Theoretical Mathematics & Applications, vol.2, no.2, 2012, 77-86
ISSN: 1792-9687 (print), 1792-9709 (online)
International Scientific Press, 2012
Existence results for a class of
semilinear elliptic systems
G.A. Afrouzi1 , M. Mirzapour2 and N.B. Zographopoulos3
Abstract
This paper deals with the existence of solutions to a class of semilinear potential elliptic systems of the form


 −div(a(x)∇u) = λFu (x, u, v) in Ω,
−div(b(x)∇v) = λFv (x, u, v) in Ω,


u=v=0
on ∂Ω,
where the domain Ω is a bounded domain, the weights a(x), b(x) are
measurable nonnegative weights and λ is a positive parameter.
Mathematics Subject Classification: 35J60, 35B30, 35B40
Keywords: Degenerate elliptic equations, Variational methods
1
2
3
Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran, e-mail: afrouzi@umz.ac.ir
Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran, e-mail: mirzapour@stu.umz.ac.ir
University of Military Education, Hellenic Army Academy, Athens Greece,
e-mail: nzograp@gmail.com
Article Info: Received : March 25, 2012. Revised : May 1, 2012
Published online : June 30, 2012
78
1
Existence results for semilinear elliptic systems
Introduction
In this paper, we deal with a class of semilinear elliptic systems of the form


 −div(a(x)∇u) = λFu (x, u, v) in Ω,
(1)
−div(b(x)∇v) = λFv (x, u, v) in Ω,


u=v=0
on ∂Ω,
where the domain Ω is a bounded domain in RN (N > 2), the weights a(x), b(x)
are measurable nonnegative weights on Ω, (Fu , Fv ) = ∇F stands for the gradients of F in the variables (u, v) ∈ R2 and λ is a positive parameter.
Recently, many authors have studied the existence of nontrivial solutions
for such problems (see [3, 6, 8, 10,12-14] and their references) because several
physical phenomena related equilibrium of continuous media are modeled by
these elliptic problems (see [5]).
In [7], N.B. Zographopoulos studied a class of degenerate potential semilinear elliptic systems of the form

γ−1
δ+1

 −div(a(x)∇u) = λµ(x)|u| |v| u in Ω,
(2)
−div(b(x)∇v) = λµ(x)|u|γ+1 |v|δ−1 v in Ω,


u=v=0
on ∂Ω,
where λ > 0, γ, δ ≥ 0 and µ(x) may change sign. He proved the existence of
at least one solution for the system (2) under suitable assumption on the data.
In this paper, we consider system (1) and prove under the suitable conditions on nonlinearities Fu and Fv , by using the Minimum principle (see [2,
p. 4, Theorem 1.2]) and the Mountain pass theorem of A. Ambrosetti and
Robinowitz [4], the system (1) has at least two nontrivial solutions.
Throughout this work, we assume the weights a, b ∈ L1loc (Ω),
a−s , b−s ∈ L1 (Ω), s ∈ ( N2 , ∞) ∩ [1, ∞). With the number s we define
2s =
N 2s
N 2s
2s
, 2∗s =
> 2.
=
s+1
N − 2s
N (s + 1) − 2s
We define the Hilbert spaces W01,2 (Ω, a) and W01,2 (Ω, b) as the clusures of
C0∞ (Ω) with respect to the norms
Z
2
kuka =
a(x)|∇u|2 dx
for all u ∈ C0∞ (Ω),
Ω
79
G.A. Afrouzi, M. Mirzapour and N.B. Zographopoulos
kvk2b
Set W =
the norm
W01,2 (Ω, a)
=
Z
b(x)|∇v|2 dx
for all
Ω
v ∈ C0∞ (Ω).
× W01,2 (Ω, b). It is clear that W is a Hilbert space under
k(u, v)kW = kuka + kvkb
for all
(u, v) ∈ W,
and with respect to the scalar product
Z
hϕ, ψiW = (a(x)∇ϕ1 ∇ψ1 + b(x)∇ϕ2 ∇ψ2 )dx
Ω
for all φ = (ϕ1 , ϕ2 ), ψ = (ψ1 , ψ2 ) ∈ W.
Then W is a uniformly convex space. Moreover, the continuous embedding
W ,→ (W 1,2s )2
2s
(cf. Example1.3, [15]) and we have the Sobolev’s ems+
1
2∗s
bedding W ,→ (L (Ω))2 . We notice that the compact embedding
holds with 2s =
W ,→ Lr (Ω) × Lt (Ω)
holds provided that 1 ≤ r, t < 2∗s .
Next, we assume that F (x, t, s) is a C 1 −functional on Ω×[0, ∞)×[0, ∞) →
R, satisfying the hypotheses below:
(F1 ) There exist positive constant c1 , c2 such that
|Ft (x, t, s)| ≤ c1 tγ sδ+1
,
|Fs (x, t, s)| ≤ c2 tγ+1 sδ
for all (t, s) ∈ R2 , a.e. x ∈ Ω and some γ, δ > 1 with
γ + 1 < p < 2∗s , δ + 1 < q < 2∗s .
γ+1
p
+
δ+1
q
(F2 ) There exist positive constant c and 2 < α, β < 2∗s such that
|F (x, t, s)| ≤ c(1 + |t|α + |s|β ).
(F3 ) There exist R > 0, θ and θ0 with
1
2∗s
< θ, θ0 <
1
2
such that
0 < F (x, t, s) ≤ θtFt (x, t, s) + θ0 sFs (x, t, s).
for all x ∈ Ω and |t|, |s| ≥ R.
= 1 and
80
Existence results for semilinear elliptic systems
(F4 ) There exist α > 2, β > 2 and > 0 such that
|F (x, t, s)| ≤ c(|t|α + |s|β )
for all x ∈ Ω and |t|, |s| ≤ .
Definition 1.1. We say that (u, v) ∈ W is a weak solution of system (1) if
and only if
Z
Z
(a(x)∇u∇ϕ + b(x)∇v∇ψ)dx = λ (Fu (x, u, v)ϕ + Fv (x, u, v)ψ)dx,
Ω
Ω
for all (ϕ, ψ) ∈ W .
The functional corresponding to problem (1) is
Z
Z
1
2
2
(a(x)|∇u| + b(x)|∇v| )dx − λ F (x, u, v)dx.
Iλ (u, v) =
2 Ω
Ω
(3)
It is easy to see that the functional I(u, v) is well defined and is of class C 1 in
W. Thus, weak solutions of (1) are exactly the critical points of the functional
Iλ .
Now, we can describe our main results as follows.
Theorem 1.2. Suppose that the condition (F1 ) is satisfied. Then there
exists a constant λ > 0 such that for all 0 ≤ λ < λ, system (1) has a weak
solution.
Theorem 1.3. In addition suppose that the condition (F1 ) − (F4 ) are satisfied. Then problem (1) has a nontrivial solution.
2
Proof of Theorem 1.2
Lemma 2.1. The functional Iλ given by (3) is weakly lower semicontinous
in W .
G.A. Afrouzi, M. Mirzapour and N.B. Zographopoulos
81
Proof. Let {(um , vm )} be a sequence that converges weakly to (u, v) in W .
By the weak lower semicontinuity of the norms in the spaces W01,2 (Ω, a) and
W01,2 (Ω, b) we deduce that
Z h
Z h
i
i
2
2
2
2
lim inf
a(x)|∇u| + b(x)|∇v| dx. (4)
a(x)|∇um | + b(x)|∇vm | dx ≥
m→∞
Ω
Ω
We shall show that
lim
m→∞
Z
F (x, um , vm )dx =
Ω
Z
F (x, u, v)dx.
(5)
Ω
Indeed, we have
Z h
i F (x, um , vm ) − F (x, u, v) dx
Ω
Z F
(x,
u
+
θ
(u
−
u),
v
+
θ
(v
−
v))
≤
|um − u|dx
u
1,m m
2,m m
Ω
Z + Fv (x, u + θ1,m (um − u), v + θ2,m (vm − v))|vm − v|dx
ZΩ
≤ c1 |u + θ1,m (um − u)|γ |v + θ2,m (vm − v)|δ+1 |um − u|dx
Ω
Z
+c2 |u + θ1,m (um − u)|γ+1 |v + θ2,m (vm − v)|δ |vm − v|dx
Ω
≤ c1 ku + θ1,m (um − u)kγLp kv + θ2,m (vm − v)kδ+1
Lq kum − ukLp
δ
+c2 ku + θ1,m (um − u)kγ+1
Lp kv + θ2,m (vm − v)kLq kvm − vkLq .
(6)
Since 2 < γ + 1 < p < 2∗s and 2 < δ + 1 < q < 2∗s , the sequence {(um , vm )}
converges strongly to (u, v) in the space Lp (Ω) × Lq (Ω). It is easy to see that
ku + θ1,m (um − u)kLp
and
kv + θ2,m (vm − v)kLq
are bounded. Thus, it follows from (6) that relation (5) holds true. Then we
have
Iλ (u, v) ≤ lim inf Iλ (um , vm ).
m→∞
82
Existence results for semilinear elliptic systems
Lemma 2.2. The functional Iλ given by (3) is coercive and bounded below
in W .
Proof. By (F1 ), there exists c3 > 0 such that for all (t, s) ∈ R2 and a.e.
x ∈ Ω we have
|F (x, t, s)| ≤ c3 |t|γ+1 |s|δ+1 .
Using Holder’s and Young’s inequalities, we obtain
Z
Z
F (x, u, v)dx ≤ c3 |u|γ+1 |v|δ+1 dx
Ω
Ω
Z
γ + 1 Z
δ+1
p
q
|u| dx +
|v| dx
≤ c3
p
q
Ω
Ω
Z
Z
γ + 1
δ+1 0
2
≤ c3
s a(x)|∇u| dx +
s
b(x)|∇v|2 dx
p
q
Ω
Ω
where s and s0 are the imbedding constants of W01,2 (Ω, a) ,→ Lp (Ω) and
W01,2 (Ω, b) ,→ Lq (Ω), respectively. Then we can write
1
γ + 1
δ + 1
2
− λc
− λc
kuka +
kvk2b ,
Iλ (u, v) ≥
2
p
2
q
n
o
p
q
where c = max{c3 s, c3 s0 }. Let λ = min 2c(γ+1)
> 0, then for all
, 2c(δ+1)
0 ≤ λ < λ we conclude that Iλ (u, v) → ∞, provided that k(u, v)k → ∞.
1
By Lemmas (2.1) and (2.2), applying the Minimum principle, the functional
Iλ attains its minimum, and thus system (1) admits at least one weak solution.
3
Proof of Theorem 1.3
Lemma 3.1. The functional Iλ given by (3) satisfies the Palais-Smale condition in W .
Proof. Let {(um , vm )} be a Palais-Smale sequence for the functional Iλ ,
thus there exists c4 > 0 such that
|Iλ (um , vm )| ≤ c4
for any
m∈N
(7)
83
G.A. Afrouzi, M. Mirzapour and N.B. Zographopoulos
and there exists a strictly decreasing sequence {m }∞
m=1 , limm→∞ m = 0, such
that
|hIλ0 (um , vm ), (ξ, η)i| ≤ m k(ξ, η)k,
(8)
for any m ∈ N and for any (ξ, η) ∈ W .
By Lemma (2.2), we deduce that Iλ is coercive, relation (7) implies that
the sequence {(um , vm )} is bounded in W . Since W is a Hilbert space, there
exists (u, v) ∈ W such that, passing to subsequence, still denote by {(um , vm )},
it converges weakly to (u, v) in W and strongly in Lp (Ω) × Lq (Ω). Choosing
(ξ, η) = (um − u, 0) in (7), we have
Z
Z
a(x)|∇um |∇(um − u) − λ Fu (x, um , vm )(um − u) ≤ m kum − uk. (9)
Ω
Ω
Using the condition (F1 ) combined with Holder’s inequality we conclude that
Z
Z
Fu (x, um , vm )|um − u|dx ≤ c1 |um |γ |vm |δ+1 |um − u|dx
Ω
Ω
≤ c1 kum kγLp kvm kδ+1
Lq kum − ukLp .
It follows from relations (9) and (10) that
Z
lim
a(x)|∇um |∇(um − u)dx = 0
m→∞
subtracting
Z
Ω
a(x)|∇u|(∇um − ∇u)dx,
Ω
we obtain
0 = lim
m→∞
Z
a(x)(|∇um | − |∇u|)(∇um − ∇u)dx ≥ lim (kum ka − kuka )2 ≥ 0
Ω
m→∞
which implies that kum ka → kuka . The uniform convexity of W01,2 (Ω, a) yields
that um converges strongly to u in W01,2 (Ω, a).
Similarly, we obtain vm → v in W01,2 (Ω, b) as n → ∞.
By Lemma (3.1), we obtain that the functional Iλ satisfies (PS)-condition
(compactness condition). Now we verify that the functional Iλ has the geometry of the Mountain pass theorem.
Lemma 3.2. Under assumption (F1 ) − (F4 ), the functional Iλ satisfies
84
Existence results for semilinear elliptic systems
(i) There exists ρ, σ > 0 such that k(u, v)kH = ρ implies I(u, v) ≥ σ > 0.
(ii) There exists (z1 , z2 ) ∈ W such that k(z1 , z2 )kH > ρ and I(z1 , z2 ) ≤ 0.
Proof. (i) From (F2 ) and (F4 ), one obtains
|F (x, u, v)| ≤ c(|u|α + |v|β + |u|α + |v|β )
for all x ∈ Ω and (u, v) ∈ R2 where 2 < α, α, β, β < 2∗s . By Sobolev embedding
we obtain
Z
F (x, u, v)dx ≤ c(kukαa + kvkβb + kukαa + kvkβb ).
Ω
So, we can estimate the functional Iλ (u, v) by
1
Iλ (u, v) ≥ (kuk2a + kvk2b ) − c(kukαa + kvkβb + kukαa + kvkβb )
2
which implies that there exist σ, ρ > 0 such that Iλ (u, v) ≥ σ > 0 for kuka +
kvkb = ρ.
(ii) Using (F3 ), we have
d
0
0
0
0
F (x, tθ u, tθ v) = θuFu (x, tθ u, tθ v)tθ−1 + θ0 vFv (x, tθ u, tθ v)tθ −1
dt
1
0
≥
F (x, tθ u, tθ v)
t
which implies that there exists some function K(x, u, v) such that
0
F (x, tθ u, tθ v) ≥ tK(x, u, v).
(10)
From (11), we obtain
Z
1 2θ
0
2
2θ 0
2
I(t u, t v) =
(t kuka + t kvkb ) − F (x, tθ u, tθ v)dx
2
Ω
Z
1 2θ
0
≤
(t kuk2a + t2θ kvk2b ) − t K(x, u, v)dx.
2
Ω
θ
θ0
Since 2θ, 2θ0 < 1, we conclude that
0
I(tθ u, tθ v) → −∞
as
t → +∞,
0
and thus there exists a constant t0 such that I(tθ0 u, tθ0 v) < 0.
Consequently, the functional Iλ has a nonzero critical point and the nonzero
critical point of Iλ is precisely the nontrivial solution of problem (1).
G.A. Afrouzi, M. Mirzapour and N.B. Zographopoulos
85
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