# The Numbers of Positive Solutions by the Lusternik-Schnirelmann Category for a Quasilinear Elliptic System Critical with Hardy Terms

Authors: Mustapha Khiddi
Journal: Abstract and Applied Analysis
 (2019)

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## Abstract

In this paper, we study the quasilinear elliptic system with Sobolev critical exponent involving both concave-convex and Hardy terms in bounded domains. By employing the technique introduced by Benci and Cerami (1991), we obtain at leastcat(Ω)+1 distinct positive solutions.


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## Body

## 1. Introduction and Main Result

In this paper, we are concerned with the multiplicity of positive solutions of the following critical problem:(1)-Δpu-νup-2uxp=1p∗∂F∂ux,u,v+fλxuq-2uinΩ,-Δpv-νvp-2vxp=1p∗∂F∂vx,u,v+gμxvq-2vinΩ,u=v=0on∂Ω,where Ω is a smooth bounded domain of RN, N≥3, 0∈Ω, 1<q<p<N, p∗=pN/N-p is the critical Sobolev exponent, 0<ν<ν¯ where ν¯=N-p/pp is the best Hardy constant, and the parameter λ>0, μ>0, we assume that fλ(x)=λf+(x)+f-(x) and gμ(x)=μg+(x)+g-(x) where the weight functions f and g satisfy the following conditions: H1f,g∈C(Ω¯) with f+∞=g+∞=1, where f±=max⁡±f,0≠0 and g±=max⁡±g,0≠0.And the functionF satisfies the following conditions:f1F∈C1(Ω¯×(R+)2,R+), such that ∀t>0(2)Fx,tu,tv=tp∗Fx,u,v∀x,u,v∈Ω¯×R+2.f2Fx,u,0=Fx,0,v=∂F/∂u(x,u,0)=(∂F/∂v)(x,0,v)=0, where u,v∈R+.f3∂Fx,u,v/∂u=∂Fx,u,v/∂v are strictly increasing functions about u and v for all u>0, v>0.f4(u,v).∇Fx,u,v=p∗F(x,u,v) with ∂Fx,u,v/∂u,∂Fx,u,v/∂v=∇F.f5F(x,u,v)≤K(up+vp)p∗/p for some constant K>0.Remark 1.
We deduce form the conditions(f1), (f2), and (f3) that the functional (u,v)→ψ(u,v)=∫ΩF(x,u,v)dx is of class C1(W01,p(Ω)×W01,p(Ω),R+) and(3)ψ′u,v,a,b=∫Ω∂Fx,u,v∂ua+∂Fx,u,v∂vbdx,where (u,v),(a,b)∈W01,p(Ω)×W01,p(Ω), and ∂F/∂u,∂F/∂v∈C1(Ω×(R+)2,R+) such that (∂F/∂u)(x,tu,tv)=tp∗-1∂F/∂u(x,u,v) and (∂F/∂v)(x,tu,tv)=tp∗-1∂F/∂v(x,u,v).
Moreover, there existsC>0 such that(4)∂F∂ux,u,v≤Cup∗-1+vp∗-1∂F∂vx,u,v≤Cup∗-1+vp∗-1∀x∈Ω¯,u,v∈R+.
The proof is almost the same as that in Chu and Tang [1].Recently, many papers have studied the multiplicity of positive solutions by way of fibering method and the notions of topological indices category for different semilinear, quasilinear, and nonlocal problems involving a critical exponent and concave and convex nonlinearities (see [2–4]). Our goal here is to give a new result for this system by linking the number of positive solutions with the topology of the domain Ω. More precisely with the Category index, let us note catY(X) is the least number of closed and contractible sets in Y which cover X. Our main result is the following.Theorem 2.
LetN>p2 and p∗-N/N-p≤q<p. Suppose that F satisfies (f1)-(f5) and the functions f, g satisfy the condition (H1). Then, there exists Λ∗>0 such that if for each λp/p-q+μp/p-q∈(0,Λ∗), problem (1) has at least cat(Ω)+1 distinct positive solutions.This paper is composed of four sections. In Section2, we give some results for the Nehari manifold associated of the energy functional and fibering maps. In Section 3, we will build homotopies between Ω and certain sublevel set of the energy functional associated with (1). Finally we prove the result in Section 4.


## 2. TheNehari Manifold Associated with the Energy Functional and Fibering Maps

Let the Sobolev spaceW01,p(Ω)×W01,p(Ω) with the usual norm: (5)u,vW=up+vp1/p,u=uW01,pΩ=∫Ω∇up-νupxpdx1/p,ν∈0,ν¯.Also, the standard norm of the space Lp(Ω) is uLp(Ω)=(∫Ω‍updx)1/p. Moreover, a pair of functions (u,v)∈W is said be to a weak solution of problem (1) if (6)∫Ω∇up-2∇u∇φ1-νup-2uxpφ1dx+∫Ω∇vp-2∇v∇φ2-νvp-2vxpφ1dx-1p∗∫Ω∂Fx,u,v∂uφ1+∂Fx,u,v∂vφ2dx-∫Ωfλuq-2uφ1dx-∫Ωgμvq-2vφ2dx=0for all (φ1,φ2)∈W.We know that looking for weak solutions of (1) is like looking for the critical points of the associated functional(7)Jλ,μu,v=1pu,vWp-1p∗∫ΩFx,u+,v+dx-1qKfλ,gμu+,v+where Kfλ,gμ(u,v)=∫Ω‍fλ(x)|u|q+gμ(x)|v|qdx.By the above Remark1, the functional Jλ,μ(u,v) is well defined on the space W and is of class C1(W,R).Therefore, the solutions of (1) correspond to critical points of Jλ,μ. Let us denote by Nλ,μ the Nehari manifold related to Jλ,μ, given by(8)Nλ,μ≔u,v∈W,u,v≠0,0:Jλ,μ′u,v,u,v=0Namely,(9)Nλ,μ≔u∈W,u,v≠0,0:u,vWp=∫ΩFx,u+,v+dx+Kfλ,gμu+,v+.Notice that the functional Jλ,μ is not bounded below on the total space for that we consider the functional on the Nehari manifold.Define(10)χλ,μu,v=Jλ,μ′u,v,u,v=u,vWp-∫ΩFx,u,vdx-Kfλ,gμu,v.Let (u,v)∈Nλ,μ, and by easy calculation we have (11)χλ,μ′u,v,u,v=pu,vWp-p∗∫ΩFx,u,vdx-qKλ,μu,v=p-p∗∫ΩFx,u,vdx-q-pKfλ,gμu,v=p-qu,vWp-p∗-q∫ΩFx,u,vdx=p-p∗u,vWp-q-p∗Kfλ,gμu,v.Lemma 3.
The functionalJλ,μ is bounded below on the Nehari manifold Nλ,μ.Proof.
Let(u,v)∈Nλ,μ, and applying the Hölder inequality and the Sobolev embedding theorem, Young inequality, and Condition (H1) we have(12)Kfλ,gμu,v≤S-q/pΩp∗-q/p∗λp/p-q+μp/p-qp-q/pu,vWq,and we deduce (13)Jλ,μu,v=p∗-pp∗pu,vWp-p∗-qp∗qKfλ,gμu,v≥p∗-pp∗pu,vWp-p∗-qp∗qS-q/pΩp∗-q/p∗λp/p-q+μp/p-qp-q/pu,vWqThus, Jλ,μ is coercive and bounded below on Nλ,μ.Now, we split theNehari manifold Nλ,μ into three parts, namely, (14)Nλ,μ+≔u∈Nλ,μ:χλ,μ′u,v,u,v>0Nλ,μ-≔u∈Nλ,μ:χλ,μ′u,v,u,v<0Nλ,μ0≔u∈Nλ,μ:χλ,μ′u,v,u,v=0Then, we have the following results.Lemma 4.
Let(u0,v0)∈Nλ,μ be a local minimizer of Jλ,μ and (u0,v0)∉Nλ,μ0. Then (u0,v0) is a critical point of Jλ,μ.Proof.
The proof is standard; you can see [4].Lemma 5.
There existsΛ∗>0 such that for all λ,μ>0 such that 0<λp/p-q+μp/p-q<Λ∗ then Nλ,μ0=∅.Proof.
Suppose the contrary; that is, there existλ,μ>0 with 0<λp/p-q+μp/p-q<Λ∗, but Nλ,μ0≠∅. Let (u,v)∈Nλ,μ0; we have(15)p-qu,vWp=p∗-q∫ΩFx,u,vdxand(16)p∗-pu,vWp=p∗-qKfλ,gμu,v.By (f5) and applying the Minkowski inequality and the Sobolev embedding theorem, we have (17)∫ΩFx,u,vdx≤K∫Ωup+vpp∗/pdxp/p∗p∗/p≤K​∫Ωup∗dxp/p∗+∫Ωvp∗dxp/p∗dxp∗/p≤KS-p∗/p∫Ω​∇updx+∫Ω∇vpdxp∗/p,so(18)∫ΩFx,u,vdx≤KS-p∗/pu,vWp∗.Combining (15) and (18), we have (19)p∗-qKS-p∗/pu,vWp∗≥p-qu,vWp,then(20)u,vW≥p-qSp∗/pp∗-qK1/p∗-p,By (12) we have (21)p∗-pu,vWp=p∗-qKfλ,gμu,v≤p∗-qS-q/pΩp∗-q/p∗λp/p-q+μp/p-qp-q/pu,vWq,then(22)u,vW≤p∗-qS-q/pΩp∗-q/p∗p∗-p1/p-qλp/p-q+θp/p-q1/p.We deduct from (20) and (22) that (23)λp/p-q+μp/p-q>Λ∗,which is a contradiction.So, we haveNλ,μ=Nλ,μ-∪Nλ,μ+, and we define(24)cλ,μ=infu,v∈Nλ,μ⁡Ju,v,cλ,μ+=infu,v∈Nλ,μ+⁡Jλ,μu,v,cλ,μ-=infu,v∈Nλ,μ-⁡Jλ,μu,v.Lemma 6.
(i) For some Λ∗>0 and for λq/p-q+μq/p-q∈]0,Λ∗[ so, there exists (PS)cλ,μ-sequence un,vn of Nλ,μ for Jλ,μ.
(ii) If 0<λq/p-q+μq/p-q<Λ∗, then there exists a (PS)cλ,μ--sequence un,vn of Nλ,μ- for Jλ,μ.Proof.
You find the same proof in the following reference [5].Denote(25)SF=infu,v∈W∖0⁡u,vWp∫ΩFx,u,vdxp/p∗:∫ΩFx,u,vdx>0.We 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 (26)uεx=ηxε+xp/p-1N-p/p.From Li Wang, Qiaoling Wei, and Dongsheng Kang [6], we have(27)∫Ωuεpdxp/p∗=ε-N-p/pULp∗RNp+Oε,∫Ω∇uεpdx=ε-N-p/p∇ULpRNp+O1,∫Ω∇uεpdx∫Ωuεpdxp/p∗=S+OεN-p/p,where U(x)=(1+|x|p/p-1)-N-p/p∈W1,p(RN), and verifying S, this (28)S=∇ULpRNpULp∗RNp=infu∈W01,pRN∖0⁡∇uLpRNpuLp∗RNp.Lemma 7.
(29) c 0,0 = 1 N S F N / p .Proof.
Setu0=e1uε and v0=e2uε and (u0,v0)∈W, where e1,e2∈R+, e1p+e2p=1, and infx∈Ω¯F(x,e1,e2)≥K. Then by (f5), the definition of SF, and (27), we have (30)c0,0≤supt≥0⁡J0,0tu0tv0=1Ne1p+e2p∫Ω∇uεpdx∫ΩFx,e1uε,e2vεdxp/p∗N/p≤1N∫Ω∇uεpdxKp/p∗∫Ωuεp∗dxp/p∗N/p≤1N1Kp/p∗N/pS+OεN-p/pN/p=1N1Kp/p∗N/pSN/p+OεN-p/p≤1NSFN/p(31)c0,0≤1NSFN/p,We use the following relation: (32)supt≥0⁡tppA-tp∗p∗B=1NABp/p∗N/p,A,B>0.For the reverse inequality, the application of the mountain pass theorem gives us a Palais-Smale sequence un,vn⊂W for I0,0 at level c0,0 and from here we can show that un,vn is bounded in W using standard arguments. Since(33)un-,vn-p=I′un,vn,un-,vn-→0.Assuming that un,vn≥0, we find (34)un,vnp→land∫ΩFx,un,vndx→l.From definition (25) of SF, we get (35)SFlp/p∗=SFlimn→+∞⁡∫ΩFx,un,vndxp/p∗≤limn→+∞⁡un,vnp=l,then(36)l≥SFN/p.Since J0,0(un,vn)→c0,0 implies l=c0,0N, we deduce from (36) that(37)c0,0≥1NSFN/p.Then from (31) and (37) we obtain (38)c0,0=1NSFN/p.Next we prove thatJλ,μ satisfies the Palais-Smale condition under some level. Before, we need the following lemma.Lemma 8.
LetF∈C1(Ω¯,(R+)2) with F(x,0,0)=0 and |∂F(x,u,v)/∂u|,|∂F(x,u,v)/∂v|≤C1|u|p-1+|v|p-1 for some ≤p<∞.  C1>0. Let uk,vk be a bounded sequence in Lp(Ω¯,(R+)2), such that (uk,vk)⇀(u,v) weakly in W. Then (39)∫ΩFx,uk,vkdx→∫ΩFx,uk-u,vk-vdx+∫ΩFx,u,vdxas k→∞.Proof.
(The idea of this proof was borrowed from [7])Lemma 9.
J λ , μ satisfies the (PS)c-condition for(40)-∞<c<c∞≔1NSFN/p-Cλp/p-q+μp/p-q,where C>0 is independent on λ and μ.Proof.
The proof is similar to that of Lemma 2.1 in [8].Let(u,v)∈W, with ∫Ω‍F(x,u,v)dx>0, and put(41)tmax=tmaxu,v,λ,μ≔p-qu,vWpp∗-q∫ΩFx,u,vdx1/p∗-p>0.Then the following lemma holds. Its proof is similar to the lemma [4] (or see Tarantello [9]).Lemma 10.
Let(u,v)∈W, with ∫Ω‍F(x,u,v)dx>0, so there are unique number positives t+ and t- such that 0<t+<tmax<t- with (42)t+u,t+v∈Nλ,μ+,t-u,t-v∈Nλ,μ-and(43)Jλ,μt+u,t+v=min0≤t≤tmax⁡Jλ,μtu,tv,Jλ,μt-u,v,λ,μu,t-u,v,λ,μv=maxt≥0⁡Jλ,μtu,tv.Lemma 11.
For someλ,μ>0, and Λ∗>0 such that 0<λp/p-q+μp/p-q<Λ∗, we have(44)cλ,μ-<c∞.Proof.
First, we claim that there exist positive constantsC1,C2>0 independent of ε such that(45)0<C1<tε=t-u0,v0,λ,μ<C2<∞Let u0=e1uε and v0=e2vε. We obtain(46)u0,v0p-tεp∗-p∫ΩFx,u0,v0dx=tεq-pKfλ,gμu0,v0.Then, by (f5) and (27) we deduct that(47)tεp∗-p≤∫Ω∇UpdxK∫ΩUp∗dxp/p∗+OεN-p/p,then tε is bounded above as ε→0. Using Lemma 10, we have (48)tε≥tmaxuε,vε,λ,μ>0,then we can also suppose that tε is bounded below. By a direct calculation we have(49)∫Ωuεqdx≥Cε-N-p/pq+Np-1/pifp∗-NN-p<q,Cε-N-p/pq+Np-1/pln⁡ε,ifq=p∗-NN-p,and the constant C is a positive. So (50)Jλ,μtεu0,tεu0≤1NSFN/p+OεN-p/p-λ+μCεp-1/pN-qN-p/p,ifp∗-NN-p<q,Cεp-1/pN-qN-p/pln⁡ε,ifq=p∗-NN-pWe have p∗-NN-p<q<p, and there exist τ>0 such that(51)p-qqp-1pN-qN-pp<τ<N-pp-p-1pN-qN-pp.Let (52)λ+μ=ετBy the following relation, for x,y>0 and s∈[0,1], we have (x+y)s<xs+ys, and we obtain (53)λp/p-q+μp/p-q<ετp/p-q.By (51) we have (54)τ+p-1pN-qN-pp<min⁡τpp-q,N-pp.Then, there exists Λ∗>0 such that λp/p-q+μp/p-q∈(0,Λ∗), and we have(55)Jλ,μtεu0,tεu0≤c∞so by definition cλ,μ- we deduct that (56)cλ,μ-<c∞.For the case q=p∗-N/N-p, so we get the same result.For the existence of the first solution of our problem (1)Lemma 12.
There existsΛ∗>0 such that if λ,μ∈(0,Λ∗), then Jλ,μ has a minimizer (uλ+,uλ,μ+)∈Nλ,μ+ and its satisfies (i)
Jλ,μ(uλ,μ+,vλ,μ+)=cλ,μ+(ii)
(uλ,μ+,vλ,μ+) is a positive solution of (1).Proof.
Taking into account the fact thatNλ,μ-⊂Nλ,μ and Lemma 11 we have (57)cλ,μ≤cλ,μ-<c∞.Hence, for the proof of (i) just use the following Lemmas 11 and 9. Now let (uλ,μ+,vλ,μ+) be solution of problem (1) such that Jλ,μ(uλ,μ+,vλ,μ+)=cλ,μ. Moreover, we have (uλ,μ+,vλ,μ+)∈Nλ,μ+. In fact, if (uλ,μ+,vλ,μ+)∈Nλ,μ-, by Lemma 10, there are unique t0+, t0- such that (t0+uλ,μ+,t0+vλ,μ+)∈Nλ,μ+ and (t0+uλ,μ-,t0+vλ,μ-)∈Nλ,μ-. In particular, we have t0+<t0-=1. Since (58)ddtJλ,μt0+uλ,μ+,t0+vλ,μ+=0andd2dt2Jλ,μt0+uλ,μ+,t0+vλ,μ+>0,there exists t0+<t¯≤t0- such that Jλ,μ(t0+uλ,μ+,t0+vλ,μ+)<Jλ,μ(t¯uλ,μ+,t¯vλ,μ+). By Lemma 10(59)Jλ,μt0+uλ,μ+,t0+vλ,μ+<Jλ,μt¯uλ,μ+,t¯vλ,μ+≤Jλ,μt0-uλ,μ+,t0-vλ,μ+=Jλ,μuλ,μ+,vλ,μ+,which is impossible and by the maximum principle, we deduct that (uλ,μ+,vλ,μ+) is a positive solution of problem (1).


## 3. Some Technical Results

Lemma 13.
Let(λn) and (μn) decreasing sequences in (0,Λ∗) for some Λ∗>0 and converging to 0, so limn→+∞cλn,μn-=c0,0.Proof.
By Lemma6 there exists a sequence un,vn⊂Nλ,μ-, un,vn≥0 such that(60)Jλn,μnun,vn=cλn,μn-andJλn,μn′un,vn=0.There exists a real number sequence tn satisfying (tnun,tnvn)∈N0,0. So(61)c0,0≤J0,0tnun,tnvn=Jλn,μntnun,tnvn+tnqqKfλn,gμnun+,vn+(62)≤cλn,μn-+tnqqKfλn,gμnun+,vn+Since, by Lemma 10 for all n we have(63)0<cλ1,μ1-≤cλn,μn-≤c0,0.Moreover, (tnun,tnvn)∈N0,0 implies that (64)un,vnW=tnq-pKfλn,gμnun+,vn+and we deduct that Jλn,μn(un,vn)=cλn,μn-≤c0,0 and Jλn,μn′(un,vn)=0. We get that un,vn is bounded in W.
We can now say thattn is a bounded sequence, if we assume by contradiction that limn→+∞tn=∞. We find limn→+∞Kfλn,gμn(un+,vn+)=0, so limn→+∞(un,vn)W=0 which implies by (60) that (65)limn→+∞cλn,μn-=0which is a contradiction with (63).We consider the following lemma. See section 5.3 in [10].Lemma 14.
Suppose thatX is Banach space and F∈C1(X,R). Assume that, for c0∈R and k∈N, (1)
F satisfies the(PS)c condition for c≤c0,(2)
cat(x∈X,Fx≤c0)≥k.
ThenF has at least k critical points in x∈X,Fx≤c0.Let us consider tow subset ofRN(66)Ωr+≔x∈RN:distx,Ω<r,Ωr-≔x∈Ω:distx,∂Ω>rNote that Ωr+ and Ωr- are homotopically equivalent to Ω for some r>0. We may assume Br≔Br(0)⊂Ω. We consider(67)Wr≔u,v∈W01,pBr×W01,pBr:u,vareradial.Recall that u∈W01,p(Br) implies that u is extension in Ω with u=0 outside of Br. Let Jλ,μ,Br:Wr→R as(68)Jλ,μ,Bru,v≔1pu,vWrp-1p∗∫ΩFx,u+,v+dx-1qKfλ,gμu+,v+.We denote by(69)c~λ,μ≔infu,v∈Nλ,μ,Br-⁡Jλ,μ,Bru,v.Similar to Jλ,μ, Jλ,μ,Br can be shown to satisfy restricted versions of the same three Lemmas 7, 9, and 11. We consider(70)Nλ,μ,c~λ,μ-≔u,v∈Nλ,μ-:Jλ,μu,v≤c~λ,μLet (71)ζu,v≔NSFN/p∫ΩxFx,u,vdx,for all (u,v)∈Nλ,μ-, that is, ζ(u,v)∈RNand the mapϖ:Ωr-→Nλ,μ,c~λ,μ- given by(72)ϖyx≔uλ,μx-y,vλ,μx-yifx∈Bry,oifx∉Bry,withuλ,μ,vλ,μ radial. For all y∈Ωr-(73)SFN/pNζ∘ϖy=∫ΩxFx,uλ,μx-y,vλ,μx-ydx=∫Ωy+zFx,uλ,μz,vλ,μzdx=∫ΩyFx,uλ,μz,vλ,μzdx.Then ζ∘ϖ can be rewritten (74)ζ∘ϖy=NSFN/p∫ΩFx,uλ,μz,vλ,μzdx≕χλ,μy.Remark 15.
(75) l i m λ , μ → 0 c ~ λ , μ = c ~ 0,0 ,and (76)limλ,μ→0χλ,μ=1.Next, we define the map Hλ,μ:[0,1]×Nλ,μ,c~λ,μ-→RN by(77)Hλ,μt,u,v≔t+1-tχλ,μζu,v.Lemma 16.
For someΛ∗ such that λp/p-q+μp/p-q∈(0,Λ∗) we have(78)Hλ,μ0,1×Nλ,μ,c~λ,μ-⊂Ωr+.Proof.
We show by the absurd that there exist(tn) sequence of [0,1], λn,μn→0, and un,vn⊂Nλ,μ,c~λ,μ- such that(79)Hλn,μntn,un,vn∉Ωr+,and let tn→t0∈[0,1] (up to a subsequence of (tn)). By Remark 15, we have (80)χλn,μn→1(81)cλn,μn-≤1pun,vnWp-1p∗∫ΩFx,un,vndx-1qKfλn,gμnun+,vn+≤c~λn,μn.and (82)un,vnWp-∫ΩFx,un,vndx-Kfλn,gμnun+,vn+=0.Standard calculations show that (un,vn) is bounded in W and by this we obtain(83)cλn,μn-+o1≤1pun,vnWp-1p∗∫ΩFx,un,vndx≤c~λn,μn+o1,and(84)un,vnWp-∫ΩFx,un,vndx=o1,as n→+∞. We have by Lemmas 13 and 7 and its restricted version for Jλ,μ,Br that(85)cλn,μn-andc~λn,μnbothconvergeto1NSFN/p, then by (83), (84) and (85)(86)un,vnWp→SFN/pand∫ΩFx,un,vndx→SFN/p.Now, it is easy to see that the sequence (u~n,v~n) given by(87)u~n,v~n=un∫ΩFx,un,vndx1/p∗,vn∫ΩFx,un,vndx1/p∗verifies (88)∫ΩFx,u~n,v~ndx=1andu~n,v~nWp→SFFor a subsequence of (u~n,v~n) we have (89)u~n,v~n→u~,v~a.e.onRN∇u~n-u~p+∇v~n-v~pdx→ωinMRN,∫ΩFx,u~n,v~ndx→τinMRN.By the same way used in [[10],Lemma 1.40] (see also [7] ), we obtain (90)SF=u~,v~p+ω,1=∫ΩFx,u~,v~dx+τand (91)τp/p∗≤SFω.Since (92)∫ΩFx,u~,v~dxp/p∗≤SF-1u~,v~Wp.It is easy to confirm that ∫Ω‍F(x,u~,v~)dx and ω are equal either to 0 or to 1. Since SF is independent of Ω and is never achieved except when Ω=RN (see also [11]), so necessarily ∫Ω‍F(x,u~,v~)dx=0. Then the measure ω is concentrated at a single point y of Ω¯,
and we have(93)ζun,vn→∫Ωxdωx=y∈Ω¯⊂Ωr+.Therefore (94)Hλn,μntn,un,vn≔tn+tn-1χλ,μζun,vn→y∈Ω¯⊂Ωr+,and this is impossible.Lemma 17.
For someΛ∗>0 such that 0<λp/p-q+μp/p-q<Λ∗, we have(95)catNλ,μ,c~λ,μ-≥catΩ.Proof.
This classic proof is omitted for brevity. An identical proof can be found in [12], Lemma 14.


## 4. The Proof of Theorem2

Denote byJNλ,μ- the restriction of Jλ,μ on Nλ,μ-.Lemma 18.
If0<λp/p-q+μp/p-q<Λ∗, for some Λ∗>0, so the functional JNλ,μ- verifies the Palais-Smale condition for c<c∞.Proof.
By [[10], Proposition 5.12], there exists a sequence σn⊂R. If (un,vn) is a (PS)c for INλ,μ- at level c, there exists a sequence σn⊂R such that(96)Jλ,μ′un=σnχλ,μ′un,vn+o1,where (97)χλ,μun,vn=Jλ,μ′un,vn,un,vn=un,vnWp-∫ΩFx,un,vndx-Kfλ,gμun,vnRecall that (un,vn)∈Nλ,μ-, so 〈χλ,μ′(un,vn),(un,vn)〉<0.
If〈χλ,μ′(un,vn),(un,vn)〉→0,(98)p-qun,vnWp=p∗-q∫ΩFx,un,vndx+o1and(99)p∗-pun,vnWp=p∗-qKfλ,gμun,vn+o1.By the same argument employed in Lemma 5, we get (100)un,vnW≥p-qSp∗/pp∗-qK1/p∗-p+o1,and(101)un,vnW≤p∗-qS-q/pΩp∗-q/p∗p∗-p1/p-qλp/p-q+μp/p-q1/pand we deduct that λp/p-q+μp/p-q>Λ∗. This is contradiction.
Moreover we assume that〈χλ,μ′(un,vn),(un,vn)〉→l, as n→+∞. Since 〈Jλ,μ′(un,vn),(un,vn)〉=0, so σn→0 as n→+∞ then Jλ,μ′(un,vn)→0. Thus,(102)Jλ,μun,vn→c∈0,cλ,μ,andJλ,μ′un,vn→0,
then by Lemma9 the proof is finished.Lemma 19.
For someΛ∗>0 such that if 0<λp/p-q+μp/p-q<Λ∗, then every critical point (u,v)∈Nλ,μ- of JNλ,μ- is a critical point of Jλ,μ in W.Proof.
For the proof of this lemma, it is similar to Lemma18.Proof of Theorem2.
Applying Lemmas9 and 12, JNλ,μ- satisfies (PS)c condition for all c∈(0,cλ,μ). Then, by Lemmas 17 and 14, JNλ,μ- admits at least cat(Ω) critical points in Nλ,μ,c~λ,μ-. Hence, we deduce from Lemma 19 that Jλ,μ has at least cat(Ω) critical points in Nλ,μ-. Moreover, Nλ,μ-∩Nλ,μ+=∅, Jλ,μ at least cat(Ω)+1 critical points in W.



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