1. Introduction In this article, we study the following fractional p-Kirchhoff problem (P) involving concave-convex nonlinearities and sign-changing weight functions: (1)M∫R2nux-uypx-yn+psdxdy-Δpsu=λaxuq-2u+αα+βfxuα-2uvβ, in Ω,M∫R2nvx-vypx-yn+psdxdy-Δpsv=μbxvq-2v+βα+βfxuαvβ-2v, in Ω,u=v=0, in Rn∖Ω,where Ω is a smooth bounded domain in Rn, n>ps, s∈(0,1), λ, μ are two real parameters, 1<q<p<p(h+1)<α+β<ps∗=np/(n-ps), M is a continuous function, given by M(t)=k+lth, k>0, l>0, h≥1, and (-Δ)psu is the fractional p-Laplacian operator, which is defined as follows:(2)-Δpsux=2limε→0+∫Rn∖Bεxux-uyp-2ux-uyx-yn+psdy, x∈Rn,where Bε(x)={y∈Rn:x-y<ε}, a(x), b(x), f(x) satisfy the following assumptions: set γ=(α+β)/(α+β-q),

( A 1 ) a(x),b(x)∈Lγ(Ω), and either a±=max{±a,0}≢0 or b±=max{±b,0}≢0;

( A 2 ) f(x)∈L(Ω¯) with f∞=1 and f≥0.

In recent years, the Kirchhoff equations have received extensive attention from many scholars because of its wide application in many fields such as mathematical finance, continuum mechanics, etc. (see [1, 2]). There are many excellent and interesting results about the existence and multiplicity of solutions for nonlocal fractional problems. We can look up the literature [3, 4] for Laplace operator and [5–8] for the p-Laplacian case.

In addition, for a single equation with sign-changing weights functions, in [9, 10], the authors studied the existence and multiplicity of nonnegative solutions in subcritical and critical cases respectively. In the special case of p=2, s=1, and M=1, Tsung-Fang Wu [11] proved that system has least two nontrivial nonnegative solutions by using the Nehari manifold. Moreover, when M is not a constants, the authors [12] investigated the fractional p-Kirchhoff system with sign-changing nonlinearities, which is given by M=a+bt. For a more general case M=k+ltτ, Yang and An [13] show the system has at least two solutions with the help of Nehari manifold, but without considering sign-changing weights functions. Hence, inspired by above works, combining [12, 13], in this paper we will consider the new multiplicity result of the problem. Our conclusions can be seen as an extension of [12, 13].

To illustrate our result, we need to introduce some notations. Set Ω be an open set in Rn, 0<s<1, and p≥1. Define the usual fractional Soblev Space Ws,p(Ω) and its norm(3)uWs,pΩ=uLpΩ+∫Ω×Ωux-uypx-yn+psdxdy1/p.Let K=R2n∖(CΩ×CΩ) and CΩ=Rn∖Ω. Define the space X as (4)X=u∣u:Rn→R is measurable, uΩ∈LpΩ,ux-uyx-yn/p+s∈LpK.Then the space X of norm is defined by(5)uX=uLpΩ+∫Kux-uypx-yn+psdxdy1/p.

Set Banach space X0 to be the completion of the space C0∞(Ω) in X, which is can be defined as the norm(6)uX0=∫Kux-uypx-yn+psdxdy1/p.

Clearly, (6) is equivalent to the (3), as u=0 a.e. in Rn∖Ω, we obtain that the integral in (3), (5), and (6) can be extended to the full space Rn. According to the literature [14, 15], we know that X0↪Lr(Ω) is a continuous embedding for any r∈[1,p∗], and compact whenever r∈[1,p∗). When α+β∈(p,p∗), then, for any u∈X0, we get that(7)uLα+βΩ≤SuX0.

More about the properties of X and X0, please consult [16] and the references therein. The reflexive Banach space H=X0×X0 is the Cartesian product of two spaces, which is endowed with the norm(8)u,v=uX0p+vX0p1/p=∫Kux-uypx-yn+psdxdy+∫Kvx-vypx-yn+psdxdy1/p.

Definition 1. A pair of functions (u,v)∈H is called weak solution of problem (P) if for all (ϕ,φ)∈H one has(9)MuX0p∫Kux-uyp-2ux-uyϕx-ϕyx-yn+psdxdy+MvX0p∫Kvx-vyp-2vx-vyφx-φyx-yn+psdxdy=λ∫Ωaxuq-2uϕxdx+μ∫Ωbxvq-2vφxdx+αα+β∫Ωfxuα-2uvβϕxdx+βα+β∫Ωfxuαvβ-2vφxdx.

We introduce the set(10)GΓ=λ,μ∈R2∖0,0:λaγp/p-q+μbγp/p-qp-q/p<Γα,β,q,S.Then we give our result as follows.

Theorem 2. Assume that the conditions (A1) and (A2) hold. If 1<q<p<p(h+1)<α+β<ps∗, then there is an explicit constant Γ>0 such that the problem (P) has at least two nonnegative solutions for (λ,μ)∈Gq/pΓ.

The rest of the paper is organized as follows. In Section 2, we give some notations and preliminaries about Nehari manifold and fibering maps. In Section 3, we prove Theorem 2.

2. The Variational Setting and Preliminaries Define energy functional associated with problem (P) as follows:(11)Tu,v=kpu,vp+lξu,vξ-1m∫Ωfxuαvβdx-1qQu,v,where ξ=p(h+1) and m=α+β and(12)Qu,v=λ∫Ωaxuqdx+μ∫Ωbxvqdx.

By a direct calculation we obtain that T(u,v)∈C1(H,R), and for all (ϕ,φ)∈H, we have(13)T′u,v,ϕ,φ=MuX0p∫Kux-uyp-2ux-uyϕx-ϕyx-yn+psdxdy+MvX0p∫Kvx-vyp-2vx-vyφx-φyx-yn+psdxdy-λ∫Ωaxuq-2uϕxdx-μ∫Ωbxvq-2vφxdx-αm∫Ωfxuα-2uvβϕxdx-βm∫Ωfxuαvβ-2vφxdx.Then, the weak solution (u,v) is equivalent to being a critical point of T. Since T is not bounded below on H, therefore, we consider the Nehari manifold(14)N=u,v∈H∖0,0∣T′u,v,ϕ,φ=0.By (13), we get(15)T′u,v,u,v=ku,vp+lu,vξ-∫Ωfxuαvβdx-Qu,v.Hence, (u,v)∈N if and only if(16)ku,vp+lu,vξ-∫Ωfxuαvβdx-Qu,v=0.Moreover, for any (u,v)∈N, the following equality holds:(17)Tu,v=1p-1mku,vp+1ξ-1mlu,vξ-1q-1mQu,v=1p-1qku,vp+1ξ-1qlu,vξ-1m-1q∫Ωfxuαvβdx.Obviously, the solution of the problem (P) depends on N. N is a set which we find that is smaller than X0, so it is easier to study on N. Therefore, define our familiar fiber maps: ψu,v:t→T(tu,tv) as follows:(18)ψu,vt=Ttu,tv=kptpu,vp+lξtξu,vξ-1mtm∫Ωfxuαvβdx-1qtqQu,v,(19)ψu,v′1=ku,vp+lu,vξ-∫Ωfxuαvβdx-Qu,v,(20)ψu,v′′1=p-1ku,vp+ξ-1lu,vξ-m-1∫Ωfxuαvβdx-q-1Qu,v.It follows from (19) that (u,v)∈N if and only if ψu,v′(1)=0. So it is natural that we divide N into three parts: local minima, local maxima, and points of inflection. For this, we let(21)N±≔u,v∈N:ψu,v′′1≷0,N0≔u,v∈N:ψu,v′′1=0.

Define(22)Φu,v=T′u,v,u,v, ∀u,v∈H.Moreover, for every (u,v)∈N, from (16), we also have(23)Φ′u,v,u,v=kp-mu,vp+lξ-mu,vξ-q-mQu,v=kp-qu,vp+lξ-qu,vξ-m-q∫Ωfxuαvβdx.Now, we give some preliminaries about main result.

Lemma 3. If (u0,v0) is a minimizer of T on N and (u0,v0)∉N0. For every (λ,μ)∈GΓ, then we have T′(u0,v0)=0 in H∗.

Proof. For the detailed process of certification we can refer to the literature ([17], Theorem 2.3). For the convenience of the reader we give its completeness. If (u0,v0) is a local minimizer on N to T, by the theory of Lagrange multipliers, there is a constant θ∈R such that(24)T′u0,v0=θΦ′u0,v0.So(25)T′u0,v0,u0,v0=θΦ′u0,v0,u0,v0=θψu,v′′1=0.But 〈Φ′(u0,v0),(u0,v0)〉≠0, because of (u0,v0)∉N0. Therefore θ=0. This completes the proof.

Lemma 4. T is coercive and bounded below on N.

Proof. By the Sobolev inequalities and Hölder inequalities, we get(26)Qu,v≤λaLγuLmq+μbLγvLmq≤Sqλaγp/p-q+μbγp/p-qp-q/pu,vq.Also, according to (17) and (26), we have (27)Tu,v=1p-1mku,vp+1ξ-1mlu,vξ-1q-1mQu,v≥1p-1mku,vp+1ξ-1mlu,vξ-1q-1mSqλaγp/p-q+μbγp/p-qp-q/pu,vq.As q<p<ξ<m, from above inequality, we can conclude T is coercive and bounded below on N. We complete this proof.

Lemma 5. Under condition (A2), there exists Γ>0, given by(28)Γ=km-pm-qSqkp-qm-qSmp-q/m-p,such that, for any (λ,μ)∈GΓ, we have N0=∅.

Proof. We argue by contradiction, moreover dividing the following two cases: assume that there exist (λ,μ)∈GΓ such that N0≠∅. Then for (u,v)∈N0, we have(29)Φu,v=0,Φ′u,v,u,v=0.

Case 1. Q ( u , v ) = 0 . From (29), (19), and (20), we have(30)0=p-1ku,vp+ξ-1lu,vξ-m-1∫Ωfxuαvβdx=p-mku,vp+ξ-mlu,vξ<0,which is a contradiction.

Case 2. Q ( u , v ) ≠ 0 , then it follows from (29), (23), and (26) that(31)u,v≤m-qSqλaΓp/p-q+μbΓp/p-qp-q/pkm-p1/p-q.In addition, by condition (A2) and Young’s inequality, we get(32)∫Ωfxuαvβdx≤αm∫Ωfxumdx+βm∫Ωfxvmdx≤αmf∞SmuX0m+βmf∞SmvX0m≤Smu,vm.By (29), (32), and (23), we have(33)kp-qu,vp≤m-q∫Ωfxuαvβdx≤m-qSmu,vm,and hence, we obtain(34)u,v≥kp-qm-qSm1/m-p.From (31) and (34), we get(35)λaγp/p-q+μbγp/p-qp-q/p≥km-pm-qSqkp-qm-qSmp-q/m-p=Γ,which contradicts (λ,μ)∈GΓ. We have completed the proof of this lemma.

According to Lemma 4 and Lemma 5, we know N=N++N- for (λ,μ)∈GΓ, and we set (36)Z+=infu,v∈N+Tu,v,Z-=infu,v∈N-Tu,v.We introduce the following lemma.

Lemma 6. Assume that (λ,μ)∈Gq/pΓ. Then, we have

(i) Z+<0,

(ii) Z->e0 for some e0>0.

Proof. (i) Set (u,v)∈N+, we know 〈Φ′(u,v),(u,v)〉>0, and from (23), we have(37)∫Ωfxuαvβdx<kp-qm-qu,vp+lξ-qm-qu,vξ.Put (37) into (17),(38)Tu,v<-kp-qmqpu,vp-lm-pp-qmqpu,vξ<0,which implies Z+=infu,v∈N+T(u,v)<0.

(ii) It follows from (17) and (23) that (39)Tu,v≥m-pkmpu,vp-m-qmqQu,v≥m-pkmpu,vp-m-qmqSqλaγp/p-q+μbγp/p-qp-q/pu,vq=u,vqm-pkmpu,vp-q-m-qmqSqλaγp/p-q+μbγp/p-qp-q/p.Since (u,v)∈N-, then 〈Φ′(u,v),(u,v)〉<0. Hence, according to (17), we obtain (34), combining above inequality and (34), and we get (40)Tu,v≥kp-qm-qSmq/m-pm-pkmpkp-qm-qSmp-q/m-p-m-qmqSqλaγp/p-q+μbγp/p-qp-q/p.Obviously, if ((λaγ)p/p-q+(μbγ)p/(p-q))(p-q)/p<q/pΓ, then there is a constant e0>0 such that Z->e0.

Under condition (A2), we study the behavior of the fibering map with respect to the sign of Q(u,v). We divide into two cases.

Case 1 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M192"><mml:mi>Q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>></mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:math></inline-formula>). Fix (u,v)∈H. Let(41)Ou,vt=ktp-qu,vp+ltξ-quvξ-tm-q∫Ωfxuαvβdx.Clearly, (tu,tv)∈N if and only if Ou,v(t)=Q(u,v).(42)Ou,v′t=ktp-q-1u,vp+ltξ-q-1u,vξ-tm-q-1∫Ωfxuαvβdx.It is obvious that Ou,v(t)→-∞ as t→∞. By (42), we know that(43)limt→0+Ou,v′t>0,limt→∞Ou,v′t<0.Thus, there is a unique t∗=t∗(u,v)>0, such that Ou,v(t) is increasing on (0,t∗), decreasing on (t∗,∞). Moreover, Ou,v′(t∗)=0, in addition,(44)Ou,vt∗=t∗-qkt∗pu,vp+lt∗ξu,vξ-t∗m∫Ωfxuαvβdx,where t∗ is the root of(45)kp-qt∗pu,vp+lξ-qt∗ξu,vξ-m-qt∗m∫Ωfxuαvβdx=0.From above the equality, we obtain(46)t∗≥kp-qu,vpm-q∫Ωfxuαvβdx1/m-p≔t0.Since (λ,μ)∈Gq/pΓ, according to (26) and (46), we get (47)Ou,vt∗≥Ou,vt0≥kt0p-qu,vp-t0m-p∫Ωfxuαvβdx≥u,vqkm-pm-qkp-qm-qSmp-q/m-p≥u,vqSqqpkm-pm-qSqkp-qm-qSmp-q/m-p≥Sqλaγp/p-q+μbγp/p-qp-q/pu,vq≥Qu,v.Hence, there are unique t+<t∗ and t->t∗ such that Ou,v(t+)=Ou,v(t-)=Q(u,v). It means that (t+u,t+v)∈N and (t-u,t-v)∈N. Moreover it is easy to see that 〈Φ′(tu,tv),(tu,tv)〉=tq+1Ou,v′(t), also Ou,v′(t+)>0 and Ou,v′(t-)<0 imply (t+u,t+v)∈N+ and (t-u,t-v)∈N-. Since ψu,v′(t)=tq(Ou,v(t)-Q(u,v)), then ψu,v′(t)<0 for any t∈[0,t+) and ψu,v′(t)>0 for any t∈(t+,t-). Thus, T(t+u,t+v)=inf0<t<t-T(tu,tv). Furthermore, ψu,v′(t)>0 for any t∈[t+,t-), ψu,v′(t-)=0, and ψu,v′(t)<0 for any t∈(t-,∞) imply that T(t-u,t-v)=supt>t∗T(tu,tv).

Case 2 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M234"><mml:mi>Q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>≤</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:math></inline-formula>). As we know that Ou,v(t)→-∞ as t→∞, therefore for any (λ,μ)∈R, there is a unique t∗>0 such that Ou,v(t∗)=Q(u,v); moreover ψu,v′(t)>0 for any t∈[0,t∗) and ψu,v′(t)<0 for any t∈(t∗,∞), which implies that (t∗u,t∗v)∈N- and T(t∗u,t∗v)=supt≥0T(tu,tv).

Thus according to the above discussion we obtain that the following lemma.

Lemma 7. Under condition (A2), for any (λ,μ)∈G(q/p)Γ, we have the following:

(i) If Q(u,v)≤0, then there is a unique t∗>0 such that (t∗u,t∗v)∈N-, and (48)Tt∗u,t∗v=supt≥0 Ttu,tv. (ii) If Q(u,v)>0, then there exist a unique t∗=t∗(u,v)>0 and unique t+(u,v)<t∗<t-(u,v) such that (t+u,t+v)∈N+, (t-u,t-v)∈N-, and(49)Tt+u,t+v=inf0<t<t-Ttu,tv,Tt-u,t-v=supt>t∗ Ttu,tv.

3. Proof of the Main Result In this section, we establish the existence of minimizers in N+ and N-.

Proposition 8. Under condition (A2), if (λ,μ)∈Gq/pΓ, then the functional T has a minimizer (u0+,v0+) in N+ and fulfills the following:

(i) T(u0+,v0+)=Z+<0,

(ii) (u0+,v0+) is a solution of problem (P).

Proof. Since T is bounded from below on N+, there is a minimizing sequence {(uk,vk)}∈N+ such that(50)limk→∞Tuk,vk=infu,v∈N+Tu,v=Z+.Hence, by Lemma 4, then (uk,vk) is bounded on H. So there exists (u0+,v0+)∈H, up to a subsequence, such that(51)uk⇀u0+,vk⇀v0+ weakly in H as k→∞.Moreover, according to ([3], lemma 8),(52)uk→u0+,vk→v0+ strongly in LrΩ as k→∞,ukx→u0+x,vkx→v0+x a.e. in Ω as k→∞.For each 1≤r<p∗, by ([18], Theorem IV-9), there exists s(x)∈Lr(Rn) such that(53)ukx≤sx,vkx≤sx, almost everywhere in Rn.

Hence, by the dominated convergence theorem, we get(54)limk→∞λ∫Ωaxukqdx+μ∫Ωbxvkqdx=∫Ωlimk→∞axukq+bxvkqdx=∫Ωaxu0+q+bxv0+qdx,and(55)limk→∞∫Ωfxukαvkβdx=∫Ωfxu0+αv0+βdx,as n→∞. Now, on N, from (17), we have(56)1q-1mQuk,vk=1p-1mk(uk,vkp+1ξ-1mluk,vkξ-Tuk,vk.Letting k→∞, since q<p<ξ<m, from Lemma 6, (50), and (54), we get(57)Qu0+,v0+>0.From Lemma 7, there exists t+<t∗ such that (t+u0+,t+v0+)∈N+ and 〈T′(t+u0+,t+v0+),(t+u0+,t+v0+)〉=0. Next we show that (uk,vk)→(u0+,v0+) strongly in H. If this is not true, then(58)u0+,v0+<liminfk→∞uk,vk.Since (59)T′t+uk,t+vk,t+uk,t+vk=kt+puk,vkp+lt+ξuk,vkξ-t+m∫Ωfxukαvkβdx-t+qQuk,vk,and(60)T′t+u0+,t+v0+,t+u0+,t+v0+=kt+pu0+,v0+p+lt+ξu0+,v0+ξ-t+m∫Ωfxu0+αv0+βdx-t+qQu0+,v0+,we obtain(61)limk→∞T′t+uk,t+vk,t+uk,t+vk>T′t+u0+,t+v0+,t+u0+,t+v0+=0.It means that 〈T′(t+uk,t+vk),(t+uk,t+vk)〉>0 for n large enough. As {(uk,vk}∈N+, it is obvious to know that 〈T′(uk,vk),(uk,vk)〉=0, and 〈T′(tuk,tvk),(tuk,tvk)〉<0 for 0<t<1. Thus we have t+>1. On the other hand, T(tu0+,tv0+) is decreasing on (0,t+), so(62)Tt+u0+,t+v0+≤Tu0+,v0+<limk→∞Tuk,vk=infu,v∈N+Tu,v=Z+,which is a contradiction. Thus (uk,vk)→(u0+,u0+) strongly in H. This means(63)Tuk,vk→Tu0+,v0+=infu,v∈N+Tu,v=Z+, k→∞.That is, (u0+,v0+) is a minimizer of T on N+; using Lemma 3, (u0+,v0+) is a solution of problem (P).

Proposition 9. Under condition (A2), if (λ,μ)∈Gq/pΓ, then the functional T has a minimizer (u0-,v0-) in N- and fulfills the following:

(i) T(u0-,v0-)=Z->0,

(ii) (u0-,v0-) is a solution of problem (P).

Proof. T is bounded from below such that(64)limk→∞Tuk~,vk~=infu,v∈N-Tu,v=Z-.It is similar to the proof of the Proposition 8, so there exists (u0-,v0-)∈H, up to a subsequence, such that(65)uk~⇀u0-,vk~⇀v0- weakly in H as k→∞.Moreover,(66)uk~→u0-,vk~→v0- strongly in LrΩ as k→∞,uk~x→u0-x,vk~x→v0-x a.e. in Ω as k→∞.For each 1≤r<p∗, by the dominated convergence theorem, we also get (67)limk→∞λ∫Ωaxuk~qdx+μ∫Ωbxvk~qdx=∫Ωlimk→∞axuk~q+bxvk~qdx=∫Ωaxu0-q+bxv0-qdx,and(68)limk→∞∫Ωfxuk~αvk~βdx=∫Ωfxu0-αv0-βdx,and similarly, by Lemma 7, there is a t- such that (t-u0-,t-v0-)∈N-. Next we show that (uk~,vk~)→(u0-,v0-) strongly in H. Suppose that this is not true, then(69)u0-,v0-<liminfk→∞uk~,vk~.Since {(uk~,vk~)}∈N- and T(uk~,vk~)≥T(tuk~,tvk~) for any t>0, we also have(70)Tt-u0-,t-v0-<limk→∞Tt-uk~,t-vk~=infu,v∈N-Tuk~,vk~=Z-,which is a contradiction. Hence (uk~,vk~)→(u0-,v0-) strongly in H. This implies(71)Tuk~,vk~→Tu0-,v0-=infu,v∈N-Tu,v=Z-, k→∞.Namely, (u0-,v0-) is a minimizer of T on N-; by Lemma 3, (u0-,v0-) is a solution of problem (P).

Proof of Theorem <xref ref-type="statement" rid="thm1.2">2</xref>. According to Propositions 8 and 9, we obtain that, for (λ,μ)∈Gq/pΓ, problem (P) has two solutions (u0+,v0+)∈N+ and (u0-,v0-)∈N- in H. Since(72)Tu0±,v0±=Tu0±,v0±,u0±,v0±∈N±,moreover N+∩N-=∅, so we get that (u0±,v0±) are distinct nonnegative solutions.