# The Nehari Manifold for a Fractional p-Kirchhoff System Involving Sign-Changing Weight Function and Concave-Convex Nonlinearities

Authors: Jiabin Zuo; Tianqing An; Libo Yang; Xiangsheng Ren
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In this paper, we are concerned with the following fractional p-Kirchhoff system with sign-changing nonlinearities:M(∫R2nux-uyp/x-yn+psdxdy)-Δpsu=λa(x)uq-2u+α/(α+β)f(x)uα-2uvβ,inΩ, M(∫R2n|v(x)-v(y)|p/|x-y|n+psdxdy)-Δpsv=μb(x)vq-2v+(β/α+β)f(x)uαvβ-2v,inΩ, and u=v=0,inRn∖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,a(x),b(x)∈L(α+β)/(α+β-q)(Ω) are sign changing and either a±=max⁡{±a,0}≢0 or b±=max⁡{±b,0}≢0, f∈L(Ω¯) with f∞=1, and f≥0. Using Nehari manifold method, we prove that the system has at least two solutions with respect to the pair of parameters (λ,μ).


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## Body

## 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,inΩ,M∫R2nvx-vypx-yn+psdxdy-Δpsv=μbxvq-2v+βα+βfxuαvβ-2v,inΩ,u=v=0,inRn∖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. SetΩ 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→Rismeasurable,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 spaceX0 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 ofX 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 Section2, 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 thatT(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 Lemma4 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 (Q(u,v)>0).
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 (Q(u,v)≤0).
As we know thatOu,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) IfQ(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 inN+ 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.
SinceT 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+weaklyinHask→∞.Moreover, according to ([3], lemma 8),(52)uk→u0+,vk→v0+stronglyinLrΩask→∞,ukx→u0+x,vkx→v0+xa.e.inΩask→∞.For each 1≤r<p∗, by ([18], Theorem IV-9), there exists s(x)∈Lr(Rn) such that(53)ukx≤sx,vkx≤sx,almosteverywhereinRn.
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-weaklyinHask→∞.Moreover,(66)uk~→u0-,vk~→v0-stronglyinLrΩask→∞,uk~x→u0-x,vk~x→v0-xa.e.inΩask→∞.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 Theorem2.
According to Propositions8 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.



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*Source: 7624373-2019-01-02.xml*