# Existence of Solutions for a Modified Nonlinear Schrödinger System

Authors: Yujuan Jiao; Yanli Wang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We are concerned with the following modified nonlinear Schrödinger system:-Δu+u-(1/2)uΔ(u2)=(2α/(α+β))|u|α-2|v|βu,  x∈Ω,  -Δv+v-(1/2)vΔ(v2)=(2β/(α+β))|u|α|v|β-2v,  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.


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## Body

## 1. Introduction

Let us consider the following modified nonlinear Schrödinger system:(1)-Δu+u-12uΔ(u2)=2αα+β|u|α-2|v|βu,x∈Ω,-Δv+v-12vΔ(v2)=2βα+β|u|α|v|β-2v,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.Solutions for the system (1) are related to the existence of the standing wave solutions of the following quasilinear Schrödinger equation:
(2)i∂tz=-Δz+V(x)z-f(|z|2)z-kΔh(|z|2)h′(|z|2)z,Δz+V(x)z-f(|z|2)z-kΔh(|z|2)h′(|z|2)zx∈ℝN,
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 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:(3)-Δu+λV(x)u-kΔ(u2)u=|u|p-2u,x∈ℝN.
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 and Tang considered the following quasilinear Schrödinger system:
(4)-Δu+(λa(x)+1)u-12(Δ|u|2)u=2αα+β|u|α-2|v|βu,(λa(x)+1)u-12(Δ|u|2)u=2αα+β|u|α-2|v|βux∈RN,-Δu+(λb(x)+1)u-12(Δ|u|2)u=2βα+β|u|α|v|β-2v,(λb(x)+1)u-12(Δ|u|2)u=2βα+β|u|α|v|β-2vx∈RN,u(x)⟶0,v(x)⟶0,|x|⟶∞
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 inta-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 Section2, we introduce a perturbation of the functional and give our main results (Theorems 1 and 2). In Section 3, we verify the Palais-Smale 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 sameC 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.The weak form of the system (1) is
(5)∫Ω‍((1+u2)∇u∇φ+(1+|∇u|2)uφ)+∫Ω‍((1+v2)∇v∇ψ+(1+|∇v|2)vψ)-2αα+β∫Ω‍|u|α-2|v|βuφ-2βα+β∫Ω‍|u|α|v|β-2vψ=0,-2βα+β∫Ω‍|u|α|v|β-2vψ=0(φ,ψ)∈C0∞(Ω)×C0∞(Ω),
which is formally the variational formulation of the following functional:
(6)I0(u,v)=12∫Ω‍((1+u2)|∇u|2+u2)+12∫Ω‍((1+v2)|∇v|2+v2)-2α+β∫Ω‍|u|α|v|β.We may define the derivative ofI0 at (u,v) in the direction of (φ,ψ)∈C0∞(Ω)×C0∞(Ω) as follows:
(7)〈I0′(u,v),(φ,ψ)〉=∫Ω‍(1+u2)∇u∇φ+∫Ω‍(1+|∇u|2)uφ+∫Ω‍(1+v2)∇v∇ψ+∫Ω‍(1+|∇v|2)vψ-2αα+β∫Ω‍|u|α-2|v|βuφ-2βα+β∫Ω‍|u|α|v|β-2vψ.
We call that (u,v) is a critical point of I0 if (u,v)∈W01,2(Ω)×W01,2(Ω), ∫Ω‍u2|∇u|2<∞, ∫Ω‍v2|∇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 method [11], and change of variables [8, 9]. In this paper, we consider a perturbed functional
(8)Iμ(u,v)=14μ∫Ω‍(|∇u|4+|∇v|4)+I0(u,v)=14μ∫Ω‍(|∇u|4+|∇v|4)+12∫Ω‍((1+u2)|∇u|2+u2)+12∫Ω‍((1+v2)|∇v|2+v2)-2α+β∫Ω‍|u|α|v|β,
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:
(9)〈Iμ′(u,v),(φ,ψ)〉=μ∫Ω‍|∇u|2∇u∇φ+μ∫Ω‍|∇v|2∇v∇ψ+∫Ω‍(1+u2)∇u∇φ+∫Ω‍(1+|∇u|2)uφ+∫Ω‍(1+v2)∇v∇ψ+∫Ω‍(1+|∇v|2)vψ-2αα+β∫Ω‍|u|α-2|v|βuφ-2βα+β∫Ω‍|u|α|v|β-2vψ
for all (φ,ψ)∈C0∞(Ω)×C0∞(Ω). 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 thatα>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 Iμn′(un,vn)=0 and Iμn(un,vn)≤C for some C independent of n. Then, up to a subsequence
(10)un⟶u,vn⟶vinW01,2(Ω),un∇un⟶u∇u,vn∇vn⟶v∇vinL2(Ω),μn∫Ω‍(|∇un|4+|∇vn|4)⟶0,Iμn′(un,vn)⟶I0′(u,v)
as n→∞ and (u,v) is a critical point of  I0.Using Theorem1, we have the following existence result.Theorem 2.
Assume thatα>2, β>2 and α+β<2·2*. Then Iμ has a positive critical point (uμ,vμ) and a negative critical point (u~μ,v~μ), and (uμ,vμ) (resp., (u~μ,v~μ)) converges to a positive (resp., negative) solution for the system (1) as μ→0.Notation. We denote by ∥·∥ the norm of W01,4(Ω) and by |·|s the norm of Ls(Ω)(1≤s<+∞).


## 3. Compactness of the Perturbed Functional

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.Proposition 3.
Forμ>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,
(11)Iμ(un,vn)⟶c,Iμ′(un,vn)⟶0stronglyin(W01,4(Ω)×W01,4(Ω))*
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).For giving the proof of Proposition3, we need the following lemma firstly.Lemma 4.
Suppose that a sequence{(un,vn)}⊂W01,4(Ω)×W01,4(Ω) satisfies (11). Then
(12)limsupn→∞∥(un,vn)∥4≤(14-1α+β)-1μ-1c.Proof.
It follows from (11) that
(13)c+o(1)-1α+βo(1)∥(un,vn)∥=Iμ(un,vn)-1α+β〈Iμ′(un,vn),(un,vn)〉=(14-1α+β)μ∫Ω‍(|∇un|4+|∇vn|4)+(12-1α+β)∫Ω‍(|∇un|2+|∇vn|2)+(12-1α+β)∫Ω‍(|un|2+|vn|2)+(12-2α+β)∫Ω‍(un2|∇un|2+vn2|∇vn|2)≥(14-1α+β)μ∫Ω‍(|∇u|4+|∇v|4).
Thus we have
(14)limsupn→∞∥(un,vn)∥4≤(14-1α+β)-1μ-1c.
This completes the proof of Lemma 4.Now we give the proof of Proposition3.Proof of Proposition3.
From Lemma4, 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
(15)(un,vn)⇀(u,v)weaklyinW01,4(Ω)×W01,4(Ω)asn⟶∞,un⟶u,vn⟶vstronglyinLs(Ω)asn⟶∞forany2<s<2·2*.
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
(16)o(1)∥(un-um,vn-vm)∥=〈Iμ′(un,vn)-Iμ′(um,vm),(un-um,vn-vm)〉=μ∫Ω‍(|∇un|2∇un-|∇um|2∇um)(∇un-∇um)+μ∫Ω‍(|∇vn|2∇vn-|∇vm|2∇vm)(∇vn-∇vm)+∫Ω‍|un-um|2+∫Ω‍|∇un-∇um|2+∫Ω‍(un2∇un-um2∇um)(∇un-∇um)+∫Ω‍|vn-vm|2+∫Ω‍|∇vn-∇vm|2+∫Ω‍(vn2∇vn-vm2∇vm)(∇vn-∇vm)+∫Ω‍(un|∇un|2-um|∇um|2)(un-um)+∫Ω‍(vn|∇vn|2-vm|∇vm|2)(vn-vm)-2αα+β∫Ω‍(|un|α-2|vn|βun-|um|α-2|vm|βum)(un-um)-2βα+β∫Ω‍(|un|α|vn|β-2vn-|um|α|vm|β-2vm)(vn-vm).
We may estimate the terms involved as follows:
(17)μ∫Ω‍(|∇un|2∇un-|∇um|2∇um)(∇un-∇um)≥μ∫Ω‍|∇un-∇um|4,μ∫Ω‍(|∇vn|2∇vn-|∇vm|2∇vm)(∇vn-∇vm)≥μ∫Ω‍|∇vn-∇vm|4,∫Ω‍(un2∇un-um2∇um)(∇un-∇um)≥∫Ω‍un2|∇un-∇um|2+∫Ω‍(un2-um2)∇um(∇un-∇um)≥-|un-um|4(|un|4+|um|4)∥um∥(∥un∥+∥um∥)⟶0asm,n⟶∞,∫Ω‍(vn2∇vn-vm2∇vm)(∇vn-∇vm)≥∫Ω‍vn2|∇vn-∇vm|2+∫Ω‍(vn2-vm2)∇vm(∇vn-∇vm)≥-|vn-vm|4(|vn|4+|vm|4)∥vm∥(∥vn∥+∥vm∥)⟶0asm,n⟶∞,|∫Ω‍(un|∇un|2-um|∇um|2)(un-um)|≤(|un|4∥un∥2+|um|4∥um∥2)|un-um|4⟶0asm,n⟶∞,|∫Ω‍(vn|∇vn|2-vm|∇vm|2)(vn-vm)|≤(|vn|4∥vn∥2+|vm|4∥vm∥2)|vn-vm|4⟶0asm,n⟶∞,2αα+β|∫Ω‍(|un|α-2|vn|βun-|um|α-2|vm|βum)(un-um)|≤2αα+β∫Ω‍(|un|α-1|vn|β+|um|α-1|vm|β)|un-um|≤2αα+β(|un|α+βα-1|vn|α+ββ+|um|α+βα-1|vm|α+ββ)|un-um|α+β⟶0asm,n⟶∞,2βα+β|∫Ω‍(|un|α|vn|β-2vn-|um|α|vm|β-2vm)(vn-vm)|≤2βα+β∫Ω‍(|un|α|vn|β-1+|um|α|vm|β-1)|vn-vm|≤2βα+β(|un|α+βα|vn|α+ββ-1+|um|α+βα|vm|α+ββ-1)|vn-vm|α+β⟶0asm,n⟶∞.
Returning to (16), we have
(18)μ∫Ω‍(|∇un-∇um|4+|∇vn-∇vm|4)≤o(1)∥(un-um,vn-vm)∥+o(1),
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.


## 4. Some Asymptotic Behavior

Proposition3 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
(19)μn⟶0,Iμn(un,vn)⟶c,∥Iμn′(un,vn)∥*⟶0.The following proposition is the key of this section.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
(20)(un,vn)⇀(u,v)inW01,2(Ω)×W01,2(Ω),(un∇un,vn∇vn)⇀(u∇u,v∇v)inL2(Ω)×L2(Ω),(un(x),vn(x))⟶(u(x),v(x))a.e.x∈Ω
as n→∞.Proof.
Similar to the proof of Lemma4, by (19), we have
(21)C≥Iμn(un,vn)-1α+β〈Iμn′(un,vn),(un,vn)〉≥(14-1α+β)μn∫Ω‍(|∇un|4+|∇vn|4)+(12-1α+β)∫Ω‍(|∇un|2+|∇vn|2)+(12-1α+β)∫Ω‍(|un|2+|vn|2)+(12-2α+β)∫Ω‍(un2|∇un|2+vn2|∇vn|2).
Thus
(22)μn∫Ω‍(|∇un|4+|∇vn|4)+∫Ω‍(|∇un|2+|∇vn|2)+∫Ω‍(|un|2+|vn|2)+∫Ω‍(un2|∇un|2+vn2|∇vn|2)≤C
for some C independent of n. Then, up to a subsequence, we have
(23)(un,vn)⇀(u,v)inW01,2(Ω)×W01,2(Ω),(un∇un,vn∇vn)⇀(u∇u,v∇v)inL2(Ω)×L2(Ω),(un(x),vn(x))⟶(u(x),v(x))a.e.x∈Ω
as n→∞. This completes the proof of Proposition 5.


## 5. Proof of Main Results

In this section, we give the proof of our main results. Firstly, we prove Theorem1.Proof of Theorem1.
Note that(un,vn) satisfies the following equation:
(24)μn∫Ω‍|∇un|2∇un∇φ+μn∫Ω‍|∇vn|2∇vn∇ψ+∫Ω‍((1+un2)∇un∇φ+(1+|∇un|2)unφ)+∫Ω‍((1+vn2)∇vn∇ψ+(1+|∇vn|2)vnψ)-2αα+β∫Ω‍|un|α-2|vn|βunφ-2βα+β∫Ω‍|un|α|vn|β-2vnψ=0
for all (φ,ψ)∈W01,4(Ω)×W01,4(Ω). Since
(25)(∫Ω‍|un|4N/(N-2))(N-2)/N≤C∫Ω‍un2|∇un|2≤C,(∫Ω‍|vn|4N/(N-2))(N-2)/N≤C∫Ω‍vn2|∇vn|2≤C.
By Moser’s iteration, we have
(26)|un|L∞≤C,|vn|L∞≤C.
Hence,
(27)|u|L∞≤C,|v|L∞≤C
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=ξexp(-un), ψ=ηexp(-vn), where ξ∈C0∞(Ω), ξ≥0, η∈C0∞(Ω),η≥0. Substituting (φ,ψ) into (24), we have
(28)0=μn∫Ω‍|∇un|2∇un(∇ξexp(-un)-ξ∇unexp(-un))+μn∫Ω‍|∇vn|2∇vn(∇ηexp(-vn)-η∇vnexp(-vn))+∫Ω‍(1+un2)∇un(∇ξexp(-un)-ξ∇unexp(-un))+∫Ω‍(1+vn2)∇vn(∇ηexp(-vn)-η∇vnexp(-vn))+∫Ω‍(1+|∇un|2)unξexp(-un)+∫Ω‍(1+|∇vn|2)vnηexp(-vn)-2αα+β∫Ω‍|un|α-2|vn|βunξexp(-un)-2βα+β∫Ω‍|un|α|vn|β-2vnηexp(-vn)≤μn∫Ω‍|∇un|2∇un∇ξexp(-un)+μn∫Ω‍|∇vn|2∇vn∇ηexp(-vn)+∫Ω‍(1+un2)∇un∇ξexp(-un)+∫Ω‍(1+vn2)∇vn∇ηexp(-vn)+∫Ω‍unξexp(-un)+∫Ω‍vnηexp(-vn)-∫Ω‍(1+un2-un)|∇un|2ξexp(-un)-∫Ω‍(1+vn2-vn)|∇vn|2ηexp(-vn)-2αα+β∫Ω‍|un|α-2|vn|βunξexp(-un)-2βα+β∫Ω‍|un|α|vn|β-2vnηexp(-vn).
Note that 1+un2-un≥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 (29)0≤∫Ω‍(1+u2)∇u∇ξexp(-u)+∫Ω‍(1+v2)∇v∇ηexp(-v)+∫Ω‍uξexp(-u)+∫Ω‍vηexp(-v)-∫Ω‍(1+u2-u)|∇u|2ξexp(-u)-∫Ω‍(1+v2-v)|∇v|2ηexp(-v)-2αα+β∫Ω‍|u|α-2|v|βuξexp(-u)-2βα+β∫Ω‍|u|α|v|β-2vηexp(-v)0=∫Ω‍(1+u2)∇u∇(ξexp(-u))+∫Ω‍(1+v2)∇v∇(ηexp(-v))+∫Ω‍(1+|∇u|2)uξexp(-u)+∫Ω‍(1+|∇v|2)vηexp(-v)-2αα+β∫Ω‍|u|α-2|v|βuξexp(-u)-2βα+β∫Ω‍|u|α|v|β-2vηexp(-v).
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)→(χexpu,ωexpv) in W01,2(Ω)×W01,2(Ω), (ξn,ηn)→(χexpu,ωexpv) a.e. x∈Ω and {(ξn,ηn)} is uniformly bounded in L∞(Ω)×L∞(Ω). Then by approximations in (29) we may obtain
(30)∫Ω‍(1+u2)∇u∇χ+∫Ω‍(1+v2)∇v∇ω+∫Ω‍(1+|∇u|2)uχ+∫Ω‍(1+|∇v|2)vω-2αα+β∫Ω‍|u|α-2|v|βuχ-2βα+β∫Ω‍|u|α|v|β-2vω≥0
for all (χ,ω)≥(0,0), (χ,ω)∈C0∞(Ω)×C0∞(Ω).
Similarly, we may obtain an opposite inequality. Thus we have(31)∫Ω‍(1+u2)∇u∇χ+∫Ω‍(1+v2)∇v∇ω+∫Ω‍(1+|∇u|2)uχ+∫Ω‍(1+|∇v|2)vω-2αα+β∫Ω‍|u|α-2|v|βuχ-2βα+β∫Ω‍|u|α|v|β-2vω=0
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)
(32)∫Ω‍((1+2u2)|∇u|2+u2)+∫Ω‍((1+2v2)|∇v|2+v2)-2∫Ω‍|u|α|v|β=0.
Setting (φ,ψ)=(un,vn) in (24), we have
(33)μn∫Ω‍(|∇un|4+|∇vn|4)+∫Ω‍((1+2un2)|∇un|2+un2)+∫Ω‍((1+2vn2)|∇vn|2+vn2)-2∫Ω‍|un|α|vn|β=0.
Using ∫Ω‍|un|α|vn|β→∫Ω‍|u|α|v|β as n→∞, (32), (33), and lower semicontinuity, we obtain
(34)∫Ω‍|∇un|2⟶∫Ω‍|∇u|2,∫Ω‍|∇vn|2⟶∫Ω‍|∇v|2,∫Ω‍un2|∇un|2⟶∫Ω‍u2|∇u|2,∫Ω‍vn2|∇vn|2⟶∫Ω‍v2|∇v|2
as n→∞.
In particular, we have(35)un⟶u,vn⟶vinW01,2(Ω),un∇un⟶u∇u,vn∇vn⟶v∇vinL2(Ω),μn∫Ω‍(|∇un|4+|∇vn|4)⟶0,Iμn′(un,vn)⟶I0′(u,v)
as n→∞. This completes the proof of Theorem 1.Next, we apply the mountain pass theorem to obtain existence of critical points ofIμ. Set
(36)Σρ={(u,v)∈W01,4(Ω)×W01,4(Ω)∣∫Ω‍((1+u2)|∇u|2+u2)+∫Ω‍((1+v2)|∇v|2+v2)≤ρ2}
for ρ>0.Let us consider the functional(37)Iμ+(u,v)=14μ∫Ω‍(|∇u|4+|∇v|4)+12∫Ω‍((1+u2)|∇u|2+u2)+12∫Ω‍((1+v2)|∇v|2+v2)-2α+β∫Ω‍(u+)α(v+)β.
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
(38)(u+)α(v+)β≤ε(u+)α+β+Cε(v+)α+β.
Since
(39)∫Ω‍|u|α+β≤C(∫Ω‍u2|∇u|2)(α+β)/4,∫Ω‍|v|α+β≤C(∫Ω‍v2|∇v|2)(α+β)/4.
Then
(40)-2α+β∫Ω‍(u+)α(v+)β≥-2α+βε∫Ω‍(u+)α+β-2α+βCε∫Ω‍(u+)α+β≥-2Cα+βε∫Ω‍(∫Ω‍u2|∇u|2)(α+β)/4-2Cεα+β(∫Ω‍v2|∇v|2)(α+β)/4≥-2Cα+βερ(α+β)/2-2Cεα+βρ(α+β)/2≥-1α+βρ2
for ε,ρ small. Thus we have
(41)Iμ+(u,v)≥12∫Ω‍((1+u2)|∇u|2+u2)+12∫Ω‍((1+v2)|∇v|2+v2)-2α+β∫Ω‍(u+)α(v+)β≥12ρ2-1α+βρ2=(12-1α+β)ρ2
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
(42)Iμ+(g(1),h(1))<0,∫Ω‍((1+g2(1))|∇g(1)|2+g2(1))+∫Ω‍((1+h2(1))|∇h(1)|2+h2(1))>ρ2,supt∈[0,1]Iμ+(g(t),h(t))≤m
for some m independent of μ∈(0,1].Define(43)cμ=inf(g,h)∈Γsupt∈[0,1]Iμ+(g(t),h(t)),
where
(44)Γ={(g,h)∈C([0,1],W01,4(Ω)×W01,4(Ω))∣(g(0),h(0))=(0,0),(g(1),h(1))=(Tφ,Tψ)W01,4}.
From the mountain pass theorem we obtain that
(45)cμ≥(12-1α+β)ρ2
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 constantsρ and m independent of μ such that Iμ has a positive critical point (uμ,vμ) satisfying
(46)(12-1α+β)ρ2≤Iμ(uμ,vμ)≤m.Finally, we give the proof of Theorem2.Proof of Theorem2.
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.



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*Source: 431672-2013-07-01.xml*