On a Class of Nonuniformly Nonlinear Systems with Dirichlet Boundary Conditions
The existence and multiplicity of weak solutions for some nonuniformly nonlinear elliptic systems are obtained by using the minimum principle and the Mountain-pass theorem.
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| author | Afrouzi, G. A. Naghizadeh, Z. Chung, N. T. Афрузі, Г. А. Нагізадех, З. Чунг, Н. Т. |
| author_facet | Afrouzi, G. A. Naghizadeh, Z. Chung, N. T. Афрузі, Г. А. Нагізадех, З. Чунг, Н. Т. |
| author_sort | Afrouzi, G. A. |
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| datestamp_date | 2019-12-05T10:26:31Z |
| description | The existence and multiplicity of weak solutions for some nonuniformly nonlinear elliptic systems are obtained by using the minimum principle and the Mountain-pass theorem. |
| first_indexed | 2026-03-24T02:20:42Z |
| format | Article |
| fulltext |
UDC 517.9
G. A. Afrouzi, Z. Naghizadeh (Univ. Mazandaran, Babolsar, Iran),
N. T. Chung (Quang Binh Univ., Vietnam)
ON A CLASS OF NONUNIFORMLY NONLINEAR SYSTEMS
WITH DIRICHLET BOUNDARY CONDITIONS
ПРО ОДИН КЛАС НЕОДНОРIДНО НЕЛIНIЙНИХ СИСТЕМ
З ГРАНИЧНИМИ УМОВАМИ ТИПУ ДIРIХЛЕ
The existence and multiplicity of weak solutions for some nonuniformly nonlinear elliptic systems are obtained by using
the minimum principle and the Mountain pass theorem.
Iснування та кратнiсть слабких розв’язкiв деяких нерiвномiрно нелiнiйних елiптичних систем дослiджено за допо-
могою принципу мiнiмуму та теореми про гiрський перевал.
1. Introduction. We study the nonuniformly nonlinear elliptic system
−∆pu− div(h1(|∇u|p)|∇u|p−2∇u) = Fu(x, u, v) in Ω,
−∆qv − div(h2(|∇v|q)|∇v|q−2∇v) = Fv(x, u, v) in Ω,
u = v = 0 in ∂Ω,
(1.1)
where Ω is a bounded smooth open set in RN , −∆pu = div(|∇u|p−2∇u) is the p-Laplacian of u,
2 ≤ p ≤ q and h1, h2 ∈ C+(R,R). If h1(t) = h2(t) = 1 +
t√
1 + t2
, t ≥ 0, then (1.1) is called a
capillarity system. Capillarity can be briefly explained by considering the effects of two opposing
forces: adhesion, i.e., the attractive (or repulsive) force between the molecules of the liquid and those
of the container; and cohesion, i.e., the attractive force between the molecules of the liquid. The study
of capillary phenomena has gained some attention recently. This increasing interest is motivated not
only by fascination in naturally-occurring phenomena such as motion of drops, bubbles, and waves
but also its importance in applied fields ranging from industrial and biomedical and pharmaceutical
to microfluidic systems, see [11, 12].
It should be noticed that the proof of the existence results for nonlinear elliptic systems is a
long-standing question, see [7] and the references therein. To our knowledge, elliptic equations of
(1.1) type has been firstly investigated by J. M. Bezerra do Ó [13], in which the author extended the
existence results by D. G. Costa et al. [4] (for the p-Laplacian) to a more general class of operators.
He also achieved a multiplicity result using Morse theory. On this topic, we refer to recent interesting
papers [5, 6, 8 – 10, 15]. There, the authors have used different methods to prove the existence of a
nontrivial solution or the existence of infinitely many solutions. In [1, 16], the authors studied the
existence of a solution for (1.1) using the minimum principle. The purpose of this note is to deal
with the multiplicity of solutions for system (1.1) by using the minimum principle combined with the
mountain pass theorem. Thus, our result is a natural extension from the previous ones [1, 13, 16].
c© G. A. AFROUZI, Z. NAGHIZADEH, N. T. CHUNG, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9 1155
1156 G. A. AFROUZI, Z. NAGHIZADEH, N. T. CHUNG
Through this paper for (u, v) ∈ R2, denote |(u, v)|2 = |u|2+|v|2.We assume that F : Ω×R2 → R
is of C1 class such that F (x, 0, 0) = 0 for all x ∈ Ω and (Fu, Fv) =
(
∂F
∂u
,
∂F
∂v
)
, Fu and Fv are
Carathéodory functions satisfying the following growth conditions:
(H1) lim|u|→∞
|Fu(x, u, v)|
|u|p−1
= 0, uniformly in (x, v) ∈ Ω × R, lim|v|→∞
|Fv(x, u, v)|
|v|q−1
= 0,
uniformly in (x, v) ∈ Ω× R;
(H2) lim|(u,v)|→0
|F (x, u, v)|
|u|δ+1|v|γ+1
= 0, lim|(u,v)|→∞
|F (x, u, v)|
|u|δ+1|v|γ+1
= ∞, uniformly in x ∈ Ω × R,
where δ, γ ≥ 0,
δ + 1
p
+
γ + 1
q
= 1;
(H3) let h1 and h2 ∈ C+(R,R); we assume that h1 and h2 are the continuous and nondecreasing
functions satisfying the following growth conditions: there exist α1, α2, β1 and β2 ∈ R such that
0 < α1 ≤ h1(t) ≤ β1,
0 < α2 ≤ h2(t) ≤ β2.
The main result of this paper is given by the following theorem:
Theorem 1.1. Suppose that (H1) – (H3) hold. Then system (1.1) has at least two nontrivial
weak solutions.
This paper is organized as follows. In Section 2, we present some notations and relevant lemmas.
We reserve the Section 3 for the proof of the main result.
2. Notations and preliminary lemmas. Let the product space H = H1,p
0 (Ω) ×H1,q
0 (Ω) with
the norm
∥∥(u, v)
∥∥
H
= ‖u‖1,p + ‖v‖1,q =
∫
Ω
|∇u|p dx
1
p
+
∫
Ω
|∇u|q dx
1
q
.
Let us define the mappings
J1(u, v) =
1
p
∫
Ω
|∇u|p dx+
1
q
∫
Ω
|∇v|q dx
and J ′1 : H → H∗ by
〈J ′1(u, v), (ξ, η)〉 =
∫
Ω
(|∇u|p−2∇u∇ξ + |∇v|q−2∇v∇η) dx
for any (u, v), (ξ, η) ∈ H.
Let us define the mappings
h(u, v) =
1
p
u∫
0
h1(s) ds+
1
q
v∫
0
h2(s) ds, J2(u, v) =
∫
Ω
h(|∇u|p, |∇v|q) dx
and J ′2 : H → H∗ by
〈J ′2(u, v), (ξ, η)〉 =
∫
Ω
[
h1(|∇u|p)|∇u|p−2∇u∇ξ + h2(|∇v|q)|∇v|q−2∇v∇η
]
dx
for any (u, v), (ξ, η) ∈ H.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
ON A CLASS OF NONUNIFORMLY NONLINEAR SYSTEMS . . . 1157
Let us define the mapping
Ŵ (u, v) =
∫
Ω
F (x, u, v) dx
and Ŵ ′ : H → H∗ by
〈Ŵ ′(u, v), (ξ, η)〉 =
∫
Ω
[
Fu(x, u, v)ξ + Fv(x, u, v)η
]
dx
for any (u, v), (ξ, η) ∈ H.
We need certain properties of the functional J = J1 + J2 : H → R defined by
J(u, v) =
1
p
∫
Ω
|∇u|p dx+
1
q
∫
Ω
|∇v|q dx+
1
p
∫
Ω
|∇u|p∫
0
h1(s) ds+
1
q
∫
Ω
|∇v|q∫
0
h2(s) ds
for all (u, v) ∈ H.
Definition 2.1. We say that w = (u, v) is a weak solution of system (1.1) if and only if
〈J ′(u, v), (ξ, η)〉 = 〈Ŵ ′(u, v), (ξ, η)〉
for any (ξ, η) ∈ H.
Definition 2.2. An operator J : H → H∗ verifies the (S+) condition if for any sequence
{(un, vn)} ∈ H such that {(un, vn)}⇀ (u, v) weakly and
lim sup
n→∞
〈
J ′(un, vn), (un − u, vn − v)
〉
≤ 0
we have that {(un, vn)} → (u, v) strongly in H.
Lemma 2.1. The functional J is weakly lower semicontinuous.
Proof. Let (u, v) ∈ H and ε > 0 be fixed. Using the properties of lower semicontinuous function
(see [3], Section I.3), it is enough to prove that there exists δ > 0 such that
J(u, v) ≥ J(u1, v1)− ε, ∀ (u, v) ∈ H : ‖(u, v)− (u1, v1)‖ < δ. (2.1)
Using the hypothesis (H3), it is easy to check that J is convex. Hence, we have
J(u, v) ≥ J(u1, v1) + 〈J ′(u1, v1), (u− u1, v − v1)〉 ∀ (u, v) ∈ H.
Using condition (H3) and Hölder’s inequality we deduce there exists a positive constant c > 0 such
that
J(u, v) ≥ J(u1, v1)−
∫
Ω
|∇u1|p−2|∇u1| |∇u−∇u1| dx−
−
∫
Ω
|∇v1|q−2|∇v1| |∇v −∇v1| dx−
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
1158 G. A. AFROUZI, Z. NAGHIZADEH, N. T. CHUNG
−
∫
Ω
|h1(|∇u1|p)| |∇u1|p−2|∇u1| |∇u−∇u1|dx−
−
∫
Ω
|h2(|∇v1|q)| |∇v1|q−2|∇v1| |∇v −∇v1|dx ≥
≥ J(u1, v1)− (β1 + 1)‖u1‖p−1
1,p ‖u− u1‖1,p − (β2 + 1)‖v1‖q−1
1,q ‖v − v1‖1,q ≥
≥ J(u1, v1)− c‖(u− u1, v − v1)‖H ∀ (u, v) ∈ H.
It is clear that taking δ =
ε
c
relation (2.1) holds true for all (u1, v1) ∈ H with ‖(u, v)− (u1, v1)‖H <
< δ. Thus we proved that J is strongly lower semicontinous. Taking into account the fact that J is
convex then by [2] (Corollary III.8) we conclude that J is weakly lower semicontinous.
Lemma 2.1 is proved.
Lemma 2.2. The functional Ŵ is weakly continuous.
Proof. Let {wn} = {(un, vn)} be a sequence that converges weakly to w = (u, v) in H. We
will show that
lim
n→∞
∫
Ω
F (x, un, vn) dx =
∫
Ω
F (x, u, v) dx. (2.2)
From (H1) and the continuity of the potential F, for any ε > 0, there exists a positive constant
M = M(ε) such that∣∣Fu(x, u, v)
∣∣ ≤ ε|u|p−1 +Mε,
∣∣Fv(x, u, v)
∣∣ ≤ ε|v|q−1 +Mε (2.3)
for all (x, u, v) ∈ Ω× R2. Hence,∫
Ω
[
F (x, un, vn)− F (x, u, v)
]
dx =
=
∫
Ω
∇F (x,w + θn(wn − w))(wn − w) dx =
=
∫
Ω
Fu(x, u+ θ1,n(un − u), v + θ2,n(vn − v)) (un − u) dx+
+
∫
Ω
Fv(x, u+ θ1,n(un − u), v + θ2,n(vn − v)) (vn − v) dx,
where θn = (θ1,n, θ2,n) and 0 ≤ θ1,n(x), θ2,n(x) ≤ 1 for all x ∈ Ω. Now, using (2.3) and Hölder’s
inequality we conclude that ∣∣∣∣∣∣
∫
Ω
[F (x, un, vn)− F (x, u, v)] dx
∣∣∣∣∣∣ ≤
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
ON A CLASS OF NONUNIFORMLY NONLINEAR SYSTEMS . . . 1159
≤
∫
Ω
∣∣Fu(x, u+ θ1,n(un − u), v + θ2,n(vn − v)
∣∣ |un − u| dx+
+
∫
Ω
∣∣Fv(x, u+ θ1,n(un − u), v + θ2,n(vn − v)
∣∣ |vn − v| dx ≤
≤
∫
Ω
(
ε |u+ θ1,n(un − u)|p−1 +Mε
)
|un − u| dx+
+
∫
Ω
(
ε| v + θ1,n(vn − v)|q−1 +Mε
)
|vn − v| dx ≤
≤Mε|Ω|
p−1
p ‖un − u‖Lp(Ω) + ε‖u+ θ1,n(un − u)‖p−1
Lp(Ω) ‖un − u‖Lp(Ω)+
+Mε|Ω|
q−1
q ‖vn − v‖Lq(Ω) + ε‖v + θ2,n(vn − v)‖q−1
Lq(Ω) ‖vn − v‖Lq(Ω). (2.4)
On the other hand, since H ↪→ Li(Ω) × Lj(Ω) is compact for all i ∈ [p, p∗) and j ∈ [p, p∗) the
sequence {wn} converges to w = (u, v) in the space Lp(Ω)× Lq(Ω), i.e., {un} converges strongly
to u in Lp(Ω) and {vn} converges strongly to v in Lq(Ω). Hence, it is easy to see that the sequences{
‖u+ θ1,n(un − u)‖Lp(Ω)} and {‖v+ θ2,n(vn − v)‖Lq(Ω)
}
are bounded. Thus, it follows from (2.4)
that relation (2.2) holds true.
Lemma 2.2 is proved.
Lemma 2.3. The functional J ′ : H → H∗ verifies the (S+) condition.
Proof. Assume that (un, vn) ⇀ (u, v) in H and
lim sup
n→∞
〈
J ′(un, vn), (un − u, vn − v)
〉
≤ 0. (2.5)
Since {(un, vn)} is weakly convergent to (u, v) in H it follows that {(un, vn)} is bounded in H. By
the condition (H3) we have
J(un, vn) ≤ 1
p
(β1 + 1)
∫
Ω
|∇un|p dx+
1
q
(β2 + 1)
∫
Ω
|∇vn|q dx.
So {J(un, vn)} is bounded. Then we may assume that J{(un, vn)} → α. Using Lemma 2.1, we
find
J(u, v) ≤ lim inf
n→∞
J(un, vn) = α.
Since J is convex the following inequality holds true:
J(u, v) ≥ J(un, vn) +
〈
J ′(un, vn), (u− un, v − vn)
〉
for all n. (2.6)
Using (2.5), (2.6), we have
J(u, v)− lim sup
n→∞
J(un, vn) = lim inf
n→∞
(J(u, v)− J(un, vn)) ≥
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
1160 G. A. AFROUZI, Z. NAGHIZADEH, N. T. CHUNG
≥ lim inf
n→∞
〈J ′(un, vn), (u− un, v − vn)〉 =
= − lim sup
n→∞
〈J ′(un, vn), (un − u, vn − v)〉 ≥ 0,
which implies that J(u, v) ≥ α and thus J(u, v) = α.
We also have
(un + u
2
,
vn + v
2
)
converges weakly to (u, v) in H. Using again Lemma 2.1 we
deduce that
α = J(u, v) ≤ lim inf
n→∞
J
(
un + u
2
,
vn + v
2
)
. (2.7)
If we suppose that {(un, vn)} does not converges to (u, v), then there exists ε > 0 and a subsequence
{(unk
, vnk
)} such that ‖(unk
− u, vnk
− v)‖H = ‖unk
− u‖
H1,p
0
+ ‖vnk
− v‖
H1,q
0
≥ ε.
If ‖unk
− u‖
H1,p
0
≥ ε
2
we know that T1(u) =
1
p
∫
Ω
|∇u|p dx is p-uniformly convex, i.e., there
exists a positive constant k1 > 0 such that
T1
(
u+ v
2
)
≤ 1
2
T1(u) +
1
2
T1(v)− k1‖u− v‖p
H1,p
0
.
Hence, we have
1
2p
∫
Ω
|∇u|p dx+
1
2p
∫
Ω
|∇unk
|p dx− 1
p
∫
Ω
∣∣∣∣∇u+∇unk
2
∣∣∣∣p dx ≥ k1
∫
Ω
|∇u−∇unk
|p dx.
That fact and convexity of J2, ‖ · ‖H1,q
0
imply
1
2
J(u, v) +
1
2
J(unk
, vnk
)− J
(
unk
+ u
2
,
vnk
+ v
2
)
=
=
1
2p
∫
Ω
|∇u|p dx+
1
2p
∫
Ω
|∇unk
|p dx− 1
p
∫
Ω
∣∣∣∣∇u+∇unk
2
∣∣∣∣p dx+
+
1
2q
∫
Ω
|∇v|q dx+
1
2q
∫
Ω
|∇vnk
|q dx− 1
q
∫
Ω
∣∣∣∣∇v +∇vnk
2
∣∣∣∣q dx+
+
1
2
J2(u, v) +
1
2
J2(unk
, vnk
)− J2
(
unk
+ u
2
,
vnk
+ v
2
)
≥
≥ k1
∫
Ω
|∇u−∇unk
|p dx ≥ k1
( ε
2
)p
.
Letting k →∞ we find
lim sup
n→∞
J
(
unk
+ u
2
,
vnk
+ v
2
)
≤ α− k1
( ε
2
)p
which contradicts (2.7).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
ON A CLASS OF NONUNIFORMLY NONLINEAR SYSTEMS . . . 1161
If ‖vnk
− v‖
H1,q
0
≥ ε
2
we know that T2(v) =
1
q
∫
Ω
|∇v|q dx is q-uniformly convex. That fact
and convexity of J2, ‖ · ‖H1,p
0
imply that
lim sup
n→∞
J
(
unk
+ u
2
,
vnk
+ v
2
)
≤ α− k1
( ε
2
)q
,
which contradicts (2.7).
Similarly if ‖un‖H1,p
0
≥ ε
2
and ‖vn‖H1,q
0
≥ ε
2
we obtain contradictions.
Lemma 2.3 is proved.
In our proof, we use the mountain pass theorem stated in [2]. For the reader’s convenience, we
recall it as follows.
Definition 2.3. Let (X, ‖ · ‖) be a real Banach space, J ∈ C1(X,R). We say that J satisfies the
(PS)c condition if any sequence {um} ⊂ X such that J(um)→ c and J ′(um)→ 0 as m→∞ has
a convergent subsequence.
Proposition 2.1 (see [2]). Let (X, ‖ · ‖) be a real Banach space, J ∈ C1(X,R) satisfies the
(PS)c condition for any c > 0, J(0) = 0 and the following conditions hold:
(i) There exists a function φ ∈ X such that ‖φ‖ > ρ and J(φ) < 0.
(ii) There exist two positive constants ρ and R such that J(u) ≥ R for any u ∈ X with ‖u‖ = ρ.
Then the functional J has a critical value c ≥ R, i.e., there exists u ∈ X such that J ′(u) = 0 and
J(u) = c.
3. Proof of the main theorem. In this section we give the proof of Theorem 1.1. Let
J(u, v) =
∫
Ω
h(|∇u|p, |∇v|q) dx as in Section 2, and let the energy E : H → R given by
E(u, v) = J(u, v)−
∫
Ω
F (x, u, v) dx
for any (u, v) ∈ H. Then weak solutions of system (1.1) are exactly the critical points of E(u, v) in
H. Lemmas 2.1 and 2.2 imply that E is weakly lower semicontinous.
By Hölder’s inequality, (2.4), we have
F (x, u, v) =
u∫
0
∂F
∂s
(x, s, v) ds+ F (x, 0, v) =
=
u∫
0
∂F
∂s
(x, s, v) ds+
v∫
0
∂F
∂s
(x, 0, s) ds+ F (x, 0, 0) ≤
≤
u∫
0
(ε|u|p−1 +Mε) ds+
v∫
0
(ε|v|q−1 +Mε) ds =
=
ε
p
|u|p +Mεu+
ε
p
|v|q +Mεv,
so
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
1162 G. A. AFROUZI, Z. NAGHIZADEH, N. T. CHUNG∣∣∣∣∣∣
∫
Ω
F (x, u, v) dx
∣∣∣∣∣∣ ≤
∫
Ω
|F (x, u, v)| dx ≤
≤ ε
1
p
∫
Ω
|u|p dx+
1
q
∫
Ω
|v|q dx
+Mε
∫
Ω
u dx+
∫
Ω
v dx
≤
≤ ε
p
Sp1
∫
Ω
|∇u|p dx+
ε
q
Sq2
∫
Ω
|∇v|q dx+
+Mε|Ω|
p−1
p S1
∫
Ω
|∇u|p dx
1
p
+Mε|Ω|
q−1
q S2
∫
Ω
|∇v|q dx
1
q
≤
≤ ε
p
Sp1 ‖u‖1,p +
ε
q
Sq2 ‖v‖
q
1,q +A (‖u‖1,p + ‖v‖1,q) ,
where S1, S2 are the embedding constants of H1,p
0 (Ω) ↪→ Lp(Ω), H1,q
0 (Ω) ↪→ Lq(Ω) and A =
= max
{
Mε|Ω|
p−1
p S1,Mε|Ω|
q−1
q S2
}
.
Hence
E(u, v) ≥ 1
p
(α1 + 1− εSp1)
∫
Ω
|∇u|p dx+
1
q
(α2 + 1− εSq2)
∫
Ω
|∇v|q dx−A‖(u, v)‖H .
Letting ε =
1
2
min
{
α1 + 1
Sp1
,
α2 + 1
Sq2
}
. Note that
∫
Ω
|∇u|p dx+
∫
Ω
|∇v|q dx ≥ 1
2p
∫
Ω
|∇u|p dx
1
p
+
∫
Ω
|∇v|q dx
1
q
p
− 1.
Since p ≤ q, we obtain that
E(u, v) ≥ 1
2q
min{α1 + 1, α2 + 1} ×
[
1
2p
‖(u, v)‖pH − 1
]
−A‖(u, v)‖H .
It follows that E is coercive in H. By (H1), (H3), E is continuously differentiable on H and〈
E′(u, v), (ε, η)
〉
=
=
∫
Ω
[
h1(|∇u|p)|∇u|p−2∇u∇ξ + h2(|∇v|q)|∇u|q−2∇v∇η − Fu(x, u, v)ξ − Fv(x, u, v)η
]
dx =
=
〈
J ′(u, v), (ε, η)
〉
−
〈
Ŵ ′(u, v), (ε, η)
〉
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
ON A CLASS OF NONUNIFORMLY NONLINEAR SYSTEMS . . . 1163
for any (u, v) ∈ H. By Lemmas 2.1, 2.2 and the coercivity of E, applying the Minimum principle
(see [14, p. 4], Theorem 1.2), the functional E has a global minimum and system (1.1) admits a weak
solution w1 = (u1, v1) ∈ H. Moreover, we shall prove that E(w1) < 0.
Indeed, let ϕ1 be first eigenfunction of −∆p associated with first eigenvalue λ1 and ψ1 be first
eigenfunction of −∆q associated with first eigenvalue µ1. In view of (H2), we get for M > 0 and
t > 0 sufficiently large,
E
(
t
1
pϕ1, t
1
qψ1
)
≤
≤
λ1
p
(β1 + 1)
∫
Ω
|ϕ1|p dx+
µ1
q
(β2 + 1)
∫
Ω
|ψ1|q dx
t−
∫
Ω
F
(
x, t
1
pϕ1, t
1
qψ1
)
dx <
<
λ1
p
(β1 + 1)
∫
Ω
|ϕ1|p dx+
µ1
q
(β2 + 1)
∫
Ω
|ψ1|q dx
t−Mt
δ+1
p +
γ+1
q
∫
Ω
|ϕ1|δ+1|ψ1|γ+1 dx.
Letting
M =
λ1
p
(β1 + 1)
∫
Ω
|ϕ1|p dx+
µ1
q
(β2 + 1)
∫
Ω
|ψ1|qdx∫
Ω
|ϕ1|δ+1|ψ1|γ+1 dx
we have E(u0, v0) < 0, where w0 = (u0, v0) =
(
t
1
pϕ1, t
1
qψ1
)
. This means that
−∞ < E(w1) = inf{E(u, v) : (u, v) ∈ H} < 0 (3.1)
and w1 6≡ 0.
In the next parts, we shall show the existence of the second weak solution w2 = (u2, v2) ∈ H
(w2 6= w2) of system (1.1) by applying the Mountain Pass theorem in [2]. To this purpose, we first
show that J has the geometry of the Mountain Pass theorem.
It is clear that E(0, 0) = 0. By using Young’s inequality, (H2) we get for ε > 0
E(u, v) ≥ 1
p
(α1 + 1)
∫
Ω
|∇u|p dx+
1
q
(α2 + 1)
∫
Ω
|∇v|qdx−
− εpSp1
(δ + 1)λ1
∫
Ω
|∇u|p dx− εqSq2
(γ + 1)µ1
∫
Ω
|∇v|qdx =
=
(
1
p
(α1 + 1)− εpSp1
(δ + 1)λ1
)∫
Ω
|∇u|pdx+
(
1
q
(α2 + 1)− εqSq2
(γ + 1)µ1
)∫
Ω
|∇v|qdx.
Letting 0 < ε <
1
2
min
{
(α1 + 1)(δ + 1)
p2Sp1
,
µ1(α2 + 1)(γ + 1)
q2Sq2
}
. Hence there exists r > 0 small
enough and such that
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
1164 G. A. AFROUZI, Z. NAGHIZADEH, N. T. CHUNG
inf
‖(u,v)‖=r
E(u, v) > 0 = E(u, v).
On the other hand, by (3.1) there exists t > 0 (large enough) that for w0 =
(
t
1
pϕ1, t
1
qψ1
)
∈ H we
have both ‖w0‖H > r and E(w0) < 0.
In order to verify the (PS)c condition we proceed as follows. Let {(un, vn)} ∈ H be a sequence
satisfying
E(un, vn)→ c,
∥∥E′(un, vn)
∥∥
H∗ → 0.
Since E is coercive, it follows that the sequence {(un, vn)} is bounded in H. Up to a subsequence,
(un, vn) ⇀ (u, v) weakly in H. From E′ = J ′ − Ŵ ′ we get〈
J ′(un, vn), (un − u, vn − v)
〉
=
〈
E′(un, vn), (un − u, vn − v)
〉
+
+
∫
Ω
[
Fu(x, un, vn)(un − u) + Fv(x, un, vn)(vn − v)
]
dx. (3.2)
Since
∥∥E′(un, vn)
∥∥
H∗ → 0 and
{
(un − u, vn − v)
}
is bounded in H, by the inequality∣∣〈E′(un, vn), (un − u, vn − v)
〉∣∣ ≤ ∥∥E′(un, vn)
∥∥
H∗‖(un − u, vn − v)‖H
it follows that 〈
E′(un, vn), (un − u, vn − v)
〉
→ 0.
By (2.3), (3.2) we get∫
Ω
(
|Fu(x, un, vn)||un − u|+ |Fv(x, un, vn)||vn − v|
)
dx ≤
≤ ε‖un‖p−1
Lp(Ω) ‖un − u‖Lp(Ω) +Mε|Ω|
p−1
p ‖un − u‖Lp(Ω)+
+ε‖vn‖q−1
Lq(Ω) ‖vn − v‖Lq(Ω) +Mε|Ω|
q−1
q ‖vn − v‖Lq(Ω).
Since (un, vn)→ (u, v) strongly in Lp(Ω)× Lq(Ω), we get
lim
n→∞
∫
Ω
(
|Fu(x, un, vn)||un − u|+ |Fv(x, un, vn)||vn − v|
)
dx = 0.
In conclusion, relation (3.2) implies
lim sup
n→∞
〈
J ′(un, vn), (un − u, vn − v)
〉
≤ 0.
Then applying Lemma 2.3 we deduce that {(un, vn)} converges strongly to (u2, v2) in H. Set
c = inf
χ∈Γ
max
w∈χ([0,1])
E(w),
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
ON A CLASS OF NONUNIFORMLY NONLINEAR SYSTEMS . . . 1165
where Γ :=
{
χ ∈ C([0, 1], H) : χ(0) = 0, χ(1) = w0
}
. We know that all assumptions of Proposi-
tion 2.1 are satisfied. Therefore, there exists 0 6≡ w2 ∈ H such that E(w2) = c and 〈E′(w2), (ξ, η)〉 =
= 0 for all (ξ, η) ∈ H or w2 is a weak solution of (1.1). Moreover w2 6= w1 since E(w2) = c > 0 >
> E(w1).
Theorem 1.1 is proved.
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Received 25.07.12,
after revision — 06.05.14
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 9
|
| id | umjimathkievua-article-2206 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:20:42Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/dd/d8c382af827e0e1207cceae98dca0bdd.pdf |
| spelling | umjimathkievua-article-22062019-12-05T10:26:31Z On a Class of Nonuniformly Nonlinear Systems with Dirichlet Boundary Conditions Про один клас неоднорідно нелінійних систем з граничними умовами типу Діріхле Afrouzi, G. A. Naghizadeh, Z. Chung, N. T. Афрузі, Г. А. Нагізадех, З. Чунг, Н. Т. The existence and multiplicity of weak solutions for some nonuniformly nonlinear elliptic systems are obtained by using the minimum principle and the Mountain-pass theorem. Існування та кратність слабких розв'язків деяких нерівномірно нєлінійних еліптичних систем досліджено за допомогою принципу мінімуму та теореми про гірський перевал. Institute of Mathematics, NAS of Ukraine 2014-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2206 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 9 (2014); 1155–1165 Український математичний журнал; Том 66 № 9 (2014); 1155–1165 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2206/1413 https://umj.imath.kiev.ua/index.php/umj/article/view/2206/1414 Copyright (c) 2014 Afrouzi G. A.; Naghizadeh Z.; Chung N. T. |
| spellingShingle | Afrouzi, G. A. Naghizadeh, Z. Chung, N. T. Афрузі, Г. А. Нагізадех, З. Чунг, Н. Т. On a Class of Nonuniformly Nonlinear Systems with Dirichlet Boundary Conditions |
| title | On a Class of Nonuniformly Nonlinear Systems with Dirichlet Boundary Conditions |
| title_alt | Про один клас неоднорідно нелінійних систем з граничними умовами типу Діріхле |
| title_full | On a Class of Nonuniformly Nonlinear Systems with Dirichlet Boundary Conditions |
| title_fullStr | On a Class of Nonuniformly Nonlinear Systems with Dirichlet Boundary Conditions |
| title_full_unstemmed | On a Class of Nonuniformly Nonlinear Systems with Dirichlet Boundary Conditions |
| title_short | On a Class of Nonuniformly Nonlinear Systems with Dirichlet Boundary Conditions |
| title_sort | on a class of nonuniformly nonlinear systems with dirichlet boundary conditions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2206 |
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