Hindawi Publishing Corporation
Boundary Value Problems
Volume 2007, Article ID 85621, 5 pages
doi:10.1155/2007/85621
Research Article
Existence and Nonexistence Results for a Class of
Quasilinear Elliptic Systems
Said El Manouni and Kanishka Perera
Received 18 June 2007; Accepted 20 August 2007
Recommended by Donal O’Regan
Using variational methods, we prove the existence and nonexistence of positive solutions
for a class of (p, q)-Laplacian systems with a parameter.
Copyright © 2007 S. El Manouni and K. Perera. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In a recent paper, Perera [1] studied the existence, multiplicity, and nonexistence of positive classical solutions of the p-Laplacian problem
−Δ p u = λ f (x,u) in Ω,
u = 0 on ∂Ω,
(1.1)
where Ω is a smooth bounded domain in Rn , n ≥ 1, Δ p u = div(|∇u| p−2 ∇u) is the pLaplacian of u, 1 < p < ∞, λ > 0 is a parameter, and f is a Carathéodory function on
Ω × [0, ∞) satisfying
f (x,t) ≤ Ct p−1
∀(x,t),
(1.2)
where C denotes a generic positive constant. Assuming
t
( f1 ) ∃δ > 0 such that F(x,t) := 0 f (x,τ)dτ ≤ 0 when t ≤ δ,
( f2 ) ∃t0 > 0 such that F(x,t0 ) > 0,
( f3 ) limsup t→∞ (F(x,t)/t p ) ≤ 0 uniformly in x
and using variational methods, the author proved that there are λ < λ such that (1.1)
has no positive solution for λ < λ and at least two positive solutions u1 > u2 for λ ≥ λ. A
similar result for the semilinear case p = 2 was proved by Maya and Shivaji [2].
2
Boundary Value Problems
In the present paper we consider the corresponding (p, q)-Laplacian system
−Δ p u = λFu (x,u,v) in Ω,
−Δq v = λFv (x,u,v) in Ω,
u = v = 0 on ∂Ω,
(1.3)
where 1 < p, q < ∞ and F is a C 1 -function on Ω × [0, ∞) × [0, ∞) satisfying
Ft (x,t,s) ≤ Ct α sβ+1 ,
Fs (x,t,s) ≤ Ct α+1 sβ
∀(x,t,s)
(1.4)
for some α,β > 0 with (α + 1)/ p + (β + 1)/q = 1. We will extend the results of Perera [1] to
this system as follows.
Theorem 1.1. There is a λ such that (1.3) has no positive solution for λ < λ.
Theorem 1.2. Assume
(F1 ) ∃δ > 0 such that F(x,t,s) ≤ 0 when t p + sq ≤ δ;
(F2 ) ∃t0 ,s0 > 0 such that F(x,t0 ,s0 ) > 0;
(F3 ) limsup |(t,s)|→∞ (F(x,t,s)/t α+1 sβ+1 ) ≤ 0 uniformly in x.
t,s>0
Then there is a λ such that (1.3) has at least two positive solutions for λ ≥ λ.
2. Proofs of Theorems 1.1 and 1.2
The first eigenvalue of the problem
−Δ p u = λ|u|α−1 u|v |β+1
−Δq v = λ|u|
α+1
u=v=0
β −1
|v |
in Ω,
in Ω,
v
(2.1)
on ∂Ω,
where α,β > 0 with (α + 1)/ p + (β + 1)/q = 1 is positive and is given by
λ1 = inf
α+1
∇u p + β + 1 ∇v q : (u,v) ∈ W 1,p (Ω) × W 1,q (Ω),
0
0
p
q
Ω
Ω
α+1 β+1
u v =1
(2.2)
(see de Thélin [3]). If (1.3) has a positive solution (u,v), testing the two equations in (1.3)
by u and v, respectively, and using (1.4) give
Ω
p
∇u = λ
Ω
q
∇v = λ
Ω
Ω
Fu (x,u,v)u ≤ λC
Fv (x,u,v)v ≤ λC
Ω
Ω
|u|α+1 |v |β+1 ,
(2.3)
|u|α+1 |v |β+1 ,
so
α+1
∇u p + β + 1 ∇v q ≤ λC
q
Ω p
Ω
|u|α+1 |v |β+1
(2.4)
S. El Manouni and K. Perera 3
and hence λ ≥ λ1 /C by (2.2), proving Theorem 1.1.
To prove Theorem 1.2, set F(x,t,s) = 0 if t < 0 or s < 0, and consider the C 1 -functional
Φλ (u,v) =
1,p
1
∇u p + 1 ∇v q − λF(x,u,v)
p
q
Ω
(2.5)
1,q
on the space W0 (Ω) × W0 (Ω) with the norm
(u,v) = u
1 + v 2 ,
(2.6)
where
u 1 =
Ω
p
∇u 1/ p
,
v 2 =
Ω
q
∇v 1/q
.
(2.7)
If (u,v) is a critical point of Φλ , denoting by u− and v− the negative parts of u and v,
respectively, we have
0 = Φλ (u,v),(u− ,v− )
=
Ω
p −2
∇u ∇u·∇u− + ∇v q−2 ∇v ·∇v −
− λ Fu (x,u,v)u− + Fv (x,u,v)v
−
(2.8)
p
q
= u − 1 + v − 2 ,
so u,v ≥ 0. Furthermore, u,v ∈ L∞ (Ω) ∩ C 1 (Ω) by Anane [4] and DiBenedetto [5], so it
follows from the Harnack inequality that either u,v > 0 or u,v ≡ 0 (see Trudinger [6]).
Thus, nontrivial critical points of Φλ are positive solutions of (1.3).
By (1.4),
F(x,t,s) ≤ C |t |α+1 |s|β+1
∀(x,t,s) ∈ Ω × R × R.
(2.9)
Let γ = 1/(max {α,β} + 1). By (F3 ), there is an Mλ > 0 such that
(t,s) ≥ Mλ =⇒ F(x,t,s) ≤ γλ1 |t |α+1 |s|β+1 .
(2.10)
2λ
Combining (2.9) and (2.10) gives
λF(x,t,s) ≤
γλ1 α+1 β+1
|t | |s| + Cλ
2
∀(x,t,s)
(2.11)
for some Cλ > 0. Hence,
Φλ (u,v) ≥
Ω
γ
α+1
∇u p + β + 1 ∇v q − λ1 |u|α+1 |v |β+1 − Cλ
p
q
2
p
q
≥ δ u
1 + v 2 − Cλ μ(Ω),
(2.12)
where δ = min {(α + 1)/ p,(β + 1)/q}γ/2 and μ denotes the Lebesgue measure in Rn . So
Φλ is bounded from below and coercive. This yields a global minimizer (u1 ,v1 ) since Φλ
is weakly lower semicontinuous.
4
Boundary Value Problems
Lemma 2.1. There is a λ such that inf Φλ < 0, and hence (u1 ,v1 )=(0,0), for λ ≥ λ.
1,p
Proof. Taking a sufficiently large compact subset Ω of Ω and (u0 ,v0 ) ∈ W0 (Ω)
1,q
×W0 (Ω) such that u0 = t0 , v0 = s0 on Ω and 0 ≤ u0 ≤ t0 ,0 ≤ v0 ≤ s0 on Ω\Ω , where t0 ,
s0 are as in (F2 ), we have
Ω
F x,u0 ,v0 ≥
Ω
β+1
F x,t0 ,s0 − Ct0α+1 s0 μ(Ω\Ω ) > 0,
so Φλ (u0 ,v0 ) < 0 for λ large enough.
(2.13)
Now we fix λ ≥ λ and obtain a critical point (u2 ,v2 ) with Φλ (u2 ,v2 ) > 0 via the mountain pass lemma, which will complete the proof since Φλ (0,0) = 0 > Φλ (u1 ,v1 ).
Lemma 2.2. The origin is a strict local minimizer of Φλ .
Proof. Set Ωu,v = {x ∈ Ω : |u(x)| p + |v(x)|q > δ }. By (F1 ), F(x,u,v) ≤ 0 on Ω\Ωu,v and
hence
Φλ (u,v) ≥
1
1
p
q
u 1 + v 2 − λ
p
q
Ωu,v
F(x,u,v).
(2.14)
By (2.9), Young’s and Hölder’s inequalities, and the Sobolev imbedding,
Ωu,v
F(x,u,v) ≤ C
α+1
Ωu,v
|u|
≤ C μ Ωu,v
β+1
|v |
1−(p/r)
≤C
p
u 1
Ωu,v
α+1 p β+1 q
|u| +
|v |
p
q
+ μ Ωu,v )
1−(q/s)
q
v 2 ,
(2.15)
where r = np/(n − p) if p < n, r > p if p ≥ n and s = nq/(n − q) if q < n, s > q if q ≥ n.
Since
μ Ωu,v ≤
1
δ
Ωu,v
p
q
|u| p + |v |q ≤ C u
1 + v 2 −→ 0
as (u,v) −→ 0,
(2.16)
the conclusion follows from (2.14) and (2.15).
Since Φλ is coercive, every Palais-Smale sequence is bounded and hence contains a
convergent subsequence as usual. So the mountain pass lemma now gives a critical point
(u2 ,v2 ) of Φλ at the level
c := inf
max
Φλ (u,v) > 0,
γ∈Γ (u,v)∈γ([0,1])
1,p
(2.17)
1,q
where Γ = {γ ∈ C([0,1],W0 (Ω) × W0 (Ω)) : γ(0) = (0,0),γ(1) = (u1 ,v1 )} is the class
of paths joining the origin to (u1 ,v1 ) (see Rabinowitz [7]).
References
[1] K. Perera, “Multiple positive solutions for a class of quasilinear elliptic boundary-value problems,” Electronic Journal of Differential Equations, no. 7, pp. 5, 2003.
[2] C. Maya and R. Shivaji, “Multiple positive solutions for a class of semilinear elliptic boundary
value problems,” Nonlinear Analysis, vol. 38, no. 4, pp. 497–504, 1999.
S. El Manouni and K. Perera 5
[3] F. de Thélin, “Première valeur propre d’un système elliptique non linéaire,” Comptes Rendus de
l’Académie des Sciences, vol. 311, no. 10, pp. 603–606, 1990.
[4] A. Anane, “Simplicité et isolation de la première valeur propre du p-laplacien avec poids,”
Comptes Rendus des Séances de l’Académie des Sciences, vol. 305, no. 16, pp. 725–728, 1987.
[5] E. DiBenedetto, “C 1+α local regularity of weak solutions of degenerate elliptic equations,” Nonlinear Analysis, vol. 7, no. 8, pp. 827–850, 1983.
[6] N. S. Trudinger, “On Harnack type inequalities and their application to quasilinear elliptic equations,” Communications on Pure and Applied Mathematics, vol. 20, no. 4, pp. 721–747, 1967.
[7] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential
Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical
Society, Washington, DC, USA, 1986.
Said El Manouni: Department of Mathematics, Faculty of Sciences, Al-Imam University,
P.O. Box 90950, Riyadh 11623, Saudi Arabia
Email address: samanouni@imamu.edu.sa
Kanishka Perera: Department of Mathematical Sciences, Florida Institute of Technology,
Melbourne, FL 32901, USA
Email address: kperera@fit.edu