Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 08, pp. 1–6.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
SIMPLICITY AND STABILITY OF THE FIRST EIGENVALUE
OF A (p; q) LAPLACIAN SYSTEM
GHASEM A. AFROUZI, MARYAM MIRZAPOUR, QIHU ZHANG
Abstract. This article concerns special properties of the principal eigenvalue
of a nonlinear elliptic system with Dirichlet boundary conditions. In particular,
we show the simplicity of the first eigenvalue of
−∆p u = λ|u|α−1 |v|β−1 v
in Ω,
−∆q v = λ|u|α−1 |v|β−1 u
in Ω,
(u, v) ∈ W01,p (Ω) × W01,q (Ω),
with respect to the exponents p and q, where Ω is a bounded domain in RN .
1. Preliminaries
Eigenvalue problems for p-Laplacian operators subject to Zero Dirichlet boundary conditions on a bounded domain have been studied extensively during the past
two decades, and many interesting results have been obtained. Most of the investigations have relied on variational methods and deduced the existence of a principal
eigenvalue as a consequence of minimization results of appropriate functionals.
In this article, we study the eigenvalue system
−∆p u = λ|u|α−1 |v|β−1 v
α−1
−∆q v = λ|u|
β−1
|v|
u
in Ω,
in Ω,
(1.1)
(u, v) ∈ W01,p (Ω) × W01,q (Ω),
where Ω ⊂ RN is a bounded domain, p, q > 1 and α, β are real numbers satisfying
α > 0,
β > 0,
α β
+ = 1.
p
q
(1.2)
We mention that problem (1.1) aries in several fields of application. For instance, in
the case where p > 2, problem (1.1) appears in the study of non-Newtonian fluids,
pseudoplastics for 1 < p < 2, and in reaction-diffusion problems, flows through
porous media, nonlinear elasticity, petroleum extraction, astronomy and glaciology
for p = 4/3 (see [3, 5]).
2000 Mathematics Subject Classification. 35J60, 35B30, 35B40.
Key words and phrases. Eigenvalue problem; quasilinear operator; simplicity; stability.
c 2012 Texas State University - San Marcos.
Submitted November 3, 2011. Published January 12, 2012.
1
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G. A. AFROUZI, M. MIRZAPOUR, Q. ZHANG
EJDE-2012/08
The principal eigenvalue λ1 (p; q) of (1.1) is obtained using the LjusternickSchnirelman theory by minimizing the functional
Z
Z
β
α
|∇u|p dx +
|∇v|q dx,
J(u, v) =
p Ω
q Ω
on C 1 -manifold: {(u, v) ∈ W01,p (Ω) × W01,q (Ω); Λ(u, v) = 1}, where
Z
Λ(u, v) =
|u|α−1 |v|β−1 uv dx.
Ω
We recall that λ1 (p, q) can be variationally characterized as
λ1 (p, q) = inf{J(u, v), (u, v) ∈ W01,p (Ω) × W01,q (Ω); Λ(u, v) = 1}
(1.3)
From the maximum principle of Vázquez, see [12], we deduce that the corresponding
eigenpair of λ1 (p; q); that is, (u; v) is such that u; v > 0. We call it positive
eigenvector.
Definition 1.1. An open subset Ω of RN is said to have the segment property if
for any x ∈ ∂Ω, there exists an open set Gx ∈ RN with x ∈ Gx and a pair yx of
RN \{0} such that if z ∈ Ω ∩ Gx and t ∈ (0, 1), then z + tyx ∈ Ω.
This property allows us to push the support of a function u in Ω by a translation
. The following results play an important role in the proof of Theorem 2.
Lemma 1.2. Let Ω be a bounded domain in RN having the segment property. If
u ∈ W 1,p (Ω) ∩ W01,s (Ω) for some s ∈ (1, p), then u ∈ W01,p (Ω).
2. Simplicity
Firstly we introduce
|u| p
dx
|∇u| dx + (p − 1)
|∇ϕ|p
ϕ
Ω
Ω
Z
|u|p−2 u
|∇ϕ|p−2 ∇ϕ∇udx
−p
p−1
Ω ϕ
Z
Z
∆p ϕ p
=
|∇u|p dx +
|u| dx.
p−1
ϕ
Ω
Ω
Z
Ap (u, ϕ) =
Z
p
(2.1)
Lemma 2.1 ([4]). For all (u, ϕ) ∈ (W01,p (Ω) ∩ C 1,γ (Ω))2 with ϕ > 0 in Ω and
γ ∈ (0, 1), we have Ap (u, ϕ); i.e.,
Z
Z
∆p ϕ p
|∇u|p dx ≥
|u| dx,
p−1
Ω ϕ
Ω
and if Ap (u, ϕ) = 0 there is c ∈ R such that u = cϕ.
Proof. Using Young’s inequality (since
1
p
+
p−1
p
= 1) we can write, for > 0,
|u| p−1
u|u|p−2
∇u|∇ϕ|p−2 ∇ϕ p−1 ≤ |∇u||∇ϕ|p−1
ϕ
ϕ
(2.2)
p
−
1 u p
p
p
|
|
|∇ϕ|
.
≤ |∇u|p +
p
pp ϕ
By choosing = 1 and integration over Ω, we have
Z
Z
Z
u|u|p−2 u
p−2
p
p
|∇ϕ| ∇ϕ∇u
dx ≤
|∇u| dx + (p − 1)
| |p |∇ϕ|p dx. (2.3)
p−1
ϕ
ϕ
Ω
Ω
Ω
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SIMPLICITY AND STABILITY
Therefore, we conclude that Ap (u, ϕ) ≥ 0.
If Ap (u, ϕ) = 0, then we obtain
Z
Z
Z
u|u|p−2 u
p
dx
=
|∇u|
dx
+
(p
−
1)
| |p |∇ϕ|p dx.
p
|∇ϕ|p−2 ∇ϕ∇u
p−1
ϕ
Ω
Ω ϕ
Ω
3
(2.4)
Letting = 1 in (2.2), we obtain
Z
Z
|u| p−1
u|u|p−2
p−2
dx.
(2.5)
|∇u||∇ϕ|p−1
∇u|∇ϕ| ∇ϕ p−1 dx =
ϕ
ϕ
Ω
Ω
Combining the two inequalities, we deduce that |∇u| = | ϕu ∇ϕ|, it follows that
∇u = η ϕu ∇ϕ, where |η| = 1. On the other hand, Ap (u, ϕ) = 0 implies that
η = 1 and ∇( ϕu ) = 0; that is, there is c ∈ R such that u = cϕ. This completes the
proof.
Theorem 2.2. Let λ1 (p, q) be defined by (1.3), then λ1 (p, q) is simple.
Proof. Let (u, v) and (ϕ, ψ) be two eigenvectors associated with λ1 (p, q). We show
that there exist real numbers k1 , k2 such that u = k1 ϕ and v = k2 ψ. Using Young’s
inequality, by (1.2) and the definition of λ1 (p, q), we can write
J(ϕ, ψ) = λ1 (p, q)Λ(ϕ, ψ)
Z
|ϕ|α |ψ|β
≤ λ1 (p, q)
uα v β α β dx
u v
ZΩ
h α |ϕ|p
β |ψ|q i
≤ λ1 (p, q)
uα v β
+
dx
p up
q vq
Ω
Z h
α uα−1 v β p β uα v β−1 q i
|ϕ| +
|ψ| dx
≤ λ1 (p, q)
p−1
q v q−1
Ω p u
Z
Z
α
−∆p u p
β
−∆q v q
≤
|ϕ| dx +
|ψ| dx.
p Ω up−1
q Ω v q−1
Due to Lemma 2.1, we obtain
Z
Z
−∆p u p
β
−∆q v q
α
|ϕ| dx +
|ψ| dx.
J(u, v) =
p Ω up−1
q Ω v q−1
Thus
Z
Ω
|∇ϕ|p dx =
Z
Ω
−∆p u p
|ϕ| dx,
up−1
Z
|∇ψ|q dx =
Ω
Z
Ω
−∆q v q
|ψ| dx.
v q−1
By Lemma 2.1, there exist real numbers k1 and k2 such that u = k1 ϕ and v = k2 ψ
and the theorem follows.
3. Stability
Theorem 3.1. Let Ω be a bounded domain in RN having the segment property.
Then, the function (p, q) → λ1 (p, q) is continuous from Iα,β to R+ , where
Iα,β = {(p, q) ∈ (1, +∞) × (1, +∞) such that (1.2) is satisfied}.
Proof. Let (tn )n≥1 , tn = (pn , qn ) be a sequence in Iα,β converging at t = (p, q) ∈
Iα,β . We will prove that
lim λ1 (pn , qn ) = λ1 (p, q).
n→∞
4
G. A. AFROUZI, M. MIRZAPOUR, Q. ZHANG
EJDE-2012/08
Indeed, let (ϕ, ψ) ∈ C0∞ (Ω) × C0∞ (Ω) such that Λ(ϕ, ψ) > 0; hence,
λ1 (pn , qn ) ≤
α
pn
pn k∇ϕkpn
+
β
qn
qn k∇ψkqn
Λ(ϕ, ψ)
,
since λ1 (pn , qn ) is the infimum. Letting n tend to infinity, we deduce from Lebesgue’s
theorem
β
α
p
q
p k∇ϕkp + q k∇ψkq
.
lim sup λ1 (pn , qn ) ≤
Λ(ϕ, ψ)
n→∞
Then
lim sup λ1 (pn , qn ) ≤ λ1 (p, q).
(3.1)
n→∞
On the other hand, let {(pnk , qnk )}k≥1 be a subsequence of (tn )n such that
lim λ1 (pnk , qnk ) = lim inf λ1 (pn , qn ).
n→∞
k→∞
Let us fix 0 > 0 small enough, so that for all ∈ (o, 0 ), we have
1 < min(p − , q − ),
∗
(3.2)
∗
max(p + , q + ) < min((p − ) , (q − ) ).
1,pn
(3.3)
1,qn
For each k ∈ N, let (u(pnk ,qnk ) , v(pnk ,qnk ) ) ∈ W0 k (Ω) × W0 k (Ω) be a principal
eigenfunction of (Spnk ,qnk ) related with λ1 (pnk , qnk ). Then, by Holder’s inequality,
for ∈ (0, 0 ), the following inequalities hold:
pn −p+
k
k∇u(pnk ,qnk ) kp− ≤ k∇u(pnk ,qnk ) kpnk |Ω| pnk (p−) ,
k∇v(pnk ,qnk ) kq− ≤ k∇v(pnk ,qnk ) kqnk |Ω|
qn −q+
k
qn (q−)
k
.
(3.4)
(3.5)
Combining these two inequalities and using the variational characterization of λ1 ,
we have
pn −p+
n p λ (p , q ) o p 1
k
nk
nk 1 nk nk
k∇u(pnk ,qnk ) kp− ≤
|Ω| pnk (p−)
(3.6)
α
qn −q+
n q λ (p , q ) o q 1
k
nk
nk 1 nk nk
k∇v(pnk ,qnk ) kq− ≤
|Ω| qnk (q−) .
(3.7)
β
Therefore, via (3.2) and (3.3), for a subsequence
(u(pnk ,qnk ) , v(pnk ,qnk ) ) * (u, v)
weakly in W01,p− (Ω) × W01,q− (Ω),
(u(pnk ,qnk ) , v(pnk ,qnk ) ) → (u, v)
strongly in Lp+ (Ω) × Lq+ (Ω)
Passing to the limit in (3.6) and (3.7), respectively as k → ∞ and as → ∞, we
have
p
k∇ukpp ≤
lim λ1 (pnk , qnk ) < ∞,
α k→∞
q
k∇vkqq ≤
lim λ1 (pnk , qnk ) < ∞.
β k→∞
Then
u ∈ W01,p− (Ω) ∩ W 1,p (Ω) = W01,p (Ω),
because Ω satisfies the segment property.
v ∈ W01,q− (Ω) ∩ W 1,q (Ω) = W01,q (Ω),
EJDE-2012/08
SIMPLICITY AND STABILITY
5
On the other hand, from the variational characterization of λ1 (pnk , qnk ), (3.4),
(3.5), and using the weak lower semi continuity of the norm; it follows that
1 β
1 α
k∇ukpp− +
k∇vkqq− ≤ lim λ1 (pnk , qnk ).
(3.8)
k→+∞
|Ω| p− p
|Ω| q− q
Letting → 0+ in (3.8), the Fatou lemma yields
β
α
k∇ukpp + k∇vkqq ≤ lim λ1 (pnk , qnk ).
k→∞
p
q
Since Λ(u(pnk ,qnk ) , v(pnk ,qnk ) ) = 1 via compactness of Λ, (u, v) is admissible in the
variational characterization of λ1 (p, q); hence
λ1 (p, q) ≤ lim λ1 (pnk , qnk ) = lim inf λ1 (pn , qn ).
n→∞
k→∞
This and (3.1) will complete the proof. Observe that the segment property is used
only to prove that
λ1 (p, q) ≤ lim inf λ1 (pn , qn ).
n→∞
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Ghasem Alizadeh Afrouzi
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
E-mail address: afrouzi@umz.ac.ir
Maryam Mirzapour
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
E-mail address: mirzapour@stu.umz.ac.ir
6
G. A. AFROUZI, M. MIRZAPOUR, Q. ZHANG
EJDE-2012/08
Qihu Zhang
Department of Mathematics and Information Science, Zhengzhou University of Light
Industry, Zhengzhou, Henan 450002, China
E-mail address: zhangqh1999@yahoo.com.cn