2004-Fez conference on Differential Equations and Mechanics
Electronic Journal of Differential Equations, Conference 11, 2004, pp. 129–134.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
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STABILITY OF THE PRINCIPAL EIGENVALUE OF A
NONLINEAR ELLIPTIC SYSTEM
ABDELOUAHED EL KHALIL, MOHAMMED OUANAN,
ABDELFATTAH TOUZANI
Abstract. This paper concerns some special properties of the principal eigenvalue of nonlinear elliptic systems with Dirichlet boundary conditions. We
study the stability with respect to the exponents p and q; and the dependence
on the domain variations.
1. Introduction
In this work, we survey two results concerning the nonlinear elliptic system
−∆p u = λ|u|α |v|β v
α
β
−∆q v = λ|u| |v| u
(u, v) ∈
W01,p (Ω)
×
in Ω
in Ω
(1.1)
W01,q (Ω),
where Ω ⊂ RN , ( N ≥ 3), is a bounded domain not necessary regular; α, β, p, q are
reel numbers such that p > 1, q > 1, α ≥ 0, β ≥ 0 satisfying
α+1 β+1
+
= 1.
p
q
The principal eigenvalue λ1 (p, q) of (1.1) is defined as
λ1 = inf A(u, v) : (u, v) ∈ W01,p (Ω) × W01,q (Ω), B(u, v) = 1 ,
(1.2)
(1.3)
where
A(u, v) =
Z
β+1
|∇u|p dx +
|∇v|q dx,
q
Ω
Ω
Z
B(u, v) =
|u|α |v|β uv dx.
α+1
p
Z
Ω
It is known (see [2]) that this eigenvalue is simple, isolated, and can be expressed
as (1.3). It is also known that the associated principal eigenfunction (u, v) can be
chosen strictly positive in Ω. This eigenfunction is a regular pair of functions where
2000 Mathematics Subject Classification. 35P20, 35J70, 35J20.
Key words and phrases. p-Laplacian operator; principal eigenvalue; stability;
segment property; dependence on the domain.
c
2004
Texas State University - San Marcos.
Published October 15, 2004.
129
130
A. EL KHALIL, M. OUANAN, A. TOUZANI
EJDE/CONF/11
the infimum of the energy functional A is attained on the C 1 -manifold {(u, v) ∈
W01,p (Ω) × W01,q (Ω) : B(u, v) = 1}.
The operator −∆p arises in problems from pure mathematics, such as the theory
of quasiregular and quasiconformal mapping, as well as in problems from a variety of
applications, e.g. non-Newtonian fluids, reaction diffusion problems, flow through
porous media, nonlinear elasticity, glaciology, petroleum extraction, astronomy, etc.
The main purpose of this work is to prove that the mapping (p, q) 7→ λ1 (p, q) is
continuous (stable) on the set
Iα,β = {(p, q) ∈ (]1, +∞[)2 such that (1.2) is satisfied},
for any bounded domain Ω having the segment property. Recall that an open subset
Ω of RN is said to have the segment property if, given any x ∈ ∂Ω, there exist an
open set Gx in RN with x ∈ Gx and yx of RN \ {0} such that, if z ∈ Ω ∩ Gx and
t ∈]0, 1[, then z + tyx ∈ Ω.
This property allows us by a translation to push the support of a function u in
Ω. This class of domains for which the boundary is sufficiently regular to guarantee
that
W 1,t (Ω) ∩1<r<t W01,r (Ω) = W01,t (Ω),
for any t > 1. See [3] for more details about this property.
The difficulty is that as the exponents (p, q) change, the appropriate spaces Lp (Ω)
and Lq (Ω) also change. In the case of equation ∆p u + λ|u|p−2 u = 0, the same result
is studied by Lindqvist [5] but in a regular bounded domain. El Khalil et al. [3]
r
studied the equation ∆p u + λg|u|p−2 u = 0 with g ∈ L∞
loc (Ω) ∩ L (Ω) and Ω having
the segment property. Note that the stability was partially studied in [4].
For the dependence of λ1 upon variations of the domain, instead of the exponents
(p,q)
(p,q)
p and q, we prove that λ1 (Ωj ) → λ1 (Ω) as j → +∞, where Ω = ∪j∈N∗ Ωj .
In the case of only one equation, see [1] for contracting domain, and see [6] for
variations of the domain.
2. Stability
Theorem 2.1. Let Ω be a bounded domain in RN having the segment property.
Then, the function (p, q) 7→ λ1 (p, q) is continuous from Iα,β to R+ . where
Iα,β = {(p, q) ∈ (]1, +∞[)2 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→+∞
Indeed, let (φ, ψ) ∈ C0∞ (Ω) × C0∞ (Ω) such that B(φ, ψ) > 0; hence,
λ1 (pn , qn ) ≤
α+1
pn
pn k∇φkpn
+
β+1
qn
qn k∇ψkqn
B(φ, ψ)
,
since λ1 (pn , qn ) being the infimum. Letting n tend to infinity, we deduce from
Lebesgue’s theorem
lim sup λ1 (pn , qn ) ≤
n→+∞
α+1
p
p k∇φkp
+
β+1
q
q k∇ψkq
B(φ, ψ)
.
(2.1)
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PRINCIPAL EIGENVALUE
131
Then,
lim sup λ1 (pn , qn ) ≤ λ1 (p, q).
(2.2)
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 ε0 > 0 small enough, so that for all ε ∈ (0, ε0 ), we have
1 < min(p − ε, q − ε),
∗
(2.3)
∗
max(p + ε, q + ε) < min((p − ε) , (q − ε) ).
1,pn
(2.4)
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 Hölder’s inequality,
for ε ∈ (0, ε0 ), the following inequalities hold
pn −p+ε
k
k∇u(pnk ,qnk ) kp−ε ≤ k∇u(pnk ,qnk ) kpnk |Ω| pnk (p−ε) ,
k∇v(pnk ,qnk ) kq−ε ≤ k∇v(pnk ,qnk ) kqnk |Ω|
qn −q+ε
k
qn (q−ε)
k
.
(2.5)
(2.6)
Combining these two inequalities and using the variational characterization of λ1 ,
we have
pn −p+ε
k
pnk λ1 (pnk , qnk ) p 1
nk
k∇u(pnk ,qnk ) kp−ε ≤
|Ω| pnk (p−ε)
(2.7)
α+1
qn −q+ε
k
qnk λ1 (pnk , qnk ) q 1
nk
k∇v(pnk ,qnk ) kq−ε ≤
|Ω| qnk (q−ε) .
(2.8)
β+1
Therefore, via (2.3) and (2.4), for a subsequence
(u(pnk ,qnk ) , v(pnk ,qnk ) ) * (u, v) weakly in W01,p−ε (Ω) × W01,q−ε (Ω),
(u(pnk ,qnk ) , v(pnk ,qnk ) ) → (u, v)
strongly in Lp+ε (Ω) × Lq+ε (Ω).
Passing to the limit in (2.7) and (2.8), respectively as k → ∞ and as → 0+ , we
have
p
k∇ukpp ≤
lim λ1 (pnk , qnk ) < +∞,
α + 1 k→+∞
q
k∇vkqq ≤
lim λ1 (pnk , qnk ) < +∞.
β + 1 k→+∞
Then,
u ∈ W01,p−ε (Ω) ∩ W 1,p (Ω) = W01,p (Ω),
v ∈ W01,q−ε (Ω) ∩ W 1,q (Ω) = W01,q (Ω),
because Ω satisfies the segment property.
On the other hand, from the variational characterization of λ1 (pnk , qnk ), (2.5),
(2.6), and using weak lower semi continuity of the norm; it follows that
1 α+1
1 β+1
k∇ukpp−ε +
k∇vkqq−ε ≤ lim λ1 (pnk , qnk ).
ε
ε
p−ε
q−ε
k→+∞
p
q
|Ω|
|Ω|
Letting ε → 0+ in (2.10), the Fatou lemma yields,
α+1
β+1
k∇ukpp +
k∇vkqq ≤ lim λ1 (pnk , qnk ).
k→+∞
p
q
132
A. EL KHALIL, M. OUANAN, A. TOUZANI
EJDE/CONF/11
Since B(u(pnk ,qnk ) , v(pnk ,qnk ) ) = 1 via compactness of B, (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 (2.2) complete the proof.
Remark 2.2. Observe that the segment property is used only for to prove
λ1 (p, q) ≤ lim inf λ1 (pn , qn ).
n→+∞
3. Dependence with variations domain
Theorem 3.1. Let Ω1 ⊂ Ω2 ⊂ Ω3 ⊂ . . . be an exhaustion of Ω such that
[
Ω=
Ωj .
j∈N∗
Then
(p,q)
(p,q)
(Ωj ) = λ1 (Ω),
Z
Z
p
lim
|∇uej − ∇u| dx = lim
|∇vej − ∇v|q dx = 0,
lim λ1
j→+∞
j→+∞
j→+∞
Ω
(3.1)
(3.2)
Ω
(p,q)
where (u, v) is a positive principal eigenfunction of (1.1) related to λ1 (Ω); (uj , vj )
(p,q)
is the principal eigenfunction related to λ1 (Ωj ) and (uej , vej ) is the extended by 0
in Ω\Ωj of (uj , vj ).
Proof. To prove (3.1), we note that
(p,q)
λ1
(p,q)
(Ω1 ) ≥ λ1
(p,q)
(Ω2 ) ≥ · · · ≥ λ1
(Ω).
C0∞ (Ω)
For each ε > 0, there is a pair of function (ϕε , ψε ) ∈
× C0∞ (Ω) such that
Z
|ϕε |α |ψε |β ϕε ψε dx > 0
Ω
R
R
α+1
|∇ϕε |p dx + β+1
|∇ψε |q dx
p
q
(p,q)
Ω
Ω
R
λ1 (Ω) >
− ε,
|ϕ |α |ψε |β ϕε ψε dx
Ω ε
(p,q)
since λ1 (Ω) is the infimum.
The support of ϕε and ψε being compact sets, are covered a finite number of the
sets Ωj ; hence, there exist j0 ∈ N∗ such that
supp ϕε ⊂ Ωj
and
supp ψε ⊂ Ωj
∀j ≥ j0 .
Then,
(p,q)
λ1 (Ωj )
R
|∇ϕε |p dx + β+1
|∇ψε |q dx
q
Ωj
R
≤
|ϕε |α |ψε |β ϕε ψε dx
Ωj
R
R
α+1
|∇ϕε |p dx + β+1
|∇ψε |q dx
p
q
Ω
Ω
R
=
|ϕ |α |ψε |β ϕε ψε dx
Ω ε
α+1
p
R
Ωj
(p,q)
≤ λ1
(Ω) + (p,q)
for all large j. It is clear that λ1
desired result.
(p,q)
(Ω) ≥ limj→+∞ λ1
(Ωj ). which proves the
EJDE/CONF/11
PRINCIPAL EIGENVALUE
133
For the strong convergence (3.2), we proceed as follows: First, we have (uej , vej ) ∈
W01,p (Ω) × W01,q (Ω); uej ≥ 0, vej ≥ 0 a.e in Ω;
Z
|uej |α |vej |β u
ej vej dx = 1,
Ω
because
Z
Ω
|uej |α |vej |β uej vej dx =
Z
|uj |α |vj |β uj vj dx = 1;
Ωj
gj and ∇vej = ∇v
gj a.e. in Ω. Hence, (uej , vej ) is admissible function in
and ∇uej = ∇u
(p,q)
definition of λ1 (Ω). Which implies
Z
Z
α+1
β+1
(p,q)
(p,q)
λ1 (Ω) ≤
|∇uej |p dx +
|∇vej |q dx = λ1 (Ωj ).
p
q
Ω
Ω
Therefore, via (3.1) and for a subsequence,
weakly in W01,p (Ω) × W01,q (Ω),
(uej , vej ) * (ϕ, ψ)
strongly in Lp (Ω) × Lq (Ω),
(uej , vej ) → (u, v)
ϕ ≥ 0, ψ ≥ 0 a.e in Ω
R
Since B is compact, we have Ω ϕα+1 ψ β+1 dx = 1. Then (ϕ, ψ) is admissible func(p,q)
tion in definition of λ1 (Ω). We deduce that
Z
Z
α+1
β+1
(p,q)
p
|∇φ| dx +
|∇ψ|q dx
λ1 (Ω) ≤
p
q
Ω
Ω
Z
α + 1 Z
β+1
p
≤ lim inf
|∇uej | dx +
|∇vej |q dx
j→∞
p
q
Ω
Ω
(p,q)
= lim inf λ1
j→∞
(Ωj ).
Then
(p,q)
λ1 (Ω)
α+1
=
p
Z
β+1
|∇φ| dx +
q
Ω
p
Z
|∇ψ|q dx
Ω
(p,q)
which impliesR (ϕ, ψ) is positive eigenfunction related to λ1 (Ω) that it is simple
(see [2]) and Ω |ϕ|α |ψ|β ϕψdx = 1, hence φ ≡ u and ψ ≡ v . Also, (u, v) is independent of the subsequence. Then {(uej , vej )}j converge to (u, v) least in Lp (Ω) × Lq (Ω).
For the strong convergence (2.3), we use Clarckson inequality and distinguish
three possible cases for (p, q) ∈ Iα,β .
case 1: p ≥ 2 and q ≥ 2. For all j ≥ j0 , we have
Z
Z
∇uej − ∇u p
∇vej − ∇v q
α+1
dx + β + 1
dx
p
2
q
2
Ω
Ω
Z
Z
Z
i
∇uej + ∇u p
α + 1h
1
dx + 1
≤
−
|∇uej |p dx +
|∇u|p dx
p
2
2 Ω
2 Ω
Ω
Z
Z
Z
i
∇vej + ∇v p
β + 1h
1
1
q
+
−
dx +
|∇vej | dx +
|∇v|q dx .
q
2
2 Ω
2 Ω
Ω
134
A. EL KHALIL, M. OUANAN, A. TOUZANI
EJDE/CONF/11
Also, we have
Z
uej + u α+1 vej + v β+1
dx
2
2
Ω
Z
Z
∇uej + ∇u p
∇vej + ∇v q
α+1
dx + β + 1
dx.
≤
p
2
q
2
Ω
Ω
Combining the last inequality and using Lebegue’s Theorem; we obtain
Z
Z
∇uej − ∇u p
∇vej − ∇v q α + 1
β+1
dx ≤ 0,
lim sup
dx +
p
2
q
2
j→+∞
Ω
Ω
(p,q)
λ1
(Ω)
Case 2: 1 < p < 2 and 2 < q or 1 < q < 2 and 2 < p. Using the second
Clarckson inequality, we prove that
Z
p
lim
|∇uej − ∇u| dx = 0.
j→+∞
Ω
Using first Clarckson inequality, we prove that
Z
q
lim
|∇vej − ∇v| dx = 0.
j→+∞
Ω
Which completes the proof.
References
[1] A. Anane, Etude des valeurs propres et de la résonance pour l’opérateur p-Laplacien, thèse
de Doctorat, U.L.B (1987 -1988).
[2] A. El Khalil, M. Ouanan, and A. Touzani; Bifurcation of nonlinear elliptic system from the
first eigenvalue, E. J. Qualitative Theory of Diff. Equ., No. 21. (2003), pp. 1-18.
[3] A. El Khalil, P. Lindqvist, and A. Touzani, On the stability of the eigenvalue of Ap u +
λg(x)|u|p−2 u = 0, with varying p, to appear in Rendiconti di Matematica, Serie VII Volume
24, Roma (2004).
[4] A. El Khalil, S. El Manouni, and M. Ouanan, Simplicity and Stability of the first eigenvalue
of a Nonlinear Elleptic System, submitted in International Journal of Mathematics and
Mathematical Sciences.
[5] P. Lindqvist, On Nonlinear Rayleigh Quotients, Potential Analysis 2, pp. 199-218, (1993).
[6] P. Lindqvist, On a Nonlinear eigenvalue problem, Ber. Univer. Jyväskylä Math. Inst. 68 ,
pp. 33-54, (1995).
Abdelouahed El Khalil
Department of Mathematics and Industrial Genie, Ecole Polytechnic School Montreal,
Montreal, Canada
E-mail address: lkhlil@hotmail.com
Mohammed Ouanan
Department of mathematics, Faculty of sciences Dhar-Mahraz, P.O. Box 1796 Atlas,
Fez 30000, Morocco
E-mail address: m ouanan@hotmail.com
Abdelfattah Touzani
Department of mathematics, Faculty of sciences Dhar-Mahraz, P.O. Box 1796 Atlas,
Fez 30000, Morocco
E-mail address: atouzani@iam.net.ma