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Research article Special Issues

Existence of solutions for the fractional pq-Laplacian equation with nonlocal Choquard reaction

  • We consider the following class of fractional pq-Laplacian differential equation with Choquard term:

    {(Δ)spu+(Δ)squ+V(x)(|u|p2u+|u|q2u)+RNg(x)|u|rdx=RNRNk(u(x))K(u(y))|xy|αdxdy,xRN,uWs,pV(RN)Ws,qV(RN),xRN,

    where s(0,1),2prq<N/s,0<α<N, (Δ)sm with m{p,q} is the fractional m-Laplacian operator, g(x):RNR, by introducing a potential term function to restore compactness in the corresponding spaces. Using variational techniques and inequalities such as Hardy–Littlewood–Sobolev, we ensure the geometric conditions of the mountain pass theorem in order to show the existence of solutions.

    Citation: Liyan Wang, Baocheng Zhang, Zhihui Lv, Kun Chi, Bin Ge. Existence of solutions for the fractional pq-Laplacian equation with nonlocal Choquard reaction[J]. AIMS Mathematics, 2025, 10(4): 9042-9054. doi: 10.3934/math.2025416

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  • We consider the following class of fractional pq-Laplacian differential equation with Choquard term:

    {(Δ)spu+(Δ)squ+V(x)(|u|p2u+|u|q2u)+RNg(x)|u|rdx=RNRNk(u(x))K(u(y))|xy|αdxdy,xRN,uWs,pV(RN)Ws,qV(RN),xRN,

    where s(0,1),2prq<N/s,0<α<N, (Δ)sm with m{p,q} is the fractional m-Laplacian operator, g(x):RNR, by introducing a potential term function to restore compactness in the corresponding spaces. Using variational techniques and inequalities such as Hardy–Littlewood–Sobolev, we ensure the geometric conditions of the mountain pass theorem in order to show the existence of solutions.



    The purpose of this paper is to investigate the existence of solutions to the following fractional pq-Laplacian equation with Choquard term:

    (Δ)spu+(Δ)squ+V(x)(|u|p2u+|u|q2u)+RNg(x)|u|rdx=RNRNk(u(x))K(u(y))|xy|αdxdy (1.1)

    where (Δ)sp and (Δ)sq are the fractional pq-Laplacian operators, s(0,1),2prq<N/s,0<α<N, g(x):RNR+.

    The nonlocal operator (Δ)sm is the fractional m-Laplacian operator, defined as

    (Δ)smu(x)=2limϵ0RNBε(x)|u(x)u(y)|m2(u(x)u(y))|xy|N+smdy,xRN,

    where m(p,q), s(0,1), 2m<Ns.

    The term on the right side of Eq (1.1) is referred to as the Choquard term, for the case s=1 and K(u)=|u|2, Eq (1.1) is called the Choquard–Pekar equation [1]. The nonlinear Choquard equation can be used to describe mathematical physical phenomena. The Choquard nonlinear term in it appears in many mathematical physical models, such as the mean field limit of weakly interacting molecules, the Pekar theory of polarons, and the Schrodinger–Newton system, etc [2,3,4,5,6].

    Ambrosio [7] explored the issue of the lack of compactness associated with the critical Sobolev exponents. He addressed this issue by making an asymptotic estimation of the minimum values. Subsequently, he applied the mountain pass theorem. By virtue of this theorem, he proved the existence of weak solutions to Eq (1.2). These weak solutions are both non-negative and non-trivial.

    (Δ)s1pu+(Δ)s2qu=μ|u|q2u+λ|u|p2u+|u|ps12u. (1.2)

    In the local case p=q2, Eq (1.2) becomes a fractional p-Laplacian equation of the form:

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

    Cheng and Tang [8] discussed that both potential function and nonlinear term are allowed to change signs, and obtained the existence of solutions to the above problem. Torres [9] uses the mountain pass theorem with (C)c conditions and other theorems; problem (1.3) is presented as the existence of solutions and radial symmetric solutions when f satisfies the p-superlinearity condition. The idea to prove their result, consists in replacing the path in the mountain pass setting by its symmetrization. By applying the fixed-point method, Souza [10] considers the existence of solutions to the above problem with mixed nonlinearity term f(x,u), where the nonlinear f(x,u)=l(x,u)+λh(x).

    In addition, under the continuity and boundedness of potential function V(x), Xu et al.[11] establish the existence of a solution via the Fountain theorem for the case f(x,u)=l(x,u)+λ|u|q2u+h(x)|u|p2u. Using variational methods, Xiang [12] prove the nonexistence and multiplicity of solutions to problem (1.3) depending on λ in the case f(x,u)=λa(x)|u|p2uh(x)|u|p2u; their results extend the previous work of the fractional p-Laplacian setting. Furthermore, they weaken one of the conditions used in their paper. Hence the results of this paper are new even in the fractional Laplacian case.

    (Δ)su+V(εx)u+Cu=f(x,u),xRN. (1.4)

    Very recently, numerous studies exist concerning the existence, multiplicity, and concentration behavior of solutions for equations of the form (1.4), particularly under varying Sobolev subcritical potential conditions; we also refer to [13,14,15,16] and references therein.

    (Δ)spu+(Δ)squ+V(εx)(up1+uq1)=f(u)+uqs1,xRN, (1.5)

    where ε>0 is a small parameter, s(0,1),1<p<q<N/s,V:RNR is a continuous function. Ambrosio [17] used the topological structure of the set of potential functions V that reach their minimum value, combined the truncation of independent variables with Morse-type iterations, the minimax theorem, and the Ljusternik–Schnirellmann category theory to obtain the multiplicity results of the solutions to Eq (1.5).

    (Δ)spu+(Δ)squ+V(x)(|u|p2u+|u|q2u)+g(x)|u|r2u=K(x)f(x,u)+h(u),xRN, (1.6)

    where V:RNR+ is a continuous function, and h:RR is a perturbation term, Wang [18] studied two cases of Eq (1.6): If f(x,u) is sublinear, by means of Clark's theorem, which takes into account the symmetric condition of the functional, infinitely many solutions can be obtained; if f(x,u) is superlinear, by using the symmetric mountain pass theorem, infinitely many solutions can be derived.

    εp(Δ)pu+εq(Δ)qu+V(x)(|u|p2u+|u|q2u)=W(x)f(u),xRN, (1.7)

    where 1<p<q<N,Δsu=div(|u|s2u), s{p,q}, is the s-Laplacian operator, and ε is a small positive parameter. Zhang et al.[19] assumed that the potential energy V, W, and nonlinear term f satisfy certain conditions. Using topology and variational methods, they established the existence of positive solutions to Eq (1.7) when the function is sufficiently small, and also proved the nonexistence of ground state solutions.

    (Δ)spu+(Δ)squ+V(εx)(|u|p2u+|u|q2u)=f(u), xRN, (1.8)

    where ε>0 is a small parameter, s(0,1),1<p<q<N/s, the above equation has a continuous potential energy function and a nonlinear term with subcritical growth. Alves et al. [20] applied theoretical tools, such as the minimax theorem, to establish the existence and multiplicity of solutions to Eq (1.8), provided that the parameter ε is sufficiently small.

    Motivated by the aforementioned papers, in this article, this paper investigates the fractional pq-Laplacian problem with a Choquard term, characterized by the following features:

    ⅰ) The problem involves the combined effects of fractional differential operators with unbalanced growth, where the associated energy corresponds to a fractional differential variational functional.

    ⅱ) Due to the unbounded nature of the domain, the Palais–Smale sequence fails to exhibit compactness.

    ⅲ) The Choquard reaction term on the right-hand side introduces a nonlocal characteristic.

    Before presenting our main result, we first introduce the assumptions on the potential V(x) and the nonlinearity k(u).

    (V1) V(x)C(RN,R+): there exists a constant V0>0, such that for all xRN, V(x)V0.

    (V2) There exist constants d,b>0 and γ>N(m1) such that

    lim|y|meas({xRN |V(x)|x|γb }Bd(y))=0,

    where meas() is the Lebesgue measure in RN.

    (k1) k(x)C(R,R), there exist pp1p2 and a positive constant C0 such that

    |k(t)|C0(|t|p11+|t|p21),tR,

    where p2((2Nα)qN,(2Nα)qN), q=NqNq.

    (k2) There exists ς>q such that

    0<ςK(t)2k(t)t,tR,

    where K(t)=t0k(x)dx.

    Theorem 1.1. Assuming that conditions (V1), (V2), (k1), and (k2) are satisfied, problem (1.1) has at least one nontrivial solution in E.

    This paper is organized as follows: In Section 2, we state the notations and main Lemma. Section 3 will be devoted to the proofs of the main results of the paper.

    There may be multiple selection methods for some spaces, but they should be considered as suitable for discussion. In order to obtain a solution to the problem (1.1), the space should be smaller, which can make the direction of the discussion clearer and eliminate some uncertainty. As for its smoothness, it is also advisable to request a lower level as much as possible to avoid increasing the complexity of argumentation and calculation. Next we introduce some relevant space and variational setting.

    The fractional Sobolev space Ws,m(RN) is defined by

    Ws,m(RN)={uLm(RN),[u]s,m<},

    with the corresponding norm

    us,m=(umm+[u]ms,m)1m,

    where [u]s,m denotes the Gagliardo seminorm and us is given by

    [u]s,m=(R2N|u(x)u(y)|m|xy|N+smdxdy)1m,m{p,q},us=(RN|u|sds)1s,

    for u,vWs,m(RN), we have

    u,vs,m=R2N|u(x)u(y)|m2(u(x)u(y))(v(x)v(y))|xy|N+smdxdy,

    let us introduce the space

    Ws,mV(RN)={uWs,m(RN),RNV(x)|u|mdx<+},

    and it is equipped with the norm

    uWs,mV=([u]ms,m+umm,V)1m,umm,V=RNV(x)|u|mdx,

    after consideration, we define the working space:

    E=Ws,pV(RN)Ws,qV(RN),

    which is a uniformly convex Banach space (similar to [7]), endowed with the norm

    uE=m=p,quWs,mV=uWs,pV+uWs,qV.

    We define the energy functional associated with problem (1.1)

    J(u)=1p(R2N|u(x)u(y)|p|xy|N+spdydx+RNV(x)|u|pdx)+1q(R2N|u(x)u(y)|q|xy|N+sqdydx+RNV(x)|u|qdx)+1rRNg(x)|u|rdx12RNRNK(u(x))K(u(y))|xy|αdydx,

    applying some standard arguments, we can see that J with the derivative given by

    J(u),v=m=p,q(R2N|u(x)u(y)|m2(u(x)u(y))(v(x)v(y))|xy|N+smdxdy+RNV(x)|u|m2uv(x)dx)+RNg(x)|u|r2uv(x)dxRNRNK(u(y))k(u(x))v(x)|xy|αdydx.

    "" means weak convergence, "" means strong convergence.

    "" means continuous embedding, "↪↪" means compact embedding.

    Definition 2.1. [21] Let (E,) be a real Banach space, φC1(E,R). We say that φC1(E,R) satisfies the (PS)c-condition if any sequence {un}E such that

    φ(un)candφ(un)0, (2.1)

    admits a strongly convergent subsequence. Such a sequence is called a (PS) sequence on level c, or a (PS)c-sequence for short.

    Now, we present the following lemmas which will play a crucial role in the proof of the main theorems.

    Lemma 2.1. (Hardy–Littlewood–Sobolev inequality [22]) Let 1<r,s<+ and 0<α<N such that 1r+1s+αN=2. If ϕLr(RN) and ψLs(RN), then there exists a sharp constant C(N,α,r,s)>0, independent of ϕ and ψ, such that

    RNRNϕ(x)ψ(y)|xy|αdydxC(N,α,r,s)|ϕ|r|ψ|s.

    Lemma 2.2.[18] Suppose that (V1) and (V2) are valid. If {uj} is a bounded sequence in E, then there exists uELϑ(RN) such that, up to a subsequence,

    ujustronglyinLϑ(RN),

    as j, for any ϑ[1,ms), ms=NmNsm.

    Lemma 2.3.[21] Let E be a real Banach space with its dual space E, φC1(E,R), for some μ<η, ρ>0, and eE with e>ρ, such that

    max{φ(0),φ(e)}μ<ηinfu=ρφ(u),

    under the above assumption, if the functional φ satisfies the (PS)c-condition, then we can see that c is a critical point of φ, where c is the mountain pass level of φ, defined as

    c=infγΓmaxt[0,1]φ(γ(t)),

    and Γ={γC([0,1],E):γ(0)=0,γ(1)=e}, Γ is the set of continuous paths joining 0 and e.

    Based on self-argumentation or utilizing existing abstract critical point theorems, provide reasonable and specific conditions for the studied problem to ensure the existence of differentiable functional critical points. Starting from the given specific conditions, verify the requirements of the Mountain Pass theorem, and obtain the existence of the solution to the problem.

    To prove Theorem 1.1, we will apply a mountain pass-type argument to find critical the point of J. Firstly, we need to verify that the functional J satisfies the geometrical assumptions of the mountain pass theorem.

    Lemma 3.1. Assume that (V1), (V2), (k1), and (k2) hold; then the functional J(u) satisfies the following conditions:

    (a) There exist σ,ϱ>0, such that J(u)σ with ||u||=ϱ.

    (b) There exist δ>0 and u0E such that ||u0||>δJ(u0)<0.

    Proof. (a) Using the Hardy–Littlewood–Sobolev inequality and (k1), we immediately obtain

    RNRNK(u(x))K(u(y))|xy|αdydxC1(u2p1rp1+u2p2rp2),

    where C1 is a constant, let uE; we can choose ||u||=ϱ(0,1), then we have uqWs,qVupWs,pV1.

    Combining the above inequality, we can infer

    J(u)=upWs,pVp+uqWs,qVq+urr,gr12RNRNK(u(x))K(u(y))|xy|αdydxupWs,pVp+uqWs,qVq+urr,grC1(u2p1rp1+u2p2rp2)uqWs,pVq+uqWs,qVqC1(u2p1rp1+u2p2rp2)C2q(uWs,pV+uWs,qV)qC1(u2p1E+u2p2E)=C2uqEqC1(u2p1E+u2p2E),

    where C2 is a constant and

    ur,g=(RNg(x)|u|rdx)1r.

    Since p2((2Nα)q/N,(2Nα)q/N), then 2p2>q; consequently, ϱ is sufficiently small and can find σ>0 such that J(u)σ.

    (b) Fix u0W1,ξV(RN){0}, with u0>0; let us define

    h(τ)=RNRNK(τu0u0)K(τu0u0)|xy|αdxdy,

    according to the (k2) condition, we immediately obtain

    h(τ)=RNRN2K(τu0u0)k(τu0u0)u0u0|xy|αdxdyλτRNRNK(τu0u0)K(τu0u0)|xy|αdxdyλτh(τ), (3.1)

    in terms of interval [1,τu0], for τ>max{1/2,1/u0}, we deduce that

    h(τu0)h(1)(τu0)λ,

    and therefore

    RNRNK(τu0)K(τu0)|xy|αdxdyRNRNK(u0u0)K(u0u0)|xy|αdxdy(τu0)λ..

    Moreover, for τ>1/u0, it follows from the above conclusion that

    J(τu0)=τpp(R2N|u0(x)u0(y)|p|xy|N+spdydx+RNV(x)|u0|pdx)+τqq(R2N|u0(x)u0(y)|q|xy|N+sqdydx+RNV(x)|u0|qdx)+τrrRNg(x)|u0|rdx12RNRNK(τu0(x))K(τu0(y))|xy|αdydxC3(τp+τq+τr)C4τς,

    where C3 and C4 are constants. Taking e=τu0, for τ sufficiently large, we deduce that (b) holds since ς>q.

    Proof of Theorem 1.1. First, under the assumptions in Theorem 1.1, we need to prove that any (PS)c sequence {un} is bounded in E. For every cR, let {un}E be a (PS)c-sequence; that is, J(un)c and J(un)0. Then we deduce that

    c+unEJ(un)1ςJ(un),un=(1p1ς)(R2N|un(x)un(y)|p|xy|N+spdydx+RNV(x)|un(x)|pdx)+(1q1ς)(R2N|un(x)un(y)|q|xy|N+sqdydx+RNV(x)|un|qdx)+(1r1ς)RNg(x)|un|rdx+RNRNK(un)|xy|α[1ςk(un)un12K(un)]dydx(1q1ς)m=p,q[R2N|un(x)un(y)|m|xy|N+smdxdy+RNV(x)|un|m2unv(x)dx]=(1q1ς)(unpWs,pV+unqWs,qV).

    Arguing by contradiction, we assume that ||un||, and then we have the following two cases:

    (1) unWs,pV.

    (2) unWs,pV is bounded, unWs,qV.

    We divide into two cases to prove the conclusion.

    Case 1. Since p<q, we can see that |unqpWs,qV1, then unqWs,qVunpWs,qV>1, for n large enough

    c+unE(1q1ς)(unpWs,pV+unqWs,qV)(1q1ς)(unpWs,pV+unpWs,qV)C5(unWs,pV+unWs,qV)p=C5unpE,

    clearly, this is impossible since p>1, and we get a contradiction.

    Case 2. In view of the above relation, we immediately obtain

    c+unWs,pV+unWs,qV(1q1ς)(unpWs,pV+unqWs,qV)(1q1ς)unpWs,qV,

    moreover, we have

    cunpWs,qV+unWs,pVunpWs,qV+unWs,qVunpWs,qV(1q1ς),

    which is a contradiction.

    Hence, the (PS)c sequence {un} of J is bounded in E. According to Lemma 2.2, we can extract a subsequence (still denoted {un}) and uE such that

    {unuinE,unuinLϑ(RN),unua.e.inRN,

    where ϑ[m,ms), ms=NmNsm.

    Observe that, by a direct calculation, we have

    J(un)J(u),unu=m=p,qR2N|un(x)un(y)|m2(un(x)un(y))((unu)(x)(unu)(y))|xy|N+smdydxm=p,qR2N|u(x)u(y)|m2(u(x)u(y))((unu)(x)(unu)(y))|xy|N+smdydx+m=p,qRNV(x)(|un|m2un|u|m2u)(unu)dx+RNg(x)(|un|r2un|u|r2u)(unu)dx+RNRNK(un(y))k(un(x))(un(x)u(x))|xy|αdydxRNRNK(u(y))k(u(x))(un(x)u(x))|xy|αdydx=on(1). (3.2)

    Next, we estimate the above each term. Applying the Minkowski inequality and Holder inequality, we can conclude that

    |RNRNK(un(y))k(un(x))(un(x)u(x))|xy|αdydx|[RN|K(un(y))|rdy]1r[RN|k(un(x))(un(x)u(x))|rdx]1rC6[RN(|un|p1+|un|p2)rdy]1r[RN[(|un|p11+|un|p21)|un(x)u(x)|]rdx]1rC7[unp1rp1+unp2rp2)(unp11rp1un(x)u(x)rp1+unp21rp2un(x)u(x)rp2)C8[un(x)u(x)rp1+un(x)u(x)rp2]=on(1). (3.3)

    Similarly, we can obtain

    |RNRNK(u(y))k(u(x))(un(x)u(x))|xy|αdydx|C9[un(x)u(x)βp1+un(x)u(x)βp2] = on(1), (3.4)

    where Ci(i=6,...9) is constant, β=2N2Nα. On the other hand, taking into account that m2, applying the following inequality

    |db|mcm(|d|m2d|b|m2b)(db),d,bRN.

    We infer that

    [|un(x)un(y)|m2(un(x)un(y))|u(x)u(y)|m2(u(x)u(y))]×(un(x)u(x)un(y)+u(y))|xy|(N+sm)dxdycmR2N|un(x)un(y)u(x)+u(y)|m|xy|N+smdydx0, (3.5)

    and

    m=p,qRNV(x)(|un|m2un|u|m2u)(unu)dxcmm=p,qRNV(x)|unu|mdx=m=p,qcmunumm,V0. (3.6)

    Similarly, we obtain

    RNg(x)(|un|r2un|u|r2u)(unu)dx0. (3.7)

    Therefore, it is easy to see that

    [|un(x)un(y)|m2(un(x)un(y))|u(x)u(y)|m2(u(x)u(y))]×(un(x)u(x)un(y)+u(y))|xy|(N+sm)dxdy=on(1), (3.8)

    we also have

    m=p,qRNV(x)(|un|m2un|u|m2u)(unu)dx=on(1), (3.9)

    and

    RNg(x)(|un|r2un|u|r2u)(unu)dx=on(1). (3.10)

    Now, we obtain

    m=p,qunumm,V=on(1), (3.11)

    moreover, there holds

    m=p,qR2NR2N|(unu)(x)(unu)(y)|m|xy|N+smdxdy=on(1). (3.12)

    Putting together (3.11) and (3.12), we have

    unuE=m=p,q(R2N|(unu)(x)(unu)(y)|m|xy|N+smdxdy+RNV(x)|unu|mdx)1m=on(1).

    We can see that the functional J(u) satisfies the (PS)c-condition. Thus, problem (1.1) possesses at least one solution. The proof is completed.

    We point out that the main novelty of the paper is the combination of both double phase fractional differential operators and nonlocal Choquard reaction. Using employing variational techniques and inequalities such as Hardy-Littlewood-Sobolev, ensuring the geometric conditions of the Mountain Pass theorem for the energy functional. We demonstrate the existence of solution for the type of problem.

    Every author made an equal contribution to this study. Conceptualization and formal analysis by Baocheng Zhang and Zhihui Lv; validation and visualization by Kun Chi and Bin Ge; and Writing–original draft, review and editing by Liyan Wang. All authors have read and approved the final version of the manuscript for publication.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work was supported by the Guangdong Basic and Applied Basic Research Foundation (2022A1515110084), the Guangzhou Municipal Science and Technology Bureau (202201010246), and the Fundamental Research Foundation for Universities of Heilongjiang Province (2024-KYYWF-0921).

    The authors declare there are no conflicts of interest.



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