Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 431672, 9 pages
http://dx.doi.org/10.1155/2013/431672
Research Article
Existence of Solutions for a Modified Nonlinear
Schrödinger System
Yujuan Jiao1 and Yanli Wang2
1
2
College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou 730124, China
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
Correspondence should be addressed to Yujuan Jiao; jsjyj@xbmu.edu.cn
Received 8 April 2013; Accepted 18 June 2013
Academic Editor: Renat Zhdanov
Copyright © 2013 Y. Jiao and Y. Wang. 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.
We are concerned with the following modified nonlinear Schrödinger system: −Δ𝑢 + 𝑢 − (1/2)𝑢Δ(𝑢2 ) = (2𝛼/(𝛼 + 𝛽))|𝑢|𝛼−2 |V|𝛽 𝑢, 𝑥 ∈
Ω, −ΔV+V−(1/2)VΔ(V2 ) = (2𝛽/(𝛼+𝛽))|𝑢|𝛼 |V|𝛽−2 V, 𝑥 ∈ Ω, 𝑢 = 0, V = 0, 𝑥 ∈ 𝜕Ω, where 𝛼 > 2, 𝛽 > 2, 𝛼+𝛽 < 2⋅2∗ , 2∗ = 2𝑁/(𝑁−2)
is the critical Sobolev exponent, and Ω ⊂ R𝑁 (𝑁 ≥ 3) is a bounded smooth domain. By using the perturbation method, we establish
the existence of both positive and negative solutions for this system.
1. Introduction
Let us consider the following modified nonlinear Schrödinger
system:
2𝛼
1
−Δ𝑢 + 𝑢 − 𝑢Δ (𝑢2 ) =
|𝑢|𝛼−2 |V|𝛽 𝑢,
2
𝛼+𝛽
𝑥 ∈ Ω,
2𝛽
1
−ΔV + V − VΔ (V2 ) =
|𝑢|𝛼 |V|𝛽−2 V,
2
𝛼+𝛽
𝑥 ∈ Ω,
𝑢 = 0,
V = 0,
(1)
−Δ𝑢 + 𝜆𝑉 (𝑥) 𝑢 − 𝑘Δ (𝑢2 ) 𝑢 = |𝑢|𝑝−2 𝑢,
𝑥 ∈ 𝜕Ω,
∗
more naturally in mathematical physics and has been derived
as models of several physical phenomena corresponding to
various types of ℎ. For instance, the case ℎ(𝑠) = 𝑠 was used for
the superfluid film equation in plasma physics by Kurihara [1]
(see also [2]); in the case of ℎ(𝑠) = (1 + 𝑠)1/2 , (2) was used as
a model of the self-changing of a high-power ultrashort laser
in matter (see [3–6] and references therein).
In recent years, much attention has been devoted to the
quasilinear Schrödinger equation of the following form:
∗
where 𝛼 > 2, 𝛽 > 2, 𝛼 + 𝛽 < 2 ⋅ 2 , 2 = 2𝑁/(𝑁 − 2) is the
critical Sobolev exponent, and Ω ⊂ R𝑁 (𝑁 ≥ 3) is a bounded
smooth domain.
Solutions for the system (1) are related to the existence
of the standing wave solutions of the following quasilinear
Schrödinger equation:
𝑖𝜕𝑡 𝑧 = −Δ𝑧 + 𝑉 (𝑥) 𝑧 − 𝑓 (|𝑧|2 ) 𝑧 − 𝑘Δℎ (|𝑧|2 ) ℎ󸀠 (|𝑧|2 ) 𝑧,
𝑥 ∈ R𝑁 ,
(2)
where 𝑉(𝑥) is a given potential, 𝑘 is a real constant, and 𝑓, ℎ
are real functions. We would like to mention that (2) appears
𝑥 ∈ R𝑁 .
(3)
See, for example, by using a constrained minimization argument, the existence of positive ground state solution was
proved by Poppenberg et al. [7]. Using a change of variables,
Liu et al. [8] used an Orlicz space to prove the existence of
soliton solution for (3) via mountain pass theorem. Colin and
Jeanjean [9] also made use of a change of variables but worked
in the Sobolev space 𝐻1 (𝑅𝑁); they proved the existence
of positive solution for (3) from the classical results given
by Berestycki and Lions [10]. Liu et al. [11] established the
existence of both one-sign and nodal ground states of soliton
type solutions for (3) by the Nehari method. In particular,
in [12], by using Nehari manifold method and concentration
compactness principle (see [13]) in the Orlicz space, Guo
2
Journal of Applied Mathematics
and Tang considered the following quasilinear Schrödinger
system:
1
2𝛼
− Δ𝑢 + (𝜆𝑎 (𝑥) + 1) 𝑢 − (Δ|𝑢|2 ) 𝑢 =
|𝑢|𝛼−2 |V|𝛽 𝑢,
2
𝛼+𝛽
𝑥 ∈ 𝑅𝑁,
− Δ𝑢 + (𝜆𝑏 (𝑥) + 1) 𝑢 −
2𝛽
1
(Δ|𝑢|2 ) 𝑢 =
|𝑢|𝛼 |V|𝛽−2 V,
2
𝛼+𝛽
The weak form of the system (1) is
∫ ((1 + 𝑢2 ) ∇𝑢∇𝜑 + (1 + |∇𝑢|2 ) 𝑢𝜑)
Ω
+ ∫ ((1 + V2 ) ∇V∇𝜓 + (1 + |∇V|2 ) V𝜓)
Ω
−
2𝛽
2𝛼
∫ |𝑢|𝛼−2 |V|𝛽 𝑢𝜑 −
∫ |𝑢|𝛼 |V|𝛽−2 V𝜓 = 0,
𝛼+𝛽 Ω
𝛼+𝛽 Ω
𝑥 ∈ 𝑅𝑁,
𝑢 (𝑥) 󳨀→ 0,
V (𝑥) 󳨀→ 0,
(𝜑, 𝜓) ∈ 𝐶0∞ (Ω) × 𝐶0∞ (Ω) ,
(5)
|𝑥| 󳨀→ ∞
(4)
with 𝑎(𝑥) ≥ 0, 𝑏(𝑥) ≥ 0 having a potential well and 𝛼 >
2, 𝛽 > 2, 𝛼 + 𝛽 < 2 ⋅ 2∗ , where 2∗ = 2𝑁/(𝑁 − 2) is the
critical Sobolev exponent, and they proved the existence of
a ground state solution for the system (4) which localizes
near the potential well int 𝑎−1 (0) for 𝜆 large enough. Guo
and Tang [14] also considered ground state solutions of the
single quasilinear Schrödinger equation corresponding to the
system (4) by the same methods and obtained similar results.
It is worth pointing out that the existence of one-bump or
multibump bound state solutions for the related semilinear
Schrödinger equation (3) for 𝑘 = 0 has been extensively
studied. One can see Bartsch and Wang [15], Ambrosetti et al.
[16], Ambrosetti et al. [17], Byeon and Wang [18], Cingolani
and Lazzo [19], Cingolani and Nolasco [20], Del Pino and
Felmer [21, 22], Floer and Weinstein [23], and Oh [24, 25]
and the references therein.
The system (1) is a kind of “limit” problem of the system
(4) as 𝜆 → ∞. The existence of solutions for the system (1)
has important physical interest. The purpose of this paper is
to study the existence of both positive and negative solutions
for the system (1). We mainly follow the idea of Liu et al. [26]
to perturb the functional and obtain our main results. We
point out that the procedure to the system (1) is not trivial
at all. Since the appearance of the quasilinear terms 𝑢Δ(𝑢2 )
and VΔ(V2 ), we need more delicate estimates.
The paper is organized as follows. In Section 2, we
introduce a perturbation of the functional and give our main
results (Theorem 1 and 2). In Section 3, we verify the PalaisSmale condition for the perturbed functional. Section 4 is
devoted to some asymptotic behavior of sequence {(𝑢𝑛 , V𝑛 )} ⊂
𝑊01,4 (Ω) × 𝑊01,4 (Ω) and {𝜇𝑛 } ⊂ (0, 1] satisfying some conditions. Finally, our main results will be proved in Section 5.
Throughout this paper, we will use the same 𝐶 to denote
various generic positive constants, and we will use 𝑜(1) to
denote quantities that tend to 0.
2. Perturbation of the Functional and
Main Results
In order to obtain the desired existence of solutions for the
system (1), in this section, we introduce a perturbation of the
functional and give our main results.
which is formally the variational formulation of the following
functional:
𝐼0 (𝑢, V) =
1
∫ ((1 + 𝑢2 ) |∇𝑢|2 + 𝑢2 )
2 Ω
+
1
2
∫ ((1 + V2 ) |∇V|2 + V2 ) −
∫ |𝑢|𝛼 |V|𝛽 .
2 Ω
𝛼+𝛽 Ω
(6)
We may define the derivative of 𝐼0 at (𝑢, V) in the direction
of (𝜑, 𝜓) ∈ 𝐶0∞ (Ω) × 𝐶0∞ (Ω) as follows:
⟨𝐼0󸀠 (𝑢, V) , (𝜑, 𝜓)⟩ = ∫ (1 + 𝑢2 ) ∇𝑢∇𝜑
Ω
+ ∫ (1 + |∇𝑢|2 ) 𝑢𝜑+∫ (1 + V2 ) ∇V∇𝜓
Ω
Ω
+ ∫ (1 + |∇V|2 ) V𝜓
Ω
−
2𝛼
∫ |𝑢|𝛼−2 |V|𝛽 𝑢𝜑
𝛼+𝛽 Ω
−
2𝛽
∫ |𝑢|𝛼 |V|𝛽−2 V𝜓.
𝛼+𝛽 Ω
(7)
We call that (𝑢, V) is a critical point of 𝐼0 if (𝑢, V) ∈
𝑊01,2 (Ω) × 𝑊01,2 (Ω), ∫Ω 𝑢2 |∇𝑢|2 < ∞, ∫Ω V2 |∇V|2 < ∞ and
⟨𝐼0󸀠 (𝑢, V), (𝜑, 𝜓)⟩ = 0 for all (𝜑, 𝜓) ∈ 𝐶0∞ (Ω) × 𝐶0∞ (Ω). That is,
(𝑢, V) is a weak solution for the system (1).
When we consider the system (1) by using the classical
critical point theory, we encounter the difficulties due to the
lack of an appropriate working space. In general it seems
there is no suitable space in which the variational functional
𝐼0 possesses both smoothness and compactness properties.
For smoothness one would need to work in a space smaller
than 𝑊01,2 (Ω) to control the term involving the quasilinear
term in the system (1), but it seems impossible to obtain
bounds for (𝑃𝑆)𝑐 sequence in this setting. There have been
several ideas and approaches used in recent years to overcome
the difficulties such as by minimizations [7, 27], the Nehari
Journal of Applied Mathematics
3
method [11], and change of variables [8, 9]. In this paper, we
consider a perturbed functional
𝐼𝜇 (𝑢, V) =
=
1
𝜇 ∫ (|∇𝑢|4 + |∇V|4 ) + 𝐼0 (𝑢, V)
4 Ω
3. Compactness of the Perturbed Functional
1
1
𝜇 ∫ (|∇𝑢|4 +|∇V|4 )+ ∫ ((1+𝑢2 ) |∇𝑢|2 +𝑢2 )
4 Ω
2 Ω
+
1
2
∫ ((1 + V2 ) |∇V|2 + V2 ) −
∫ |𝑢|𝛼 |V|𝛽 ,
2 Ω
𝛼+𝛽 Ω
(8)
where 𝜇 ∈ (0, 1] is a parameter. Then it is easy to see that 𝐼𝜇 is
a 𝐶1 -functional on 𝑊01,4 (Ω)×𝑊01,4 (Ω). We also can define the
derivative of 𝐼𝜇 at (𝑢, V) in the direction of (𝜑, 𝜓) as follows:
⟨𝐼𝜇󸀠 (𝑢, V) , (𝜑, 𝜓)⟩ = 𝜇 ∫ |∇𝑢|2 ∇𝑢∇𝜑 + 𝜇 ∫ |∇V|2 ∇V∇𝜓
Proposition 3. For 𝜇 > 0 fixed, the functional 𝐼𝜇 (𝑢, V) satisfies
(PS)𝑐 condition for all 𝑐 ∈ R. That is, any sequence {(𝑢𝑛 , V𝑛 )} ⊂
𝑊01,4 (Ω) × 𝑊01,4 (Ω) satisfying, for 𝑐 ∈ R,
𝐼𝜇 (𝑢𝑛 , V𝑛 ) 󳨀→ 𝑐,
∗
𝐼𝜇󸀠 (𝑢𝑛 , V𝑛 ) 󳨀→ 0 𝑠𝑡𝑟𝑜𝑛𝑔𝑙𝑦 𝑖𝑛 (𝑊01,4 (Ω) × 𝑊01,4 (Ω))
(11)
+ ∫ (1+𝑢2 ) ∇𝑢∇𝜑 +∫ (1+|∇𝑢|2 ) 𝑢𝜑
+ ∫ (1+V2 ) ∇V∇𝜓 +∫ (1+|∇V|2 ) V𝜓
For giving the proof of Proposition 3, we need the
following lemma firstly.
Ω
Ω
Ω
Ω
Ω
−
2𝛼
∫ |𝑢|𝛼−2 |V|𝛽 𝑢𝜑
𝛼+𝛽 Ω
Lemma 4. Suppose that a sequence {(𝑢𝑛 , V𝑛 )} ⊂ 𝑊01,4 (Ω) ×
𝑊01,4 (Ω) satisfies (11). Then
−
2𝛽
∫ |𝑢|𝛼 |V|𝛽−2 V𝜓
𝛼+𝛽 Ω
1
1
󵄩
󵄩4
lim sup󵄩󵄩󵄩(𝑢𝑛 , V𝑛 )󵄩󵄩󵄩 ≤ ( −
) 𝜇−1 𝑐.
4 𝛼+𝛽
𝑛→∞
−1
(9)
𝐶0∞ (Ω)
𝐶0∞ (Ω).
for all (𝜑, 𝜓) ∈
×
The idea is to obtain the
existence of the critical points of 𝐼𝜇 for 𝜇 > 0 small and to
establish suitable estimates for the critical points as 𝜇 → 0
so that we may pass to the limit to get the solutions for the
original system (1).
Our main results are as follows.
Theorem 1. Assume that 𝛼 > 2, 𝛽 > 2 and 𝛼 + 𝛽 < 2 ⋅ 2∗ . Let
𝜇𝑛 → 0 and let {(𝑢𝑛 , V𝑛 )} ⊂ 𝑊01,4 (Ω) × 𝑊01,4 (Ω) be a sequence
of 𝐼𝜇𝑛 satisfying 𝐼𝜇󸀠 𝑛 (𝑢𝑛 , V𝑛 ) = 0 and 𝐼𝜇𝑛 (𝑢𝑛 , V𝑛 ) ≤ 𝐶 for some 𝐶
independent of 𝑛. Then, up to a subsequence
𝑢𝑛 ∇𝑢𝑛 󳨀→ 𝑢∇𝑢,
In this section, we verify the Palais-Smale condition ((PS)𝑐
condition in short) for the perturbed functional 𝐼𝜇 (𝑢, V). We
have the following proposition.
has a strongly convergent subsequence in 𝑊01,4 (Ω) × 𝑊01,4 (Ω),
∗
where (𝑊01,4 (Ω) × 𝑊01,4 (Ω)) is the dual space of 𝑊01,4 (Ω) ×
𝑊01,4 (Ω).
Ω
𝑢𝑛 󳨀→ 𝑢,
Notation. We denote by ‖ ⋅ ‖ the norm of 𝑊01,4 (Ω) and by | ⋅ |𝑠
the norm of 𝐿𝑠 (Ω) (1 ≤ 𝑠 < +∞).
V𝑛 󳨀→ V
𝑖𝑛 𝑊01,2 (Ω) ,
V𝑛 ∇V𝑛 󳨀→ V∇V
Proof. It follows from (11) that
𝑐 + 𝑜 (1) −
1
󵄩
󵄩
𝑜 (1) 󵄩󵄩󵄩(𝑢𝑛 , V𝑛 )󵄩󵄩󵄩
𝛼+𝛽
= 𝐼𝜇 (𝑢𝑛 , V𝑛 ) −
1
⟨𝐼󸀠 (𝑢 , V ) , (𝑢𝑛 , V𝑛 )⟩
𝛼+𝛽 𝜇 𝑛 𝑛
1
1
󵄨4 󵄨 󵄨4
󵄨
=( −
) 𝜇 ∫ (󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 + 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 )
4 𝛼+𝛽
Ω
1
1
󵄨2 󵄨 󵄨2
󵄨
+( −
) ∫ (󵄨󵄨∇𝑢 󵄨󵄨 + 󵄨󵄨∇V 󵄨󵄨 )
2 𝛼 + 𝛽 Ω 󵄨 𝑛󵄨 󵄨 𝑛󵄨
(13)
1
1
󵄨 󵄨2 󵄨 󵄨2
+( −
) ∫ (󵄨󵄨𝑢 󵄨󵄨 + 󵄨󵄨V 󵄨󵄨 )
2 𝛼 + 𝛽 Ω 󵄨 𝑛󵄨 󵄨 𝑛󵄨
𝑖𝑛 𝐿2 (Ω) ,
󵄨4 󵄨 󵄨4
󵄨
𝜇𝑛 ∫ (󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 + 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 ) 󳨀→ 0,
Ω
(12)
(10)
𝐼𝜇󸀠 𝑛 (𝑢𝑛 , V𝑛 ) 󳨀→ 𝐼0󸀠 (𝑢, V)
as 𝑛 → ∞ and (𝑢, V) is a critical point of 𝐼0 .
Using Theorem 1, we have the following existence result.
Theorem 2. Assume that 𝛼 > 2, 𝛽 > 2 and 𝛼 + 𝛽 < 2 ⋅ 2∗ .
Then 𝐼𝜇 has a positive critical point (𝑢𝜇 , V𝜇 ) and a negative
𝑢𝜇 , ̃V𝜇 )) converges to
critical point (̃
𝑢𝜇 , ̃V𝜇 ), and (𝑢𝜇 , V𝜇 ) (resp., (̃
a positive (resp., negative) solution for the system (1) as 𝜇 → 0.
1
2
󵄨
󵄨2
󵄨 󵄨2
+( −
) ∫ (𝑢2 󵄨󵄨∇𝑢 󵄨󵄨 + V𝑛2 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 )
2 𝛼 + 𝛽 Ω 𝑛󵄨 𝑛󵄨
1
1
≥( −
) 𝜇 ∫ (|∇𝑢|4 + |∇V|4 ) .
4 𝛼+𝛽
Ω
Thus we have
−1
1
1
󵄩
󵄩4
lim sup󵄩󵄩󵄩(𝑢𝑛 , V𝑛 )󵄩󵄩󵄩 ≤ ( −
) 𝜇−1 𝑐.
4 𝛼+𝛽
𝑛→∞
This completes the proof of Lemma 4.
Now we give the proof of Proposition 3.
(14)
4
Journal of Applied Mathematics
Proof of Proposition 3. From Lemma 4, we know that
{(𝑢𝑛 , V𝑛 )} is bounded in 𝑊01,4 (Ω) × 𝑊01,4 (Ω). So there exists a
subsequence of {(𝑢𝑛 , V𝑛 )}, still denoted {(𝑢𝑛 , V𝑛 )}, such that
(𝑢𝑛 , V𝑛 ) ⇀ (𝑢, V)
V𝑛 󳨀→ V
as 𝑛 󳨀→ ∞
strongly in 𝐿𝑠 (Ω)
for any 2 < 𝑠 < 2 ⋅ 2∗ .
󵄨2
󵄨2
󵄨
󵄨
= 𝜇 ∫ (󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 ∇𝑢𝑛 − 󵄨󵄨󵄨∇𝑢𝑚 󵄨󵄨󵄨 ∇𝑢𝑚 ) (∇𝑢𝑛 − ∇𝑢𝑚 )
Ω
󵄨2
󵄨 󵄨2
󵄨
+ 𝜇 ∫ (󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 ∇V𝑛 − 󵄨󵄨󵄨∇V𝑚 󵄨󵄨󵄨 ∇V𝑚 ) (∇V𝑛 − ∇V𝑚 )
Ω
󵄨
󵄨2
󵄨
󵄨2
+ ∫ 󵄨󵄨󵄨𝑢𝑛 − 𝑢𝑚 󵄨󵄨󵄨 + ∫ 󵄨󵄨󵄨∇𝑢𝑛 − ∇𝑢𝑚 󵄨󵄨󵄨
Ω
Ω
Ω
−
󵄨
󵄨2
− ∇𝑢𝑚 ) + ∫ 󵄨󵄨󵄨V𝑛 − V𝑚 󵄨󵄨󵄨
Ω
󵄨
󵄨2
2
+ ∫ 󵄨󵄨󵄨∇V𝑛 −∇V𝑚 󵄨󵄨󵄨 +∫ (V𝑛2 ∇V𝑛 −V𝑚
∇V𝑚 ) (∇V𝑛 −∇V𝑚 )
Ω
Ω
󵄨
󵄨2
󵄨
󵄨2
+ ∫ (𝑢𝑛 󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 − 𝑢𝑚 󵄨󵄨󵄨∇𝑢𝑚 󵄨󵄨󵄨 ) (𝑢𝑛 − 𝑢𝑚 )
Ω
󵄨 󵄨2
󵄨
󵄨2
+ ∫ (V𝑛 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 − V𝑚 󵄨󵄨󵄨∇V𝑚 󵄨󵄨󵄨 ) (V𝑛 − V𝑚 )
Ω
−
−
2
∇V𝑚 ) (∇V𝑛 − ∇V𝑚 )
∫ (V𝑛2 ∇V𝑛 − V𝑚
Ω
󵄨
󵄨2
2
) ∇V𝑚 (∇V𝑛 −∇V𝑚 )
≥ ∫ V𝑛2 󵄨󵄨󵄨∇V𝑛 −∇V𝑚 󵄨󵄨󵄨 +∫ (V𝑛2 −V𝑚
Ω
Ω
󵄨 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩
󵄨
≥ −󵄨󵄨󵄨V𝑛 − V𝑚 󵄨󵄨󵄨4 (󵄨󵄨󵄨V𝑛 󵄨󵄨󵄨4 + 󵄨󵄨󵄨V𝑚 󵄨󵄨󵄨4 ) 󵄩󵄩󵄩V𝑚 󵄩󵄩󵄩 (󵄩󵄩󵄩V𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩V𝑚 󵄩󵄩󵄩)
󳨀→ 0 as 𝑚, 𝑛 󳨀→ ∞,
= ⟨𝐼𝜇󸀠 (𝑢𝑛 , V𝑛 ) − 𝐼𝜇󸀠 (𝑢𝑚 , V𝑚 ) , (𝑢𝑛 − 𝑢𝑚 , V𝑛 − V𝑚 )⟩
2
𝑢𝑚
∇𝑢𝑚 ) (∇𝑢𝑛
󳨀→ 0 as 𝑚, 𝑛 󳨀→ ∞,
(15)
󵄩
󵄩
𝑜 (1) 󵄩󵄩󵄩(𝑢𝑛 − 𝑢𝑚 , V𝑛 − V𝑚 )󵄩󵄩󵄩
+∫
󵄨
󵄨2
2
) ∇𝑢𝑚 (∇𝑢𝑛 −∇𝑢𝑚 )
≥ ∫ 𝑢𝑛2 󵄨󵄨󵄨∇𝑢𝑛 −∇𝑢𝑚 󵄨󵄨󵄨 +∫ (𝑢𝑛2 −𝑢𝑚
Ω
Ω
󵄨 󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩
󵄨
≥ −󵄨󵄨󵄨𝑢𝑛 − 𝑢𝑚 󵄨󵄨󵄨4 (󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨4 + 󵄨󵄨󵄨𝑢𝑚 󵄨󵄨󵄨4 ) 󵄩󵄩󵄩𝑢𝑚 󵄩󵄩󵄩 (󵄩󵄩󵄩𝑢𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑢𝑚 󵄩󵄩󵄩)
Now we prove that (𝑢𝑛 , V𝑛 ) → (𝑢, V) in 𝑊01,4 (Ω) × 𝑊01,4 (Ω).
In (9), choosing (𝜑, 𝜓) = (𝑢𝑛 − 𝑢𝑚 , V𝑛 − V𝑚 ), we have
(𝑢𝑛2 ∇𝑢𝑛
Ω
weakly in 𝑊01,4 (Ω) × 𝑊01,4 (Ω)
as 𝑛 󳨀→ ∞,
𝑢𝑛 󳨀→ 𝑢,
2
∇𝑢𝑚 ) (∇𝑢𝑛 − ∇𝑢𝑚 )
∫ (𝑢𝑛2 ∇𝑢𝑛 − 𝑢𝑚
2𝛼
󵄨 󵄨𝛼−2 󵄨 󵄨𝛽
󵄨 󵄨𝛼−2 󵄨 󵄨𝛽
∫ (󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 󵄨󵄨󵄨V𝑛 󵄨󵄨󵄨 𝑢𝑛 − 󵄨󵄨󵄨𝑢𝑚 󵄨󵄨󵄨 󵄨󵄨󵄨V𝑚 󵄨󵄨󵄨 𝑢𝑚 ) (𝑢𝑛 −𝑢𝑚 )
𝛼+𝛽 Ω
2𝛽
󵄨 󵄨𝛼 󵄨 󵄨𝛽−2
󵄨 󵄨𝛼 󵄨 󵄨𝛽−2
∫ (󵄨󵄨𝑢 󵄨󵄨 󵄨󵄨V 󵄨󵄨 V𝑛 − 󵄨󵄨󵄨𝑢𝑚 󵄨󵄨󵄨 󵄨󵄨󵄨V𝑚 󵄨󵄨󵄨 V𝑚 ) (V𝑛 −V𝑚 ) .
𝛼+𝛽 Ω 󵄨 𝑛 󵄨 󵄨 𝑛 󵄨
(16)
󵄨󵄨
󵄨
󵄨󵄨∫ (𝑢 󵄨󵄨󵄨∇𝑢 󵄨󵄨󵄨2 − 𝑢 󵄨󵄨󵄨∇𝑢 󵄨󵄨󵄨2 ) (𝑢 − 𝑢 )󵄨󵄨󵄨
󵄨󵄨
𝑛󵄨
𝑛󵄨
𝑚󵄨
𝑚󵄨
𝑛
𝑚 󵄨󵄨
󵄨 Ω
󵄨
󵄨 󵄨 󵄩 󵄩2 󵄨 󵄨 󵄩 󵄩2 󵄨
󵄨
≤ (󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨4 󵄩󵄩󵄩𝑢𝑛 󵄩󵄩󵄩 + 󵄨󵄨󵄨𝑢𝑚 󵄨󵄨󵄨4 󵄩󵄩󵄩𝑢𝑚 󵄩󵄩󵄩 ) 󵄨󵄨󵄨𝑢𝑛 − 𝑢𝑚 󵄨󵄨󵄨4
󳨀→ 0 as 𝑚, 𝑛 󳨀→ ∞,
󵄨
󵄨󵄨
󵄨󵄨∫ (V 󵄨󵄨󵄨∇V 󵄨󵄨󵄨2 − V 󵄨󵄨󵄨∇V 󵄨󵄨󵄨2 ) (V − V )󵄨󵄨󵄨
󵄨󵄨
𝑛󵄨
𝑛󵄨
𝑚󵄨
𝑚󵄨
𝑛
𝑚 󵄨󵄨
󵄨
󵄨 Ω
󵄨 󵄨 󵄩 󵄩2 󵄨 󵄨 󵄩 󵄩2 󵄨
󵄨
≤ (󵄨󵄨󵄨V𝑛 󵄨󵄨󵄨4 󵄩󵄩󵄩V𝑛 󵄩󵄩󵄩 + 󵄨󵄨󵄨V𝑚 󵄨󵄨󵄨4 󵄩󵄩󵄩V𝑚 󵄩󵄩󵄩 ) 󵄨󵄨󵄨V𝑛 − V𝑚 󵄨󵄨󵄨4
󳨀→ 0 as 𝑚, 𝑛 󳨀→ ∞,
󵄨󵄨
2𝛼 󵄨󵄨󵄨
󵄨󵄨∫ (󵄨󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨󵄨𝛼−2 󵄨󵄨󵄨󵄨V𝑛 󵄨󵄨󵄨󵄨𝛽 𝑢𝑛 − 󵄨󵄨󵄨󵄨𝑢𝑚 󵄨󵄨󵄨󵄨𝛼−2 󵄨󵄨󵄨󵄨V𝑚 󵄨󵄨󵄨󵄨𝛽 𝑢𝑚 ) (𝑢𝑛 − 𝑢𝑚 )󵄨󵄨󵄨
󵄨󵄨
𝛼 + 𝛽 󵄨󵄨 Ω
≤
2𝛼
󵄨
󵄨 󵄨𝛼−1 󵄨 󵄨𝛽 󵄨 󵄨𝛼−1 󵄨 󵄨𝛽 󵄨
∫ (󵄨󵄨𝑢 󵄨󵄨 󵄨󵄨V 󵄨󵄨 + 󵄨󵄨𝑢 󵄨󵄨 󵄨󵄨V 󵄨󵄨 ) 󵄨󵄨𝑢 − 𝑢𝑚 󵄨󵄨󵄨
𝛼 + 𝛽 Ω 󵄨 𝑛󵄨 󵄨 𝑛󵄨 󵄨 𝑚󵄨 󵄨 𝑚󵄨 󵄨 𝑛
≤
2𝛼 󵄨󵄨 󵄨󵄨𝛼−1 󵄨󵄨 󵄨󵄨𝛽
󵄨
󵄨 󵄨𝛼−1 󵄨 󵄨𝛽
󵄨
(󵄨𝑢 󵄨 󵄨V 󵄨 + 󵄨󵄨𝑢 󵄨󵄨 󵄨󵄨V 󵄨󵄨 ) 󵄨󵄨𝑢 −𝑢 󵄨󵄨
𝛼 + 𝛽 󵄨 𝑛 󵄨𝛼+𝛽 󵄨 𝑛 󵄨𝛼+𝛽 󵄨 𝑚 󵄨𝛼+𝛽 󵄨 𝑚 󵄨𝛼+𝛽 󵄨 𝑛 𝑚 󵄨𝛼+𝛽
󳨀→ 0 as 𝑚, 𝑛 󳨀→ ∞,
󵄨󵄨
2𝛽 󵄨󵄨󵄨
󵄨󵄨∫ (󵄨󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨󵄨𝛼 󵄨󵄨󵄨󵄨V𝑛 󵄨󵄨󵄨󵄨𝛽−2 V𝑛 − 󵄨󵄨󵄨󵄨𝑢𝑚 󵄨󵄨󵄨󵄨𝛼 󵄨󵄨󵄨󵄨V𝑚 󵄨󵄨󵄨󵄨𝛽−2 V𝑚 ) (V𝑛 − V𝑚 )󵄨󵄨󵄨
󵄨󵄨
𝛼 + 𝛽 󵄨󵄨 Ω
≤
2𝛽
󵄨
󵄨 󵄨𝛼 󵄨 󵄨𝛽−1 󵄨 󵄨𝛼 󵄨 󵄨𝛽−1 󵄨
∫ (󵄨󵄨𝑢 󵄨󵄨 󵄨󵄨V 󵄨󵄨 + 󵄨󵄨󵄨𝑢𝑚 󵄨󵄨󵄨 󵄨󵄨󵄨V𝑚 󵄨󵄨󵄨 ) 󵄨󵄨󵄨V𝑛 − V𝑚 󵄨󵄨󵄨
𝛼 + 𝛽 Ω 󵄨 𝑛󵄨 󵄨 𝑛󵄨
≤
2𝛽 󵄨󵄨 󵄨󵄨𝛼 󵄨󵄨 󵄨󵄨𝛽−1 󵄨󵄨 󵄨󵄨𝛼 󵄨󵄨 󵄨󵄨𝛽−1 󵄨󵄨
󵄨
(󵄨𝑢 󵄨 󵄨V 󵄨 + 󵄨𝑢 󵄨 󵄨V 󵄨 ) 󵄨V −V 󵄨󵄨
𝛼 + 𝛽 󵄨 𝑛 󵄨𝛼+𝛽 󵄨 𝑛 󵄨𝛼+𝛽 󵄨 𝑚 󵄨𝛼+𝛽 󵄨 𝑚 󵄨𝛼+𝛽 󵄨 𝑛 𝑚 󵄨𝛼+𝛽
We may estimate the terms involved as follows:
󵄨2
󵄨2
󵄨
󵄨
𝜇 ∫ (󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 ∇𝑢𝑛 − 󵄨󵄨󵄨∇𝑢𝑚 󵄨󵄨󵄨 ∇𝑢𝑚 ) (∇𝑢𝑛 − ∇𝑢𝑚 )
Ω
󵄨
󵄨4
≥ 𝜇 ∫ 󵄨󵄨󵄨∇𝑢𝑛 − ∇𝑢𝑚 󵄨󵄨󵄨 ,
Ω
󵄨2
󵄨 󵄨2
󵄨
𝜇 ∫ (󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 ∇V𝑛 − 󵄨󵄨󵄨∇V𝑚 󵄨󵄨󵄨 ∇V𝑚 ) (∇V𝑛 − ∇V𝑚 )
Ω
󵄨
󵄨4
≥ 𝜇 ∫ 󵄨󵄨󵄨∇V𝑛 − ∇V𝑚 󵄨󵄨󵄨 ,
Ω
󳨀→ 0 as 𝑚, 𝑛 󳨀→ ∞.
(17)
Returning to (16), we have
󵄨4 󵄨
󵄨4
󵄨
𝜇 ∫ (󵄨󵄨󵄨∇𝑢𝑛 − ∇𝑢𝑚 󵄨󵄨󵄨 + 󵄨󵄨󵄨∇V𝑛 − ∇V𝑚 󵄨󵄨󵄨 )
Ω
󵄩
󵄩
≤ 𝑜 (1) 󵄩󵄩󵄩(𝑢𝑛 − 𝑢𝑚 , V𝑛 − V𝑚 )󵄩󵄩󵄩 + 𝑜 (1) ,
(18)
Journal of Applied Mathematics
5
which implies that ‖(𝑢𝑛 − 𝑢𝑚 , V𝑛 − V𝑚 )‖ → 0, that is,
(𝑢𝑛 , 𝑢𝑚 ) → (𝑢, V) in 𝑊01,4 (Ω) × 𝑊01,4 (Ω). This completes the
proof of Proposition 3.
5. Proof of Main Results
4. Some Asymptotic Behavior
Proof of Theorem 1. Note that (𝑢𝑛 , V𝑛 ) satisfies the following
equation:
Proposition 3 enables us to apply minimax argument to the
functional 𝐼𝜇 (𝑢, V). In this section, we also study the behavior
of sequence {(𝑢𝑛 , V𝑛 )} ⊂ 𝑊01,4 (Ω) × 𝑊01,4 (Ω) and {𝜇𝑛 } ⊂ (0, 1]
satisfying
𝜇𝑛 󳨀→ 0,
In this section, we give the proof of our main results. Firstly,
we prove Theorem 1.
󵄨
󵄨2
󵄨 󵄨2
𝜇𝑛 ∫ 󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 ∇𝑢𝑛 ∇𝜑 + 𝜇𝑛 ∫ 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 ∇V𝑛 ∇𝜓
Ω
Ω
󵄨2
󵄨
+ ∫ ((1 + 𝑢𝑛2 ) ∇𝑢𝑛 ∇𝜑 + (1 + 󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 ) 𝑢𝑛 𝜑)
𝐼𝜇𝑛 (𝑢𝑛 , V𝑛 ) 󳨀→ 𝑐,
Ω
(19)
󵄩󵄩 󸀠
󵄩∗
󵄩󵄩𝐼𝜇 (𝑢𝑛 , V𝑛 )󵄩󵄩󵄩 󳨀→ 0.
󵄩 𝑛
󵄩
The following proposition is the key of this section.
󵄨 󵄨2
+ ∫ ((1 + V𝑛2 ) ∇V𝑛 ∇𝜓 + (1 + 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 ) V𝑛 𝜓)
Ω
Proposition 5. Assume sequence {(𝑢𝑛 , V𝑛 )} ⊂ 𝑊01,4 (Ω) ×
𝑊01,4 (Ω) and {𝜇𝑛 } ⊂ (0, 1] satisfy (19). Then after extracting
a sequence, still denoted by 𝑛, one has
(𝑢𝑛 , V𝑛 ) ⇀ (𝑢, V)
𝑖𝑛 𝑊01,2 (Ω) × 𝑊01,2 (Ω) ,
(𝑢𝑛 ∇𝑢𝑛 , V𝑛 ∇V𝑛 ) ⇀ (𝑢∇𝑢, V∇V)
𝑖𝑛 𝐿2 (Ω) × 𝐿2 (Ω) , (20)
(𝑢𝑛 (𝑥) , V𝑛 (𝑥)) 󳨀→ (𝑢 (𝑥) , V (𝑥))
𝑎.𝑒. 𝑥 ∈ Ω
−
2𝛼
󵄨 󵄨𝛼−2 󵄨 󵄨𝛽
∫ 󵄨󵄨𝑢 󵄨󵄨 󵄨󵄨V 󵄨󵄨 𝑢 𝜑
𝛼 + 𝛽 Ω 󵄨 𝑛󵄨 󵄨 𝑛󵄨 𝑛
−
2𝛽
󵄨 󵄨𝛼 󵄨 󵄨𝛽−2
∫ 󵄨󵄨𝑢 󵄨󵄨 󵄨󵄨V 󵄨󵄨 V𝑛 𝜓 = 0
𝛼 + 𝛽 Ω 󵄨 𝑛󵄨 󵄨 𝑛󵄨
for all (𝜑, 𝜓) ∈ 𝑊01,4 (Ω) × 𝑊01,4 (Ω). Since
󵄨 󵄨4𝑁/(𝑁−2)
(∫ 󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨
)
Ω
as 𝑛 → ∞.
(𝑁−2)/𝑁
Proof. Similar to the proof of Lemma 4, by (19), we have
𝐶 ≥ 𝐼𝜇𝑛 (𝑢𝑛 , V𝑛 ) −
󵄨 󵄨4𝑁/(𝑁−2)
(∫ 󵄨󵄨󵄨V𝑛 󵄨󵄨󵄨
)
Ω
1
⟨𝐼󸀠 (𝑢 , V ) , (𝑢𝑛 , V𝑛 )⟩
𝛼 + 𝛽 𝜇𝑛 𝑛 𝑛
󵄨󵄨 󵄨󵄨
󵄨󵄨𝑢𝑛 󵄨󵄨𝐿∞ ≤ 𝐶,
(21)
1
2
󵄨
󵄨2
󵄨 󵄨2
+( −
) ∫ (𝑢2 󵄨󵄨∇𝑢 󵄨󵄨 + V𝑛2 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 ) .
2 𝛼 + 𝛽 Ω 𝑛󵄨 𝑛󵄨
Thus
󵄨4 󵄨 󵄨4
󵄨2 󵄨 󵄨2
󵄨
󵄨
𝜇𝑛 ∫ (󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 + 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 ) + ∫ (󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 + 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 )
Ω
Ω
󵄨
󵄨2
󵄨 󵄨2
󵄨 󵄨2 󵄨 󵄨2
+ ∫ (󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 + 󵄨󵄨󵄨V𝑛 󵄨󵄨󵄨 ) + ∫ (𝑢𝑛2 󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 + V𝑛2 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 ) ≤ 𝐶
Ω
Ω
(22)
for some 𝐶 independent of 𝑛. Then, up to a subsequence, we
have
(𝑢𝑛 , V𝑛 ) ⇀ (𝑢, V)
in
(𝑢𝑛 ∇𝑢𝑛 , V𝑛 ∇V𝑛 ) ⇀ (𝑢∇𝑢, V∇V)
(Ω) ×
𝑊01,2
(Ω) ,
in 𝐿2 (Ω) × 𝐿2 (Ω) , (23)
(𝑢𝑛 (𝑥) , V𝑛 (𝑥)) 󳨀→ (𝑢 (𝑥) , V (𝑥))
𝑎.𝑒. 𝑥 ∈ Ω
as 𝑛 → ∞. This completes the proof of Proposition 5.
(25)
󵄨 󵄨2
≤ 𝐶 ∫ V𝑛2 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 ≤ 𝐶.
Ω
󵄨󵄨 󵄨󵄨
󵄨󵄨V𝑛 󵄨󵄨𝐿∞ ≤ 𝐶.
(26)
|V|𝐿∞ ≤ 𝐶
(27)
Hence,
|𝑢|𝐿∞ ≤ 𝐶,
1
1
󵄨 󵄨2 󵄨 󵄨2
+( −
) ∫ (󵄨󵄨𝑢 󵄨󵄨 + 󵄨󵄨V 󵄨󵄨 )
2 𝛼 + 𝛽 Ω 󵄨 𝑛󵄨 󵄨 𝑛󵄨
𝑊01,2
(𝑁−2)/𝑁
󵄨
󵄨2
≤ 𝐶 ∫ 𝑢𝑛2 󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 ≤ 𝐶,
Ω
By Moser’s iteration, we have
1
1
󵄨4 󵄨 󵄨4
󵄨
≥( −
) 𝜇 ∫ (󵄨󵄨∇𝑢 󵄨󵄨 + 󵄨󵄨∇V 󵄨󵄨 )
4 𝛼 + 𝛽 𝑛 Ω 󵄨 𝑛󵄨 󵄨 𝑛󵄨
1
1
󵄨2 󵄨 󵄨2
󵄨
+( −
) ∫ (󵄨󵄨∇𝑢 󵄨󵄨 + 󵄨󵄨∇V 󵄨󵄨 )
2 𝛼 + 𝛽 Ω 󵄨 𝑛󵄨 󵄨 𝑛󵄨
(24)
for some 𝐶 independent of 𝑛. To show that (𝑢, V) is a critical
point of 𝐼0 we use some arguments in [28, 29] (see more
references therein). In (24) we choose 𝜑 = 𝜉 exp(−𝑢𝑛 ), 𝜓 =
𝜂 exp(−V𝑛 ), where 𝜉 ∈ 𝐶0∞ (Ω), 𝜉 ≥ 0, 𝜂 ∈ 𝐶0∞ (Ω), 𝜂 ≥ 0.
Substituting (𝜑, 𝜓) into (24), we have
󵄨
󵄨2
0 = 𝜇𝑛 ∫ 󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 ∇𝑢𝑛 (∇𝜉 exp (−𝑢𝑛 ) − 𝜉∇𝑢𝑛 exp (−𝑢𝑛 ))
Ω
󵄨 󵄨2
+ 𝜇𝑛 ∫ 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 ∇V𝑛 (∇𝜂 exp (−V𝑛 ) − 𝜂∇V𝑛 exp (−V𝑛 ))
Ω
+ ∫ (1 + 𝑢𝑛2 ) ∇𝑢𝑛 (∇𝜉 exp (−𝑢𝑛 ) − 𝜉∇𝑢𝑛 exp (−𝑢𝑛 ))
Ω
+ ∫ (1 + V𝑛2 ) ∇V𝑛 (∇𝜂 exp (−V𝑛 ) − 𝜂∇V𝑛 exp (−V𝑛 ))
Ω
󵄨2
󵄨
+ ∫ (1 + 󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 ) 𝑢𝑛 𝜉 exp (−𝑢𝑛 )
Ω
6
Journal of Applied Mathematics
󵄨 󵄨2
+ ∫ (1 + 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 ) V𝑛 𝜂 exp (−V𝑛 )
= ∫ (1 + 𝑢2 ) ∇𝑢∇ (𝜉 exp (−𝑢))
Ω
−
−
Ω
2𝛼
󵄨 󵄨𝛼−2 󵄨 󵄨𝛽
∫ 󵄨󵄨𝑢 󵄨󵄨 󵄨󵄨V 󵄨󵄨 𝑢 𝜉 exp (−𝑢𝑛 )
𝛼 + 𝛽 Ω 󵄨 𝑛󵄨 󵄨 𝑛󵄨 𝑛
+ ∫ (1 + V2 ) ∇V∇ (𝜂 exp (−V))
2𝛽
󵄨 󵄨𝛼 󵄨 󵄨𝛽−2
∫ 󵄨󵄨𝑢 󵄨󵄨 󵄨󵄨V 󵄨󵄨 V𝑛 𝜂 exp (−V𝑛 )
𝛼 + 𝛽 Ω 󵄨 𝑛󵄨 󵄨 𝑛󵄨
+ ∫ (1+|∇𝑢|2 ) 𝑢𝜉 exp (−𝑢)+∫ (1+|∇V|2 ) V𝜂 exp (−V)
Ω
Ω
󵄨
󵄨2
≤ 𝜇𝑛 ∫ 󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 ∇𝑢𝑛 ∇𝜉 exp (−𝑢𝑛 )
Ω
󵄨 󵄨2
+ 𝜇𝑛 ∫ 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 ∇V𝑛 ∇𝜂 exp (−V𝑛 )
Ω
Ω
−
2𝛼
∫ |𝑢|𝛼−2 |V|𝛽 𝑢𝜉 exp (−𝑢)
𝛼+𝛽 Ω
−
2𝛽
∫ |𝑢|𝛼 |V|𝛽−2 V𝜂 exp (−V) .
𝛼+𝛽 Ω
(29)
+ ∫ (1 + 𝑢𝑛2 ) ∇𝑢𝑛 ∇𝜉 exp (−𝑢𝑛 )
Ω
+ ∫ 𝑢𝑛 𝜉 exp (−𝑢𝑛 ) + ∫ V𝑛 𝜂 exp (−V𝑛 )
Let (𝜒, 𝜔) ≥ (0, 0), (𝜒, 𝜔) ∈ 𝐶0∞ (Ω) × 𝐶0∞ (Ω). We may
choose a sequence of nonnegative functions {(𝜉𝑛 , 𝜂𝑛 )} ⊂
𝐶0∞ (Ω) × 𝐶0∞ (Ω) such that (𝜉𝑛 , 𝜂𝑛 ) → (𝜒 exp 𝑢, 𝜔 exp V) in
𝑊01,2 (Ω) × 𝑊01,2 (Ω), (𝜉𝑛 , 𝜂𝑛 ) → (𝜒 exp 𝑢, 𝜔 exp V) a.e. 𝑥 ∈ Ω
and {(𝜉𝑛 , 𝜂𝑛 )} is uniformly bounded in 𝐿∞ (Ω) × 𝐿∞ (Ω). Then
by approximations in (29) we may obtain
󵄨2
󵄨
− ∫ (1 + 𝑢𝑛2 − 𝑢𝑛 ) 󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 𝜉 exp (−𝑢𝑛 )
∫ (1 + 𝑢2 ) ∇𝑢∇𝜒 + ∫ (1 + V2 ) ∇V∇𝜔 + ∫ (1 + |∇𝑢|2 ) 𝑢𝜒
+ ∫ (1 + V𝑛2 ) ∇V𝑛 ∇𝜂 exp (−V𝑛 )
Ω
Ω
Ω
Ω
Ω
󵄨 󵄨2
− ∫ (1 + V𝑛2 − V𝑛 ) 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 𝜂 exp (−V𝑛 )
+ ∫ (1 + |∇V|2 ) V𝜔 −
Ω
−
Ω
2𝛼
󵄨 󵄨𝛼−2 󵄨 󵄨𝛽
∫ 󵄨󵄨𝑢 󵄨󵄨 󵄨󵄨V 󵄨󵄨 𝑢 𝜉 exp (−𝑢𝑛 )
𝛼 + 𝛽 Ω 󵄨 𝑛󵄨 󵄨 𝑛󵄨 𝑛
−
(28)
0, 1+V𝑛2 −V𝑛
≥
≥ 0. By Fatou’s Lemma, the
weak convergence of {(𝑢𝑛 , V𝑛 )} and the fact that 𝜇𝑛 ∫Ω (|∇𝑢𝑛 |4 +
|∇V𝑛 |4 ) is bounded, we have
0 ≤ ∫ (1 + 𝑢2 ) ∇𝑢∇𝜉 exp (−𝑢) + ∫ (1 + V2 ) ∇V∇𝜂 exp (−V)
Ω
Ω
Ω
− ∫ (1 + V2 − V) |∇V|2 𝜂 exp (−V)
Ω
−
2𝛼
∫ |𝑢|𝛼−2 |V|𝛽 𝑢𝜉 exp (−𝑢)
𝛼+𝛽 Ω
−
2𝛽
∫ |𝑢|𝛼 |V|𝛽−2 V𝜂 exp (−V)
𝛼+𝛽 Ω
2𝛽
∫ |𝑢|𝛼 |V|𝛽−2 V𝜔 ≥ 0
𝛼+𝛽 Ω
∫ (1 + 𝑢2 ) ∇𝑢∇𝜒 + ∫ (1 + V2 ) ∇V∇𝜔 + ∫ (1 + |∇𝑢|2 ) 𝑢𝜒
Ω
Ω
+ ∫ (1 + |∇V|2 ) V𝜔 −
Ω
−
Ω
− ∫ (1 + 𝑢2 − 𝑢) |∇𝑢|2 𝜉 exp (−𝑢)
2𝛼
∫ |𝑢|𝛼−2 |V|𝛽 𝑢𝜒
𝛼+𝛽 Ω
for all (𝜒, 𝜔) ≥ (0, 0), (𝜒, 𝜔) ∈ 𝐶0∞ (Ω) × 𝐶0∞ (Ω).
Similarly, we may obtain an opposite inequality. Thus we
have
Ω
+ ∫ 𝑢𝜉 exp (−𝑢) + ∫ V𝜂 exp (−V)
Ω
(30)
2𝛽
󵄨 󵄨𝛼 󵄨 󵄨𝛽−2
−
∫ 󵄨󵄨𝑢 󵄨󵄨 󵄨󵄨V 󵄨󵄨 V𝑛 𝜂 exp (−V𝑛 ) .
𝛼 + 𝛽 Ω 󵄨 𝑛󵄨 󵄨 𝑛󵄨
Note that 1+𝑢𝑛2 −𝑢𝑛
Ω
Ω
2𝛼
∫ |𝑢|𝛼−2 |V|𝛽 𝑢𝜒
𝛼+𝛽 Ω
2𝛽
∫ |𝑢|𝛼 |V|𝛽−2 V𝜔 = 0
𝛼+𝛽 Ω
(31)
for all (𝜒, 𝜔) ∈ 𝐶0∞ (Ω) × 𝐶0∞ (Ω). That is, (𝑢, V) is a critical
point of 𝐼0 and a solution for the system (1). By doing
approximations again, we have (𝑢, V) in the place of (𝜒, 𝜔) of
(31)
∫ ((1 + 2𝑢2 ) |∇𝑢|2 + 𝑢2 ) + ∫ ((1 + 2V2 ) |∇V|2 + V2 )
Ω
Ω
𝛼
𝛽
− 2 ∫ |𝑢| |V| = 0.
Ω
(32)
Journal of Applied Mathematics
7
Setting (𝜑, 𝜓) = (𝑢𝑛 , V𝑛 ) in (24), we have
Since
󵄨4 󵄨 󵄨4
󵄨2
󵄨
󵄨
𝜇𝑛 ∫ (󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 + 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 ) + ∫ ((1 + 2𝑢𝑛2 ) 󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 + 𝑢𝑛2 )
Ω
Ω
+ ∫ ((1 +
Ω
󵄨 󵄨2
2V𝑛2 ) 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨
+
∫ |𝑢|𝛼+𝛽 ≤ 𝐶(∫ 𝑢2 |∇𝑢|2 )
(33)
󵄨 󵄨𝛼 󵄨 󵄨𝛽
− 2 ∫ 󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 󵄨󵄨󵄨V𝑛 󵄨󵄨󵄨 = 0.
Ω
V𝑛2 )
𝑢𝑛 ∇𝑢𝑛 󳨀→ 𝑢∇𝑢,
≤ 𝐶(∫ V |∇V| )
(𝛼+𝛽)/4
Ω
(39)
.
≥−
2
2
𝛼+𝛽
𝛼+𝛽
−
𝜀 ∫ (𝑢+ )
𝐶𝜀 ∫ (𝑢+ )
𝛼+𝛽 Ω
𝛼+𝛽
Ω
≥−
(𝛼+𝛽)/4
2𝐶
𝜀 ∫ (∫ 𝑢2 |∇𝑢|2 )
𝛼+𝛽 Ω Ω
(𝛼+𝛽)/4
2𝐶𝜀
−
(∫ V2 |∇V|2 )
𝛼+𝛽 Ω
in 𝑊01,2 (Ω) ,
V𝑛 ∇V𝑛 󳨀→ V∇V
2
2
𝛼
𝛽
∫ (𝑢+ ) (V+ )
𝛼+𝛽 Ω
−
󵄨 󵄨2
∫ V𝑛2 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 󳨀→ ∫ V2 |∇V|2
Ω
Ω
(34)
V𝑛 󳨀→ V
2
,
Then
as 𝑛 → ∞.
In particular, we have
𝑢𝑛 󳨀→ 𝑢,
𝛼+𝛽
Ω
󵄨 󵄨2
∫ 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 󳨀→ ∫ |∇V|2 ,
Ω
Ω
󵄨
󵄨2
∫ 𝑢𝑛2 󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 󳨀→ ∫ 𝑢2 |∇𝑢|2 ,
Ω
Ω
Ω
∫ |V|
Using ∫Ω |𝑢𝑛 |𝛼 |V𝑛 |𝛽 → ∫Ω |𝑢|𝛼 |V|𝛽 as 𝑛 → ∞, (32), (33), and
lower semicontinuity, we obtain
󵄨
󵄨2
∫ 󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 󳨀→ ∫ |∇𝑢|2 ,
Ω
Ω
Ω
(𝛼+𝛽)/4
in 𝐿2 (Ω) ,
≥−
2𝐶𝜀 (𝛼+𝛽)/2
2𝐶
𝜀𝜌(𝛼+𝛽)/2 −
𝜌
𝛼+𝛽
𝛼+𝛽
≥−
1
𝜌2
𝛼+𝛽
(35)
󵄨4 󵄨 󵄨4
󵄨
𝜇𝑛 ∫ (󵄨󵄨󵄨∇𝑢𝑛 󵄨󵄨󵄨 + 󵄨󵄨󵄨∇V𝑛 󵄨󵄨󵄨 ) 󳨀→ 0,
Ω
𝐼𝜇󸀠 𝑛 (𝑢𝑛 , V𝑛 ) 󳨀→ 𝐼0󸀠 (𝑢, V)
for 𝜀, 𝜌 small. Thus we have
as 𝑛 → ∞. This completes the proof of Theorem 1.
Next, we apply the mountain pass theorem to obtain
existence of critical points of 𝐼𝜇 . Set
Σ𝜌 ={(𝑢, V) ∈ 𝑊01,4 (Ω)×𝑊01,4 (Ω) | ∫ ((1+𝑢2 ) |∇𝑢|2 +𝑢2 )
Ω
+ ∫ ((1 + V2 ) |∇V|2 + V2 ) ≤ 𝜌2 }
𝐼𝜇+ (𝑢, V)
≥
1
∫ ((1 + 𝑢2 ) |∇𝑢|2 + 𝑢2 )
2 Ω
+
1
2
𝛼
𝛽
∫ ((1 + V2 ) |∇V|2 + V2 ) −
∫ (𝑢+ ) (V+ )
2 Ω
𝛼+𝛽 Ω
1
1
1
1
≥ 𝜌2 −
𝜌2 = ( −
) 𝜌2
2
𝛼+𝛽
2 𝛼+𝛽
Ω
(41)
(36)
for 𝜌 > 0.
Let us consider the functional
𝐼𝜇+ (𝑢, V) =
for (𝑢, V) ∈ 𝜕Σ𝜌 and for 𝜌 > 0 small enough. Choose
(𝜑, 𝜓) ≥ (0, 0), (𝜒, 𝜔) ∈ 𝐶0∞ (Ω) × 𝐶0∞ (Ω) and 𝑇 > 0. Define
a path (𝑔, ℎ): [0, 1] → 𝑊01,4 (Ω) × 𝑊01,4 (Ω) by (𝑔(𝑡), ℎ(𝑡)) =
(𝑡𝑇𝜑, 𝑡𝑇𝜓). When 𝑇 is large enough, we have
1
𝜇 ∫ (|∇𝑢|4 + |∇V|4 )
4 Ω
+
1
∫ ((1 + 𝑢2 ) |∇𝑢|2 + 𝑢2 )
2 Ω
1
+ ∫ ((1 + V2 ) |∇V|2 + V2 )
2 Ω
𝐼𝜇+ (𝑔 (1) , ℎ (1)) < 0,
(37)
+ ∫ ((1 + ℎ2 (1)) |∇ℎ (1)|2 + ℎ2 (1)) > 𝜌2 ,
𝛽
𝛼+𝛽
𝛼+𝛽
+ 𝐶𝜀 (V+ )
(42)
Ω
Here and in the following we denote 𝑢+ = max{𝑢, 0}. The
functional 𝐼𝜇 satisfies (PS)𝑐 condition. Similarly, we may
verify that 𝐼𝜇+ satisfies (PS)𝑐 condition. By 𝜀-Young inequality,
for any 𝜀 > 0, there exists 𝐶𝜀 > 0 such that
𝛼
󵄨2
󵄨
∫ ((1 + 𝑔2 (1)) 󵄨󵄨󵄨∇𝑔 (1)󵄨󵄨󵄨 + 𝑔2 (1))
Ω
2
𝛼
𝛽
−
∫ (𝑢+ ) (V+ ) .
𝛼+𝛽 Ω
(𝑢+ ) (V+ ) ≤ 𝜀(𝑢+ )
(40)
.
(38)
sup 𝐼𝜇+ (𝑔 (𝑡) , ℎ (𝑡)) ≤ 𝑚
𝑡∈[0,1]
for some 𝑚 independent of 𝜇 ∈ (0, 1].
Define
𝑐𝜇 = inf sup 𝐼𝜇+ (𝑔 (𝑡) , ℎ (𝑡)) ,
(𝑔,ℎ)∈Γ𝑡∈[0,1]
(43)
8
Journal of Applied Mathematics
where
Γ = {(𝑔, ℎ) ∈ 𝐶 ([0, 1] , 𝑊01,4 (Ω) × 𝑊01,4 (Ω)) | (𝑔 (0) , ℎ (0))
= (0, 0) , (𝑔 (1) , ℎ (1)) = (𝑇𝜑, 𝑇𝜓) } .
(44)
From the mountain pass theorem we obtain that
1
1
𝑐𝜇 ≥ ( −
) 𝜌2
2 𝛼+𝛽
(45)
is a critical value of 𝐼𝜇+ .
Let (𝑢𝜇 , V𝜇 ) be a critical point corresponding to 𝑐𝜇 . We
have (𝑢𝜇 , V𝜇 ) ≥ (0, 0). Thus (𝑢𝜇 , V𝜇 ) is a positive critical point
of 𝐼𝜇 by the strong maximum principle. In summary, we have
the following.
Proposition 6. There exist positive constants 𝜌 and 𝑚 independent of 𝜇 such that 𝐼𝜇 has a positive critical point (𝑢𝜇 , V𝜇 )
satisfying
1
1
( −
) 𝜌2 ≤ 𝐼𝜇 (𝑢𝜇 , V𝜇 ) ≤ 𝑚.
2 𝛼+𝛽
(46)
Finally, we give the proof of Theorem 2.
Proof of Theorem 2. For a positive solution of the system
(1), the proof follows from Proposition 6 and Theorem 1. A
similar argument gives a negative solution of the system (1).
This completes the proof of Theorem 2.
Acknowledgments
This paper was finished while Y. Jiao visited School of
Mathematical Sciences of Beijing Normal University as a
visiting fellow, and she would like to express her gratitude
for their hospitality during her visit. Y. Jiao is supported
by the National Science Foundation of China (11161041 and
31260098) and Fundamental Research Funds for the Central
Universities (zyz2012074).
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