# On the Existence of Infinitely Many Solutions for Nonlocal Systems with Critical Exponents

Authors: M. Khiddi; R. Echarghaoui
Journal: Abstract and Applied Analysis
 (2016)

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## Abstract

We study a class of semilinear nonlocal elliptic systems posed on settings without compact Sobolev embedding. By employing critical point theory and concentration estimates, we prove the existence of infinitely many solutions for values of the dimensionN, where N>6s, provided 0<s<1.


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## Body

## 1. Introduction and Main Result

Nonlocal problems and operators have enjoyed much attention from mathematicians in recent years probably due to their interesting analytical structure and their numerous physical applications in many fields such as flame propagation, conservation laws, ultrarelativistic limits of quantum mechanics, quasi-geostrophic flows, and the thin obstacle problem (see [1–3]). In this paper, we are concerned with the study of the infinitely many solutions for the following system: (1)-Δsu=2αα+βuα-2uvβ+λuinΩ,-Δsv=2βα+βuαvβ-2v+μvinΩ,u=v=0on∂Ω,where Ω is an open bounded domain in RN, λ,μ>0 and α>1, β>1 satisfy α+β=2s∗=2N/N-2s, s∈(0,1) and N>2s. When α=β, λ=μ, and u=v, problem (1) reduces to the Brézis-Nirenberg type problem with fractional Laplacian:(2)-Δsu=u2s∗-2u+λuinΩ,u=0on∂Ω.In [4], Brézis and Nirenberg considered the existence of positive solutions for problem (2) with s=1. Such a problem involves the critical Sobolev exponent 2∗=2N/N-2 for N≥3, and it is well known that the Sobolev embedding H01(Ω)↪L2∗(Ω) is not compact even if Ω is bounded. Hence, the associated functional of problem (2) does not satisfy the Palais-Smale condition, and critical point theory can not be applied directly to find solutions of the problem. However, it is found in [4] that the functional satisfies the (PS)c condition for c∈(0,1/NSN/2), where S is the best Sobolev constant and 1/NSN/2 is the least energy level at which the Palais-Smale condition fails. So a positive solution can be found if the mountain pass value corresponding to problem (2) is strictly less than 1/NSN/2. In [5], a concentration-compactness principle was developed to treat noncompact critical variational problems. In the study of the existence of multiple solutions for critical problems, to retain the compactness, it is necessary to have a full description of energy levels at which the associated functional does not satisfy the Palais-Smale condition. A global compactness result is found in [6], which describes precisely the obstacles of the compactness for critical semilinear elliptic problems. This compactness result shows that above a certain energy level it is impossible to prove the Palais-Smale condition. For this reason, to obtain infinitely many solutions for the critical problem, it is essential to find a condition that can replace the standard Palais-Smale condition. In [7], Devillanova and Solimini considered (2) with s=1 and obtained infinitely many solutions for every λ>0 if N≥7. They proved this latter result by employing the concentration estimates and lower bound of augmented Morse index on min–max points (see [8]), which seem unapplicable to the case of (1) directly. This work was extended to an analogous problem involvingp-Laplacian for 1<p<∞ by Cao et al. [9]. They proved that if N>p2+p, the following problem, (3)-Δpu=up∗-2u+λup-2uinΩ,u=0on∂Ω,where μ>0 and p∗=pN/N-p, has infinitely many solutions. Recently, Yan et al. extended the result in [10] to problem (2) and obtained infinitely many solutions for every λ>0 if N>6s, where 0<s<1. Similar problems to (1) for the Laplacian operator have been studied extensively in recent years; see [6, 11–14] and the references therein. In particular, Alves et al. [11] studied thep-Laplacian system with critical growth and obtained the existence of infinitely many solutions if N>p2+p. An important cornerstone for these works has been laid out in a remarkable paper [15], where Caffarelli and Silvestre gave a new formulation of the fractional Laplacian through Dirichlet-Neumann maps. This is extensively used in the recent literature since it allows transforming nonlocal problems to local ones, which permits the use of variational methods. This will come in handy for this work. A more general form of the fractional operator has been studied and a multiplicity of solutions has been shown in several cases. For further reading, we refer the reader to these recently published papers on the fractional Laplacian [16–22].In this paper, we prove that (1) has infinitely many solutions under the following conditions: N>6s, where 0<s<1, by studying system (1) in the critical case α+β=2s∗. We adapt the original idea used in [7] which was described in the above paragraph (see also [9–11, 13]). After perturbing problem (1) into a subcritical case with a (PS) functional in all energy levels, estimates on the set of the solution sequences to the subcritical case of (1), a global compactness argument, and a (local) Pohozaev identity are used to establish the strong convergence and finally use min–max theorems on a genus homotopic class to produce infinitely many critical values. The paper is organized as follows: in Section 2, some notations and preliminary results are established. In Section 3, we establish a local Pohozaev identity which allows us to prove the E0s(CΩ)-strong convergence of solutions for the subcritical case of problem (18), and, finally, we show how this technique allows the application of classical min–max arguments to (1) and to prove, in this way, the existence of infinitely many solutions. Our main result is the following.Theorem 1.
Let0<s<1 and λ≥0, μ≥0 with λ+μ>0. Then problem (1) has infinitely many solutions when N>6s.


## 2. Notation and Preliminaries

The powers(-Δ)s of the positive Laplacian operator -Δ, in Ω, with zero Dirichlet boundary conditions are defined via its spectral decomposition, namely, (4)-Δsux≔∑j=1kajρjsφjx,where (ρj,φj) is the sequence of eigenvalues and eigenfunctions of the operator (-Δ) in Ω under zero Dirichlet boundary data and aj are the coefficients of u for the base {φj}j=1∞ in L2(Ω). In fact, the fractional Laplacian (-Δ)s is well defined in the space of functions (5)H0sΩ≔u=∑j=1kajφj∈L2Ω:uH0s=∑j=1kaj2ρjs1/2<∞,and uH0s=(-Δ)s/2uL2(Ω).A pair of functions(u,v)∈H0s(Ω)×H0s(Ω) is said to be a weak solution of problem (1) if (6)∫Ω-Δs/2u-Δs/2φ1+-Δs/2v-Δs/2φ2dx-∫Ωλuφ1+μvφ2dx-2αα+β∫Ωuα-2uvβφ1dx-2βα+β∫Ωuαvβ-2vφ2dx=0,for all (φ1,φ2)∈H0s(Ω)×H0s(Ω). Solutions to problem (1) will be obtained as critical points of the corresponding energy functional (7)Jλ,μu,v≔12∫Ω-Δs/2u2+-Δs/2v2dx-12∫Ωλu2+μv2dx-2α+β∫Ωuαvβdx.The functional is well defined in H0s(Ω)×H0s(Ω), and, moreover, the critical points of the functional Jλ,μ correspond to solutions of (1). We now conclude the main ingredients of a recently developed technique use in order to deal with fractional powers of the Laplacian operator. Motivated by the work of Caffarelli and Silvestre [15], several authors have considered an equivalent definition of the operator (-Δ)s/2 in a bounded domain with zero Dirichlet boundary data means of auxiliary variable, see [2, 23]. Associated with the bounded domain Ω, let us consider the cylinder CΩ≔Ω×(0,+∞)⊂R+N+1. Now, for a function u∈H0s, we define thes-harmonic extension w=Es(u) to the cylinder CΩ as the solution of the problem: (8)div⁡y1-2s∇w=0inCΩw=0on∂LΩw=uonΩ×0,-Δsux=-kslimy→0+⁡y1-2s∂w∂yx,y,where ks=21-2sΓ(1-s)/Γ(s) is a normalization constant. The extension function w(x,y) belongs to the space (9)H0,L1CΩ≔w∈L2CΩ:w=0on∂LΩ,∫CΩy1-2s∇w2dxdy<∞endowed with the norm (10)
z
H
0
,
L
1
C
Ω
≔
∫
C
Ω
y
1
-
2
s
∇
z
2
d
x
d
y
1
/
2
.The extension operator is an isometry between H0s(Ω) and H0,L1(CΩ); namely, (11)uH0sΩ=EsuH0,L1CΩ,∀u∈H0sΩ.With this extension, we can reformulate (1) as the following local problem: (12)-div⁡y1-2s∇w1=0,-div⁡y1-2s∇w2=0inCΩ,w1=w2=0,on∂LΩ,y1-2s∂w1∂ν=2αα+βw1x,0α-2w1x,0w2x,0β+λw1x,0onΩ×0,y1-2s∂w2∂ν=2βα+βw1x,0αw2x,0β-2w2x,0+μw2x,0onΩ×0,where (13)∂wi∂ν≔-kslimy→0+⁡y1-2s∂wi∂y,i=1,2and w1,w2∈H0,L1(CΩ) are thes-harmonic extension of u,v∈H0s(Ω), respectively. Let (14)E0sCΩ≔H0,L1CΩ×H0,L1CΩbe equipped with the norm (15)w1,w2E0sCΩ=w1H0,L1CΩ+w2H0,L1CΩ.An energy solution to this problem is a function (w1,w2)∈E0s(CΩ) satisfying (16)∫CΩy1-2s∇w1∇φ1dxdy+∫CΩy1-2s∇w2∇φ2dxdy=λ∫Ωw1φ1dx+2αα+β∫Ωw1α-2w1w2βφ1dx+μ∫Ωw2φ2dx+2βα+β∫Ωw1αw2β-2w2φ2dxfor all (φ1,φ2)∈E0s(CΩ). If (w1,w2)∈E0s(CΩ) satisfies (12), then (u,v)=(w1(·,0),w2(·,0)), defined in the sense of traces, belongs to the space H0s(Ω)×H0s(Ω) and it is a solution of the original problem (1). The associated energy functional to problem (12) is denoted by (17)Iλ,μw≔Iλ,μw1,w2=12∫CΩy1-2s∇w12+∇w22dxdy-12∫Ωλw12+μw22dx-2α+β∫Ωw1αw2βdx.Critical points of Iλ,μ in E0s(CΩ) correspond to the critical points of Jλ,μ:H0s(Ω)×H0s(Ω)→R. Since the problem is critical, the functional Iλ,μ does not satisfy the Palais-Smale condition. Thus the min–max theorems can not be applied directly to obtain infinitely many solutions for (1). Following the original idea in [7], employed in a closer setting in [10], we deal first with the following perturbed problem: (18)-div⁡y1-2s∇w1=0,-div⁡y1-2s∇w2=0inCΩw1=w2=0,on∂LΩy1-2s∂ω1∂ν=2α-εα+β-2εw1x,0α-2-εw1x,0w2x,0β-ε+λw1x,0onΩ×0y1-2s∂ω2∂ν=2β-εα+β-2εw1x,0α-εw2x,0β-2-εw2x,0+μw2x,0onΩ×0,where (w1,w2)∈E0s(CΩ) and 0<ε<min⁡{α-1,β-1,2s∗-1}.The functional corresponding to (18) becomes (19)Iλ,μεw,φ=12∫CΩy1-2s∇w2+∇φ2dxdy-12∫Ωλw2+μφ2dx-2α+β-2ε∫Ωwα-εφβ-εdx,where (w,φ)∈E0s(CΩ). Now Iλ,με is even and satisfies the Palais-Smale condition at all energy levels. It follows from the symmetric mountain pass lemma [24] that (18) has infinitely many solutions. More precisely, it follows from [25, Theorem  6.1] that there are positive numbers cε,l,l=1,2,…, with cε,l→+∞, as l→+∞ and a solution (wε,l,φε,l) for (18), satisfying Iε(wε,l,φε,l)=cε,l. To obtain the existence of infinitely many solutions for (1), the first step is to investigate whether (wε,l,φε,l) converges strongly in E0s(CΩ) as ε→0. That is, we need to study the compactness of the set of solutions for (18), with ε>0 small enough.


## 3.E0s(CΩ)-Strong Convergence of Solutions for the Subcritical Case of Problem (18)

In this section, we establish the strong convergence of solutions for the perturbed problem (18). Throughout this paper, we denote the norms of E0s(CΩ),Ll(Ω)(1≤l<∞) by w1,w2E0s(CΩ)=w1H0,L1(CΩ)+w2H0,L1(CΩ), uLl(Ω)=∫Ω‍|u|ldx1/l, respectively, and positive constants (possibly different) by C.Proposition 2.
Suppose thatN>6s with 0<s<1 and λ,μ≥0 satisfy λ+μ>0. Then for any sequence (un,vn) (n=1,2,…) which is a solution of (18) with ε=εn→0, and for (un,vn) satisfies un,vnE0s(CΩ)≤C for some constant C>0 independent of n, then (un,vn)n≥1 has a subsequence which converges strongly in E0s(CΩ) as n→+∞.Before giving the proof of Proposition2, we give some estimates for (un,vn). Let D be a bounded domain such that Ω⋐D and (un(x,0),vn(x,0))=(0,0) in D∖Ω. We choose A>0 large enough so that, for all x,y∈R, the following is verified: (20)2αnαn+βnxαn-2xyβn+λx≤2x+y2s∗-1+A,2βnαn+βnxαnyβn-2y+μy≤2x+y2s∗-1+A,for all x,y∈R. Let (ωn) (n=1,2,…) with ωn≥0 be a solution of(21)-div⁡y1-2s∇ω=0,inCD,ω=0,on∂LD,y1-2s∂ω∂ν=2unx,0+vnx,02s∗-1+AonD×0.By choosing D and A, we find(22)-div⁡y1-2s∇ωn±un=0,inCΩ,ωn±un=0,on∂LΩ,y1-2s∂ωn±un∂ν≥0onΩ×0.Multiplying (22) by (ωn±un)- and integrating by part, we see that(23)un≤ωn.Similarly, for vn, we have(24)vn≤ωn.Then by (23) and (24) we have(25)12un+vn≤ωn.So in the following sections we can consider just the estimates of wn in CΩ.Let(26)Ss,α,β≔infw1,w2∈E0sCΩ⁡∫CΩy1-2s∇w12+∇w22dxdy∫Ωw1αw2βdx2/2s∗.Then we have (see Lemma  2.5 [23]) (27)Ss,α,β=αββ/2s∗+βαα/2s∗Ss,N,where S(s,N) is the best Sobolev constant defined by(28)Ss,N=infz∈H0,L1CΩ∖0∫CΩy1-2s∇zx,y2dxdy∫Ωzx,02s∗dx2/2s∗,which is achieved if and only if Ω=RN by functions wε are thes-harmonic extensions of (29)uεx≔εN-2s/2ε2+x2N-2s/2,ε>0,x∈RN.Now we introduce the “problem at infinity”:(30)-div⁡y1-2s∇u=0,-div⁡y1-2s∇v=0inR+N+1,y1-2s∂u∂ν=2αα+βuα-1x,0vβx,0onRN×0,y1-2s∂v∂ν=2βα+βuαx,0vβ-1x,0onRN×0,U>0,V>0inRN×0.Here α>1, β>1, and α+β=2s∗. Set (31)U=2αα+β1/α+β-2βαβ/2α+β-2u,V=2βα+β1/α+β-2αβα/2α+β-2v.Then problem (30) can be rewritten as(32)-div⁡y1-2s∇U=0,-div⁡y1-2s∇V=0inR+N+1,y1-2s∂U∂ν=Uα-1x,0Vβx,0onRN×0,y1-2s∂V∂ν=Uαx,0Vβ-1x,0onRN×0,U>0,V>0inRN×0.Lemma 3.
Let(U,V) be a solution of (32). Then U(x,0)=V(x,0) for all x∈RN.Proof.
Let(U,V) be a solution of (32). Then (33)-div⁡y1-2s∇U-V=0,inR+N+1,y1-2s∂U-V∂ν=Uα-1x,0Vβ-1x,0V-UonRN×0,U>0,V>0inRN×0.Multiplying (33) by (U-V) and integrating by part, we obtain (34)∫R+N+1y1-2s∇U-V2dxdy=-∫RNUα-1x,0Vβ-1x,0U-V2dx≤0,which implies that (35)Ux,y-Vx,y=const.Note that U>0, V>0, in RN. It follows from (30) that Uα-1(x,0)Vβ(x,0)=y1-2s∂U/∂ν=Uα(x,0)Vβ-1(x,0) in RN×{0}. Hence U(x,0)=V(x,0).From Proposition6 in the Appendix, we have the global compactness result on (1). Let (un,vn) (n=1,2,…) be a solution of problem (18) with ε=εn→0 as n→+∞, satisfying un,vnE0s(CΩ)≤C for some constant independent of n. Then replacing the solution if necessary with a subsequence, there exist k sequences mutually diverging scaling σni with respective concentration points xni such that, as n→+∞,(36)un=u0+a0-1∑i=1kσniN-2s/2Uσnix-xni,0+ωn1,vn=v0+b0-1∑i=1kσniN-2s/2Uσnix-xni,0+ωn2,where (u0,v0)∈E0s(CΩ) is a weak solution of problem (12), ωn1,ωn2E0s(CΩ)→0 as n→+∞, U achieves the constant S(s,N), which is given in (28), and(37)a0=2βα+β2βαβ-21/2α+β-2,b0=2αα+β2αβα-21/2α+β-2.As in [7, 9, 10] we shall introduce the following facts which are essentials to prove the strong convergence of {(un,vn)} in E0s(CΩ). Among all the bubbles in (28), we can choose the slowest concentration rate, denoted by σn, which concentrates in xn the slowest rate. That is, the corresponding σ is the lowest order infinity among the ones appearing in the bubbles. Note that the number of the bubbles in (un,vn) is finite; we can always choose a constant C¯>0, independent of n, such that the region, (38)An1≔Z=x,y:Z∈BC¯+5σn-1/2xn,0∖BC¯σn-1/2xn,0∩CΩ,does not contain any concentration point of (un,vn) for every n. We set two thinner subsets as follows: (39)An2≔Z=x,y:Z∈BC¯+4σn-1/2xn,0∖BC¯+1σn-1/2xn,0∩CΩ,An3≔Z=x,y:Z∈BC¯+3σn-1/2xn,0∖BC¯+2σn-1/2xn,0∩CΩ.Then the following integral estimates hold (see Propositions  4.1 and  4.2 in [10]).Lemma 4.
Let(un,vn)n≥1 be solution of (18) with ε=εn→0. Then, there exists a constant C>0 independent of n, such that, for all p≥1, one has(40)∫An2y1-2sunp+vnpdxdy≤Cσn-N+2s∗/2s∗,∫An2∩y=0unp+vnpdx≤Cσn-N/2.
Moreover, one has(41)∫An3y1-2s∇un2+∇vn2dxdy≤Cσn-N/2s∗.Proof.
For the proof of this lemma, we use inequality (25) and we refer to [10, Propositions  4.1, 4.2].Note thatX=(x,y)∈R+N+1, and we have the following local Pohozaev identity.Lemma 5.
Letαn=α-εn, βn=β-εn, and ε=εn→0 as n→+∞ and Bn be any bounded set in CΩ. Then one has the local Pohozaev identity on Bn⊂CΩ  associated for equations of (18):(42)εnN-2s2N-εnN-2s∫Bn∩y=0unαnvnβndx+s∫Bn∩y=0λun2+μvn2dx=∫∂Bn∩y=02αn+βnunαnvnβnx-x0·νdσ+12∫∂Bn∩y=0λun2+μvn2x-x0·νdσ+N-2s2∫∂Bn∩y>0y1-2sun∂un∂ν+vn∂vn∂νdσ-12∫∂Bn∩y>0y1-2s∇un2+∇vn2X-z0·νdσ+∫∂Bn∩y>0y1-2s∇un·X-z0∂un∂ν+∇vn·X-z0∂vn∂νdσ,where ν is the outward unit normal to ∂Bn and z0=(x0,0), with x0 being a point in RN.Proof.
Using the divergence theorem, we get(43)N-2s2∫∂Bny1-2sun∂un∂ν+vn∂vn∂νdσ=12∫∂Bny1-2s∇un2+∇vn2X-z0·νdσ-∫∂Bny1-2s∇un·X-z0∂un∂ν+∇vn·X-z0∂vn∂νdσ,where ν is the outward unit normal to ∂Bn and z0=(x0,0), with x0 being a point in RN.
Since(un,vn) is solution of (18), we have the following equations:(44)y1-2sun∂un∂ν=2αnαn+βnunαnvnβn+λun2ony=0,y1-2svn∂vn∂ν=2βnαn+βnunαnvnβn+μvn2ony=0.
We obtain, from (43) and (44),(45)N-2s2∫Bn∩y=02unαnvnβn+λun2+μvn2dx+N-2s2∫∂Bn∩y>0y1-2sun∂un∂ν+vn∂vn∂νdσ=12∫Bn∩y=0y1-2s∇un2+∇vn2x-x0·νdx+12∫∂Bn∩y>0y1-2s∇un2+∇vn2X·νdσ-∫Bn∩y=0y1-2s∇un·x-x0∂un∂ν+∇vn·x-x0∂vn∂νdx-∫∂Bn∩y>0y1-2s∇un·X-z0∂un∂ν+∇vn·X-z0∂vn∂νdσ.Noting that the dot product ((x-x0)·ν)=0, on Bn∩{y=0}, we obtain(46)12∫Bn∩y=0y1-2s∇un2+∇vn2x-x0·νdx=0.Moreover, we have(47)∫Bn∩y=0y1-2s∇un·x-x0∂un∂ν+∇vn·x-x0∂vn∂νdx=∫Bn∩y=0∇un·x-x02αnαn+βnunαnvnβn+λun2dx+∫Bn∩y=0∇vn·x-x02βnαn+βnunαnvnβn+μvn2dx=∫Bn∩y=0∇2αn+βnunαnvnβn+12λun2+μvn2·x-x0dx=-N∫Bn∩y=02αn+βnunαnvnβn+12λun2+μvn2dx+∫∂Bn∩y=02αn+βnunαnvnβn+12λun2+μvn2x-x0·νdσ.Therefore, from (45), (46), and (47), we infer that (44) holds.Proof of Proposition2.
Let(un,vn)n≥1 be a bounded sequence in E0s(CΩ) composed of solutions for (18). Thus, in order to prove the E0s(CΩ)-strong convergence in (36), we just need to show that the bubbles (σni)N-2s/2U(σni(x-xni),0)(1≤i≤k) in (36) will not appear in the decomposition of un and vn. Since the proof is similar to that of Lemma  6.1 in [9], here we only give a sketch of it. From Lemma 5, for the solution concentrating sequence (un,vn) of (18) with ε=εn→0, we have the local Pohozaev identity on (48)Bn≔Btnσn-1/2xn,0∩CΩ⊂RN+1(49)4εnN2s∗2s∗-2εn∫Bn∩y=0unαnvnβndx+s∫Bn∩y=0λun2+μvn2dx=2αn+βn∫∂Bn∩y=0unαnvnβnX-z0·νdσ+12∫∂Bn∩y=0λun2+μvn2X-z0·νdσ+N-2s2∫∂Bn∩y>0y1-2sun∂un∂ν+vn∂vn∂νdσ-12∫∂Bn∩y>0y1-2s∇un2+∇vn2X-z0·νdσ+∫∂Bn∩y>0y1-2s∇un·X-z0∂un∂ν+∇vn·X-z0∂vn∂νdσ,where ν is the outward unit normal to ∂Bn and z0=(x0,0), x0∈RN. We decompose (50)∂Bn∩y>0=∂iBn∪∂eBn,where ∂iBn≔∂Bn∩CΩ and ∂eBn≔Bn∩∂LCΩ.
We consider two different cases:(i)
Btnσn-1/2xn,0∩0∩RN+1∖CΩ≠∅.(ii)
Btnσn-1/2((xn,0))∩{0}⊂CΩ.
Incase (i), we take x0∈RN∖Ω with |x0-xn|≤2tnσn-1/2 and ν·(X-(x0,0))≤0 in ∂eBn, where ν is the outward unit normal to ∂LCΩ.
Since(un,vn)=(0,0) on ∂LCΩ, we find(51)-12∫∂Bn∩y>0y1-2s∇un2+∇vn2X-z0·νdσ+∫∂Bn∩y>0y1-2s∇un·X-z0∂un∂ν+∇vn·X-z0∂vn∂νdσ=12∫∂Bn∩y>0y1-2s∇un2+∇vn2X-z0·νdσ≤0.Incase (ii), ∂eBn=∅; we take a point x0=xn.
Since(un,vn)=(0,0) on ∂LCΩ, and, for n large enough 4εnN/2s∗(2s∗-2εn)>0, we obtain(52)s∫Bn∩y=0λun2+μvn2dx≤∫∂iBn∩02αn+βnunαnvnβnX-z0·νdσ+12∫∂iBn∩y=0λun2+μvn2X-z0·νdσ+N-2s2∫∂iBny1-2sun∂un∂ν+vn∂vn∂νdσ-12∫∂iBny1-2s∇un2+∇vn2X-z0·νdσ+∫∂iBny1-2s∇un·X-z0∂un∂ν+∇vn·X-z0∂vn∂νdσ.By assumption, λ,μ≥0 and λ+μ>0; we may assume that λ>0. Set Bn′=Bσn-1(xn,0)∩CΩ. Recalling that, by (36), we have the decomposition un=un0+un1+un2, where un1=u0, un2=a0-1∑j=1k(σni)N-2s/2U(σni(x-xni),0), and un0=un-un1-un2. un0E0s(CΩ)→0 as n→∞. Then we deduce that, for n large enough, Bn′⊂Bn(53)∫Bn∩y=0un2dx≥∫Bn′∩y=0un2dx≥12∫Bn′∩y=0un22dx-2∫Bn′∩y=0un12+un02dx.After a direct calculation, we have(54)∫Bn′∩y=0un22dx≥Cσn-2s,∫Bn′∩y=0un12dx≤Cσn-N,(55)∫Bn′∩y=0un02dx≤Cun0L2s∗Ω2σn-2s.Note that un0E0s(CΩ)→0, as n→∞. Inserting (54) into (53), we get for n large enough(56)∫Bn∩y=0λun2dx≥Cσn-2s.By the choice of z0=(x0,0), as in [10], we only need to consider the right-hand side of (52) on ∂iBn. By applying Hölder inequality and using Lemma 4, we get(57)
2
α
n
+
β
n
∫
∂
i
B
n
∩
y
=
0
u
n
α
n
v
n
β
n
X
-
z
0
·
ν
d
σ
+
1
2
∫
∂
i
B
n
∩
y
=
0
λ
u
n
2
+
μ
v
n
2
X
-
z
0
·
ν
d
σ
+
N
-
2
s
2
∫
∂
i
B
n
y
1
-
2
s
u
n
∂
u
n
∂
ν
+
v
n
∂
v
n
∂
ν
d
σ
-
1
2
∫
∂
i
B
n
y
1
-
2
s
∇
u
n
2
+
∇
v
n
2
X
-
z
0
·
ν
d
σ
+
∫
∂
i
B
n
y
1
-
2
s
∇
u
n
·
X
-
z
0
∂
u
n
∂
ν
+
∇
v
n
·
X
-
z
0
∂
v
n
∂
ν
d
σ
≤
C
σ
n
-
1
/
2
∫
∂
i
B
n
∩
y
=
0
u
n
α
n
+
β
n
d
σ
α
n
/
α
n
+
β
n
∫
∂
i
B
n
∩
y
=
0
v
n
α
n
+
β
n
d
σ
β
n
/
α
n
+
β
n
+
∫
∂
i
B
n
∩
y
=
0
∇
u
n
2
+
∇
v
n
2
x
-
x
0
d
σ
+
∫
∂
i
B
n
∩
y
=
0
∇
u
n
2
+
∇
v
n
2
d
σ
1
/
2
∫
∂
i
B
n
∩
y
=
0
u
n
2
+
v
n
2
d
σ
1
/
2
≤
C
σ
n
-
N
-
2
s
/
2
.Inserting (56) and (57) into (52), we obtain (58)σn-2s≤Cσn-N-2s/2,which is a contradiction for n large enough due to N>6s.


## 4. Proof of Theorem1

For any positive integerk, define the Z2-homotopy class Fk as follows: (59)Fk≔A:A⊂E0sCΩis  compact,Z2-invariant,γA≥k,where the genus γ(A) is the smallest integer m, such that there exists an odd map ϕ∈C(A,Rm∖{0}). For k=1,2,…, we define the min–max value as (e.g., see p. 134 in [26]) (60)ck,ε=minA∈Fkmaxu,v∈A⁡Iλ,μεu,v.It follows from Corollary  7.12 in [26] that, for each small ε>0,ck,ε is critical value of Iλ,με, since Iλ,με satisfies the Palais-Smale condition. Thus problem (18) has a solution (uk,ε,vk,ε) such that Iλ,με(uk,ε,vk,ε)=ck,ε. Note that Iλ,με(tuk,ε,tvk,ε)→-∞ uniformly with respect to ε>0 as t→+∞; hence, ck,ε is uniformly bounded with respect to ε for each fixed k. By a direct calculation, we find uk,ε,vk,εE0s(CΩ)≤C uniformly with respect to ε for each fixed k. So now, we can apply Proposition 2 and obtain a subsequence of (uk,εn,vk,εn)n≥1, such that, as n→+∞,(uk,εn,vk,εn)→(uk,vk) strongly in E0s(CΩ) for some (uk,vk) and ck,εn→ck. Then (uk,vk) is solution of (1) and Iλ,μ(uk,vk)=ck. We are now ready to show that Iλ,μ has infinitely many critical point solutions. Note that ck is nondecreasing in k. By an argument similar to the one used in the proof of Theorem  1.1 in [9] we distinguish two cases.Case 1.
Suppose that there are1<k1<⋯ki<⋯, satisfying (61)ck1<⋯<cki<⋯.In this case, we have infinitely many distinct critical points and, therefore, infinitely many solutions.Case 2.
We assume, in this case, that for some positive integerm, ck=c for all k≥m. Suppose that, for any δ>0,Iλ,μ has a critical point (u,v) with Iλ,μ(u,v)∈(c-δ,c+δ) and Iλ,μ(u,v)≠c. In this case, we are done. So from now on we assume that there exists δ>0 such that Iλ,μ has no critical point (u,v) with Iλ,μ(u,v)∈(c-δ,c)∪(c,c+δ). In this case, using the deformation argument, we can prove that (62)γKc≥2,where Kc={(u,v)∈X:Iλ,μ(u,v)=c,Iλ,μ′(u,v)=0}. As a consequence, Iλ,μ has infinitely many critical points. Thus we can obtain infinitely many solutions for problem (1).



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*Source: 7197542-2016-08-31.xml*