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|p−2u=K(x)|u|p∗s−2u+Fu(x,u,v),x∈RN,εps(−Δ)spv+V(x)|v|p−2v=K(x)|v|p∗s−2v+Fv(x,u,v),x∈RN, | (1.1) |
where ε is a positive parameter, N>ps,s∈(0,1),p∗s=NpN−ps and (−Δ)sp is the fractional p-Laplacian operator, which is defined as
(−Δ)spu(x)=limε→0∫RN∖Bε(x)|u(x)−u(y)|p−2(u(x)−u(y))|x−y|N+psdy,x∈RN, |
where Bε(x)={y∈RN:|x−y|<ε}. The functions V(x),K(x) and F(x,u,v) satisfy the following conditions:
(V0)V∈C(RN,R),minx∈RNV(x)=0andthereisaconstantb>0suchthattheset Vb:={x∈RN:V(x)<b}hasfiniteLebesguemeasure;
(K0)K∈C(RN,R),0<infK≤supK<∞;
(F1)F∈C1(RN×R2,R)andFs(x,s,t),Ft(x,s,t)=o(|s|p−1+|t|p−1) uniformlyinx∈RNas|s|+|t|→0;
(F2)thereexistC0>0andp<κ<p∗ssuchthat |Fs(x,s,t)|,|Ft(x,s,t)|≤C0(1+|s|κ−1+|t|κ−1);
(F3)thereexistl0>0,d>pandμ∈(p,p∗s)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
iεωt−ε2(−Δ)sω−V(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|p−2u=f(x,u). | (1.3) |
When f(x,u)=A(x)|u|p∗s−2u+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|q−2u+g(x)|u|r−2u, 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|p−2u=|u|p∗s−2u+h(x,|u|p)|u|p−2u,x∈RN,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|p−2u=K(x)|u|p∗−2u+Hu(u,v),x∈RN,−εpΔpv+V(x)|v|p−2v=K(x)|v|p∗−2v+Hv(u,v),x∈RN. | (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|p−2u=Hu(x,u,v),x∈RN,(−Δ)sqv+b(x)|v|p−2v=Hv(x,u,v),x∈RN. | (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 ε→0, where W is stated later, satisfying:
μ−pμp[∫∫R2Nεps(|uε(x)−uε(y)|p|x−y|N+ps+|vε(x)−vε(y)|p|x−y|N+ps)dxdy+∫RNV(x)(|uε|p+|vε|p)dx]≤τεN |
and
sN∫RNK(x)(|uε|p∗s+|vε|p∗s)dx+μ−pp∫RNF(x,uε,vε)dx≤τεN. |
Theorem 1.2. Let (V0), (K0) and (F1)–(F4) hold. Then for any m∈N 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 ε→0.
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 ε→0, 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 ε−ps 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=ε−ps, then system (1.1) can be expressed as
{(−Δ)spu+λV(x)|u|p−2u=λK(x)|u|p∗s−2u+λFu(x,u,v),x∈RN,(−Δ)spv+λV(x)|v|p−2v=λK(x)|v|p∗s−2v+λFv(x,u,v),x∈RN, | (2.1) |
for λ→+∞.
We introduce the usual fractional Sobolev space
Ws,p(RN):={u∈Lp(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|x−y|N+psdxdy)1p |
is the Gagliardo seminorm of a measurable function u:RN→R. In this paper, we continue to work in the following subspace of Ws,p(RN) which is defined by
Wλ:={u∈Ws,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)|p−2(u(x)−u(y))(ϕ(x)−ϕ(y))|x−y|N+psdxdy+λ∫RNV(x)|u|p−2uϕdx+∫∫R2N|v(x)−v(y)|p−2(v(x)−v(y))(ψ(x)−ψ(y))|x−y|N+psdxdy+λ∫RNV(x)|v|p−2vψdx=λ∫RNK(x)(|u|p∗s−2uϕ+|v|p∗s−2vψ)dx+λ∫RN(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)=1p∫∫R2N|u(x)−u(y)|p|x−y|N+psdxdy+1p∫RNλV(x)|u|pdx+1p∫∫R2N|v(x)−v(y)|p|x−y|N+psdxdy+1p∫RNλV(x)|v|pdx−λp∗s∫RNK(x)(|u|p∗s+|v|p∗s)dx−λ∫RNF(x,u,v)dx=1p||(u,v)||p−λp∗s∫RNK(x)(|u|p∗s+|v|p∗s)dx−λ∫RNF(x,u,v)dx. |
Clearly, it is easy to check that Φλ∈C1(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 λ≥Λτ, system (2.1) has at least one solution (uλ,vλ)→(0,0) in W as λ→∞, satisfying:
μ−pμp[∫∫R2N(|uλ(x)−uλ(y)|p|x−y|N+ps+|vλ(x)−vλ(y)|p|x−y|N+ps)dxdy+∫RNλV(x)(|uλ|p+|vλ|p)dx]≤τλ1−Nps | (2.2) |
and
sN∫RNK(x)(|uλ|p∗s+|vλ|p∗s)dx+μ−pp∫RNF(x,uλ,vλ)dx≤τλ−Nps. | (2.3) |
Theorem 2.2. Assume that (V0), (K0) and (F1)–(F4) hold. Then for any m∈N and τ>0 there is Λmτ>0 such that if λ≥Λmτ, 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 Φ′λ(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 c−Φλ(u,v)≥αλ1−Nps.
Corollary 3.1. Under the assumptions of Proposition 3.1, Φλ satisfies the (PS)c condition for all c<αλ1−Nps.
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 c≥0 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,Φ′λ(un,vn)→0,n→∞. |
By (K0) and (F3), we deduce that
c+o(1)||(un,vn)||=Φλ(un,vn)−1μ⟨Φ′λ(un,vn),(un,vn)⟩=(1p−1μ)||(un,vn)||p+λ(1μ−1p∗s)∫RNK(x)(|u|p∗s+|v|p∗s)dx+λ∫RN[1μ(Fu(x,un,vn)un+Fv(x,un,vn)vn)−F(x,un,vn)]dx≥(1p−1μ)||(un,vn)||p, | (3.1) |
which implies that there exists M>0 such that
||(un,vn)||p≤M. |
Thus, {(un,vn)} is bounded in W. Taking the limit in (3.1), we show that c≥0. 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 un→u and vn→v in Lγloc(RN) for any γ∈[p,p∗s) 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,p∗s). Then there exists a subsequence {(unj,vnj)} such that for any ε>0, there is rε>0 with
limsupj→∞∫Bj∖Br|unj|γdx≤ε,limsupj→∞∫Bj∖Br|vnj|γdx≤ε, |
for all r≥rε, where, Br:={x∈RN:|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 t≤1, σ(t)=0 if t≥2. Define ¯uj(x)=σ(2|x|j)u(x), ¯vj(x)=σ(2|x|j)v(x). It is clear that
||u−¯uj||λ→0and||v−¯vj||λ→0asj→∞. | (3.2) |
Lemma 3.3. Let {(unj,vnj)} be stated as in Lemma 3.2, then
limj→∞∫RN[Fu(x,unj,vnj)−Fu(x,unj−¯uj,vnj−¯vj)−Fu(x,¯uj,¯vj)]ϕdx=0 |
and
limj→∞∫RN[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,
limj→∞∫Br[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
limsupj→∞∫Bj∖Br|¯uj|γdx≤∫RN∖Br|u|γdx≤ε, |
for all r≥rε, see [Lemma 3.2, 11]. From (F1) and (F2), we obtain
|Fu(x,u,v)|≤C0(|u|p−1+|v|p−1+|u|κ−1+|v|κ−1). | (3.4) |
Thus, from (3.3), (3.4) and the Hölder inequality, we have
limsupj→∞∫RN[Fu(x,unj,vnj)−Fu(x,unj−¯uj,vnj−¯vj)−Fu(x,¯uj,¯vj)]ϕdx≤limsupj→∞∫Bj∖Br[Fu(x,unj,vnj)−Fu(x,unj−¯uj,vnj−¯vj)−Fu(x,¯uj,¯vj)]ϕdx≤C1limsupj→∞∫Bj∖Br[(|unj|p−1+|¯uj|p−1+|vnj|p−1+|¯vj|p−1)]ϕdx+≤C2limsupj→∞∫Bj∖Br[(|unj|κ−1+|¯uj|κ−1+|vnj|κ−1+|¯vj|κ−1)]ϕdx≤C1limsupj→∞[|unj|p−1Lp(Bj∖Br)+|¯uj|p−1Lp(Bj∖Br)+|vnj|p−1Lp(Bj∖Br)+|¯vj|p−1Lp(Bj∖Br)]|ϕ|p+C2limsupj→∞[|unj|κ−1Lκ(Bj∖Br)+|¯uj|κ−1Lκ(Bj∖Br)+|vnj|κ−1Lκ(Bj∖Br)+|¯vj|κLκ(Bj∖Br)]|ϕ|κ≤C3εp−1p+C4εκ−1κ, |
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)→c−Φλ(u,v);
(ii)Φ′λ(unj−¯uj,vnj−¯vj)→0inW−1(thedualspaceofW).
Proof. (i) We have
Φλ(unj−¯uj,vnj−¯vj)=Φλ(unj,vnj)−Φλ(¯uj,¯vj)+λp∗s∫RNK(x)(|unj|p∗s−|unj−¯uj|p∗s−|¯uj|p∗s+|vnj|p∗s−|vnj−¯vj|p∗s−|¯vj|p∗s)dx+λ∫RN(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
limj→∞∫RNK(x)(|unj|p∗s−|unj−¯uj|p∗s−|¯uj|p∗s+|vnj|p∗s−|vnj−¯vj|p∗s−|¯vj|p∗s)dx=0 |
and
limj→∞∫RN(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)→c−Φλ(u,v). |
(ii) We observe that for any (ϕ,ψ)∈W satisfying ||(ϕ,ψ)||≤1,
⟨Φ′λ(unj−¯uj,vnj−¯vj),(ϕ,ψ)⟩=⟨Φ′λ(unj,vnj),(ϕ,ψ)⟩−⟨Φ′λ(¯uj,¯vj),(ϕ,ψ)⟩+λ∫RNK(x)[(|unj|p∗s−2unj−|unj−¯uj|p∗s−2(unj−¯uj)−|¯uj|p∗s−2¯uj)ϕ+(|vnj|p∗s−2vnj−|vnj−¯vj|p∗s−2(vnj−¯vj)−|¯vj|p∗s−2¯vj)ψ]dx+λ∫RN[(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
limj→∞∫RNK(x)(|unj|p∗s−2unj−|unj−¯uj|p∗s−2(unj−¯uj)−|¯uj|p∗s−2¯uj)ϕdx=0 |
and
limj→∞∫RNK(x)(|vnj|p∗s−2vnj−|vnj−¯vj|p∗s−2(vnj−¯vj)−|¯vj|p∗s−2¯vj)ψdx=0 |
uniformly in ||(ϕ,ψ)||≤1. By Lemma 3.3, we obtain Φ′λ(unj−¯uj,vnj−¯vj)→0. We complete this proof.
Set u1j=unj−¯uj, v1j=vnj−¯vj, then unj−u=u1j+(¯uj−u), vnj−v=v1j+(¯vj−v). 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)→c−Φλ(u,v) and Φ′λ(u1j,v1j)→0.
Note that ⟨Φ′λ(u1j,v1j),(u1j,v1j)⟩=0, by computation, we get
∫∫R2N|u1j(x)−u1j(y)|p|x−y|N+psdxdy+∫RNλV(x)|u1j|pdx+∫∫R2N|v1j(x)−v1j(y)|p|x−y|N+psdxdy+∫RNλV(x)|v1j|pdx−λ∫RNK(x)(|u1j|p∗s+|v1j|p∗s)dx−λ∫RNF(x,u1j,v1j)dx=0 | (3.5) |
Hence, by (F3) and (3.5), we have
Φλ(u1j,v1j)−1p⟨Φ′λ(u1j,v1j),(u1j,v1j)⟩=(1p−1p∗s)λ∫RNK(x)(|u1j|p∗s+|v1j|p∗s)dx+λ∫RN[1p(Fu(x,u1j,v1j)u1j+Fu(x,u1j,v1j)v1j)−F(x,u1j,v1j)]dx≥λsKminN∫RN(|u1j|p∗s+|v1j|p∗s)dx, |
where Kmin=infx∈RNK(x)>0. So, it is easy to see that
|u1j|p∗sp∗s+|v1j|p∗sp∗s≤N(c−Φλ(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|p∗s+|v1j|p∗s)dx+∫RN(Fu(x,u1j,v1j)u1j+Fv(x,u1j,v1j)v1j)dx≤b(|u1j|pp+|v1j|pp)+Cb(|u1j|p∗sp∗s+|v1j|p∗sp∗s). | (3.8) |
Let S is fractional Sobolev constant which is defined by
S|u|pp∗s≤∫∫R2N|u(x)−u(y)|p|x−y|N+psdxdyforallu∈Ws,p(RN). | (3.9) |
Proof of Proposition 3.1. Assume that (unj,vnj)↛(u,v), then liminfj→∞||(u1j,v1j)||>0 and c−Φλ(u,v)>0.
From (3.5), (3.7), (3.8) and (3.9), we deduce
S(|u1j|pp∗s+|v1j|pp∗s)≤∫∫R2N|u1j(x)−u1j(y)|p|x−y|N+psdxdy+∫RNλV(x)|u1j|pdx+∫∫R2N|v1j(x)−v1j(y)|p|x−y|N+psdxdy+∫RNλV(x)|v1j|pdx−∫RNλV(x)(|u1j|p+|v1j|p)dx=λ∫RNK(x)(|u1j|p∗s+|v1j|p∗s)dx+λ∫RN(Fu(x,u1j,v1j)u1j+Fv(x,u1j,v1j)v1j)dx−λ∫RNVb(x)(|u1j|p+|v1j|p)dx≤λCb(|u1j|p∗sp∗s+|v1j|p∗sp∗s)+o(1). |
Thus, by (3.6), we have
S≤λCb(|u1j|p∗sp∗s+|v1j|p∗sp∗s)p∗s−pp∗s+o(1)≤λCb(N(c−Φλ(u,v))λsKmin)sN+o(1), |
or equivalently
αλ1−Nps≤c−Φλ(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 Y⊂W,
Φλ(u,v)→−∞,as(u,v)∈W,||(u,v)||→+∞. |
(iii) for any τ>0 there exists Λτ>0 such that each λ≥Λτ, there exists ˜ωλ∈Y with ||˜ωλ||>ρ, Φλ(˜ωλ)≤0 and
maxt≥0Φλ(t˜ωλ)≤τλ1−Nps. |
Proof. (i) From (F1),(F2), we have for any ε>0, there is Cε>0 such that
1p∗s∫RNK(x)(|u|p∗s+|v|p∗s)dx+∫RNF(x,u,v)dx≤ε|(u,v)|pp+Cε|(u,v)|p∗sp∗s. | (4.1) |
Thus, combining with (4.1) and Sobolev inequality, we deduce that
Φλ(u,v)=1p||(u,v)||p−λp∗s∫RNK(x)(|u|p∗s+|v|p∗s)dx−λ∫RNF(x,u,v)dx≥1p||(u,v)||p−λεC5||(u,v)||p−λC6Cε||(u,v)||p∗s, |
where ε is small enough and C5,C6>0, thus (i) is proved because p∗s>p.
(ii) By (F3), we define the functional Ψλ∈C1(W,R) by
Ψλ(u,v)=1p||(u,v)||p−λl0∫RN(|u|d+|v|d)dx. |
Then
Φλ(u,v)≤Ψλ(u,v),forall(u,v)∈W. |
For any finite-dimensional subspace Y⊂W, we only need to prove
Ψλ(u,v)→−∞,as(u,v)∈Y,||(u,v)||→+∞. |
In fact, we have
Ψλ(u,v)=1p||(u,v)||p−λl0|(u,v)|dd. |
Since all norms in a finite dimensional space are equivalent and p<d<p∗s, 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|x−y|N+psdxdy:φ∈C∞0(RN),|φ|d=1}=0,p<d<p∗s, |
see [40] for this proof. For any 0<ε<1, we can take φε∈C∞0(RN) with |φε|d=1, supp φε⊂Brε(0) and [φε]pp,s<ε.
Let
¯ωλ(x):=(ωλ(x),ωλ(x))=(φε(λ1psx),φε(λ1psx)). |
For t≥0, (F3) imply that
Φλ(t¯ωλ)≤2tpp∫∫R2N|ωλ(x)−ωλ(y)|p|x−y|N+psdxdy+2tpp∫RNλV(x)|ωλ|pdx−λ∫RNF(x,tωλ,tωλ)dx≤λ1−Nps{2tpp∫∫R2N|φε(x)−φε(y)|p|x−y|N+psdxdy+2tpp∫RNV(λ−1psx)|φε|pdx−2l0td∫RN|φε|ddx}≤λ1−Nps2l0(d−p)p(∫∫R2N|φε(x)−φε(y)|p|x−y|N+psdxdy+∫RNV(λ−1psx)|φε|pdxl0d)dd−p. |
Indeed, for t>0, define
g(t)=2tpp∫∫R2N|φε(x)−φε(y)|p|x−y|N+psdxdy+2tpp∫RNλV(λ−1psx)|φε|pdx−2l0td∫RN|φε|ddx. |
It is easy to show that t0=(∫∫R2N|φε(x)−φε(y)|p|x−y|N+psdxdy+∫RNV(λ−1psx)|φε|pdxl0d)1d−p is a maximum point of g and
maxt≥0g(t)=g(t0)=2l0(d−p)p(∫∫R2N|φε(x)−φε(y)|p|x−y|N+psdxdy+∫RNV(λ−1psx)|φε|pdxl0d)dd−p. |
Since V(0)=0 and supp φε⊂Brε(0), there exists Λε>0 such that
V(λ−1psx)<ε|φε|pp,∀|x|≤rε,λ>Λε. |
Hence, we have
maxt≥0Φλ(t¯ωλ)≤2l0(d−p)p(1l0d)dd−p(2ε)dd−pλ1−Nps,∀λ>Λε. |
Choose ε>0 such that
2l0(d−p)p(1l0d)dd−p(2ε)dd−p≤τ, |
and taking Λτ=Λε, from (ii), we can take ¯t large enough and define ˜ωλ=¯t¯ωλ, then we have
Φλ(˜ωλ)<0andmax0≤t≤1Φλ(t˜ωλ)≤τλ1−Nps. |
Proof of Theorem 2.1. From Lemma 4.1, for any 0<τ<α, there exists Λτ>0 such that for λ≥Λτ, we have
c=infη∈Γλmaxt∈[0,1]Φλ(η(t))≤τλ1−Nps, |
where Γλ={η∈C([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 Φ′λ(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,p∗s]
τλ1−Nps≥Φλ(uλ,vλ)−1θ⟨Φ′λ(uλ,vλ),(uλ,vλ)⟩≥(1p−1θ)||(uλ,vλ)||p+λ(1θ−1p∗s)∫RNK(x)(|uλ|p∗s+|vλ|p∗s)dx+λ(μθ−1)∫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)=mink∈Υgen(k(A)⋂∂Bρ), |
where Υ is the set of all odd homeomorphisms k∈C(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),1≤j≤m. |
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<τλ1−Nps.
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 m∈N, ε>0 and j=1,2,⋅⋅⋅,m, one can choose m functions φjε∈C∞0(RN) with supp φiε⋂ supp φjε=∅ if i≠j, |φjε|d=1 and [φjε]pp,s<ε.
Let rmε>0 be such that supp φjε⊂Brmε(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¯ωjλ∈Fmλ,
Φλ(˜ω)=m∑j=1Φλ(tj¯ωjλ) |
and for tj>0
Φλ(tj¯ωjλ)≤2tpjp∫∫R2N|ωjλ(x)−ωjλ(y)|p|x−y|N+psdxdy+2tpjp∫RNλV(x)|ωjλ|pdx−λ∫RNF(x,tjωjλ,tjωjλ)dx≤λ1−Nps{2tpjp∫∫R2N|φjε(x)−φjε(y)|p|x−y|N+psdxdy+2tpjp∫RNV(λ−1psx)|φjε|pdx−2l0tdj∫RN|φjε|ddx}. |
Set
ηε:=max{|φjε|pp:j=1,2,⋅⋅⋅,m}. |
Since V(0)=0 and supp φjε⊂Brmε(0), there exists Λmε>0 such that
V(λ−1psx)<εηε,∀|x|≤rmε,λ>Λmε. |
Consequently, there holds
sup˜w∈FmλΦλ(˜w)≤ml0(2ε)dd−pλ1−Nps,∀λ>Λmε. |
Choose ε>0 small that ml0(2ε)dd−p<τ. Thus for any m∈N and τ∈(0,α), there exists Λmτ=Λmε such that λ>Λmτ, we can choose a m-dimensional subspace Fmλ with maxΦλ(Fmλ)≤τλ1−Nps and
cλ1≤cλ2≤⋅⋅⋅≤sup˜w∈FmλΦλ(˜w)≤τλ1−Nps. |
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<τλ1−Nps. By the usual critical point theory, all cλj are critical levels and Φλ has at least m pairs of nontrivial critical points satisfying
a≤Φλ(u,v)≤τλ1−Nps. |
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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