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Detailed tectonic geomorphology of the Dras fault zone, NW Himalaya

  • Our recent mapping of the Dras fault zone in the NW Himalaya has answered one of the most anticipated searches in recent times where strike-slip faulting was expected from the geodetic studies. Therefore, the discovery of the fault is a leap towards the understanding of the causes of active faulting in the region, and how the plate tectonic convergence between India and Eurasia is compensated in the interior portions of the Himalayan collision zone, and what does that imply about the overall convergence budget and the associated earthquake hazards. The present work is an extended version of our previous studies on the mapping of the Dras fault zone, and we show details that were either not available or briefly touched. We have used the 30 m shuttle radar topography to map the tectonic geomorphological features that includes the fault scarps, deflected drainage, triangular facets, ridge crests, faulted Quaternary landforms and so on. The results show that oblique strike-slip faulting is active in the suture zone, which suggests that the active crustal deformation is actively compensated in the interior portions of the orogen, and it is not just restricted to the frontal portions. The Dras fault is a major fault that we have interpreted either as a south dipping oblique backthrust or an oblique north dipping normal fault. The fieldwork was conducted in Leh, but it did not reveal any evidence for active faulting, and the fieldwork in the Dras region was not possible because of the politically sensitive nature of border regions where fieldwork is always an uphill task.

    Citation: AA Shah, A Rajasekharan, N Batmanathan, Zainul Farhan, Qibah Reduan, JN Malik. Detailed tectonic geomorphology of the Dras fault zone, NW Himalaya[J]. AIMS Geosciences, 2021, 7(3): 390-414. doi: 10.3934/geosci.2021023

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  • Our recent mapping of the Dras fault zone in the NW Himalaya has answered one of the most anticipated searches in recent times where strike-slip faulting was expected from the geodetic studies. Therefore, the discovery of the fault is a leap towards the understanding of the causes of active faulting in the region, and how the plate tectonic convergence between India and Eurasia is compensated in the interior portions of the Himalayan collision zone, and what does that imply about the overall convergence budget and the associated earthquake hazards. The present work is an extended version of our previous studies on the mapping of the Dras fault zone, and we show details that were either not available or briefly touched. We have used the 30 m shuttle radar topography to map the tectonic geomorphological features that includes the fault scarps, deflected drainage, triangular facets, ridge crests, faulted Quaternary landforms and so on. The results show that oblique strike-slip faulting is active in the suture zone, which suggests that the active crustal deformation is actively compensated in the interior portions of the orogen, and it is not just restricted to the frontal portions. The Dras fault is a major fault that we have interpreted either as a south dipping oblique backthrust or an oblique north dipping normal fault. The fieldwork was conducted in Leh, but it did not reveal any evidence for active faulting, and the fieldwork in the Dras region was not possible because of the politically sensitive nature of border regions where fieldwork is always an uphill task.



    In this article, we investigate the following Kirchhoff-Schrödinger-Poisson type systems on the Heisenberg group:

    {(a+bΩ|Hu|pdξ)ΔH,puμϕ|u|p2u=λ|u|q2u+|u|Q2uin Ω,ΔHϕ=|u|pin Ω,u=ϕ=0on Ω, (1.1)

    where a,b are positive real numbers, ΩHN is a bounded region with smooth boundary, 1<p<Q, Q=2N+2 is the homogeneous dimension of the Heisenberg group HN, Q=pQQp, q(2p,Q), ΔH,pu=div(|Hu|p2Hu) is known as the p-horizontal Laplacian, and μ and λ are some positive real parameters.

    In recent years, geometrical analysis of the Heisenberg group has found significant applications in quantum mechanics, partial differential equations and other fields, which has attracted the attention of many scholars who tried to establish the existence and multiplicity of solutions of partial differential equations on the Heisenberg group. For instance, in the subcase of problem (1.1), when p=2 and b=μ=0, the existence of solutions for some nonlinear elliptic problems in bounded domains has been established. Tyagi [1] studied a class of singular boundary value problem on the Heisenberg group:

    {ΔHu=μg(ξ)u(|z|4+t2)12+λf(ξ,t),ξΩ,u|Ω=0, (1.2)

    and under appropriate conditions, obtained some existence results using Bonanno's three critical point theorem. Goel and Sreenadh [2] dealt with a class of Choquard type equation on the Heisenberg group, and they established regularity of solutions and nonexistence of solutions invoking the mountain pass theorem, the linking theorem and iteration techniques and boot-strap method.

    In the case b0 and μ=0, problem (1.1) becomes the Kirchhoff problem, which has also been widely studied. For example, Sun et al. [3] dealt with the following Choquard-Kirchhoff problem with critical growth:

    M(u2)(ΔHu+u)=HN|u(η)|Qλ|η1ξ|λdη|u|Qλ2u+μf(ξ,u),

    where f is a Carathéodory function, M is the Kirchhoff function, ΔH is the Kohn Laplacian on the Heisenberg group HN, μ>0 is a parameter and Qλ=2QλQ2 is the critical exponent. In their paper, a new version of the concentration-compactness principle on the Heisenberg group was established for the first time. Moreover, the existence of nontrivial solutions was obtained even under nondegenerate and degenerate conditions. Zhou et al. [4] proved the existence of solutions of Kirchhoff type nonlocal integral-differential operators with homogeneous Dirichlet boundary conditions on the Heisenberg group using the variational method and the mountain pass theorem. Deng and Xian [5] obtained the existence of solutions for Kirchhoff type systems involving the Q-Laplacian operator on the Heisenberg group with the help of the Trudinger-Moser inequality and the mountain pass theorem. For more related results, see [6,7,8,9,10,11,12,13].

    When b=0, p=2 and μ0, problem (1.1) becomes the Schrödinger-Poisson system. This is a very interesting subject and has recently witnessed very profound results. For example, An and Liu [14] dealt with the following forms of Schrödinger-Poisson type system on the Heisenberg group:

    {ΔHu+μϕu=λ|u|q2u+|u|2uin Ω,ΔHϕ=u2in Ω,u=ϕ=0on Ω,

    where μR and λ>0 are some real parameters and 1<q<2. By applying the concentration compactness and the critical point theory, they found at least two positive solutions and a positive ground state solution.

    Liang and Pucci [15] studied the following critical Kirchhoff-Poisson system on the Heisenberg group:

    {M(Ω|Hu|2dξ)ΔHu+ϕ|u|q2u=h(ξ,u)+λ|u|2uin Ω,ΔHϕ=|u|qin Ω,u=ϕ=0on Ω,

    where ΩH1 is a smooth bounded domain, ΔH is the Kohn-Laplacian on the first Heisenberg group H1 and 1<q<2. By applying the symmetric mountain pass lemma, they obtained the multiplicity of solutions with λ sufficiently small.

    The Kirchhoff-Poisson system on the Heisenberg group with logarithmic and critical nonlinearity was considered by Pucci and Ye [13]:

    {M(Ω|Hu|2dξ)ΔHu+ϕu=|u|2u+λ|u|q2uln|u|2in Ω,ΔHϕ=|u|2in Ω,u=ϕ=0on Ω.

    Under suitable assumptions on the Kirchhoff function M covering the degenerate case, they showed that for a sufficiently large λ>0, there exists a nontrivial solution to the above problem.

    When p2 and μ0, as far as we know, for Kirchhoff-Schrodinger-Poisson systems (1.1) with critical nonlinearities on the Heisenberg group, existence and multiplicity results are not yet available. In the Euclidean case, Du et al. [16] first studied the existence results for the Kirchhoff-Poisson systems with p-Laplacian under the subcritical case using the mountain pass theorem. Later, Du et al. [17] studied quasilinear Schrödinger-Poisson systems. For the critical case, Du et al. [18] also obtained the existence of ground state solutions with the variational approach.

    Inspired by the above achievements, we aim to establish some results on the existence and multiplicity of nontrivial solutions of the Kirchhoff-Schrödinger-Poisson systems (1.1). The major difficulties in dealing with problem (1.1) are the presence of a nonlocal term and critical nonlinearities making the study of this problem very challenging.

    Before presenting the main results of this article, we first present some concepts of the Heisenberg group. The Heisenberg group is represented by HN. If ξ=(x,y,t)HN, then the definition of this group operation is

     τξ(ξ)=ξξ=(x+x,y+y,t+t+2(xyyx)), for every ξ,ξHN,

    ξ1=ξ is the inverse, and therefore (ξ)1ξ1=(ξξ)1.

    The definition of a natural group of dilations on HN is δs(ξ)=(sx,sy,s2t), for every s>0. Hence, δs(ξ0ξ)=δs(ξ0)δs(ξ). It can be easily proved that the Jacobian determinant of dilatations δs:HNHN is constant and equal to sQ, for every ξ=(x,y,t)HN. The critical exponent is Q:=pQQp, where the natural number Q=2N+2 is called the homogeneous dimension of HN. We define the Korányi norm as follows

    |ξ|H=[(x2+y2)2+t2]14, for every ξHN,

    and we derive this norm from the Heisenberg group's anisotropic dilation. Hence, the homogeneous degree of the Korányi norm is equal to 1, in terms of dilations

    δs:(x,y,t)(sx,sy,s2t), for every s>0.

    The set

    BH(ξ0,r)={ξHN:dH(ξ0,ξ)<r},

    denotes the Korányi open ball of radius r centered at ξ0. For the sake of simplicity, we shall denote Br=Br(O), where O=(0,0) is the natural origin of HN.

    The following vector fields

    T=t, Xj=xj+2yjt, Yj=yj2xjt,

    generate the real Lie algebra of left invariant vector fields for j=1,,n, which forms a basis satisfying the Heisenberg regular commutation relation on HN. This means that

    [Xj,Yj]=4δjkT, [Yj,Yk]=[Xj,Xk]=[Yj,T]=[Xj,T]=0.

    The so-called horizontal vector field is just a vector field with the span of [Xj,Yj]nj=1.

    The Heisenberg gradient on HN is

    H=(X1,X2,,Xn,Y1,Y2,,Yn),

    and the Kohn Laplacian on HN is given by

    ΔH=Nj=1X2j+Y2j=Nj=1[2x2j+2y2j+4yj2xjt4xj2xjt+4(x2j+y2j)2t2].

    The Haar measure is invariant under the left translations of the Heisenberg group and is Q-homogeneous in terms of dilations. More precisely, it is consistent with the (2n+1)-dimensional Lebesgue measure. Hence, as shown Leonardi and Masnou [19], the topological dimension 2N+1 of HN is strictly less than its Hausdorff dimension Q=2N+2. Next, |Ω| denotes the (2N + 1)-dimensional Lebesgue measure of any measurable set ΩHN. Hence,

    |δs(Ω)|=sQ|Ω|, d(δsξ)=sQdξ and |BH(ξ0,r)|=αQrQ, where αQ=|BH(0,1)|.

    Now, we can state the main result of the paper.

    Theorem 1.1. Let q(2p,Q). Then there exist positive constants μ1 and λ1 such that for every μ(0,μ1) and λ(λ1,+), the following assertions hold:

    (I) Problem (1.1) has a nontrivial weak solution;

    (II) Problem (1.1) has infinitely many nontrivial weak solutions if parameter a is large enough.

    We can give the following example for problem (1.1) with p=3 and ΩH1:

    {(a+Ω|Hu|2dξ)ΔHu+μϕu=λ|u|6u+|u|10uin Ω,ΔHϕ=|u|3in Ω,u=ϕ=0on Ω.

    In this case, N=1, p=3 and q=8, then Q=2N+2=4, Q=12. If positive parameters μ small enough and λ large enough, by Theorem 1.1, we know that problem (1.1) has a nontrivial weak solution. Moreover, if in addition the parameter a is large enough, problem (1.1) has infinitely many nontrivial weak solutions. It should be noted that the methods in An and Liu [14] and Liang and Pucci [15] do not seem to apply to problem (1.1).

    Remark 1.1. Compared with previous results, this paper has the following key new features:

    1) The presence of the nonlocal term ϕ|u|p2u;

    2) The lack of compactness caused by critical index;

    3) The presence of the p-Laplacian makes this problem more complex and interesting.

    It is worth stressing that the nonlocal term and the critical exponent lead to the lack of compactness condition, and we use the concentration-compactness principle to overcome this difficulty. Moreover, we shall use some more refined estimates to overcome the presence of the p-Laplacian.

    We need to emphasize here that despite the similarity of some properties between the classical Laplacian Δ and Kohn Laplacian ΔH, similarities can be misleading (see Garofalo and Lanconelli [20]), so there are still many properties that deserve further study. Moreover, for the case p2, it is difficult to prove the boundedness of Palais-Smale sequences. In order to overcome these difficulties, we use some more accurate estimates of relevant expression. Additionally, we use the concentration-compactness principle on the Heisenberg group to prove the compactness condition.

    The paper is organized as follows. In Section 2, we introduce some notations and known facts. Moreover, we introduce some key estimates. In addition, we define the corresponding energy functional Iλ and its derivative at u, that is, Iλ(u). In Section 3, we prove Theorem 1.1.

    First of all, we collected some known facts, useful in the sequel. For additional background material, readers are advised to refer to Papageorgiou et al. [21].

    Let

    uss=Ω|u|sdξ, for every uLs(Ω),

    represent the usual Ls-norm.

    Following Folland and Stein [22], we define the space ˚S21(Ω) as the closure of C0(Ω) in S21(HN). Then ˚S21(Ω) is a Hilbert space with the norm

    u2˚S21(Ω)=Ω|Hu|2dξ.

    We define the Folland-Stein space S1,p(Ω) as the closure of C0(Ω) with the norm

    u=(Ω|Hu|pdξ)1p.

    Then the embedding

    S1,p(Ω)Ls(Ω), for every s(1,Q),

    is compact. However, if s=Q, the embedding is only continuous (see Vassiliev [23]).

    Additionally, we say that (u,ϕ)S1,p(Ω)×S1,p(Ω) is a solution of problem (1.1) if and only if

    aΩ|Hu|p2HuHvdξ+bupΩ|Hu|p2HuHvdξ μΩϕ|u|p2uvdξλΩ|u|q2uvdξΩ|u|Q2uvdξ=0

    and

    ΩHϕHωdξΩ|u|pωdξ=0,

    for every v,ωS1,p(Ω)×S1,p(Ω). Moreover, (u,ϕ)S1,p(Ω)×S1,p(Ω) is a positive solution of problem (1.1) if u and ϕ are both positive. Therefore, in order to apply the critical point theory, we need to define the functional J(u,ϕ):S1,p(Ω)×S1,p(Ω)R as follows

    J(u,ϕ)=apup+b2pu2p+μ2pΩ|Hϕ|2dξμpΩϕ|u|pdξλqΩ|u|qdξ1QΩ|u|Qdξ,

    for every (u,ϕ)S1,p(Ω)×S1,p(Ω). Then J is C1 on S1,p(Ω)×S1,p(Ω) and its critical points are the solutions of problem (1.1). Indeed, the partial derivatives of J at (u,ϕ) are denoted by Ju(u,ϕ), Jϕ(u,ϕ), namely for every v,ωS1,p(Ω)×S1,p(Ω),

    Ju(u,ϕ)[v]=aΩ|Hu|p2HuHvdξ+bupΩ|Hu|p2HuHvdξ μΩϕ|u|p2uvdξλΩ|u|q2uvdξΩ|u|Q2uvdξ=0

    and

    Jϕ(u,ϕ)=μpΩHϕHωdξμpΩ|u|pωdξ.

    Standard computations show that Ju (respectively Jϕ) continuously maps S1,p(Ω)×S1,p(Ω) into the dual of S1,p(Ω). Moreover, the functional J is C1 on S1,p(Ω)×S1,p(Ω) and

    Ju(u,ϕ)=Jϕ(u,ϕ)=0

    if and only if (u,ϕ) is a solution of problem (1.1).

    Lemma 2.1. Let uS1,p(Ω). Then there is a unique nonnegative function ϕu˚S21(Ω) such that

    {ΔHϕ=|u|pin Ω,ϕ=0on Ω. (2.1)

    Furthermore, ϕu0 and ϕu>0 if u0. Also,

    (i) ϕtu=tpϕu, for every t>0;

    (ii) ϕu˚S21(Ω)ˆCup, where ˆC>0;

    (iii) Let unu in S1,p(Ω). Then, ϕunϕu in ˚S21(Ω), and

    Ωϕun|un|p2unvdξΩϕu|u|p2uvdξ, forevery vS1,p(Ω). (2.2)

    Proof. For any u˚S21(Ω), we define W:˚S21(Ω)R,

    W(v)=Ωv|u|pdξ, for every v˚S21(Ω).

    Let vnv˚S21(Ω), as n. It follows by the Hölder inequality that

    |W(vn)W(v)|Ω(vnv)|u|pdξ(Ω|vnv|Qdξ)1Q(Ω|u|pQQ1dξ)Q1QS1pvnv|u|ppQQ10, as n,

    where

    S=infuS1,p(Ω){0}Ω|Hu|pdξ(Ω|u|Qdξ)pQ (2.3)

    is the best Sobolev constant. This implies that W is a continuous linear functional. Using the Lax-Milgram theorem, we see that there is a unique ϕu˚S21(Ω) satisfying

    ΩHϕuHvdξ=Ωv|u|pdξ, for every vS1,p(Ω). (2.4)

    Thus, ϕu˚S21(Ω) is the unique solution of problem (2.1). Moreover, applying the maximum principle, one has ϕu0 and ϕu>0 if u0. Indeed, for every t>0, one has

    ΔHϕtu=tpup=tp(ΔHϕu)=ΔH(tpϕu).

    Hence ϕtu=tpϕu due to the uniqueness of ϕu.

    Furthermore, since ϕuS1,p(Ω), we can view it as a text function in problem (2.1). Then by (2.4), the Sobolev inequality and the Hölder inequality, we have (henceforth C0, C1, C2 will denote positive constants)

    Ω|Hϕu|2dξ=Ωϕu|u|pdξ|ϕu|L2(Ω)|u|pL2p(Ω)C1ϕu˚S21(Ω)up.

    Therefore, we get ϕu˚S21(Ω)C1up.

    Since unu in S1,p(Ω), we can conclude that unu a.e. in Ω and {|un|p} is bounded in L2(Ω). Moreover, we have |un|p|u|p in L2(Ω). Then for every v˚S21(Ω), it follows that

    Ωv|un|pdξΩv|u|pdξ, as n.

    Therefore, ϕunϕu in ˚S21(Ω). By the Hölder inequality, the Sobolev inequality and (ⅱ), one has

    Ω|ϕun|un|p2un|2p2p1dξ|ϕun|2p2p1L2(Ω)(Ω|un|2pdξ)p12p1C0ϕun2p2p1˚S21(Ω)(Ω|un|2pdξ)p12p1C2ϕun2p22p1(Ω|un|2pdξ)p12p1.

    Hence, {ϕun|un|p2un} is bounded in L2p2p1(Ω). Since

    ϕun|un|p2unϕu|u|p2u, a.e. in Ω,

    we get

    Ωϕun|un|p2unvdξΩϕu|u|p2uvdξ, for every vS1,p(Ω).

    The proof of Lemma 2.1 is complete.

    By similar arguments as in An and Liu [14], we can get the following result.

    Lemma 2.2. Let Ψ(u)=ϕu for every uS1,p(Ω), where ϕu is as in Lemma 2.1, and let

    Υ={(u,ϕ)S1,p(Ω)×S1,p(Ω):Jϕ(u,ϕ)=0}.

    Then Ψ is C1 and Υ is the graph of Ψ.

    We define the corresponding energy functional Iλ(u)=J(u,ϕu) of problem (1.1) by

    Iλ(u)=apup+b2pu2pμ2pΩϕu|u|pdξλqΩ|u|qdξ1QΩ|u|Qdξ, for every uS1,p(Ω). (2.5)

    Based on the definition of J and Lemma 2.2, we can conclude that Iλ is of C1.

    Lemma 2.3. (see An and Liu [14]) Let (u,ϕ)S1,p(Ω)×S1,p(Ω). Then (u,ϕ) is a critical point of J if and only if u is a critical point of Iλ and ϕ=Ψ(u), where Ψ was defined in Lemma 2.2.

    According to Lemma 2.3, we know that a solution (u,ϕu) of problem (1.1) corresponds to a critical point u of the functional Iλ with ϕ=Ψ(u) and

    Iλ(u),v=aΩ|Hu|p2HuHvdξ+bupΩ|Hu|p2HuHvdξ μΩϕu|u|p2uvdξλΩ|u|q2uvdξΩ|u|Q2uvdξ, for every vS1,p(Ω). (2.6)

    Therefore, based on the above arguments, we shall strive to use critical point theory and some analytical techniques to prove the existence of critical points of functional Iλ.

    In this subsection, our main focus will be on proving that the functional Iλ satisfies the Palais-Smale condition.

    Lemma 2.4. Let q(2p,Q). Then there exists μ1>0 such that for any μ<μ1, the energy functional Iλ satisfies (PS)c condition, where

    c(0, (1q1Q)(aS)Qp) (2.7)

    and S is the best Sobolev constant given by (2.3).

    Proof. Let us assume that {un}nS1,p(Ω) is a (PS)c sequence related to the functional Iλ, that is,

    Iλ(un)c and Iλ(un)0,as n. (2.8)

    It follows that

    c+o(1)un=Iλ(un)1qIλ(un)una(1p1q)unp+b(12p1q)un2pμ(12p1q)Ωϕun|un|pdξ+(1q1Q)Ω|un|Qdξa(1p1q)unp+(bμˆC)(12p1q)un2p, (2.9)

    where ˆC is a positive constant given by Lemma 2.1(ⅱ). Let μ1=bˆC. By (2.9), we know that (PS)c sequence {un}nS1,p(Ω) is bounded for every μ<μ1. Thus, we may assume that unu weakly in S1,p(Ω), and unu in Ls(Ω) with 1<s<Q. Furthermore, since Iλ(un)=Iλ(|un|), we may also assume that un0 and u0. Therefore, invoking the concentration compactness principle on the Heisenberg group (see Vassiliev [23, Lemma 3.5]), we obtain

    |Hun|pdξdω|Hu|pdξ+ΣjΛωjδxj,|un|Qdξdν=|u|Qdξ+ΣjΛνjδxj, (2.10)

    where {xj}jΛΩ is the most a countable set of distinct points, ω and ν in HN are two positive Radon measures, and {ωj}jΛ, {νj}jΛ are nonnegative numbers. Moreover, we have

    ωjSνpQj. (2.11)

    Next, we shall show that Λ=. Indeed, assume that the hypothesis ωj0 holds for some jΛ. Then when ε>0 is sufficiently small, we can find 0ψε,j1 satisfying the following

    {ψε,j=1in BH(ξj,ε2),ψε,j=0in ΩBH(ξj,ε),|Hψε,j|2ε, (2.12)

    where ψε,jC0(BH(ξj,ε)) is a cut-off function. Clearly, (unψε,j)n is bounded in S1,p(Ω). It follows from (2.8) and the boundedness of (unψε,j)n that

    Iλ(un),unψε,j0, as n,

    that is,

    a(Ω|Hun|pψε,j+Ωun|Hun|p2HunHψε,jdξ)+bunp(Ω|Hun|pψε,j+Ωun|Hun|p2HunHψε,jdξ)μΩϕun|un|pψε,jdξ=λΩuqnψε,jdξ+ΩuQnψε,jdξ+o(1). (2.13)

    It follows from the dominated convergence theorem that

    BH(ξj,ε)|un|qψε,jdξBH(ξj,ε)|u|qψε,jdξ, as n.

    Hence, letting ε0, we get

    limε0limnBH(ξj,ε)|un|qψε,jdξ=0. (2.14)

    By Lemma 2.1,

    limnΩϕun|un|p2unudξ=Ωϕu|u|pdξ, (2.15)

    and since unu in Ls(Ω) with 1<s<Q, one has

    Ω(ϕun|un|pϕun|un|p1u)dξΩ|ϕun||un|p1|unu|dξ(Ω|ϕun|un|p1|pp1dξ)p1p|unu|p0. (2.16)

    Combining (2.15) with (2.16), we obtain that

    limnΩϕun|un|pdξ=Ωϕu|u|pdξ,

    thus

    limε0limnBH(ξj,ε)ϕun|un|pψε,jdξ=limε0BH(ξj,ε)ϕu|u|pψε,jdξ=0. (2.17)

    Since

    BH(ξj,ε)dξ=BH(0,ε)dξ=|BH(0,1)|εQ,

    applying the Hölder inequality, we obtain

    limε0limnΩun|Hun|p1Hψε,jdξlimε0limn(BH(ξj,ε)|Hun|pdξ)p1p(BH(ξj,ε)|unHψε,j|pdξ)1pClimε0(BH(ξj,ε)|un|p|Hψε,j|pdξ)1pClimε0(BH(ξj,ε)|un|Qdξ)1Q(BH(ξj,ε)|Hψε,j|Qdξ)1Q=0.

    By (2.10), we have

    limε0limnΩ|Hun|pψε,jdξlimε0(ωj+BH(ξj,ε)|Hu|pψε,jdξ)=ωj (2.18)

    and

    limε0limnΩuQnψε,jdξ=limε0(νj+BH(ξj,ε)uQnψε,jdξ)=νj. (2.19)

    Therefore, by (2.13)–(2.19), one gets νjaωj. It follows from (2.11) that

    νj=0orνj(aS)Qp.

    In fact, if νj(aS)Qp holds, therefore by (2.8) and (2.10), for every μ<μ1, we have

    c=limn{Iλ(un)1qIλ(un)un}limn(1q1Q)Ω|un|Qdξ(1q1Q)νj(1q1Q)(aS)Qp, (2.20)

    which contradicts (2.7). Thus, Λ=. By (2.10) and Λ=, we have

    Ω|un|QdξΩ|u|Qdξ. (2.21)

    Let

    limnunp=A.

    If A=0, then un0 in S1,p(Ω). So assume now that A>0. By (2.8), we get

    aΩ|Hun|p2HunHvdξ+bAΩ|Hun|p2HunHvdξ μΩϕun|un|p2unvdξλΩ|un|q2unvdξΩ|un|Q2unvdξ=o(1). (2.22)

    Let v=u in (2.22). Then

    aup+bAupμΩϕu|u|pdξλΩ|u|qdξΩ|u|Qdξ=0. (2.23)

    By (2.8), (2.10), (2.21) and Lemma 2.1, one also has

    limnaunp+bAunpμΩϕu|u|pdξλΩ|u|qdξΩ|u|Qdξ=0. (2.24)

    Thus, combining (2.23) and (2.24), we get limnunp=up. Thus, we see that unu in S1,p(Ω) by the uniform convexity of S1,p(Ω). This completes the proof of Lemma 2.4.

    We need the following auxiliary lemmas to prove our main result.

    Lemma 3.1. Let q(2p,Q) and μ(0,μ1). Then functional Iλ satisfies the mountain pass geometry, that is,

    (i) There exist constants ρ,α>0 satisfying Iλ(u)|Bρα, for every uS1,p(Ω);

    (ii) There exists eS1,p(Ω)¯Bρ satisfying Iλ(e)<0.

    Proof. First, applying the Hölder inequality, we get

    Iλ(u)=apup+b2pu2pμ2pΩϕu|u|pdξλqΩ|u|qdξ1QΩ|u|Qdξapup+bμˆC2pu2pλqΩ|u|qdξ1QΩ|u|QdξapupλqSqp|Ω|QqQuq1QSQpuQ=up{apλqSqp|Ω|QqQuqp1QSQpuQp}. (3.1)

    Let

    f(t)=apλqSqp|Ω|QqQtqp1QSQptQp, for every t0.

    We now show that there exists a constant ρ>0 satisfying f(ρ)ap. We see that f is a continuous function on [0,+) and limt0+f(t)=ap. Hence there exists ρ such that f(t)apε1, for every 0tρ, where ρ is small enough such that u=ρ. If we choose ε1=a2p, we have f(t)a2p, for every 0tρ. In particular, f(ρ)a2p and we obtain Iλ(u)a2pρp=α for u=ρ. Hence assertion (i) of Lemma 3.1 holds.

    Next, we shall show that assertion (ⅱ) of Lemma 3.1 also holds:

    Iλ(su)=asppup+bs2p2pu2pμs2p2pΩϕu|u|pdξλsqqΩ|u|qdξsQQΩ|u|Qdξasppup+bs2p2pu2psQQΩ|u|Qdξ as s+. (3.2)

    Thus, we can deduce that Iλ(s0u)<0 and s0u>ρ, for every s0 large enough. Let e=s0u. Then e is the desired function and the proof of (ⅱ) of Lemma 3.1 is complete.

    Proof of Theorem 1.1(I). We claim that

    0<cλ=infhΓmax0s1Iλ(h(s))<(1q1Q)(aS)Qp, (3.3)

    where

    Γ={hC([0,1],S1,p(Ω)):h(0)=1,h(1)=e}.

    Indeed, we can choose v1S1,p(Ω){0} with v1=1. From (3.2), we have lims+Iλ(sv1)=. Then

    sups0Iλ(sv1)=Iλ(sλv1),  for some sλ>0.

    So sλ satisfies

    aspλv1p+bs2pλv12p=μs2pλΩϕv1|v1|pdξ+λsqλΩ|v1|qdξ+sQλΩ|v1|Qdξ. (3.4)

    Next, we shall prove that {sλ}μ>0 is bounded. In fact, suppose that the following hypothesis sλ1 is satisfied for every λ>0. Then it follows from (3.4) that

    (a+b)s2pλaspλv1p+bs2pλv12p=μs2pλΩϕv1|v1|pdξ+λsqλΩ|v1|qdξ+sQλΩ|v1|QdξsQλΩ|v1|Qdξ. (3.5)

    Since 2p<q<Q, we can deduce that {sλ}λ>0 is bounded.

    Next, we shall demonstrate that sλ0, as λ. Suppose to the contrary, that there exist sλ>0 and a sequence (λn)n with λn, as n, satisfying sλnsλ, as n. Invoking the Lebesgue dominated convergence theorem, we see that

    Ω|sλnv1|qdξΩ|sλv1|qdξ, as n.

    It now follows that

    λnΩ|sλv1|qdξ, as n.

    Thus, invoking (3.4), we can show that this cannot happen. Therefore, sλ0, as λ.

    Furthermore, (3.4) implies that

    limλλΩ|sλv1|qdξ=0

    and

    limλΩ|sλv1|Qdξ=0.

    Hence based on the definition of Iλ and sλ0, as λ, we get that

    limλ(sups0Iλ(sv1))=limλIλ(sλv1)=0.

    So, there is λ1>0, satisfying for every λ>λ1,

    sups0Iλ(sv1)<(1q1Q)(aS)Qp.

    Letting e=t1v1 with t1 large enough for Iλ(e)<0, we get

    0<cλmax0s1Iλ(h(s)), where h(s)=st1v1.

    Therefore

    0<cλsups0Iλ(sv1)<(1q1Q)(aS)Qp,

    for λ large enough. This completes the proof of Theorem 1.1(Ⅰ).

    In this subsection, we shall use the Krasnoselskii genus theory to prove Theorem 1.1(Ⅱ). To this end, let E be a Banach space and denote by Λ the class of all closed subsets AE{0} that are symmetric with respect to the origin, that is, uE implies uE. Moreover, suppose that X is k-dimensional and X=span{z1,,zk}. For every nk, inductively select zn+1Xn=span{z1,,zn}. Let Rn=R(Zn) and Υn=BRnZn. Define

    Wn={φC(Υn,E):φ|BRnZn=id and φ is odd}

    and

    Γi={φ(¯ΥnV):φWn,ni,VΛ,Λ is closed,γ(V)ni},

    where γ(V) is the Krasnoselskii genus of V.

    Theorem 3.1. (see Rabinowitz [24, Theorem 9.12]) Let IC1(E,R) be even with I(0)=0 and let E be an infinite-dimensional Banach space. Assume that X is a finite-dimensional space, E=XY and that I satisfies the following properties:

    (i) There exists θ>0 such that I satisfies (PS)c condition, for every c(0,θ);

    (ii) There exist ρ,α>0 satifying I(u)α, for every uBρY;

    (iii) For every finite-dimensional subspace ˜EE, there exists R=R(˜E)>ρ such that I(u)0 on ˜EBR.

    For every iN, let ci=infXΓimaxuZI(u), hence, 0cici+1 and ci<θ, for every i>k. Then every ci is a critical value of I. Moreover, if ci=ci+1==ci+p=c<θ for i>k, then γ(Kc)p+1, where

    Kc={uE:I(u)=c and I(u)=0}.

    Lemma 3.2. There is a nondecreasing sequence {sn} of positive real numbers, independent of λ, such that for every λ>0, we have

    cλn=infWΓnmaxuWIλ(u)<sn,

    where Γn was defined in Theorem 3.1.

    Proof. By the definition of Γn, one has

    cλninfWΓnmaxuW{apunp+b2pun2pμ2pΩϕunupndξ1QΩ|un|Qdξ}=sn,

    therefore sn< and snsn+1.

    Proof of Theorem 1.1(II). We note that Iλ satisfies Iλ(0)=0 and Iλ(u)=Iλ(u). In the sequel, we shall divide the proof into the following three steps:

    Step 1. We shall prove that Iλ satisfies hypothesis (ⅱ) of Theorem 3.1. Indeed, similar to the proof of (ⅰ) in Lemma 3.1, we can easily prove that the energy functional Iλ satisfies the hypothesis (ii) of Theorem 3.1.

    Step 2. We shall prove that Iλ satisfies hypothesis (ⅲ) of Theorem 3.1. Indeed, let Y be a finite-dimensional subspace of S1,p(Ω). Since all norms in finite-dimensional space are equivalent, it follows that for every uY, we have

    Iλ(u)apup+b2pu2p1QΩ|u|Qdξapup+b2pu2p1QCuQ, (3.6)

    for some positive constant C>0. Also, because of 2p<Q, we can choose a large R>0 such that Iλ(u)0 on S1,p(Ω)BR. This fact implies that the energy functional Iλ satisfies the hypothesis (iii) of Theorem 3.1.

    Step 3. We shall prove that problem (1.1) has infinitely many nontrivial weak solutions. Indeed, applying the argument in Wei and Wu [25], we can choose a1 large enough so that for every a>a1,

    supsn<(1q1Q)(aS)Qp,

    that is,

    cλn<sn<(1q1Q)(aS)Qp.

    Thus, one has

    0<cλ1cλ2cλn<sn<(1q1Q)(aS)Qp.

    From Lemma 2.4, we know that Iλ satisfies (PS)cλi(i=1,2,,n) condition. This fact implies that the levels cλ1cλ2cλn are critical values of Iλ, which be guaranteed by an application of the Rabinowitz result [24, Proposition 9.30].

    If cλi=cλi+1 where i=1,2,,k1, then applying the Ambrosetti and Rabinowitz result [26, Remark 2.12 and Theorem 4.2], we see that the set Kcλi consists of infinite number of different points, so problem (1.1) has infinite number of weak solutions. Hence, problem (1.1) has at least k pairs of solutions. Since k is arbitrary, we can conclude that problem (1.1) has infinitely many solutions. This completes the proof of Theorem 1.1(Ⅱ).

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

    Song was supported by the National Natural Science Foundation of China (No. 12001061), the Science and Technology Development Plan Project of Jilin Province, China (No. 20230101287JC) and Innovation and Entrepreneurship Talent Funding Project of Jilin Province (No. 2023QN21). Repovš was supported by the Slovenian Research Agency program No. P1-0292 and grants Nos. J1-4031, J1-4001, N1-0278, N1-0114 and N1-0083.

    The authors declare there is no conflict of interest.



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