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Research article

Infinitely many sign-changing solutions for a semilinear elliptic equation with variable exponent

  • This paper is devoted to study a class of semilinear elliptic equations with variable exponent. By means of perturbation technique, variational methods and a priori estimation, the existence of infinitely many sign-changing solutions to this class of problem is obtained.

    Citation: Changmu Chu, Yuxia Xiao, Yanling Xie. Infinitely many sign-changing solutions for a semilinear elliptic equation with variable exponent[J]. AIMS Mathematics, 2021, 6(6): 5720-5736. doi: 10.3934/math.2021337

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  • This paper is devoted to study a class of semilinear elliptic equations with variable exponent. By means of perturbation technique, variational methods and a priori estimation, the existence of infinitely many sign-changing solutions to this class of problem is obtained.



    In this paper, we discuss the existence and multiplicity of standing wave solutions for the following perturbed fractional p-Laplacian systems with critical nonlinearity

    {εps(Δ)spu+V(x)|u|p2u=K(x)|u|ps2u+Fu(x,u,v),xRN,εps(Δ)spv+V(x)|v|p2v=K(x)|v|ps2v+Fv(x,u,v),xRN, (1.1)

    where ε is a positive parameter, N>ps,s(0,1),ps=NpNps and (Δ)sp is the fractional p-Laplacian operator, which is defined as

    (Δ)spu(x)=limε0RNBε(x)|u(x)u(y)|p2(u(x)u(y))|xy|N+psdy,xRN,

    where Bε(x)={yRN:|xy|<ε}. The functions V(x),K(x) and F(x,u,v) satisfy the following conditions:

    (V0)VC(RN,R),minxRNV(x)=0andthereisaconstantb>0suchthattheset Vb:={xRN:V(x)<b}hasfiniteLebesguemeasure;

    (K0)KC(RN,R),0<infKsupK<;

    (F1)FC1(RN×R2,R)andFs(x,s,t),Ft(x,s,t)=o(|s|p1+|t|p1) uniformlyinxRNas|s|+|t|0;

    (F2)thereexistC0>0andp<κ<pssuchthat |Fs(x,s,t)|,|Ft(x,s,t)|C0(1+|s|κ1+|t|κ1);

    (F3)thereexistl0>0,d>pandμ(p,ps)suchthatF(x,s,t)l0(|s|d+|t|d)and 0<μF(x,s,t)Fs(x,s,t)s+Ft(x,s,t)tforall(x,s,t)RN×R2;

    (F4)Fs(x,s,t)=Fs(x,s,t)andFt(x,s,t)=Ft(x,s,t)forall(x,s,t)RN×R2.

    Conditions (V0),(K0), suggested by Ding and Lin [11] in studying perturbed Schrödinger equations with critical nonlinearity, and then was used in [28,32,33].

    In recent years, a great deal of attention has been focused on the study of standing wave solutions for perturbed fractional Schrödinger equation

    ε2s(Δ)su+V(x)u=f(u)inRN, (1.2)

    where s(0,1), N>2s and ε>0 is a small parameter. It is well known that the solution of (1.2) is closely related to the existence of solitary wave solutions for the following eqation

    iεωtε2(Δ)sωV(x)ω+f(ω)=0,(x,t)RN×R,

    where i is the imaginary unit. (Δ)s is the fractional Laplacian operator which arises in many areas such as physics, phase transitions, chemical reaction in liquids, finance and so on, see [1,6,18,22,27]. Additionally, Eq (1.2) is a fundamental equation of fractional quantum mechanics. For more details, please see [17,18].

    Equation (1.2) was also investigated extensively under various hypotheses on the potential and the nonlinearity. For example, Floer and Weinstein [12] first considered the existence of single-peak solutions for N=1 and f(t)=t3. They obtained a single-peak solution which concentrates around any given nondegenerate critical point of V. Jin, Liu and Zhang [16] constructed a localized bound-state solution concentrating around an isolated component of the positive minimum point of V, when the nonlinear term f(u) is a general critical nonlinearity. More related results can be seen in [5,7,10,13,14,26,43] and references therein. Recently, Zhang and Zhang [46] obtained the multiplicity and concentration of positive solutions for a class of fractional unbalanced double-phase problems by topological and variational methods. Related to (1.2) with s=1, see [31,39] for quasilinear Schrödinger equations.

    On the other hand, fractional p-Laplacian operator can be regarded as an extension of fractional Laplacian operator. Many researchers consider the following equation

    εps(Δ)spu+V(x)|u|p2u=f(x,u). (1.3)

    When f(x,u)=A(x)|u|ps2u+h(x,u), Li and Yang [21] obtained the existence and multiplicity of weak solutions by variational methods. When f(x,u)=λf(x)|u|q2u+g(x)|u|r2u, under suitable assumptions on nonlinearity and weight functions, Lou and Luo [19] established the existence and multiplicity of positive solutions via variational methods. With regard to the p-fractional Schrödinger-Kirchhoff, Song and Shi [29] considered the following equation with electromagnetic fields

    {εpsM([u]ps,Aε)(Δ)sp,Aεu+V(x)|u|p2u=|u|ps2u+h(x,|u|p)|u|p2u,xRN,u(x)0,as. (1.4)

    They obtained the existence and multiplicity solutions for (1.4) by using the fractional version of concentration compactness principle and variational methods, see also [24,25,34,35,38,41] and references therein. Related to (1.3) with s=1, see [15,23].

    Recently, from a mathematical point of view, (fractional) elliptic systems have been the focus for many researchers, see [2,8,9,20,30,37,42,44,45]. As far as we know, there are few results concerned with the (fractional) p-Laplacian systems with a small parameter. In this direction, we cite the work of Zhang and Liu [40], who studied the following p-Laplacian elliptic systems

    {εpΔpu+V(x)|u|p2u=K(x)|u|p2u+Hu(u,v),xRN,εpΔpv+V(x)|v|p2v=K(x)|v|p2v+Hv(u,v),xRN. (1.5)

    By using variational methods, they proved the existence of nontrivial solutions for (1.5) provided that ε is small enough. In [36], Xiang, Zhang and Wei investigated the following fractional p-Laplacian systems without a small parameter

    {(Δ)spu+a(x)|u|p2u=Hu(x,u,v),xRN,(Δ)sqv+b(x)|v|p2v=Hv(x,u,v),xRN. (1.6)

    Under some suitable conditions, they obtained the existence of nontrivial and nonnegative solutions for (1.6) by using the mountain pass theorem.

    Motivated by the aforementioned works, it is natural to ask whether system (1.5) has a nontrivial solution when the p-Laplacian operator is replaced by the fractional p-Laplacian operator. As far as we know, there is no related work in this direction so far. In this paper, we give an affirmative answer to this question considering the existence and multiplicity of standing wave solutions for (1.1).

    Now, we present our results of this paper.

    Theorem 1.1. Assume that (V0), (K0) and (F1)(F3) hold. Then for any τ>0, there is Γτ>0 such that if ε<Γτ, system (1.1) has at least one solution (uε,vε)(0,0) in W as ε0, where W is stated later, satisfying:

    μpμp[R2Nεps(|uε(x)uε(y)|p|xy|N+ps+|vε(x)vε(y)|p|xy|N+ps)dxdy+RNV(x)(|uε|p+|vε|p)dx]τεN

    and

    sNRNK(x)(|uε|ps+|vε|ps)dx+μppRNF(x,uε,vε)dxτεN.

    Theorem 1.2. Let (V0), (K0) and (F1)(F4) hold. Then for any mN and τ>0 there is Γmτ>0 such that if ε<Γmτ, system (1.1) has at least m pairs of solutions (uε,vε), which also satisfy the above estimates in Theorem 1.1. Moreover, (uε,vε)(0,0) in W as ε0.

    Remark 1.1. On one hand, our results extend the results in [40], in which the authors considered the existence of solutions for perturbed p-Laplacian system, i.e., system (1.1) with s=1. On the other hand, our results also extend the results in [21] to a class of perturbed fractional p-Laplacian system (1.1).

    Remark 1.2. Compared with the results obtained by [12,13,14,15,16], when ε0, the solutions of Theorems 1.1 and 1.2 are close to trivial solutions.

    In this paper, our goal is to prove the existence and multiplicity of standing wave solutions for (1.1) by variational approach. The main difficulty lies on the lack of compactness of the energy functional associated to system (1.1) because of unbounded domain RN and critical nonlinearity. To overcome this difficulty, we adopt some ideas used in [11] to prove that (PS)c condition holds.

    The rest of this article is organized as follows. In Section 2, we introduce the working space and restate the system in a equivalent form by replacing εps with λ. In Section 3, we study the behavior of (PS)c sequence. In Section 4, we complete the proof of Theorems 2.1 and 2.2, respectively.

    To obtain the existence and multiplicity of standing wave solutions of system (1.1) for small ε, we rewrite (1.1) in a equivalent form. Let λ=εps, then system (1.1) can be expressed as

    {(Δ)spu+λV(x)|u|p2u=λK(x)|u|ps2u+λFu(x,u,v),xRN,(Δ)spv+λV(x)|v|p2v=λK(x)|v|ps2v+λFv(x,u,v),xRN, (2.1)

    for λ+.

    We introduce the usual fractional Sobolev space

    Ws,p(RN):={uLp(RN):[u]s,p<}

    equipped with the norm

    ||u||s,p=(|u|p+[u]ps,p)1p,

    where ||p is the norm in Lp(RN) and

    [u]s,p=(R2N|u(x)u(y)|p|xy|N+psdxdy)1p

    is the Gagliardo seminorm of a measurable function u:RNR. In this paper, we continue to work in the following subspace of Ws,p(RN) which is defined by

    Wλ:={uWs,p(RN):RNλV(x)|u|pdx<,λ>0}

    with the norm

    ||u||λ=([u]ps,p+RNλV(x)|u|pdx)1p.

    Notice that the norm ||||s,p is equivalent to ||||λ for each λ>0. It follows from (V0) that Wλ continuously embeds in Ws,p(RN). For the fractional system (2.1), we shall work in the product space W=Wλ×Wλ with the norm ||(u,v)||p=||u||pλ+||v||pλ for any (u,v)W.

    We recall that (u,v)W is a weak solution of system (2.1) if

    R2N|u(x)u(y)|p2(u(x)u(y))(ϕ(x)ϕ(y))|xy|N+psdxdy+λRNV(x)|u|p2uϕdx+R2N|v(x)v(y)|p2(v(x)v(y))(ψ(x)ψ(y))|xy|N+psdxdy+λRNV(x)|v|p2vψdx=λRNK(x)(|u|ps2uϕ+|v|ps2vψ)dx+λRN(Fu(x,u,v)ϕ+Fv(x,u,v)ψ)dx

    for all (ϕ,ψ)W.

    Note that the energy functional associated with (2.1) is defined by

    Φλ(u,v)=1pR2N|u(x)u(y)|p|xy|N+psdxdy+1pRNλV(x)|u|pdx+1pR2N|v(x)v(y)|p|xy|N+psdxdy+1pRNλV(x)|v|pdxλpsRNK(x)(|u|ps+|v|ps)dxλRNF(x,u,v)dx=1p||(u,v)||pλpsRNK(x)(|u|ps+|v|ps)dxλRNF(x,u,v)dx.

    Clearly, it is easy to check that ΦλC1(W,R) and its critical points are weak solution of system (2.1).

    In order to prove Theorem 1.1 and 1.2, we only need to prove the following results.

    Theorem 2.1. Assume that (V0), (K0) and (F1)(F3) hold. Then for any τ>0, there is Λτ>0 such that if λΛτ, system (2.1) has at least one solution (uλ,vλ)(0,0) in W as λ, satisfying:

    μpμp[R2N(|uλ(x)uλ(y)|p|xy|N+ps+|vλ(x)vλ(y)|p|xy|N+ps)dxdy+RNλV(x)(|uλ|p+|vλ|p)dx]τλ1Nps (2.2)

    and

    sNRNK(x)(|uλ|ps+|vλ|ps)dx+μppRNF(x,uλ,vλ)dxτλNps. (2.3)

    Theorem 2.2. Assume that (V0), (K0) and (F1)(F4) hold. Then for any mN and τ>0 there is Λmτ>0 such that if λΛmτ, system (2.1) has at least m pairs of solutions (uλ,vλ), which also satisfy the estimates in Theorem 2.1. Moreover, (uλ,vλ)(0,0) in W as λ.

    In this section, we are focused on the compactness of the functional Φλ.

    Recall that a sequence {(un,vn)}W is a (PS)c sequence at level c, if Φλ(un,vn)c and Φλ(un,vn)0. Φλ is said to satisfy the (PS)c condition if any (PS)c sequence contains a convergent subsequence.

    Proposition 3.1. Assume that the conditions (V0),(K0) and (F1)(F3) hold. Then there exists a constant α>0 independent of λ such that, for any (PS)c sequence {(un,vn)}W for Φλ with (un,vn)(u,v), either (un,vn)(u,v) or cΦλ(u,v)αλ1Nps.

    Corollary 3.1. Under the assumptions of Proposition 3.1, Φλ satisfies the (PS)c condition for all c<αλ1Nps.

    The proof of Proposition 3.1 consists of a series of lemmas which will occupy the rest of this section.

    Lemma 3.1. Assume that (V0),(K0) and (F3) are satisfied. Let {(un,vn)}W be a (PS)c sequence for Φλ. Then c0 and {(un,vn)} is bounded in W.

    Proof. Let {(un,vn)} be a (PS)c sequence for Φλ, we obtain that

    Φλ(un,vn)c,Φλ(un,vn)0,n.

    By (K0) and (F3), we deduce that

    c+o(1)||(un,vn)||=Φλ(un,vn)1μΦλ(un,vn),(un,vn)=(1p1μ)||(un,vn)||p+λ(1μ1ps)RNK(x)(|u|ps+|v|ps)dx+λRN[1μ(Fu(x,un,vn)un+Fv(x,un,vn)vn)F(x,un,vn)]dx(1p1μ)||(un,vn)||p, (3.1)

    which implies that there exists M>0 such that

    ||(un,vn)||pM.

    Thus, {(un,vn)} is bounded in W. Taking the limit in (3.1), we show that c0. This completes the proof.

    From the above lemma, there exists (u,v)W such that (un,vn)(u,v) in W. Furthermore, passing to a subsequence, we have unu and vnv in Lγloc(RN) for any γ[p,ps) and un(x)u(x) and vn(x)v(x) a.e. in RN. Clearly, (u,v) is a critical point of Φλ.

    Lemma 3.2. Let {(un,vn)} be stated as in Lemma 3.1 and γ[p,ps). Then there exists a subsequence {(unj,vnj)} such that for any ε>0, there is rε>0 with

    limsupjBjBr|unj|γdxε,limsupjBjBr|vnj|γdxε,

    for all rrε, where, Br:={xRN:|x|r}.

    Proof. The proof is similar to the one of Lemma 3.2 of [11]. We omit it here.

    Let σ:[0,)[0,1] be a smooth function satisfying σ(t)=1 if t1, σ(t)=0 if t2. Define ¯uj(x)=σ(2|x|j)u(x), ¯vj(x)=σ(2|x|j)v(x). It is clear that

    ||u¯uj||λ0and||v¯vj||λ0asj. (3.2)

    Lemma 3.3. Let {(unj,vnj)} be stated as in Lemma 3.2, then

    limjRN[Fu(x,unj,vnj)Fu(x,unj¯uj,vnj¯vj)Fu(x,¯uj,¯vj)]ϕdx=0

    and

    limjRN[Fv(x,unj,vnj)Fv(x,unj¯uj,vnj¯vj)Fv(x,¯uj,¯vj)]ψdx=0

    uniformly in (ϕ,ψ)W with ||(ϕ,ψ)||1.

    Proof. By (3.2) and the local compactness of Sobolev embedding, we know that for any r>0,

    limjBr[Fu(x,unj,vnj)Fu(x,unj¯uj,vnj¯vj)Fu(x,¯uj,¯vj)]ϕdx=0, (3.3)

    uniformly for ||ϕ||1. For any ε>0, there exists rε>0 such that

    limsupjBjBr|¯uj|γdxRNBr|u|γdxε,

    for all rrε, see [Lemma 3.2, 11]. From (F1) and (F2), we obtain

    |Fu(x,u,v)|C0(|u|p1+|v|p1+|u|κ1+|v|κ1). (3.4)

    Thus, from (3.3), (3.4) and the Hölder inequality, we have

    limsupjRN[Fu(x,unj,vnj)Fu(x,unj¯uj,vnj¯vj)Fu(x,¯uj,¯vj)]ϕdxlimsupjBjBr[Fu(x,unj,vnj)Fu(x,unj¯uj,vnj¯vj)Fu(x,¯uj,¯vj)]ϕdxC1limsupjBjBr[(|unj|p1+|¯uj|p1+|vnj|p1+|¯vj|p1)]ϕdx+C2limsupjBjBr[(|unj|κ1+|¯uj|κ1+|vnj|κ1+|¯vj|κ1)]ϕdxC1limsupj[|unj|p1Lp(BjBr)+|¯uj|p1Lp(BjBr)+|vnj|p1Lp(BjBr)+|¯vj|p1Lp(BjBr)]|ϕ|p+C2limsupj[|unj|κ1Lκ(BjBr)+|¯uj|κ1Lκ(BjBr)+|vnj|κ1Lκ(BjBr)+|¯vj|κLκ(BjBr)]|ϕ|κC3εp1p+C4εκ1κ,

    where C1,C2,C3 and C4 are positive constants. Similarly, we can deduce that the other equality also holds.

    Lemma 3.4. Let {(unj,vnj)} be stated as in Lemma 3.2, the following facts hold:

    (i)Φλ(unj¯uj,vnj¯vj)cΦλ(u,v);

    (ii)Φλ(unj¯uj,vnj¯vj)0inW1(thedualspaceofW).

    Proof. (i) We have

    Φλ(unj¯uj,vnj¯vj)=Φλ(unj,vnj)Φλ(¯uj,¯vj)+λpsRNK(x)(|unj|ps|unj¯uj|ps|¯uj|ps+|vnj|ps|vnj¯vj|ps|¯vj|ps)dx+λRN(F(x,unj,vnj)F(x,unj¯uj,vnj¯vj)F(x,¯uj,¯vj))dx.

    Using (3.2) and the Brézis-Lieb Lemma [4], it is easy to get

    limjRNK(x)(|unj|ps|unj¯uj|ps|¯uj|ps+|vnj|ps|vnj¯vj|ps|¯vj|ps)dx=0

    and

    limjRN(F(x,unj,vnj)F(x,unj¯uj,vnj¯vj)F(x,¯uj,¯vj))dx=0.

    Using the fact that Φλ(unj,vnj)c and Φλ(¯uj,¯vj)Φλ(u,v) as j, we have

    Φλ(unj¯uj,vnj¯vj)cΦλ(u,v).

    (ii) We observe that for any (ϕ,ψ)W satisfying ||(ϕ,ψ)||1,

    Φλ(unj¯uj,vnj¯vj),(ϕ,ψ)=Φλ(unj,vnj),(ϕ,ψ)Φλ(¯uj,¯vj),(ϕ,ψ)+λRNK(x)[(|unj|ps2unj|unj¯uj|ps2(unj¯uj)|¯uj|ps2¯uj)ϕ+(|vnj|ps2vnj|vnj¯vj|ps2(vnj¯vj)|¯vj|ps2¯vj)ψ]dx+λRN[(Fu(x,unj,vnj)Fu(x,unj¯uj,vnj¯vj)Fu(x,¯uj,¯vj))ϕ+(Fv(x,unj,vnj)Fv(x,unj¯uj,vnj¯vj)Fv(x,¯uj,¯vj))ψ]dx.

    It follows from a standard argument that

    limjRNK(x)(|unj|ps2unj|unj¯uj|ps2(unj¯uj)|¯uj|ps2¯uj)ϕdx=0

    and

    limjRNK(x)(|vnj|ps2vnj|vnj¯vj|ps2(vnj¯vj)|¯vj|ps2¯vj)ψdx=0

    uniformly in ||(ϕ,ψ)||1. By Lemma 3.3, we obtain Φλ(unj¯uj,vnj¯vj)0. We complete this proof.

    Set u1j=unj¯uj, v1j=vnj¯vj, then unju=u1j+(¯uju), vnjv=v1j+(¯vjv). From (3.2), we have (unj,vnj)(u,v) if and only if (u1j,v1j)(0,0). By Lemma 3.4, one has along a subsequence that Φλ(u1j,v1j)cΦλ(u,v) and Φλ(u1j,v1j)0.

    Note that Φλ(u1j,v1j),(u1j,v1j)=0, by computation, we get

    R2N|u1j(x)u1j(y)|p|xy|N+psdxdy+RNλV(x)|u1j|pdx+R2N|v1j(x)v1j(y)|p|xy|N+psdxdy+RNλV(x)|v1j|pdxλRNK(x)(|u1j|ps+|v1j|ps)dxλRNF(x,u1j,v1j)dx=0 (3.5)

    Hence, by (F3) and (3.5), we have

    Φλ(u1j,v1j)1pΦλ(u1j,v1j),(u1j,v1j)=(1p1ps)λRNK(x)(|u1j|ps+|v1j|ps)dx+λRN[1p(Fu(x,u1j,v1j)u1j+Fu(x,u1j,v1j)v1j)F(x,u1j,v1j)]dxλsKminNRN(|u1j|ps+|v1j|ps)dx,

    where Kmin=infxRNK(x)>0. So, it is easy to see that

    |u1j|psps+|v1j|pspsN(cΦλ(u,v))λsKmin+o(1). (3.6)

    Denote Vb(x)=max{V(x),b}, where b is the positive constant from assumption of (V0). Since the set Vb has finite measure and (u1j,v1j)(0,0) in Lploc×Lploc, we obtain

    RNV(x)(|u1j|p+|v1j|p)dx=RNVb(x)(|u1j|p+|v1j|p)dx+o(1). (3.7)

    By (K0),(F1) and (F2), we can find a constant Cb>0 such that

    RNK(x)(|u1j|ps+|v1j|ps)dx+RN(Fu(x,u1j,v1j)u1j+Fv(x,u1j,v1j)v1j)dxb(|u1j|pp+|v1j|pp)+Cb(|u1j|psps+|v1j|psps). (3.8)

    Let S is fractional Sobolev constant which is defined by

    S|u|ppsR2N|u(x)u(y)|p|xy|N+psdxdyforalluWs,p(RN). (3.9)

    Proof of Proposition 3.1. Assume that (unj,vnj)(u,v), then liminfj||(u1j,v1j)||>0 and cΦλ(u,v)>0.

    From (3.5), (3.7), (3.8) and (3.9), we deduce

    S(|u1j|pps+|v1j|pps)R2N|u1j(x)u1j(y)|p|xy|N+psdxdy+RNλV(x)|u1j|pdx+R2N|v1j(x)v1j(y)|p|xy|N+psdxdy+RNλV(x)|v1j|pdxRNλV(x)(|u1j|p+|v1j|p)dx=λRNK(x)(|u1j|ps+|v1j|ps)dx+λRN(Fu(x,u1j,v1j)u1j+Fv(x,u1j,v1j)v1j)dxλRNVb(x)(|u1j|p+|v1j|p)dxλCb(|u1j|psps+|v1j|psps)+o(1).

    Thus, by (3.6), we have

    SλCb(|u1j|psps+|v1j|psps)pspps+o(1)λCb(N(cΦλ(u,v))λsKmin)sN+o(1),

    or equivalently

    αλ1NpscΦλ(u,v),

    where α=sKminN(SCb)Nps. The proof is complete.

    Lemma 4.1. Suppose that (V0), (K0),(F1),(F2) and (F3) are satisfied, then the functional Φλ satisfies the following mountain pass geometry structure:

    (i) there exist positive constants ρ and a such that Φλ(u,v)a for ||(u,v)||=ρ;

    (ii) for any finite-dimensional subspace YW,

    Φλ(u,v),as(u,v)W,||(u,v)||+.

    (iii) for any τ>0 there exists Λτ>0 such that each λΛτ, there exists ˜ωλY with ||˜ωλ||>ρ, Φλ(˜ωλ)0 and

    maxt0Φλ(t˜ωλ)τλ1Nps.

    Proof. (i) From (F1),(F2), we have for any ε>0, there is Cε>0 such that

    1psRNK(x)(|u|ps+|v|ps)dx+RNF(x,u,v)dxε|(u,v)|pp+Cε|(u,v)|psps. (4.1)

    Thus, combining with (4.1) and Sobolev inequality, we deduce that

    Φλ(u,v)=1p||(u,v)||pλpsRNK(x)(|u|ps+|v|ps)dxλRNF(x,u,v)dx1p||(u,v)||pλεC5||(u,v)||pλC6Cε||(u,v)||ps,

    where ε is small enough and C5,C6>0, thus (i) is proved because ps>p.

    (ii) By (F3), we define the functional ΨλC1(W,R) by

    Ψλ(u,v)=1p||(u,v)||pλl0RN(|u|d+|v|d)dx.

    Then

    Φλ(u,v)Ψλ(u,v),forall(u,v)W.

    For any finite-dimensional subspace YW, we only need to prove

    Ψλ(u,v),as(u,v)Y,||(u,v)||+.

    In fact, we have

    Ψλ(u,v)=1p||(u,v)||pλl0|(u,v)|dd.

    Since all norms in a finite dimensional space are equivalent and p<d<ps, thus (ii) holds.

    (iii) From Corollary 3.1, for λ large and c small enough, Φλ satisfies (PS)c condition. Thus, we will find a special finite dimensional-subspace by which we construct sufficiently small minimax levels for Φλ when λ large enough.

    Recall that

    inf{R2N|φ(x)φ(y)|p|xy|N+psdxdy:φC0(RN),|φ|d=1}=0,p<d<ps,

    see [40] for this proof. For any 0<ε<1, we can take φεC0(RN) with |φε|d=1, supp φεBrε(0) and [φε]pp,s<ε.

    Let

    ¯ωλ(x):=(ωλ(x),ωλ(x))=(φε(λ1psx),φε(λ1psx)).

    For t0, (F3) imply that

    Φλ(t¯ωλ)2tppR2N|ωλ(x)ωλ(y)|p|xy|N+psdxdy+2tppRNλV(x)|ωλ|pdxλRNF(x,tωλ,tωλ)dxλ1Nps{2tppR2N|φε(x)φε(y)|p|xy|N+psdxdy+2tppRNV(λ1psx)|φε|pdx2l0tdRN|φε|ddx}λ1Nps2l0(dp)p(R2N|φε(x)φε(y)|p|xy|N+psdxdy+RNV(λ1psx)|φε|pdxl0d)ddp.

    Indeed, for t>0, define

    g(t)=2tppR2N|φε(x)φε(y)|p|xy|N+psdxdy+2tppRNλV(λ1psx)|φε|pdx2l0tdRN|φε|ddx.

    It is easy to show that t0=(R2N|φε(x)φε(y)|p|xy|N+psdxdy+RNV(λ1psx)|φε|pdxl0d)1dp is a maximum point of g and

    maxt0g(t)=g(t0)=2l0(dp)p(R2N|φε(x)φε(y)|p|xy|N+psdxdy+RNV(λ1psx)|φε|pdxl0d)ddp.

    Since V(0)=0 and supp φεBrε(0), there exists Λε>0 such that

    V(λ1psx)<ε|φε|pp,|x|rε,λ>Λε.

    Hence, we have

    maxt0Φλ(t¯ωλ)2l0(dp)p(1l0d)ddp(2ε)ddpλ1Nps,λ>Λε.

    Choose ε>0 such that

    2l0(dp)p(1l0d)ddp(2ε)ddpτ,

    and taking Λτ=Λε, from (ii), we can take ¯t large enough and define ˜ωλ=¯t¯ωλ, then we have

    Φλ(˜ωλ)<0andmax0t1Φλ(t˜ωλ)τλ1Nps.

    Proof of Theorem 2.1. From Lemma 4.1, for any 0<τ<α, there exists Λτ>0 such that for λΛτ, we have

    c=infηΓλmaxt[0,1]Φλ(η(t))τλ1Nps,

    where Γλ={ηC([0,1],W):η(0)=0,η(1)=˜ωλ}. Furthermore, in virtue of Corollary 3.1, we obtain that (PS)c condition hold for Φλ at c. Therefore, by the mountain pass theorem, there is (uλ,vλ)W such that Φλ(uλ,vλ)=0 and Φλ(uλ,vλ)=c.

    Finally, we prove that (uλ,vλ) satisfies the estimates in Theorem 2.1.

    Since (uλ,vλ) is a critical point of Φλ, there holds for θ[p,ps]

    τλ1NpsΦλ(uλ,vλ)1θΦλ(uλ,vλ),(uλ,vλ)(1p1θ)||(uλ,vλ)||p+λ(1θ1ps)RNK(x)(|uλ|ps+|vλ|ps)dx+λ(μθ1)RNF(x,uλ,vλ)dx.

    Taking θ=μ, we get the estimate (2.2) and taking θ=p yields the estimate (2.3).

    To obtain the multiplicity of critical points, we will adopt the index theory defined by the Krasnoselski genus.

    Proof of Theorem 2.2. Denote the set of all symmetric (in the sense that A=A) and closed subsets of A by . For any A let gen (A) be the Krasnoselski genus and

    i(A)=minkΥgen(k(A)Bρ),

    where Υ is the set of all odd homeomorphisms kC(W,W) and ρ is the number from Lemma 4.1. Then i is a version of Benci's pseudoindex [3]. (F4) implies that Φλ is even. Set

    cλj:=infi(A)jsup(u,v)AΦλ(u,v),1jm.

    If cλj is finite and Φλ satisfies (PS)cλj condition, then we know that all cλj are critical values for Φλ.

    Step 1. We show that Φλ satisfies (PS)cλj condition at all levels cλj<τλ1Nps.

    To complete the claim, we need to estimate the level cλj in special finite-dimensional subspaces.

    Similar to proof in Lemma 4.1, for any mN, ε>0 and j=1,2,,m, one can choose m functions φjεC0(RN) with supp φiε supp φjε= if ij, |φjε|d=1 and [φjε]pp,s<ε.

    Let rmε>0 be such that supp φjεBrmε(0). Set

    ¯ωjλ(x):=(ωjλ(x),ωjλ(x))=(φjε(λ1psx),φjε(λ1psx))

    and define

    Fmλ:=Span{¯ω1λ,¯ω2λ,,¯ωmλ}.

    Then i(Fmλ)=dimFmλ=m. Observe that for each ˜ω=mj=1tj¯ωjλFmλ,

    Φλ(˜ω)=mj=1Φλ(tj¯ωjλ)

    and for tj>0

    Φλ(tj¯ωjλ)2tpjpR2N|ωjλ(x)ωjλ(y)|p|xy|N+psdxdy+2tpjpRNλV(x)|ωjλ|pdxλRNF(x,tjωjλ,tjωjλ)dxλ1Nps{2tpjpR2N|φjε(x)φjε(y)|p|xy|N+psdxdy+2tpjpRNV(λ1psx)|φjε|pdx2l0tdjRN|φjε|ddx}.

    Set

    ηε:=max{|φjε|pp:j=1,2,,m}.

    Since V(0)=0 and supp φjεBrmε(0), there exists Λmε>0 such that

    V(λ1psx)<εηε,|x|rmε,λ>Λmε.

    Consequently, there holds

    sup˜wFmλΦλ(˜w)ml0(2ε)ddpλ1Nps,λ>Λmε.

    Choose ε>0 small that ml0(2ε)ddp<τ. Thus for any mN and τ(0,α), there exists Λmτ=Λmε such that λ>Λmτ, we can choose a m-dimensional subspace Fmλ with maxΦλ(Fmλ)τλ1Nps and

    cλ1cλ2sup˜wFmλΦλ(˜w)τλ1Nps.

    From Corollary 3.1, we know that Φλ satisfies the (PS) condition at all levels cλj. Then all cλj are critical values.

    Step 2. We prove that (2.1) has at least m pairs of solutions by the mountain-pass theorem.

    By Lemma 4.1, we know that Φλ satisfies the mountain pass geometry structure. From step 1, we note that Φλ also satisfies (PS)cλj condition at all levels cλj<τλ1Nps. By the usual critical point theory, all cλj are critical levels and Φλ has at least m pairs of nontrivial critical points satisfying

    aΦλ(u,v)τλ1Nps.

    Thus, (2.1) has at least m pairs of solutions. Finally, as in the proof of Theorem 2.1, we know that these solutions satisfy the estimates (2.2) and (2.3).

    In this paper, we have obtained the existence and multiplicity of standing wave solutions for a class of perturbed fractional p-Laplacian systems involving critical exponents by variational methods. In the next work, we will extend the study to the case of perturbed fractional p-Laplacian systems with electromagnetic fields.

    The author is grateful to the referees and the editor for their valuable comments and suggestions.

    The author declares no conflict of interest.



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