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Research article Special Issues

Congestion behavior and tolling strategies in a bottleneck model with exponential scheduling preference

  • Received: 03 October 2022 Revised: 24 November 2022 Accepted: 29 November 2022 Published: 13 December 2022
  • The bottleneck model has been widely used in the past fifty years to analyze the morning commute. To reduce the complexity of analysis, most previous studies adopted discontinuous scheduling preference (DSP). However, this handling destroys the continuity in departure rate and differentiability in travel time and cumulative departures. This paper considers an exponential scheduling preference (ESP), which supposes the unit schedule delay cost for commuters exponentially changes with time. With this scheduling preference, we analytically derive solutions and economic properties of user equilibrium and social optimum in the bottleneck model. The first-best, time-varying toll and the optimal single-step toll scheme with ESP are also studied. Results indicate that ESP eliminates the discontinuity in departure rate and non-differentiability in travel time and cumulative departures, which makes the process of morning commute smooth. The ignorance of ESP will lead to underestimation in the queueing time and bias in travel behavior analysis and policymaking.

    Citation: Chuanyao Li, Yichao Lu, Yuqiang Wang, Gege Jiang. Congestion behavior and tolling strategies in a bottleneck model with exponential scheduling preference[J]. Electronic Research Archive, 2023, 31(2): 1065-1088. doi: 10.3934/era.2023053

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  • The bottleneck model has been widely used in the past fifty years to analyze the morning commute. To reduce the complexity of analysis, most previous studies adopted discontinuous scheduling preference (DSP). However, this handling destroys the continuity in departure rate and differentiability in travel time and cumulative departures. This paper considers an exponential scheduling preference (ESP), which supposes the unit schedule delay cost for commuters exponentially changes with time. With this scheduling preference, we analytically derive solutions and economic properties of user equilibrium and social optimum in the bottleneck model. The first-best, time-varying toll and the optimal single-step toll scheme with ESP are also studied. Results indicate that ESP eliminates the discontinuity in departure rate and non-differentiability in travel time and cumulative departures, which makes the process of morning commute smooth. The ignorance of ESP will lead to underestimation in the queueing time and bias in travel behavior analysis and policymaking.



    Consider with the following fourth-order elliptic Navier boundary problem

    {Δ2u+cΔu=λa(x)|u|s2u+f(x,u)inΩ,u=Δu=0onΩ, (1.1)

    where Δ2:=Δ(Δ) denotes the biharmonic operator, ΩRN(N4) is a smooth bounded domain, c<λ1 (λ1 is the first eigenvalue of Δ in H10(Ω)) is a constant, 1<s<2,λ0 is a parameter, aL(Ω),a(x)0,a(x)0, and fC(ˉΩ×R,R). It is well known that some of these fourth order elliptic problems appear in different areas of applied mathematics and physics. In the pioneer paper Lazer and Mckenna [13], they modeled nonlinear oscillations for suspensions bridges. It is worth mentioning that problem (1.1) can describe static deflection of an elastic plate in a fluid, see [21,22]. The static form change of beam or the motion of rigid body can be described by the same problem. Equations of this type have received more and more attentions in recent years. For the case λ=0, we refer the reader to [3,7,11,14,16,17,20,23,27,29,34,35,36,37] and the reference therein. In these papers, existence and multiplicity of solutions have been concerned under some assumptions on the nonlinearity f. Most of them considered the case f(x,u)=b[(u+1)+1] or f having asymptotically linear growth at infinity or f satisfying the famous Ambrosetti-Rabinowitz condition at infinity. Particularly, in the case λ0, that is, the combined nonlinearities for the fourth-order elliptic equations, Wei [33] obtained some existence and multiplicity by using the variational method. However, the author only considered the case that the nonlinearity f is asymptotically linear. When λ=1, Pu et al. [26] did some similar work. There are some latest works for problem (1.1), for example [10,18] and the reference therein. In this paper, we study problem (1.1) from two aspects. One is that we will obtain two multiplicity results when the nonlinearity f is superlinear at infinity and has the standard subcritical polynomial growth but not satisfy the Ambrosetti-Rabinowitz condition, the other is we can establish some existence results of multiple solutions when the nonlinearity f has the exponential growth but still not satisfy the Ambrosetti-Rabinowitz condition. In the first case, the standard methods for the verification of the compactness condition will fail, we will overcome it by using the functional analysis methods, i.e., Hahn-Banach Theorem combined the Resonance Theorem. In the last case, although the original version of the mountain pass theorem of Ambrosetti-Rabinowitz [1] is not directly applied for our purpose. Therefore, we will use a suitable version of mountain pass theorem and some new techniques to finish our goal.

    When N>4, there have been substantial lots of works (such as [3,7,11,16,17,26,34,35,36,37]) to study the existence of nontrivial solutions or the existence of sign-changing for problem (1.1). Furthermore, almost all of the works involve the nonlinear term f(x,u) of a standard subcritical polynomial growth, say:

    (SCP): There exist positive constants c1 and q(1,p1) such that

    |f(x,t)|c1(1+|t|q)for alltRandxΩ,

    where p=2NN4 expresses the critical Sobolev exponent. In this case, people can deal with problem (1.1) variationally in the Sobolev space H2(Ω)H10(Ω) owing to the some critical point theory, such as, the method of invariant sets of descent flow, mountain pass theorem and symmetric mountain pass theorem. It is worth while to note that since Ambrosetti and Rabinowitz presented the mountain pass theorem in their pioneer paper [1], critical point theory has become one of the main tools on looking for solutions to elliptic equation with variational structure. One of the important condition used in many works is the so-called Ambrosetti-Rabinowitz condition:

    (AR) There exist θ>2 and R>0 such that

    0<θF(x,t)f(x,t)t,forxΩand|t|R,

    where F(x,t)=t0f(x,s)ds. A simple computation explains that there exist c2,c3>0 such that F(x,t)c2|t|θc3 for all (x,t)ˉΩ×R and f is superlinear at infinity, i.e., limtf(x,t)t=+ uniformly in xΩ. Thus problem (1.1) is called strict superquadratic if the nonlinearity f satisfies the (AR) condition. Notice that (AR) condition plays an important role in ensuring the boundedness of Palais-Smale sequences. However, there are many nonlinearities which are superlinear at infinity but do not satisfy above (AR) condition such as f(x,t)=tln(1+|t|2)+|sint|t.

    In the recent years many authors tried to study problem (1.1) with λ=0 and the standard Laplacian problem where (AR) is not assumed. Instead, they regard the general superquadratic condition:

    (WSQC) The following limit holds

    lim|t|+F(x,t)t2=+,uniformly forxΩ

    with additional assumptions (see [3,5,7,11,12,15,17,19,24,26,31,37] and the references therein). In the most of them, there are some kind of monotonicity restrictions on the functions F(x,t) or f(x,t)t, or some convex property for the function tf(x,t)2F(x,t).

    In the case N=4 and c=0, motivated by the Adams inequality, there are a few works devoted to study the existence of nontrivial solutions for problem (1.1) when the nonlinearity f has the exponential growth, for example [15] and the references therein.

    Now, we begin to state our main results: Let μ1 be the first eigenvalue of (Δ2cΔ,H2(Ω)H10(Ω)) and suppose that f(x,t) satisfies:

    (H1) f(x,t)t0,(x,t)Ω×R;

    (H2) limt0f(x,t)t=f0 uniformly for a.e. xΩ, where f0[0,+);

    (H3) limtF(x,t)t2=+ uniformly for a.e. xΩ, where F(x,t)=t0f(x,s)ds.

    In the case of N>4, our results are stated as follows:

    Theorem 1.1. Assume that f has the standard subcritical polynomial growth on Ω (condition (SCP)) and satisfies (H1)(H3). If f0<μ1 and a(x)a0 (a0 is a positive constant ), then there exists Λ>0 such that for λ(0,Λ), problem (1.1) has five solutions, two positive solutions, two negative solutions and one nontrivial solution.

    Theorem 1.2. Assume that f has the standard subcritical polynomial growth on Ω (condition (SCP)) and satisfies (H2) and (H3). If f(x,t) is odd in t.

    a) For every λR, problem (1.1) has a sequence of solutions {uk} such that Iλ(uk),k, definition of the functional Iλ will be seen in Section 2.

    b) If f0<μ1, for every λ>0, problem (1.1) has a sequence of solutions {uk} such that Iλ(uk)<0 and Iλ(uk)0,k.

    Remark. Since our the nonlinear term f(x,u) satisfies more weak condition (H3) compared with the classical condition (AR), our Theorem 1.2 completely contains Theorem 3.20 in [32].

    In case of N=4, we have p=+. So it's necessary to introduce the definition of the subcritical exponential growth and critical exponential growth in this case. By the improved Adams inequality (see [28] and Lemma 2.2 in Section 2) for the fourth-order derivative, namely,

    supuH2(Ω)H10(Ω),Δu21Ωe32π2u2dxC|Ω|.

    So, we now define the subcritical exponential growth and critical exponential growth in this case as follows:

    (SCE): f satisfies subcritical exponential growth on Ω, i.e., limt|f(x,t)|exp(αt2)=0 uniformly on xΩ for all α>0.

    (CG): f satisfies critical exponential growth on Ω, i.e., there exists α0>0 such that

    limt|f(x,t)|exp(αt2)=0,uniformly onxΩ,α>α0,

    and

    limt|f(x,t)|exp(αt2)=+,uniformly onxΩ,α<α0.

    When N=4 and f satisfies the subcritical exponential growth (SCE), our work is still to consider problem (1.1) where the nonlinearity f satisfies the (WSQC)-condition at infinity. As far as we know, this case is rarely studied by other people for problem (1.1) except for [24]. Hence, our results are new and our methods are technique since we successfully proved the compactness condition by using the Resonance Theorem combined Adams inequality and the truncated technique. In fact, the new idea derives from our work [25]. Our results are as follows:

    Theorem 1.3. Assume that f satisfies the subcritical exponential growth on Ω (condition (SCE)) and satisfies (H1)(H3). If f0<μ1 and a(x)a0 (a0 is a positive constant ), then there exists Λ>0 such that for λ(0,Λ), problem (1.1) has five solutions, two positive solutions, two negative solutions and one nontrivial solution.

    Remark. Let F(x,t)=t2e|t|,(x,t)Ω×R. Then it satisfies that our conditions (H1)(H3) but not satisfy the condition (AR). It's worth noting that we do not impose any monotonicity condition on f(x,t)t or some convex property on tf(x,t)2F(x,t). Hence, our Theorem 1.3 completely extends some results contained in [15,24] when λ=0 in problem (1.1).

    Theorem 1.4. Assume that f satisfies the subcritical exponential growth on Ω (condition (SCE)) and satisfies (H2) and (H3). If f0<μ1 and f(x,t) is odd in t.

    a) For λ>0 small enough, problem (1.1) has a sequence of solutions {uk} such that Iλ(uk),k.

    b) For every λ>0, problem (1.1) has a sequence of solutions {uk} such that Iλ(uk)<0 and Iλ(uk)0,k.

    When N=4 and f satisfies the critical exponential growth (CG), the study of problem (1.1) becomes more complicated than in the case of subcritical exponential growth. Similar to the case of the critical polynomial growth in RN(N3) for the standard Laplacian studied by Brezis and Nirenberg in their pioneering work [4], our Euler-Lagrange functional does not satisfy the Palais-Smale condition at all level anymore. For the class standard Laplacian problem, the authors [8] used the extremal function sequences related to Moser-Trudinger inequality to complete the verification of compactness of Euler-Lagrange functional at some suitable level. Here, we still adapt the method of choosing the testing functions to study problem (1.1) without (AR) condition. Our result is as follows:

    Theorem 1.5. Assume that f has the critical exponential growth on Ω (condition (CG)) and satisfies (H1)(H3). Furthermore, assume that

    (H4) limtf(x,t)exp(α0t2)tβ>64α0r40, uniformly in (x,t), where r0 is the inner radius of Ω, i.e., r0:= radius of the largest open ball Ω. and

    (H5) f is in the class (H0), i.e., for any {un} in H2(Ω)H10(Ω), if un0 in H2(Ω)H10(Ω) and f(x,un)0 in L1(Ω), then F(x,un)0 in L1(Ω) (up to a subsequence).

    If f0<μ1, then there exists Λ>0 such that for λ(0,Λ), problem (1.1) has at least four nontrivial solutions.

    Remark. For standard biharmonic problems with Dirichlet boundary condition, Lam and Lu [15] have recently established the existence of nontrivial nonnegative solutions when the nonlinearity f has the critical exponential growth of order exp(αu2) but without satisfying the Ambrosetti- Rabinowitz condition. However, for problem (1.1) with Navier boundary condition involving critical exponential growth and the concave term, there are few works to study it. Hence our result is new and interesting.

    The paper is organized as follows. In Section 2, we present some necessary preliminary knowledge and some important lemmas. In Section 3, we give the proofs for our main results. In Section 4, we give a conclusion.

    We let λk (k=1,2,) denote the eigenvalue of Δ in H10(Ω), then 0<μ1<μ2<<μk< be the eigenvalues of (Δ2cΔ,H2(Ω)H10(Ω)) and φk(x) be the eigenfunction corresponding to μk. Let Xk denote the eigenspace associated to μk. In fact, μk=λk(λkc). Throughout this paper, we denote by p the Lp(Ω) norm, c<λ1 in Δ2cΔ and the norm of u in H2(Ω)H10(Ω) will be defined by the

    u:=(Ω(|Δu|2c|u|2)dx)12.

    We also denote E=H2(Ω)H10(Ω).

    Definition 2.1. Let (E,||||E) be a real Banach space with its dual space (E,||||E) and IC1(E,R). For cR, we say that I satisfies the (PS)c condition if for any sequence {xn}E with

    I(xn)c,I(xn)0 in E,

    there is a subsequence {xnk} such that {xnk} converges strongly in E. Also, we say that I satisfy the (C)c condition if for any sequence {xn}E with

    I(xn)c, ||I(xn)||E(1+||xn||E)0,

    there exists subsequence {xnk} such that {xnk} converges strongly in E.

    Definition 2.2. We say that uE is the solution of problem (1.1) if the identity

    Ω(ΔuΔφcuφ)dx=λΩa(x)|u|s2uφdx+Ωf(x,u)φdx

    holds for any φE.

    It is obvious that the solutions of problem (1.1) are corresponding with the critical points of the following C1 functional:

    Iλ(u)=12u2λsΩa(x)|u|sdxΩF(x,u)dx,uE.

    Let u+=max{u,0},u=min{u,0}.

    Now, we concern the following problem

    {Δ2u+cΔu=λa(x)|u+|s2u++f+(x,u)inΩ,u=Δu=0onΩ, (2.1)

    where

    f+(x,t)={f(x,t)t0,0,t<0.

    Define the corresponding functional I+λ:ER as follows:

    I+λ(u)=12u2λsΩa(x)|u+|sdxΩF+(x,u)dx,

    where F+(x,u)=u0f+(x,s)ds. Obviously, the condition (SCP) or (SCE) ((CG)) ensures that I+λC1(E,R). Let u be a critical point of I+λ, which means that u is a weak solution of problem (2.1). Furthermore, since the weak maximum principle (see [34]), it implies that u0 in Ω. Thus u is also a solution of problem (1.1) and I+λ(u)=Iλ(u).

    Similarly, we define

    f(x,t)={f(x,t)t0,0,t>0,

    and

    Iλ(u)=12u2λsΩa(x)|u|sdxΩF(x,u)dx,

    where F(x,u)=u0f(x,s)ds. Similarly, we also have IλC1(E,R) and if v is a critical point of Iλ then it is a solution of problem (1.1) and Iλ(v)=Iλ(v).

    Prosition 2.1. ([6,30]). Let E be a real Banach space and suppose that IC1(E,R) satisfies the condition

    max{I(0),I(u1)}α<βinf||u||=ρI(u),

    for some α<β, ρ>0 and u1E with ||u1||>ρ. Let cβ be characterized by

    c=infγΓmax0t1I(γ(t)),

    where Γ={γC([0,1],E),γ(0)=0,γ(1)=u1} is the set of continuous paths joining 0 and u1. Then, there exists a sequence {un}E such that

    I(un)cβ and (1+||un||)||I(un)||E0 as n.

    Lemma 2.1. ([28]). Let ΩR4 be a bounded domain. Then there exists a constant C>0 such that

    supuE,Δu21Ωe32π2u2dx<C|Ω|,

    and this inequality is sharp.

    Next, we introduce the following a revision of Adams inequality:

    Lemma 2.2. Let ΩR4 be a bounded domain. Then there exists a constant C>0 such that

    supuE,u1Ωe32π2u2dx<C|Ω|,

    and this inequality is also sharp.

    Proof. We will give a summarize proof in two different cases. In the case of c0 in the definition of ., if u1, we can deduce that Δu21 and by using Lemma 2.1 combined with the Proposition 6.1 in [28], the conclusion holds.

    In the case of 0<c<λ1 in the definition of ., from Lemma 2.1, the proof and remark of Theorem 1 in [2] and the proof of Proposition 6.1 in [28], we still can establish this revised Adams inequality.

    Lemma 2.3. Assume (H1) and (H3) hold. If f has the standard subcritical polynomial growth on Ω (condition (SCP)), then I+λ (Iλ) satisfies (C)c.

    Proof. We only prove the case of I+λ. The arguments for the case of Iλ are similar. Let {un}E be a (C)c sequence such that

    I+λ(un)=12||un||2λsΩa(x)|u+n|sdxΩF+(x,un)dx=c+(1), (2.2)
    (1+||un||)||I+λ(un)||E0 as n. (2.3)

    Obviously, (2.3) implies that

    I+λ(un),φ=un,φλΩa(x)|u+n|s2u+nφdxΩf+(x,un(x))φdx=(1). (2.4)

    Step 1. We claim that {un} is bounded in E. In fact, assume that

    un,  as n.

    Define

    vn=unun.

    Then, vn=1, nN and then, it is possible to extract a subsequence (denoted also by {vn}) converges weakly to v in E, converges strongly in Lp(Ω)(1p<p) and converges v a.e. xΩ.

    Dividing both sides of (2.2) by un2, we get

    ΩF+(x,un)un2dx12. (2.5)

    Set

    Ω+={xΩ:v(x)>0}.

    By (H3), we imply that

    F+(x,un)u2nv2n,xΩ+. (2.6)

    If |Ω+| is positive, since Fatou's lemma, we get

    limnΩF+(x,un)un2dxlimnΩ+F+(x,un)u2nv2ndx=+,

    which contradicts with (2.5). Thus, we have v0. In fact, we have v=0. Indeed, again using (2.3), we get

    (1+un)|I+λ(un),v|(1)v.

    Thus, we have

    Ω(ΔunΔvcunv)dxΩ(ΔunΔvcuv)dxλΩa(x)|u+n|s2u+nvdxΩf+(x,un)vdx(1)v1+un,

    by noticing that since v0, f+(x,un)v0 a.e. xΩ, thus Ωf+(x,un)vdx0. So we get

    Ω(ΔvnΔvcvnv)dx0.

    On the other hand, from vnv in E, we have

    Ω(ΔvnΔvcvnv)dxv2

    which implies v=0.

    Dividing both sides of (2.4) by un, for any φE, then there exists a positive constant M(φ) such that

    |Ωf+(x,un)unφdx|M(φ),nN. (2.7)

    Set

    fn(φ)=Ωf+(x,un)unφdx,φE.

    Thus, by (SCP), we know that {fn} is a family bounded linear functionals defined on E. Combing (2.7) with the famous Resonance Theorem, we get that {|fn|} is bounded, where |fn| denotes the norm of fn. It means that

    |fn|C. (2.8)

    Since ELppq(Ω), using the Hahn-Banach Theorem, there exists a continuous functional ˆfn defined on Lppq(Ω) such that ˆfn is an extension of fn, and

    ˆfn(φ)=fn(φ),φE, (2.9)
    ˆfnpq=|fn|, (2.10)

    where ˆfnpq denotes the norm of ˆfn(φ) in Lpq(Ω) which is defined on Lppq(Ω).

    On the other hand, from the definition of the linear functional on Lppq(Ω), we know that there exists a function Sn(x)Lpq(Ω) such that

    ˆfn(φ)=ΩSn(x)φ(x)dx,φLppq(Ω). (2.11)

    So, from (2.9) and (2.11), we obtain

    ΩSn(x)φ(x)dx=Ωf+(x,un)unφdx,φE,

    which implies that

    Ω(Sn(x)f+(x,un)un)φdx=0,φE.

    According to the basic lemma of variational, we can deduce that

    Sn(x)=f+(x,un)una.e.xΩ.

    Thus, by (2.8) and (2.10), we have

    ˆfnpq=Snpq=|fn|<C. (2.12)

    Now, taking φ=vnv in (2.4), we get

    A(vn),vnvλΩa(x)|u+n|s2u+nvndxΩf+(x,un)unvndx0, (2.13)

    where A:EE defined by

    A(u),φ=ΩΔuΔφdxcΩuφdx, u,φE.

    By the H¨older inequality and (2.12), we obtain

    Ωf+(x,un)unvndx0.

    Then from (2.13), we can conclude that

    vnvinE.

    This leads to a contradiction since vn=1 and v=0. Thus, {un} is bounded in E.

    Step 2. We show that {un} has a convergence subsequence. Without loss of generality, we can suppose that

    unu  in  E,unu in Lγ(Ω), 1γ<p,un(x)u(x) a.e. xΩ.

    Now, it follows from f satisfies the condition (SCP) that there exist two positive constants c4,c5>0 such that

    f+(x,t)c4+c5|t|q, (x,t)Ω×R,

    then

    |Ωf+(x,un)(unu)dx|c4Ω|unu|dx+c5Ω|unu||un|qdxc4Ω|unu|dx+c5(Ω(|un|q)pqdx)qp(Ω|unu|ppqdx)pqp.

    Similarly, since unu in E, Ω|unu|dx0 and Ω|unu|ppqdx0.

    Thus, from (2.4) and the formula above, we obtain

    A(un),unu0,asn.

    So, we get unu. Thus we have unu in E which implies that I+λ satisfies (C)c.

    Lemma 2.4. Let φ1>0 be a μ1-eigenfunction with φ1=1 and assume that (H1)(H3) and (SCP) hold. If f0<μ1, then:

    (i)  For λ>0 small enough, there exist ρ,α>0 such that I±λ(u)α for all uE with u=ρ,

    (ii) I±λ(tφ1) as t+.

    Proof. Since (SCP) and (H1)(H3), for any ε>0, there exist A=A(ε), M large enough and B=B(ε) such that for all (x,s)Ω×R,

    F±(x,s)12(f0+ϵ)s2+A|s|q, (2.14)
    F±(x,s)M2s2B. (2.15)

    Choose ε>0 such that (f0+ε)<μ1. By (2.14), the Poincaré inequality and the Sobolev embedding, we obtain

    I±λ(u)12u2λasΩ|u|sdxΩF±(x,u)dx12u2λasΩ|u|sdxf0+ε2u22AΩ|u|qdx12(1f0+εμ1)u2λKusCuqu2(12(1f0+εμ1)λKus2Cuq2),

    where K,C are constant.

    Write

    h(t)=λKts2+Ctq2.

    We can prove that there exists t such that

    h(t)<12(1f0+εμ1).

    In fact, letting h(t)=0, we get

    t=(λK(2s)C(q2))1qs.

    According to the knowledge of mathematical analysis, h(t) has a minimum at t=t. Denote

    ϑ=K(2s)C(q2), ˆϑ=s2qs, ˉϑ=q2qs, ν=12(1f0+εμ1).

    Taking t in h(t), we get

    h(t)<ν,0<λ<Λ,

    where Λ=(νKϑˆϑ+Cϑˉϑ)1ˉϑ. So, part (i) holds if we take ρ=t.

    On the other hand, from (2.15), we get

    I+λ(tφ1)12(1Mμ1)t2tsλsΩa(x)|φ1|sdx+B|Ω| as t+.

    Similarly, we have

    Iλ(t(φ1)), as t+.

    Thus part (ii) holds.

    Lemma 2.5. Let φ1>0 be a μ1-eigenfunction with φ1=1 and assume that (H1)(H3) and (SCE)(or (CG)) hold. If f0<μ1, then:

    (i)  For λ>0 small enough, there exist ρ,α>0 such that I±λ(u)α for all uE with u=ρ,

    (ii) I±λ(tφ1) as t+.

    Proof. From (SCE) (or (CG)) and (H1)-(H3), for any ε>0, there exist A1=A1(ε), M1 large enough, B1=B1(ε), κ1>0 and q1>2 such that for all (x,s)Ω×R,

    F±(x,s)12(f0+ϵ)s2+A1exp(κ1s2)|s|q1, (2.16)
    F±(x,s)M12s2B1. (2.17)

    Choose ε>0 such that (f0+ε)<μ1. By (2.16), the Hölder inequality and the Adams inequality (see Lemma 2.2), we obtain

    I±λ(u)12u2λasΩ|u|sdxΩF±(x,u)dx12u2λasΩ|u|sdxf0+ε2u22A1Ωexp(κ1u2)|u|q1dx12(1f0+εμ1)u2λKusA1(Ωexp(κ1r1u2(|u|u)2)dx)1r1(Ω|u|r1qdx)1r112(1f0+εμ1)u2λKusˆCuq1,

    where r1>1 sufficiently close to 1, uσ and κ1r1σ2<32π2. Remained proof is completely similar to the proof of part (ⅰ) of Lemma 2.4, we omit it here. So, part (ⅰ) holds if we take u=ρ>0 small enough.

    On the other hand, from (2.17), we get

    I+λ(tφ1)12(1M1μ1)t2tsλsΩa(x)|φ1|sdx+B1|Ω| as t+.

    Similarly, we have

    Iλ(t(φ1)), as t+.

    Thus part (ii) holds.

    Lemma 2.6. Assume (H1) and (H3) hold. If f has the subcritical exponential growth on Ω (condition (SCE)), then I+λ (Iλ) satisfies (C)c.

    Proof. We only prove the case of I+λ. The arguments for the case of Iλ are similar. Let {un}E be a (C)c sequence such that the formulas (2.2)–(2.4) in Lemma 2.3 hold.

    Now, according to the previous section of Step 1 of the proof of Lemma 2.3, we also obtain that the formula (2.7) holds. Set

    fn(φ)=Ωf+(x,un)unφdx,φE.

    Then from for any uE, eαu2L1(Ω) for all α>0, we can draw a conclusion that {fn} is a family bounded linear functionals defined on E. Using (2.7) and the famous Resonance Theorem, we know that {|fn|} is bounded, where |fn| denotes the norm of fn. It means that the formula (2.8) (see the proof of Lemma 2.3) holds.

    Since ELq0(Ω) for some q0>1, using the Hahn-Banach Theorem, there exists a continuous functional ˆfn defined on Lq0(Ω) such that ˆfn is an extension of fn, and

    ˆfn(φ)=fn(φ),φE, (2.18)
    ˆfnq0=|fn|, (2.19)

    where ˆfnq0 is the norm of ˆfn(φ) in Lq0(Ω) which is defined on Lq0(Ω) and q0 is the dual number of q0.

    By the definition of the linear functional on Lq0(Ω), we know that there is a function Sn(x)Lq0(Ω) such that

    ˆfn(φ)=ΩSn(x)φ(x)dx,φLq0(Ω). (2.20)

    Similarly to the last section of the Step 1 of the proof of Lemma 2.3, we can prove that (C)c sequence {un} is bounded in E. Next, we show that {un} has a convergence subsequence. Without loss of generality, assume that

    unβ,unu  in E,unu in Lγ(Ω), γ1,un(x)u(x) a.e. xΩ.

    Since f has the subcritical exponential growth (SCE) on Ω, we can find a constant Cβ>0 such that

    |f+(x,t)|Cβexp(32π2k(β)2t2), (x,t)Ω×R.

    Thus, from the revised Adams inequality (see Lemma 2.2),

    |Ωf+(x,un)(unu)dx|Cβ(Ωexp(32π2(β)2u2n)dx)1k|unu|kC|unu|k0,

    where k>1 and k is the dual number of k. Similar to the last proof of Lemma 2.3, we have unu in E which means that I+λ satisfies (C)c.

    Lemma 2.7. Assume (H3) holds. If f has the standard subcritical polynomial growth on Ω (condition (SCP)), then Iλ satisfies (PS)c.

    Proof. Let {un}E be a (PS)c sequence such that

    un22λsΩa(x)|un|sdxΩF(x,un)dxc, (2.21)
    ΩΔunΔφdxcΩunφdxλΩa(x)|un|s2unφdxΩf(x,un)φdx=(1)φ, φE. (2.22)

    Step 1. To prove that {un} has a convergence subsequence, we first need to prove that it is a bounded sequence. To do this, argue by contradiction assuming that for a subsequence, which is still denoted by {un}, we have

    un.

    Without loss of generality, assume that un1 for all nN and let

    vn=unun.

    Clearly, vn=1, nN and then, it is possible to extract a subsequence (denoted also by {vn}) converges weakly to v in E, converges strongly in Lp(Ω)(1p<p) and converges v a.e. xΩ.

    Dividing both sides of (2.21) by un2, we obtain

    ΩF(x,un)un2dx12. (2.23)

    Set

    Ω0={xΩ:v(x)0}.

    By (H3), we get that

    F(x,un)u2nv2n,xΩ0. (2.24)

    If |Ω0| is positive, from Fatou's lemma, we obtain

    limnΩF(x,un)un2dxlimnΩ0F(x,un)u2nv2ndx=+,

    which contradicts with (2.23).

    Dividing both sides of (2.22) by un, for any φE, then there exists a positive constant M(φ) such that

    |Ωf(x,un)unφdx|M(φ),nN. (2.25)

    Set

    fn(φ)=Ωf(x,un)unφdx,φE.

    Thus, by (SCP), we know that {fn} is a family bounded linear functionals defined on E. By (2.25) and the famous Resonance Theorem, we get that {|fn|} is bounded, where |fn| denotes the norm of fn. It means that

    |fn|˜C. (2.26)

    Since ELppq(Ω), using the Hahn-Banach Theorem, there exists a continuous functional ˆfn defined on Lppq(Ω) such that ˆfn is an extension of fn, and

    ˆfn(φ)=fn(φ),φE, (2.27)
    ˆfnpq=|fn|, (2.28)

    where ˆfnpq denotes the norm of ˆfn(φ) in Lpq(Ω) which is defined on Lppq(Ω).

    Remained proof is completely similar to the last proof of Lemma 2.3, we omit it here.

    Lemma 2.8. Assume (H3) holds. If f has the subcritical exponential growth on Ω (condition (SCE)), then Iλ satisfies (PS)c.

    Proof. Combining the previous section of the proof of Lemma 2.7 with slightly modifying the last section of the proof of Lemma 2.6, we can prove it. So we omit it here.

    To prove the next Lemma, we firstly introduce a sequence of nonnegative functions as follows. Let Φ(t)C[0,1] such that

    Φ(0)=Φ(0)=0,
    Φ(1)=Φ(1)=0.

    We let

    H(t)={1nΦ(nt),ift1n,t,if1n<t<11n,11nΦ(n(1t)),if11nt1,1,if1t,

    and ψn(r)=H((lnn)1ln1r). Notice that ψn(x)E, B the unit ball in RN, ψn(x)=1 for |x|1n and, as it was proved in [2],

    Δψn2=22π(lnn)12An=ψn+(1),asn.

    where 0limnAn1. Thus, we take x0Ω and r0>0 such that B(x0,r)Ω, denote

    Ψn(x)={ψn(|xx0|)ψn,ifxB(x0,r0),0,ifxΩB(x0,r0).

    Lemma 2.9. Assume (H1) and (H4) hold. If f has the critical exponential growth on Ω (condition (CG)), then there exists n such that

    max{I±λ(±tΨn):t0}<16π2α0.

    Proof. We only prove the case of I+λ. The arguments for the case of Iλ are similar. Assume by contradiction that this is not the case. So, for all n, this maximum is larger or equal to 16π2α0. Let tn>0 be such that

    I+λ(tnΨn)16π2α0. (2.29)

    From (H1) and (2.29), we conclude that

    t2n32π2α0. (2.30)

    Also at t=tn, we have

    tnts1nλΩa(x)|Ψn|sdxΩf(x,tnΨn)Ψndx=0,

    which implies that

    t2ntsnλΩa(x)|Ψn|sdx+B(x0,r0)f(x,tnΨn)tnΨndx. (2.31)

    Since (H4), for given ϵ>0 there exists Rϵ>0 such that

    tf(x,t)(βϵ)exp(α0t2), tRϵ.

    So by (2.31), we deduce that, for large n

    t2ntsnλΩa(x)|Ψn|sdx+(βϵ)π22r40exp[((tnAn)2α032π21)4lnn]. (2.32)

    By (2.30), the inequality above is true if, and only if

    limnAn=1  and  tn(32π2α0)12. (2.33)

    Set

    An={xB(x0,r0):tnΨn(x)Rϵ},Bn=B(x0,r0)An,

    and break the integral in (2.31) into a sum of integrals over An and Bn. By simple computation, we have

    [32π2α0](βϵ)limnB(x0,r0)exp[α0t2n|Ψn(x)|2]dx(βϵ)r40π22. (2.34)

    The last integral in (2.34), denote In is evaluated as follows:

    In(βϵ)r40π2.

    Thus, finally from (2.34) we get

    [32π2α0](βϵ)r40π22,

    which means β64α0r40. This results in a contradiction with (H4).

    To conclude this section we state the Fountain Theorem of Bartsch [32].

    Define

    Yk=kj=1Xj,  Zk=¯jkXj. (2.35)

    Lemma 2.10. (Dual Fountain Theorem). Assume that IλC1(E,R) satisfies the (PS)c condition (see [32]), Iλ(u)=Iλ(u). If for almost every kN, there exist ρk>rk>0 such that

    (i) ak:=infuZk,u=ρkIλ(u)0,

    (ii) bk:=maxuYk,u=rkIλ(u)<0,

    (iii) bk=infuZk,u=ρkIλ(u)0, as k,

    then Iλ has a sequence of negative critical values converging 0.

    Proof of Theorem 1.1. For I±λ, we first demonstrate that the existence of local minimum v± with I±λ(v±)<0. We only prove the case of I+λ. The arguments for the case of Iλ are similar.

    For ρ determined in Lemma 2.4, we write

    ˉB(ρ)={uE, uρ},  B(ρ)={uE, u=ρ}.

    Then ˉB(ρ) is a complete metric space with the distance

    dist(u,v)=uv,u,vˉB(ρ).

    From Lemma 2.4, we have for 0<λ<Λ,

    I+λ(u)|B(ρ)α>0.

    Furthermore, we know that I+λC1(ˉB(ρ),R), hence I+λ is lower semi-continuous and bounded from below on ˉB(ρ). Set

    c1=inf{I+λ(u),uˉB(ρ)}.

    Taking ˜ϕC0(Ω) with ˜ϕ>0, and for t>0, we get

    I+λ(t˜ϕ)=t22˜ϕ2λtssΩa(x)|˜ϕ|sdxΩF+(x,t˜ϕ)dxt22˜ϕ2λtssΩa(x)|˜ϕ|sdx<0,

    for all t>0 small enough. Hence, c1<0.

    Since Ekeland's variational principle and Lemma 2.4, for any m>1, there exists um with um<ρ such that

    I+λ(um)c1,I+λ(um)0.

    Hence, there exists a subsequence still denoted by {um} such that

    umv+,I+λ(v+)=0.

    Thus v+ is a weak solution of problem (1.1) and I+λ(v+)<0. In addition, from the maximum principle, we know v+>0. By a similar way, we obtain a negative solution v with Iλ(v)<0.

    On the other hand, from Lemmas 2.3 and 2.4, the functional I+λ has a mountain pass-type critical point u+ with I+λ(u+)>0. Again using the maximum principle, we have u+>0. Hence, u+ is a positive weak solution of problem (1.1). Similarly, we also obtain a negative mountain pass-type critical point u for the functional Iλ. Thus, we have proved that problem (1.1) has four different nontrivial solutions. Next, our method to obtain the fifth solution follows the idea developed in [33] for problem (1.1). We can assume that v+ and v are isolated local minima of Iλ. Let us denote by bλ the mountain pass critical level of Iλ with base points v+,v:

    bλ=infγΓmax0t1Iλ(γ(t)),

    where Γ={γC([0,1],E),γ(0)=v+,γ(1)=v}. We will show that bλ<0 if λ is small enough. To this end, we regard

    Iλ(tv±)=t22v±2λtssΩa(x)|v±|sdxΩF(x,tv±)dx.

    We claim that there exists δ>0 such that

    Iλ(tv±)<0, t(0,1), λ(0,δ). (3.1)

    If not, we have t0(0,1) such that Iλ(t0v±)0 for λ small enough. Similarly, we also have Iλ(tv±)<0 for t>0 small enough. Let ρ0=t0v± and ˇc±=inf{I±λ(u),uˉB(ρ0)}. Since previous arguments, we obtain a solution v± such that Iλ(v±)<0, a contradiction. Hence, (3.1) holds.

    Now, let us consider the 2-dimensional plane Π2 containing the straightlines tv and tv+, and take vΠ2 with v=ϵ. Note that for such v one has vs=csϵ. Then we get

    Iλ(v)ϵ22λscssh0ϵs.

    Thus, for small ϵ,

    Iλ(v)<0. (3.2)

    Consider the path ˉγ obtained gluing together the segments {tv:ϵv1t1},{tv+:ϵv+1t1} and the arc {vΠ2:v=ϵ}. by (3.1)and (3.2), we get

    bλmaxvˉγIλ(v)<0,

    which verifies the claim. Since the (PS) condition holds because of Lemma 2.3, the level{Iλ(v)=bλ} carries a critical point v3 of Iλ, and v3 is different from v±.

    Proof of Theorem 1.2. We first use the symmetric mountain pass theorem to prove the case of a). It follows from our assumptions that the functional Iλ is even. Since the condition (SCP), we know that (I1) of Theorem 9.12 in [30] holds. Furthermore, by condition (H3), we easily verify that (I2) of Theorem 9.12 also holds. Hence, by Lemma 2.7, our theorem is proved.

    Next we use the dual fountain theorem (Lemma 2.10) to prove the case of b). Since Lemma 2.7, we know that the functional Iλ satisfies (PS)c condition. Next, we just need to prove the conditions (ⅰ)-(ⅲ) of Lemma 2.10.

    First, we verify (ⅰ) of Lemma 2.10. Define

    βk:=supuZk,u=1us.

    From the conditions (SCP) and (H2), we get, for uZk,uR,

    Iλ(u)u22λβskussf0+ϵ2u22c6uq14(1f0+ϵμ1)u2λβskuss. (3.3)

    Here, R is a positive constant and ϵ>0 small enough. We take ρk=(4μ1λβsk/[(μ1f0ϵ)s])12s. Since βk0,k, it follows that ρk0,k. There exists k0 such that ρkR when kk0. Thus, for kk0,uZk and u=ρk, we have Iλ(u)0 and (ⅰ) holds. The verification of (ⅱ) and (ⅲ) is standard, we omit it here.

    Proof of Theorem 1.3. According to our assumptions, similar to previous section of the proof of Theorem 1.1, we obtain that the existence of local minimum v± with I±λ(v±)<0. In addition, by Lemmas 2.5 and 2.6, for I±λ, we obtain two mountain pass type critical points u+ and u with positive energy. Similar to the last section of the proof of Theorem 1.1, we can also get another solution u3, which is different from v± and u±. Thus, this proof is completed.

    Proof of Theorem 1.4. We first use the symmetric mountain pass theorem to prove the case of a). It follows from our assumptions that the functional Iλ is even. Since the condition (SCE), we know that (I1) of Theorem 9.12 in [30] holds. In fact, similar to the proof of (ⅰ) of Lemma 2.5, we can conclude it. Furthermore, by condition (H3), we easily verify that (I2) of Theorem 9.12 also holds. Hence, by Lemma 2.8, our theorem is proved.

    Next we use the dual fountain theorem (Lemma 2.10) to prove the case of b). Since Lemma 2.8, we know that the functional Iλ satisfies (PS)c condition. Next, we just need to prove the conditions (ⅰ)-(ⅲ) of Lemma 2.10.

    First, we verify (ⅰ) of Lemma 2.10. Define

    βk:=supuZk,u=1us.

    From the conditions (SCE), (H2) and Lemma 2.2, we get, for uZk,uR,

    Iλ(u)u22λβskussf0+ϵ2u22c7uq14(1f0+ϵμ1)u2λβskuss. (3.4)

    Here, R is a positive constant small enough and ϵ>0 small enough. We take ρk=(4μ1λβsk/[(μ1f0ϵ)s])12s. Since βk0,k, it follows that ρk0,k. There exists k0 such that ρkR when kk0. Thus, for kk0,uZk and u=ρk, we have Iλ(u)0 and (ⅰ) holds. The verification of (ⅱ) and (ⅲ) is standard, we omit it here.

    Proof of Theorem 1.5. According to our assumptions, similar to previous section of the proof of Theorem 1.1, we obtain that the existence of local minimum v± with I±λ(v±)<0. Now, we show that I+λ has a positive mountain pass type critical point. Since Lemmas 2.5 and 2.9, then there exists a (C)cM sequence {un} at the level 0<cM16π2α0. Similar to previous section of the proof of Lemma 2.6, we can prove that (C)cM sequence {un} is bounded in E. Without loss of generality, we can suppose that

    unu+inE.

    Following the proof of Lemma 4 in [9], we can imply that u+ is weak of problem (1.1). So the theorem is proved if u+ is not trivial. However, we can get this due to our technical assumption (H5). Indeed, assume u+=0, similarly as in [9], we obtain f+(x,un)0 in L1(Ω). Since (H5), F+(x,un)0 in L1(Ω) and we get

    limnun2=2cM<32π2α0,

    and again following the proof in [9], we get a contradiction.

    We claim that v+ and u+ are distinct. Since the previous proof, we know that there exist sequence {un} and {vn} in E such that

    unv+, I+λ(un)c+<0, I+λ(un),un0, (3.5)

    and

    vnu+, I+λ(vn)cM>0, I+λ(vn),vn0. (3.6)

    Now, argue by contradiction that v+=u+. Since we also have vnv+ in E, up to subsequence, limnvnv+>0. Setting

    wn=vnvn,  w0=v+limnvn,

    we know that wn=1 and wnw0 in E.

    Now, we consider two possibilities:

    (i) w0=1,(ii) w0<1.

    If (ⅰ) happens, we have vnv+ in E, so that I+λ(vn)I+λ(v+)=c+. This is a contradiction with (3.5) and (3.6).

    Now, suppose that (ⅱ) happens. We claim that there exists δ>0 such that

    hα0vn232π21w02δ (3.7)

    for n large enough. In fact, by the proof of v+ and Lemma 2.9, we get

    0<cM<c++16π2α0. (3.8)

    Thus, we can choose h>1 sufficiently close to 1 and δ>0 such that

    hα0vn216π2cMI+λ(v+)vn2δ.

    Since vnv+, by condition (H5), up to a subsequence, we conclude that

    12vn2=cM+λsΩa(x)vs+dx+ΩF+(x,v+)dx+(1). (3.9)

    Thus, for n sufficiently large we get

    hα0vn232π2cM+λsΩa(x)vs+dx+ΩF+(x,v+)dx+(1)cMI+λ(v+)δ. (3.10)

    Thus, from (3.9) and the definition of w0, (3.10) implies (3.7) for n large enough.

    Now, taking ˜h=(h+ϵ)α0vn2, it follows from (3.7) and a revised Adams inequality (see [28]), we have

    Ωexp((h+ϵ)α0vn2|wn|2dxC (3.11)

    for ϵ>0 small enough. Thus, from our assumptions and the Hölder inequality we get vnv+ and this is absurd.

    Similarly, we can find a negative mountain pass type critical point u which is different that v. Thus, the proof is completed.

    In this research, we mainly studied the existence and multiplicity of nontrivial solutions for the fourth-order elliptic Navier boundary problems with exponential growth. Our method is based on the variational methods, Resonance Theorem together with a revised Adams inequality.

    The authors would like to thank the referees for valuable comments and suggestions in improving this article. This research is supported by the NSFC (Nos. 11661070, 11764035 and 12161077), the NSF of Gansu Province (No. 22JR11RE193) and the Nonlinear mathematical physics Equation Innovation Team (No. TDJ2022-03).

    There is no conflict of interest.



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