# Multiplicity of Positive Solutions for a Second-Order Elliptic System of Kirchhoff Type

Authors: S. Khademloo; E. Valipour; A. Babakhani
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study elliptic problems of Kirchhoff type inΩ⊂RN(N≥3). Using variational tools, we establish the existence of at least two nontrivial and nonnegative solutions.


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## Body

## 1. Introduction and Preliminaries

In this paper, we are concerned with the following problem:(1)-a1+b1∫Ω‍∇u2+axu2dxΔu+axu=Fu(x,u,v)+λbxuq-2uinΩ,-(a2+b2(∫Ω‍∇v2+a(x)v2)dx)(Δv+a(x)v)=Fv(x,u,v)+μc(x)vq-2vinΩ,u=v=0on⁡∂Ω,
where Ω⊂RN  (N≥3) is a bounded domain with the smooth boundary ∂Ω such that 0∈Ω, Δu=div⁡(∇u) is the Laplacian operator, 1<q<2, λ,μ>0, ai,bi>0  (i=1,2), and a,b,c∈C(Ω,R+), the function F∈C1(Ω¯×(R+)2,R+), is positively homogeneous of degree α=4N/(N-2) which is the Sobolev critical exponent; that is, F(x,tu,tv)=tαF(x,u,v), (t>0) holds for all (x,u,v)∈Ω¯×(R+)2, (Fu,Fv)=∇F.In recent years, there have been many papers concerned with the existence of the positive solutions for Kirchhoff equation(2)-M∫Ω‍∇u2dxΔu=λfx,uinΩu=Δu=0on⁡∂Ω,
which is related to the stationary analogue of the Kirchhoff equation
(3)utt-M∫Ω‍∇u2dxΔu=f(x,u),
where M(s)=a+bs, a,b>0. It was proposed by Kirchhoff [1] as an extension of the classical D’Alembert wave equation for free vibrations of elastic strings.Some interesting studies on these problems by variational methods can be found in [2–6]. As for perturbed fourth-order Kirchhoff-type elliptic problems, in [7] the following equation,
(4)ΔΔup-2Δu-M∫Ω‍∇updxp-1Δpu+ρup-2u=λfx,uinΩu=Δu=0on⁡∂Ω,
where p>max⁡1,N/2, λ>0, f:Ω×R→R is an L1-Carathéodory function, and M:[0,+∞[→R is a continuous function, has been investigated. The authors proved (4) has multiple nontrivial weak solutions.In [8] the authors established the existence of a weak solution for the following system equation:
(5)-M1∫Ω‍∇updxp-1Δpu=f(u,v)+ρ1xinΩ-M2∫Ω‍∇vpdxp-1Δpv=f(u,v)+ρ2xinΩ∂u∂η=∂v∂η=0on⁡∂Ω,
where M1(t),M2(t)≥m0>0.Motivated by the results of the above cited papers, we will attempt to treat problem (1) and extend the results for our problem.In this paper we make the following assumptions.LetS be the best Sobolev embedding constant defined by
(6)S=inf⁡u∈W01,2(Ω)∖{0}∫Ω‍∇u2+axu2dx∫Ω‍uαdx2/α
and let |Ω| be the Lebesgue measure of Ω; ·∞ denotes the L∞(Ω) norm, β=min⁡{b1,b2}, and
(7)C(q,N,K,S,Ω,a1,a2)=Aα-2Sq/2α-qΩ(α-q)/αA2-qSα/2Kα-q(2-q)/(α-2),C0=q2C(q,N,K,S,Ω,a1,a2),
where A=min⁡{a1,a2}.Also the following hold:(F1)
F:Ω¯×R+×R+→R+ is a C1 function and F(x,tu,tv)=tαF(x,u,v),(F2)
F(x,u,0)=F(x,0,v)=Fu(x,u,0)=Fv(x,0,v)=0 where u,v∈R+,(F3)
Fu(x,u,v),Fv(x,u,v) are strictly increasing function about u, v for all u>0, v>0.In addition, using assumption (F1), we have the so-called Euler identity(8)u,v·∇F(x,u,v)=αF(x,u,v)
and, for a positive constant K,
(9)F(x,u,v)≤Ku2+v2α/2.LetW01,2(Ω) be the completion of C0∞(Ω) with respect to the norm (∫Ω‍|∇u|2)1/2.It is easy to show that, for everyu∈W01,2(Ω), the above norm is equivalent with u=(∫Ω‍|∇u|2+a(x)|u|2)1/2. Problem (1) is posed in the framework of the Sobolev space
(10)E={(u,v)∈W01,2Ω2∣∫Ω‍a(x)(u2+v2)dx<+∞},
with the standard norm
(11)u,vE=∫Ω‍∇u2+axu2dx+∫Ω‍∇v2+axv2dx1/2.
We will look for solutions of (1) by finding critical points of the energy functional Jλ,μ:E→R given by
(12)Jλ,μu,v=a12u2+a22v2+b14u4+b24v4-1α∫Ω‍Fx,u,vdx-1qKλ,μu,v,
where Kλ,μ:E→R is the functional defined by
(13)Kλ,μ=∫Ω‍(λb(x)uq+μc(x)vq)dx.
It is well known that the functional Jλ,μ∈C1(E,R). For any (φ1,φ2)∈E, there holds
(14)Jλ,μ′u,v,u,v=a1+b1∫Ω‍∇u2+axu2dx×∫Ω‍∇u∇φ1+axuφ1dx+a2+b2∫Ω‍∇v2+axv2dx×∫Ω‍∇v∇φ2+axuφ2dx-1α∫Ω‍Fux,u,vφ1+Fvx,u,vφ2dx-λ∫Ω‍bxuq-2uφ1dx-μ∫Ω‍cxvq-2vφ2dx.
Consider the Nehari manifold
(15)Nλ,μ=u,v∈E∖0,0Jλ,μ′u,v,u,v=0.
Note that (u,v)∈Nλ,μ if and only if
(16)a1u2+a2v2+b1u4+b2v4-∫Ω‍F(x,u,v)dx-Kλ,μ(u,v)=0.
So Nλ,μ contains all nontrivial weak solutions of (1).DefineΦλ,μ(u,v)=〈Jλ,μ′(u,v),(u,v)〉. Then, for (u,v)∈Nλ,μ,
(17)Φλ,μ′u,v,u,v=2a1u2+2a2v2+4b1u4+4b2v4-α∫Ω‍F(x,u,v)dx-qKλ,μ(u,v)=2b1u4+2b2v4-(α-2)∫Ω‍F(x,u,v)dx-(q-2)Kλ,μ(u,v)=-2a1u2-2a2v2-(α-4)∫Ω‍F(x,u,v)dx-(q-4)Kλ,μ(u,v)=(2-α)(a1u2+a2v2)+(4-α)(b1u4+b2v4)-(q-α)Kλ,μ(u,v)=(2-q)(a1u2+a2v2)+4-qb1u4+b2v4-α-q∫Ω‍Fx,u,vdx.
Now, we split Nλ,μ into three parts:
(18)Nλ,μ+={(u,v)∈Nλ,μ∣〈Φλ,μ′(u,v),(u,v)〉>0};Nλ,μ0={(u,v)∈Nλ,μ∣〈Φλ,μ′(u,v),(u,v)〉=0};Nλ,μ-={(u,v)∈Nλ,μ∣〈Φλ,μ′(u,v),(u,v)〉<0}.


## 2. Statement of the Main Results

Let us first defineϱλ,μ=(λ|b|∞)2/(2-q)+(μ|c|∞)2/(2-q) and the main results read as follows.Theorem 1.
If(λ,μ) satisfy 0<ϱλ,μ<C(q,N,K,S,|Ω|,a1,a2) and (F1)–(F3) hold, then problem (1) has at least one positive solution.Theorem 2.
If(λ,μ) satisfy 0<ϱλ,μ<C0* and (F1)–(F3) hold, then problem (1) has at least two positive solutions.Note that, using assumption (F3), we have thatFu,Fv∈C(Ω¯×(R+)2,R+) are positively homogeneous of degree α-1. This implies that
(19)Fux,u,v≤M(uα-1+vα-1),|Fv(x,u,v)|≤M(uα-1+vα-1),∀x∈Ω¯,u,v∈R+,
for some positive constant M. Similar to Willem [9, Theorem A.2], we consider the continuity of the superposition operator
(20)A:L2(Ω)⟶Lq(Ω):(u,v)⟼f(x,u,v).Lemma 3.
Assume that|Ω|<∞, r<∞, f∈C(Ω¯×R2,R), and
(21)|f(x,u,v)|≤c(1+u2/r+|v|2/r).
Then, for every (u,v)∈L2(Ω), f(·,u,v)∈Lr(Ω) and the operator A:Lp(Ω)→Lr(Ω):(u,v)↦f(x,u,v) is continuous.Now, we consider the functionalψ(u,v)=∫Ω‍F(x,u,v)dx; then we have the following result.Lemma 4.
Assume that|Ω|<∞, F∈C(Ω¯×(R+)2,R+) satisfying (F3); then the functional ψ is of class C1(E,R+) and
(22)〈ψ′(u,v),(φ1,φ2)〉=∫Ω‍(Fu(x,u,v)φ1+Fv(x,u,v)φ2)dx,
where (u,v),(φ1,φ2)∈E.Proof.
The proof is almost the same as that in [10].Lemma 5.
The energy functionalJλ,μ is coercive and bounded below on Nλ,μ.Proof.
If(u,v)∈Nλ,μ, then by the Hölder inequality and the Sobolev embedding theorem
(23)Jλ,μu,v=α-22α(a1u2+a2v2)+α-44α(b1u4+b2v4)-α-qαqKλ,μ(u,v)≥α-22α(a1u2+a2v2)+α-44α(b1u4+b2v4)-α-qαqϱλ,μ(2-q)/2S-q/2|Ω|(α-q)/α(u,v)Eq.
Thus, Jλ,μ is coercive and bounded below on Nλ,μ.Lemma 6.
Suppose that(u0,v0) is a local minimizer for Jλ,μ on Nλ,μ and that (u0,v0)∉Nλ,μ0. Then Jλ,μ′(u0,v0)=0 in E-1 (the dual space of the Sobolev space E).Proof.
If(u0,v0) is a local minimizer for Jλ,μ on Nλ,μ, then (u0,v0) is a solution of the optimization problem minimizer Jλ,μ(u,v) subject to Φλ,μ(u,v)=0. Hence, by the theory of Lagrange multipliers, there exists ξ1∈R, such that
(24)Jλ,μ′(u0,v0)=ξ1Φλ,μ′u0,v0inE-1(Ω),
and thus,
(25)〈Jλ,μ′(u0,v0),(u0,v0)〉=ξ1〈Φλ,μ′(u0,v0),(u0,v0)〉.
Since (u0,v0)∈Nλ,μ, we have 〈Jλ,μ′(u0,v0),(u0,v0)〉=0. Moreover, 〈Φλ,μ′(u0,v0),(u0,v0)〉≠0, so ξ1=0. This completes the proof.Lemma 7.
If0<ϱλ,μ<C(q,N,K,S,|Ω|,a1,a2), then Nλ,μ0=∅.Proof.
Suppose otherwise that0<ϱλ,μ<C(q,N,K,S,|Ω|,a1,a2) such that Nλ,μ0≠∅.
Then, for(u,v)∈Nλ,μ0,
(26)0=〈Φλ,μ′(u,v),(u,v)〉=(2-α)(a1u2+a2v2)+4-αb1u4+b2v4-q-αKλ,μu,v=(2-q)(a1u2+a2v2)+b1u4+b2v4-(α-q)∫Ω‍F(x,u,v)dx.
By the Holder inequality and the Sobolev embedding theorem
(27)(α-2)(a1u2+a2v2)+(α-4)(b1u4+b2v4)=α-qKλ,μu,v≤(α-q)ϱλ,μ(2-q)/2S-q/2Ω(α-q)/α(u,v)Eq.
So,
(28)Aα-2u,vE2≤(α-2)(a1u2+a2v2)≤(q-α)ϱλ,μ(2-q)/2S-q/2|Ω|(α-q)/α(u,v)Eq.
Thus,
(29)u,vE≤α-qϱλ,μ2-q/2S-q/2Ωα-q/αAα-21/(2-q)
and, by the Minkowski inequality, the Sobolev embedding theorem, and (9),
(30)∫Ω‍Fx,u,vdx≤K∫Ω‍u2+v2α/2dx2/α·α/2≤K∫Ω‍uαdx2/α+∫Ω‍vαdx2/αα/2≤KS-α/2(u,v)Eα.
Thus,
(31)u,vE≥A2-qSα/2Kα-q1/α-2.
This implies that
(32)ϱλ,μ≥C(q,N,K,S,Ω,a1,a2),
which is a contradiction. Thus, we can conclude that if
(33)0<ϱλ,μ<C(q,N,K,S,Ω,a1,a2),
we have Nλ,μ0=∅.By Lemma7, we write Nλ,μ=Nλ,μ+∪Nλ,μ- and define
(34)θλ,μ=inf⁡(u,v)∈Nλ,μJλ,μ(u,v);θλ,μ+=inf⁡(u,v)∈Nλ,μ+Jλ,μ(u,v);θλ,μ-=inf⁡u,v∈Nλ,μ-Jλ,μu,v.
Then we have the following result.Lemma 8.
Consider the following.(i)
If0<ϱλ,μ<C(q,N,K,S,|Ω|,a1,a2), then one has θλ,μ≤θλ,μ+<0.(ii)
If0<ϱλ,μ<C0, then θλ,μ->d0 for some constant
(35)d0=d0(q,N,K,S,Ω,ϱλ,μ,a1,a2,b1,b2)>0.Proof.
(i) Let(u,v)∈Nλ,μ+. Then
(36)(2-q)(a1u2+a2v2)+(4-q)(b1u4+b2v4)-(α-q)∫Ω‍F(x,u,v)dx>0
and so
(37)Jλ,μu,v=(12-1q)(a1u2+a2v2)+(14-1q)(b1u4+b2v4)+(1q-1α)∫Ω‍F(x,u,v)dx<q-2q12-1αa1u2+a2v2+q-4q(14-1α)(b1u4+b2v4)<0.
Thus, from the definition of θλ,μ and θλ,μ+, we can deduce that θλ,μ≤θλ,μ+<0.
(ii) Let(u,v)∈Nλ,μ-. Then
(38)(2-q)(a1u2+a2v2)+(4-q)(b1u4+b2v4)-(α-q)∫Ω‍F(x,u,v)dx<0.
By (8) we have
(39)(2-q)(a1u2+a2v2)+(4-q)(b1u4+b2v4)<α-q∫Ω‍Fx,u,vdx≤α-qKS-α/q(u,v)Eα.
This implies that
(40)u,vE>A2-qSα/qKα-q1/(α-2).
By (23) in the proof of Lemma 5,
(41)Jλ,μu,v≥A(12-1α)(u,v)E2-(1q-1α)ϱλ,μ(2-q)/2S-q/2Ω(α-q)/α(u,v)Eq>A(12-1α)Aq-2Sα/qKα-q2/(α-2)-(1q-1α)ϱλ,μ(2-q)/2S(N-q)/2Ω(α-q)/α×Aq-2Sα/qKα-qq/(α-2).
Thus, if 0<λ<C0, then
(42)Jλ,μ(u,v)>d0∀(u,v)∈Nλ,μ-,
for some d0=d0  (λ,μ,q,N,K,S,|Ω|,|b|∞,|c|∞,a1,a2,b1,b2)>0. This completes the proof.For each(u,v)∈E with ∫Ω‍F(x,u,v)dx>0, set
(43)tmax⁡=α-q∫Ω‍Fx,u,vdx-1TT2-qa1u2+a2v2=mm+4-qb1u4+b2v4TT=m×α-q∫Ω‍Fx,u,vdx-11/α-2>0.
Then we have the following.Lemma 9 (see [11, Lemma 2.6]).
For each(u,v)∈E with ∫Ω‍F(x,u,v)dx>0, there are unique 0<t+<tmax⁡<t- such that (t+u,t+v)∈Nλ,μ+,(t-u,t-v)∈Nλ,μ- and Jλ,μ(t+u,t+v)=inf⁡0≤t≤tmax⁡⁡Jλ,μ(tu,tv); Jλ,μ(t-u,t-v)=sup⁡t≥0⁡Jλ,μ(tu,tv).


## 3. Proof of the Main Theorems

We will need the following lemma.Lemma 10 (see [12]).
Consider the following.(i)
If0<ϱλ,μ(2-q)/2<C(q,N,K,S,|Ω|,ϱλ,μ,a1,a2), then there exists a (PS)θλ,μ-sequence {(un,vn)}⊂Nλ,μ in E for Jλ,μ.(ii)
If0<ϱλ,μ<C0, then there exists a (PS)θλ,μ--sequence {(un,vn)}⊂Nλ,μ- in E for Jλ,μ.Theorem 11.
If0<ϱλ,μ<C(q,N,K,S,|Ω|,ϱλ,μ,a1,a2) and (F1)–(F3) hold, then Jλ,μ has a minimizer (u0+,v0+) in Nλ,μ+ and it satisfies the following: (i)
Jλ,μ(u0+,v0+)=θλ,μ=θλ,μ+;(ii)
(u0+,v0+) is a positive solution of (1).Proof.
By Lemma10(i), there exists a minimizing sequence {(un,vn)} for Jλ,μ on Nλ,μ such that
(44)Jλ,μun,vn=θλ,μ+o1,Jλ,μ′un,vn=o1.
Then by Lemma 5 and the compact imbedding theorem, there exist a subsequence {(un,vn)} and (u0+,v0+)∈E such that
(45)un⇀u0+weaklyinW01,2,un⟶u0+stronglyinLq(Ω),vn⇀v0+weaklyinW01,2,vn⟶v0+stronglyinLq(Ω).
This implies that Kλ,μ(un,vn)→Kλ,μ(u0+,v0+) as n→∞. By (44) and (45), it is easy to prove that (u0+,v0+) is s weak solution of (1). Since
(46)Jλ,μun,vn=N+24N(a1u2+a2v2)+12N(b1u4+b2v4)-α-qαqKλ,μ(un,vn)≥-α-qαqKλ,μ(un,vn)
and by Lemma 8(i),
(47)Jλ,μ(un,vn)⟶θλ,μ<0asn⟶∞.
Letting n→∞, we see that Kλ,μ(u0+,v0+)>0. Thus, (u0+,v0+) is a nontrivial solution of problem (1). Now it follows that un→u0+ strongly in W01,2 and vn→v0+ strongly in W01,2 and Jλ,μ(u0+,v0+)=θλ,μ. By (u0+,v0+)∈Nλ,μ and applying Fatou’s lemma, we get
(48)θλ,μ≤Jλ,μu0+,v0+=N+24N(a1u0+2+a2v0+2)×12N(b1u0+4+b2v0+4)-α-qαqKλ,μu0+,v0+≤lim⁡inf⁡n→∞N+24N22N+24Na1un2+a2vn2≤lim⁡inf⁡n→∞vvv+12Nb1u4+b2v4≤lim⁡inf⁡n→∞vvv-α-qαqKλ,μun,vnN+24N22≤lim⁡inf⁡n→∞Jλ,μ(un,vn)=θλ,μ.
This implies that
(49)Jλ,μ(u0+,v0+)=θλ,μ;lim⁡n→∞un2=u0+2,lim⁡n→∞vn2=v0+2.
Let u~n=un-u0+, v~n=vn-v0+; then by the Brezis-Lieb lemma [13], this implies
(50)u~n2=un2-u0+2,v~n2=vn2-v0+2.
Therefore, un→u0+ strongly in W01,2 and vn→v0+ strongly in W01,2. Moreover, we have (u0+,v0+)∈Nλ,μ+. In fact, if (u0+,v0+)∈Nλ,μ-, by Lemma 9, there are unique t0+ and t0- such that (t0+u0+,t0+v0+)∈Nλ,μ+ and (t0-u0+,t0-v0+)∈Nλ,μ-. In particular, we have t0+<t0-=1. Since
(51)ddtJλ,μt0+u0+,t0+v0+=0,d2dt2Jλ,μ(t0+u0+,t0+v0+)>0,
there exists t0+<t¯≤t0- such that Jλ,μ(t0+u0+,t0+v0+)<Jλ,μ(t¯u0+,t¯v0+). By Lemma 9,
(52)Jλ,μt0+u0+,t0+v0+<Jλ,μ(t¯u0+,t¯v0+)≤Jλ,μ(t0-u0+,t0-v0+)=Jλ,μ(u0+,v0+)
which is a contradiction. It follows from the maximum principle that (u0+,v0+) is a positive solution of problem (1). This completes the proof.The following two lemmas are similar to those in [14].Lemma 12.
If{(un,vn)}⊂E is a (PS)c-sequence for Jλ,μ with (un,vn)⇀(u,v) in E, then Jλ,μ′(u,v)=0, and there exists a positive constant Λ, such that Jλ,μ(u,v)≥-Λϱλ,μ.Lemma 13.
If{(un,vn)}⊂E is a (PS)c-sequence for Jλ,μ, then {(un,vn)} is bounded in E.Define(53)SF≔inf⁡u,v∈Ea1u2+a2v2+b1u4+b2v4∫Ω‍Fx,u,vdx2/α:∫Ω‍Fx,u,vdx>0+b2v4∫Ω‍2/α.
In addition, we need the following version of the Brezis-Lieb lemma [13].Lemma 14.
ConsiderF∈C1(Ω¯,(R+)2) with F(x,0,0)=0 and
(54)∂Fx,u,v∂u,∂Fx,u,v∂v≤C1(u+v)
for some C1>0. Let {(uk,vk)} be a bounded sequence in L2(Ω¯,(R+)2) such that (uk,vk)⇀(u,v) weakly in E. Then as k→∞,
(55)∫Ω‍Fx,uk,vkdx⟶∫Ω‍F(x,uk-u,vk-v)dx+∫Ω‍F(x,u,v)dx.Lemma 15.
J
λ
,
μ satisfies the (PS)c condition with c satisfying
(56)-∞<c<c∞=(14-1α)SF2N/(N+2)-Λϱλ,μ.Proof.
Let{(un,vn)}⊂E be a (PS)c-sequence for Jλ,μ with c∈(-∞,c∞). It follows from Lemma 13 that {(un,vn)} is bounded in E, and then (un,vn)⇀(u,v) up to a subsequence, where (u,v) is a critical point of Jλ,μ. Furthermore, we may assume
(57)un⇀u,vn⇀vinW01,2un⟶u,vn⟶vinLq(Ω)un⟶u,vn⟶va.e.on⁡Ω.
Hence we have that Jλ,μ′(u,v)=0 and
(58)Kλ,μ(un,vn)⟶Kλ,μ(u,v).
Let u~n=un-u, v~n=vn-v. Then by the Brezis-Lieb lemma [13], we obtain
(59)u~n2⟶un2-u2,v~n2⟶vn2-v2asn⟶∞,
and by Lemma 14,
(60)∫Ω‍F(x,u~n,v~n)dx⟶∫Ω‍F(x,un,vn)dx-∫Ω‍F(x,u,v)dx.
Since Jλ,μ(un,vn)=c+o(1), Jλ,μ′(un,vn)=o(1) and (58)–(60), we can deduce that
(61)a12u~n2+a22v~n2+b14u~n4+b24v~n4-1α∫Ω‍Fx,u~n,v~ndx=c-Jλ,μ(u,v)+o(1).
So
(62)14(a1u~n2+a2v~n2+b1u~n4+b2v~n4)-1α∫Ω‍F(x,u~n,v~n)dx≤a12u~n2+a22v~n2+b14u~n4+b24v~n4-1α∫Ω‍Fx,u~n,v~ndx=c-Jλ,μu,v+o1,a1u~n2+a2v~n2+b1u~n4+b2v~n4-∫Ω‍F(x,u~n,v~n)dx=o(1).
Hence, we may assume that
(63)a1u~n2+a2v~n2+b1u~n4+b2v~n4⟶l,∫Ω‍F(x,u~n,v~n)dx⟶l.
If l=0, the proof is complete. Assume l>0; then from (63), we obtain
(64)SFl2/α=SFlim⁡n→∞∫Ω‍Fx,u~n,v~ndx2/α≤lim⁡n→∞(a1u~n2+a2v~n2+b1u~n4+b2v~n4)=l
which implies that l≥SF2N/(N+2). In addition, from Lemma 12, (61), and (63), we get
(65)c≥(14-1α)l+Jλ,μ(u,v)≥(14-1α)SF2N/(N+2)-Λϱλ,μ,
which contradicts c<(1/4-1/α)SF2N/(N+2)-Λϱλ,μ.Lemma 16.
There exists a nonnegative function(u,v)∈E∖{(0,0)} and C*>0 such that, for ϱλ,μ∈(0,C*), one has
(66)sup⁡t≥0Jλ,μ(tu,tv)<c∞.
In particular, θλ,μ-<c∞ for all ϱλ,μ∈(0,C*).Proof.
Since0∈Ω, there is ρ0>0 such that BN(0;2ρ0)⊂Ω. Now, we consider the functional I:E→R defined by
(67)Iu,v=a12u2+a22v2-1α∫Ω‍Fx,u,vdx∀(u,v)∈E
and define a cut-off function η(x)∈C0∞(Ω) such that η(x)=1 for |x|<ρ0, η(x)=0 for |x|>2ρ0, 0≤η≤1, and |∇η|≤C. For ε>0, let
(68)uε(x)=η(x)(ε+x2)(N-2)/2.
From [14], we have the following estimates:
(69)∫Ω‍uεαdx2/α=ε-(N-2)/2ULα(RN)2+O(ε),∫Ω‍∇uε2dx=ε-(N-2)/2∇UL2(RN)2+O(1),∫Ω‍(∇uε2+a(x)uε2)dx∫Ω‍uεαdx2/α=S+O(ε(N-2)/2),
where U(x)=(1+x2)-(N-2)/2∈W1,2(RN) is a minimizer of
(70)∫Ω‍∇u2+axu2dxuLαRN2u∈W1,2(RN)∖{0};
that is,
(71)∫Ω‍(∇U2+a(x)U2)dxULα(RN)2=S=inf⁡u∈W1,2(RN)∖{0}∫Ω‍(∇u2+a(x)u2)dxuLα(RN)2.
Set u0=e1uε, v0=e2uε and (u0,v0)∈E, where (e1,e2)∈(R+)2 and inf⁡x∈Ω¯F(x,e1,e2)≥K. Then, by (F1), (9), the definition of SF, and (69), we obtain that
(72)sup⁡t≥0I(te1uε,te2uε)≤N+24Na1e1uε2+a2e2uε2∫Ω‍Fx,e1uε,e2uεdx2/α2N/(N+2)≤N+24Na1e12+a2e22uε2∫Ω‍uεαFx,e1,e2dx2/α2N/(N+2)≤N+24Na1e12+a2e22K2/α2N/(N+2)×uε2∫Ω‍uεαdx2/α2N/(N+2)≤N+24Na1e12+a2e22K2/α2N/(N+2)×S+OεN-2/22N/(N+2)=N+24Na1e12+a2e22K2/α2N/N+2×S2N/N+2+OεN-2/2≤N+24NSF2N/(N-2)+O(ε(N-2)/2),
where the following fact has been used:
(73)sup⁡t≥0t22A-tααB=N+24NAB2/α(N-2)/2A,B>0.
We can choose δ1>0 such that, for all ϱλ,μ∈(0,δ1), we have
(74)c∞=14-1αSF2N/(N+2)-Λϱλ,μ>0.
Using the definitions of Jλ,μ and (u0,v0), we get
(75)Jλ,μtu0,tv0≤t2max⁡{a1,a2}2(u0,v0)E2∀t≥0,λ,μ>0,
which implies that there exists t0∈(0,1) satisfying
(76)sup⁡0≤t≤t0Jλ,μ(t0u0,t0v0)<c∞,∀ϱλ,μ∈(0,δ1).
On the other hand,
(77)sup⁡t≥t0Jλ,μ(tu0,tv0)=sup⁡t≥t0tqqI(tu0,tv0)+b14tu04+b24tv04=supt≥t0vvv-tqqKλ,μu0,v0≤N+24NSF2N/N-2+OεN-2/2+max⁡b1,b2t4e14+e242uε4-t0qqe1qλb0+e2qμc0∫BN(0,ρ0)‍uεqdx≤N+24NSF2N/N-2+OεN-2/2+max⁡b1,b2t4e14+e242uε4-t0qqmλb0+μc0∫BN(0,ρ0)‍uεqdx,
where m=min⁡{e1q,e2q}, b0=min⁡b(x), and c0=min⁡c(x) on B(0,ρ0).
Let0<ε≤ρ02; we get
(78)∫BN(0,ρ0)‍uεqdx=∫BN(0,ρ0)‍1ε+x2((N-2)/2)qdx≥∫BN(0,ρ0)‍12ρ02((N-2)/2)qdx=C2(N,q,ρ0).
Combining with (77) and the above inequality, for all ε=ϱλ,μ2/(N-2p)∈(0,ρ02), we have the following.
According to properties ofuε and F((F1)–(F3),4.4), we can conclude that there exists the positive constant C3 such that uε4≤C3, so we have
(79)sup⁡t≥t0Jλ,μtu0,tv0≤N+24NSF2N/(N+2)+O(ϱλ,μ)+max⁡{b1,b2}t4(e14+e24)2C3-t0qqϱλ,μmC2.
There exists a constant Λ1, such that (N+2/4N)SF2N/(N+2)-Λ1ϱλ,μ≤c∞.
Hence, we can chooseδ2>0 such that, for all ϱλ,μ∈(0,δ2), we obtain
(80)O(ϱλ,μ)+max⁡{b1,b2}t4(e14+e24)2C3-t0qqϱλ,μmC2<-Λ1ϱλ,μ.
If we see C*=min⁡{δ1,ρ0N-2,δ2} and ε=ϱλ,μ2/(N-2p), then for λ∈(0,C*) we have
(81)sup⁡t≥t0Jλ,μ(tu0,tv0)<c∞.
Finally, we prove that θλ,μ-<c∞ for all ϱλ,μ∈(0,C*). Recall that (u0,v0)=(e1uε,e2uε). It is easy to see that
(82)∫‍F(x,u0,v0)dx>0.
Combining this with Lemma 9, from the definition of θλ,μ- and (81), we get that there exists t0>0 such that (t0u0,t0v0)∈Nλ,μ- and
(83)θλ,μ-≤Jλ,μt0u0,t0v0≤sup⁡t≥0Jλ,μt0u0,t0v0<c∞
for all ϱλ,μ∈(0,C*).Theorem 17.
Ifϱλ,μ∈(0,C0*) and (F1)–(F3) hold, then Jλ,μ has a minimizer (u0-,v0-) in Nλ,μ- and it satisfies the following: (i)
Jλ,μ(u0-,v0-)=θλ,μ-,(ii)
(u0-,v0-) is a positive solution of problem (1),
whereC0*=min⁡{C*,C0}.Proof.
By Lemma10(ii), there is a (PS)θλ,μ--sequence {(un,vn)}⊆Nλ,μ- in E for Jλ,μ for all ϱλ,μ∈(0,C0). From Lemmas 15, 16, and 8(ii), for ϱλ,μ∈(0,C*),Jλ,μ satisfies (PS)θλ,μ- condition and θλ,μ->0. Since Jλ,μ is coercive on Nλ,μ, we get that (un,vn) is bounded in E. Therefor, there exists a subsequence still denoted by (un,vn) and (u0-,v0-)∈Nλ,μ- such that (un,vn)→(u0-,v0-) strongly in E and Jλ,μ(u0-,v0-)=θλ,μ->0 for all ϱλ,μ∈(0,C0*). Finally, by the same arguments as in the proof of Theorem 11, for all ϱλ,μ∈(0,C0*), we have that u0- is a positive solution of problem (1).Now, we complete the proof of Theorems1 and 2. By Theorem 11, we obtain that, for all ϱλ,μ∈C(q,N,K,S,|Ω|,a1,a2), problem (1) has a positive solution (u0+,v0+)∈Nλ,μ+. On the other hand, from Theorem 17, we get the second positive solution u0-,v0-∈Nλ,μ- for all 0<ϱλ,μ<C0*<C(q,N,K,S,|Ω|,a1,a2). Since Nλ,μ+∩Nλ,μ-=∅, this implies that (u0+,v0+) and (u0-,v0-) are distinct. This completes the proof of Theorems 1 and 2.



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*Source: 280130-2014-11-12.xml*