v² ) ≥ (t² φ1 , t² φ2 ).
(3.5)
(u² , v² ) ≥ ²
Thus, there is a largest number t² > 0 such that
(3.5) implies
Pk
i=1
PN
1 p−1−α1
i p−1−αi
ai (x)uα
≥ a1 (x)tp−1
φα
≥ C0p−1−α1 t²p−1 a1 (x)φ1p−1 ,
² v²
²
1 φ2
q−1−βj βj
v²
j=1 bj (x)u²
≥
1 β1
b1 (x)tq−1
φq−1−β
φ2
²
1
≥
C0q−1−β1 tq−1
b1 (x)φ2q−1 ,
²
(3.6)
(3.7)
By the weak comparison principle (Lemma 2.1 ), it follows from (3.6) and (3.7) that
α
1
1− p−1
λ² (u² , v² ) ≥ T (u² , v² ) ≥ (C0
β
1
1− q−1
t² φ1 , C0
t² φ2 ).
(3.8)
But
t² is a largest number satisfying (3.5), (??) implies
α
1
1− p−1
t² ≥ min{C0
α
1
1− p−1
so that λ² ≥ min{C0
β
1
1− q−1
, C0
}t² λ−1
² ,
β
1
1− q−1
, C0
} > 0. Consequently λ1 > 0, as required.
Denote by λ0 = λ11 , so (u0 , v0 ) is a nonnegative eigenfunction for λ0 . Combining (1) (or (2))
with the fact that |u0 |0,Ω̄ + |v0 |0,Ω̄ = 1 implies u0 6≡ 0, v0 6≡ 0. By Lemma 2.3 and Lemma ?? we
arrive at u0 ∈ C01+β (Ω̄), v0 ∈ C01+β (Ω̄) for some β ∈ (0, 1), and u0 > 0, v0 > 0 in Ω. Hence (u0 , v0 )
is a positive eigenfunction of (1.3) for λ0 .
Now, we prove that λ0 is unique. Suppose that (u1 , v1 ), (u2 , v2 ) are two linearly independent positive eigenfunctions associated with the eigenvalues Λ1 , Λ2 , respectively, for the eigenvalue
problem (1.3). From Lemma 2.1 and Lemma 2.3, we have Λ1 , Λ2 > 0 and
∂u2
∂v1
∂v2
∂u1
< 0,
< 0,
< 0, and
< 0, on ∂Ω.
∂ν
∂ν
∂ν
∂ν
(3.9)
We may assume that (0, 0) < (u1 , v1 ) ≤ (u2 , v2 ) and Λ1 ≤ Λ2 . We claim Λ1 = Λ2 . On the
contrary, suppose Λ1 < Λ2 . Let us consider the following elliptic system:

Pk
αi p−1−αi
−∆p u + aψp (u) = Λp−1

1
i=1 ai (x)u v

Pk

p−1
p−1
p−1−α

α
i
i

in Ω,
i (x)u2 v2
 +(Λ2 − Λ1 ) i=1 aP
N
q−1
q−1−βj βj
(3.10)
v
−∆q v + bψq (v) = Λ1
j=1 bj (x)u


q−1−βj βj
q−1 PN
q−1

)
b
(x)u
v
in
Ω,
−
Λ
+(Λ

1
2
2
2
j=1 j


u = v = 0 on ∂Ω.
5
Observe that (u2 , v2 ) is a solution of (3.10). By Lemma 2.1, we have (ξu1 , ξv1 ) ≤ T̃ (ξu1 , ξv1 )
for all ξ ∈ (0, +∞), and T̃ (ξu2 , ξv2 ) ≤ (ξu2 , ξv2 ) for all ξ ∈ (1. + ∞), where T̃ be the mapping
from V+ to V+ definied by (2.2) for the special functions
f˜(x, u, v) =
Λp−1
1
k
X
ai (x)uαi v p−1−αi
i=1
+ (Λp−1
− Λp−1
)
2
1
k
X
i p−1−αi
ai (x)uα
, in Ω,
2 v2
i=1
g̃(x, u, v) =
N
X
q−1
Λ1
bj (x)uq−1−βj v βj
j=1
− Λq−1
)
+ (Λq−1
1
2
N
X
q−1−βj βj
v2
bj (x)u2
in Ω.
j=1
Making use of (3.9) and the fact that (0, 0) < (u1 , v1 ) ≤ (u2 , v2 ), we can pick ξ ≥ 1 so large that
(u2 , v2 ) ≤ ξ(u1 , v1 ). Then, by Lemma 2.1 we arrive at
(u2 , v2 ) ≤ ξ(u1 , v1 ) ≤ T̃ (ξu1 , ξv1 ) ≤ ... ≤ T̃ k (ξu1 , ξv1 ) ≤
... ≤ T̃ k (ξu2 , ξv2 ) ≤ ... ≤ T̃ (ξu2 , ξv2 ) ≤ ξ(u2 , v2 )
(3.11)
The compactness of T̃ k implies T̃ k (ξu2 , ξv2 ) 7−→ (u3 , v3 ) in V+ as k 7−→ +∞, hence
(u2 , v2 ) ≤ (ξu1 , ξv1 ) ≤ T̃ (u3 , v3 ) = (u3 , v3 ) ≤ ξ(u2 , v2 ),
(3.12)
and (u3 , v3 ) is a positive solution of (3.10). We claim that (u3 , v3 ) = (u2 , v2 ). If this is the case,
(3.12) implies that
(u2 , v2 ) = ξ(u1 , v1 ),
a contradiction to our assumption that (u1 , v1 ) and (u2 , v2 ) are not colinear. Then we have Λ1 = Λ2
as desired.
On the contrary, suppose (u3 , v3 ) 6≡ (u2 , v2 ). Without loss of generality, we assume that
(u3 , v3 ) ≤ (u2 , v2 ) is false. Consequently, by (3.9) we can pick t ∈ (1, +∞) which is the smallest
number satisfying
t−1 (u3 , v3 ) ≤ (u2 , v2 )
(3.13)
t1−p f˜(u3 , u3 ) < f˜(t−1 u3 , t−1 v3 ) ≤ f˜(u2 , v2 ),
(3.14)
We have
and
−∆p (t−1 u3 ) + aψp (t−1 u3 ) = t1−p f˜(u3 , v3 ) in Ω,
−∆p (u2 ) + aψp (u2 ) = f˜(u2 , v2 ) in Ω,
u3 = u2 = 0 on ∂Ω.
(3.15)
By (3.14) and (3.15), the maximum principle and comparison principle near the boundary ∂Ω
force
0 < t−1 u3 < u2 in Ω̄δ − ∂Ω and
6
∂u2
∂t−1 u3
<
< 0 on ∂Ω.
∂ν
∂ν
(3.16)
Making use of (3.16), we find t̄ ∈ (1, t) such that
0 < t̄−1 u3 ≤ u2 in Ω̄δ − ∂Ω and
∂u2
∂ t̄−1 u3
≤
< 0 on ∂Ω,
∂ν
∂ν
(3.17)
and
t̄1−p f˜(u3 , v3 ) ≤ f˜(u2 , v2 ), for x ∈ Ω − Ω̄δ
(3.18)
by compactness of the set Ω − Ωδ .
We now consider the following elliptic equations:
−∆p (t̄−1 u3 ) + aψp (t̄−1 u3 ) = t̄1−p f˜(u3 , v3 ) in Ω \ Ω̄δ ,
−∆p (u2 ) + aψp (u2 ) = f˜(u2 , v2 ) in Ω \ Ω̄δ ,
t̄−1 u3 ≤ u2 on ∂(Ω \ Ω̄δ ).
(3.19)
By (3.17) ,(3.18) and (3.19), the weak comparison principle implies
t̄−1 u3 ≤ u2 in Ω \ Ω̄δ ,
and
t̄−1 u3 ≤ u2 in Ω.
(3.20)
t̄−1 v3 ≤ v2 in Ω.
(3.21)
Similarly, we have
(3.19) and (3.20) force, for some t̄ ∈ (1, t),
t̄−1 (u1 , v1 ) ≤ (u2 , v2 ),
a contradiction to our choice of t.
2
Remark It is obvious that if α1 = p − 1, (1.3) has a unique eigenvalue for which eigenfunctions
are the form (u, 0) with u > 0, and if β1 = q − 1, (1.3) has a unique eigenvalue for which
the eigenfunctions have the form (0, v) with v > 0. If (1.3) has an eigenvalue with positive
eigenfunctions, from the proof of Theorem 4.1 we know that the eigenvalue is unique. Hence, (1.3)
has at most three eigenvalues with nonnegative eigenfunctions. It is possible that (1.3) may have
no eigenvalue with positive eigenfunctions.
Assume that p = q > 1, a = b > 0, and that there exist positive constants C1 , C2 , C1 ≤ C2 ,
such that
N
N
k
X
X
X
bj (x)tp−1−βj
C1p−1
bj (x)tp−1−βj ≤
ai (x)tαi ≤ C2p−1
j=1
j=1
i=1
for all (t, x) ∈ [C1 , C2 ] × Ω̄. Then the eigenvalue problem (1.3) has a unique eigenvalue λ0 for
which the eigenfunctions are positive in Ω.
Proof: We denote by
k
N
X
X
f (x, u, v) = max{
ai (x)uαi v p−1−αi , C1p−1
bj (x)up−1−βj v βj },
i=1
j=1
7
g(x, u, v) = max{C21−p
k
X
ai (x)uαi v p−1−αi ,
i=1
N
X
bj (x)up−1−βj v βj },
j=1
for all (x, u, v) ∈ Ω̄ × [0, +∞) × [0, +∞). Consider the following eigenvalue problem
−∆p u + aψp (u) = λp−1 f (x, u, v) in Ω,
−∆q u + aψq (u) = λp−1 g(x, u, v) in Ω,
u = v = 0 on ∂Ω.
(3.22)
(3.23)
(3.24)
Using the analogous proof to that of Theorem 3.1, we infer that (3.24) has a positive eigenvalue
λ0 for which (3.24) possesses a nonnegative eigenfunction (u0 , v0 ).
Making use of the weak comparison principle, we have
C1 v0 ≤ u0 ≤ C2 v0 for all x ∈ Ω̄.label3.23
(3.25)
A combination of (??) with the strong maximum principle implies (u0 , v0 ) is a positive eigenfunction
for λ0 . Thus, λ0 is a positive eigenvalue of (1.3) by the definition of the functions f (x, u, v) and
g(x, u, v).
The proof of the uniqueness of λ0 is the same as that in Theorem 3.1, we omit it.
2
Corollary 3.2 Assume that p = q > 1, a = b > 0, and one of the following conditions is satisfied.
Then (1.3) has a unique eigenvalue with positive eigenfunctions.
there exist α1 and β1 such that
a1 (x) > 0, b1 (x) > 0 in Ω̄,
p − 1 ≤ α1 + βj < 2(p − 1) for all j ∈ {1, 2, ..., N },
p − 1 ≤ αi + β1 < 2(p − 1) for all i ∈ {1, 2, ..., k},
α1 < p − 1 ( or β1 < p − 1) , ai (x) ≡ ai > 0, bj (x) ≡ bj > 0 for i=1,2,...,k, j=1,2,...,N, here ai
Pk
PN
and bj are positive numbers, and i=1 ai (x) = j=1 bj (x) for all x ∈ Ω̄.
If αi and βj satisfy the condition (1), we can prove that the conditions in Theorem 3 hold.
More precisely, we choose positive constant C1 small enough such that
C1p−1
N
X
bj (x)up−1−βj v βj ≤ a1 (x)uα1
j=1
for (x, u, v) ∈ Ω̄ × (0. + ∞) × (0, +∞) with C1 v ≤ u, and a positive number C2 large enough such
that
k
X
ai (x)uαi v p−1−αi ≤ C2p−1 b1 (x)up−1−β1 v β1
i=1
for (x, u, v) ∈ Ω̄ × (0, +∞) × (0, +∞) with u ≤ C2 v. Therefore, Theorem 3.2 implies that (1.3) has
a unique eigenvalue with positive eigenfunctions.
2
8
4
Uniqueness of Eigenfunctions
In this section, we prove that, under some reasonable hypotheses, the eigenfunction associated to
λ0 is unique up to a scalar multiple. Consider the following eigenvalue problem
Pk
−∆p u + aψp (u) = λp−1 A(x) i=1 ai uαi v p−1−αi in Ω,
PN
−∆p v + aψp (v) = λp−1 A(x) j=1 bi uq−1−βj v βj in Ω,
u = v = 0 on ∂Ω
(4.1)
where ai and bj are positive numbers, A(x) is a continuous, nonnegative, never vanishing function
in Ω.
Theorem 4.1 Assume that α1 < p − 1(or β1 < p − 1). Then, the eigenvalue problem (4.1) has a
unique eigenvalue λ0 for which the eigenfunctions are positive, the eigenfunction associated to λ0
is unique up to a scalar multiple.
Proof: It is obvious that we can choose a positive constant C such that
k
X
C p−1−αi ai ≡
i=1
N
X
bj C βj −p+1 .
j=1
We denote by λ1 , φ1 (x) the first eigenvalue and the eigenfunction, respectively, of the following
eigenvalue problem:
Pk
−∆p φ1 + aψp (φ1 ) = λp−1
A(x) i=1 ai C p−1−αi φ1p−1 in Ω,
(4.2)
1
φ1 = 0 on ∂Ω.
(4.3)It is well-known that
Z
Z
k
X
p
p
ai C p−1−αi |u|p dx = 1 , u ∈ W01,p }
λp−1
=
inf
{
[|
5
u|
+
a|u|
]dx
:
A(x)
1
Ω
Ω
i=1
and the eigenfunction for λ1 is a minimizer, such a minimizer φ1 exists and is unique up to a scalar
multiple. Obviously, λ1 is an eigenvalue of (4.1), and (C −1 φ1 , φ1 ) is a positive eigenfunction of
(4.1) associated to λ1 .
In analogy with our proof of Theorem 3.1, we infer that (4.1) has a unique eigenvalue with
positive eigenfunctions, and hence λ0 = λ1 .
Now we prove that the eigenfunction for λ0 is unique up to a scalar multiple. Without loss of
generality, we assume C = 1. Suppose (u1 , v1 ) is another positive eigenfunction for λ0 , that is,
(u1 , v1 ) satisfies
−∆p u1 + aψp (u1 ) = λp−1
A(x)
1
Pk
αi p−1−αi
i=1 ai u1 v1
PN
q−1−βj βj
λp−1
A(x) j=1 bj u1
v1
1
−∆p v1 + aψp (v1 ) =
u1 = v1 = 0 on ∂Ω.
9
in Ω,
(4.4)
in Ω,
(4.5)
(4.6)
Case (i) α1 < p−1 and β1 < p−1 , or αi = p−1 and βj = p−1 for i = 1, 2, ..., k, j = 1, 2, ..., N .
Making use of the H ölder0s inequality, we deduce from (4.6) that
Z
Ω
Z
k Z
X
αi +1
−1
( A(x)ai up1 dx) p ( A(x)ai v1p dx)1−p (αi +1) ,
[| 5 u1 |p + a|u1 |p ]dx ≤ λp−1
1
[| 5 v1 |p + a|v1 |p ]dx ≤ λp−1
1
ku1 kpA(x)ai ,p
Ai =
ku1 kp
A(x)
here kukB(x),p denotes (
Ω
Pk
i=1
ku1 kpA(x)bj ,p
Cj =
R
kv1 kpA(x)bj ,p
p
p−1
B(x)u dx)
,
kv1 kpA(x)ai ,p
Bi =
ku1 kpA(x)ai ,p
ai ,p
,
kv1 kpA(x)bj ,p
Dj =
kv1 kp
A(x)
PN
j=1
(4.7)
(4.8)
,
.
bj ,p
. Thus, we have
Z
Ω
Ω
Ω
j=1
Denote by
Ω
Z
N Z
X
−1
−1
( A(x)bj up1 dx)1−p (βj +1) ( A(x)bj v1p dx)p βj .
Z
Ω
Ω
i=1
(| 5 u1 |p + a|u1 |p )dx ≤ λp−1
1
K
X
1−p−1 (αi +1)
Ai Bi
Z X
K
A(x)ai |ui |p dx,
(4.9)
A(x)bj |vj |p dx,
(4.10)
Ω i=1
i=1
and
Z
Ω
(| 5 v1 |p + a|v1 |p )dx ≤ λp−1
1
N
X
1−p−1 (βj +1)
Dj Cj
Z X
N
Ω j=1
j=1
with
k
X
Ai = 1,
i=1
N
X
Dj = 1, Bi =
j=1
kv1 kpA(x),p
1
=
.
Cj
ku1 kpA(x),p
(4.11)
By the definition of λ1 it follows from (4.9) and (4.10) that
k
X
1−p−1 (αi +1)
Ai Bi
≥ 1 and
i=1
N
X
1−p−1 (βj +1)
Dj Cj
≥ 1.
(4.12)
j=1
Combining(4.11) with (4.12) we deduce that
Bi = Cj = 1,
and
X
1−p−1 (αi +1)
Ai Bi
X
= 1,
i=1
1−p−1 (βj +1)
Dj Cj
= 1.
(4.13)
j=1
A combibation of (4.9),(4.10) and (4.13) with the fact that the minimizer definied in (4.3) is unique
up to a scalar multiple yields
(u1 , v1 ) = d(φ1 , φ1 )
10
for some d ∈ (0, +∞).
Case (ii)
α1 < p − 1, βj = p − 1 for all j ∈ {1, 2, ..., N }. In this case, from (4.7) we have
Z
Ω
(| 5 v1 |p + a|v1 |p dx ≤ λp−1
1
Z X
N
A(x)bj |vj |p dx,
Ω j=1
which implies
v1 = dφ1 for some d ∈ (0, +∞).
It is obvious that dφ1 and u1 are two positive solutions of
Pk
A(x) i=1 ai uαi v1p−1−αi in Ω,
−∆p u + aψp (u) = λp−1
1
u = 0 on ∂Ω.
(4.14)
(4.15)
The fact that (4.15) possesses only a positve solution (cf. [1, Theorem 1]) implies u1 = dφ1 .
Case (iii)
β1 < p − 1, αi = p − 1 for all i ∈ {1, 2, ..., k}. In this case, the proof is the same
as that in case (ii), we omit it.
2
Remark When αi = p − 1(i = 1, 2, ..., k), βj = p − 1(j = 1, 2, ..., N ), we easily deduce that (4.1)
has a unique eigenvalue with positive eigenfunctions if and only if ai and bj satisfy the equality
k
X
ai =
i=1
N
X
bj .
j=1
References
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[3] Vazquez J.L., A strong maximum principle for some quasilinear elliptic equations, Appl. Math.
Optim.,12(1984),191-202.
[4] Berger M.S.,Nonlinearity and functional analysis, Academic press,New York-San FranciscoLondon, 1977.
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