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Research article

Simplifying coefficients in differential equations associated with higher order Bernoulli numbers of the second kind

  • Received: 21 December 2018 Accepted: 14 February 2019 Published: 19 February 2019
  • MSC : Primary 34A05; Secondary 11A25, 11B68, 11B73, 11B83

  • In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second kind.

    Citation: Feng Qi, Da-Wei Niu, Bai-Ni Guo. Simplifying coefficients in differential equations associated with higher order Bernoulli numbers of the second kind[J]. AIMS Mathematics, 2019, 4(2): 170-175. doi: 10.3934/math.2019.2.170

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  • In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second kind.


    Let Ω be a bounded domain of RN(N3) with smooth boundary Ω. In this paper, we consider the existence of W1,10(Ω)solutions to the following elliptic problem

    {div(M(x,u)u)+|u|2uθ=f,xΩ,u=0,xΩ, (1.1)

    where NN1θ<2, M:Ω×RRN2 is a symmetric Carathéodory matrix function, which satisfies the following assumptions: for some real constants γ>0, α>0,β>0,

    |M(x,s)|β,M(x,s)ξξα(a(x)+|s|)γ|ξ|2, (1.2)

    for almost every xΩ, (s,ξ)R×RN, where a(x) is a measurable function, such that

    0<ζa(x)ρ, (1.3)

    for some positive constants ζ,ρ.

    We note that there are two difficulties in dealing with (1.1), the first one is the fact that, due to hypothesis (1.2), the differential operator A(u)=div(M(x,u)u) is well defined in H10(Ω), but it not coercive on H10(Ω) when u is large enough. Therefore, the classical Leray-Lions theorem cannot be applied even if f is sufficiently regular. The second difficulty is dealing with lower order term which singular natural growth with respect to the gradient. In order to overcome these difficulties, we approximate problem (1.1) by means of truncations in M(x,s) to get a coercive differential operator on H10(Ω).

    The existence of W1,10(Ω) solution to elliptic problem has been studied by many authors. Boccardo and Croce [4] proved the existence of W1,10(Ω) solutions to problem

    {div(a(x)u(1+|u|)γ)=f,xΩ,u=0,xΩ,

    where a:ΩR is a measurable function which satisfies (1.3), fLm(Ω) with

    m=NN+1γ(N1),1N1<γ<1.

    In the literature [6], the authors considered the existence and regularity of solutions to the following elliptic equation with noncoercivity

    {div(a(x,u)u)=f,xΩ,u=0,xΩ, (1.4)

    where Ω is an open bounded subset of RN(N3), fLm(Ω) and a(x,s):Ω×RR is a Carathéodory function which satisfies

    α(1+|s|)γa(x,s)β,

    where 0γ<1.The existence results of solutions to problem (1.4)are as following:

    ● There exists a weak solution uH10(Ω)L(Ω) to (1.4) if m>N2.

    ● There exists a weak solution uH10(Ω)Lr(Ω) to (1.4) with r=Nm(1γ)N2m if

    2NN+2γ(N2)m<N2.

    ● There exists a distributional solution uW1,q0(Ω) to (1.4) with q=Nm(1γ)Nm(1+γ)<2 if

    NN+1γ(N1)<m<2NN+2γ(N2).

    In [15], the under the assumption (1.2)-(1.3), Souilah proved the existence results of solutions to problem

    {div(M(x,u)u)+|u|2uθ=f+λur,xΩ,u=0,xΩ, (1.5)

    where 0<θ<1,0<r<2θ,λ>0,fLm(Ω)(m1). There exists at least a solution to problem (1.5):

    ● If 2N2Nθ(N2)m<N2, then uH10(Ω)L(Ω).

    ● If 1<m<2N2Nθ(N2), then uW1,q0(Ω) with q=Nm(2θ)Nmθ.

    ● If mN2, then uH10(Ω)L(Ω).

    Moreover, the existence of solutions uH10(Ω) to problem (1.5) with λ=0 have been obtained in [9]. Some other related results see [1,3,5,7,10,11,12,14,16].

    Based on the above research results, the aim of this article is to study the existence of W1,10(Ω) solution to problem (1.1).

    In order to state the main results of this paper, the following definition need to be introduced. We use the following notion of distributional solution to problem (1.1).

    Definition 1.1. We say that uW1,10(Ω) is a distributional solution to problem (1.1) if u>0 in Ω, |u|2uθL1(Ω) and

    ΩM(x,u)uφ+Ω|u|2uθφ=Ωfφ,

    for every φC0(Ω).

    Our main results are following:

    Theorem 1.2. Assume that (1.2)-(1.3) hold, fLm(Ω) is a nonnegative function with

    m=N2Nθ(N1),NN1<θ<2. (1.6)

    Then there exists a distributional solution uW1,10(Ω) to problem (1.1).

    Remark 1.3. Notice that the result of previous theorem do not depend on γ.

    Remark 1.4. Observe that, m>1 if and only if θ>NN1.

    For fL1(Ω), we have the following theorem.

    Theorem 1.5. Assume (1.2)-(1.3) hold, fL1(Ω) is a nonnegative function and θ=NN1. Then there exists a distributional solution uW1,10(Ω) to problem (1.1).

    The paper is organized as follows. In section 2, we collect some definitions and useful tools. The proof of Theorem 1.2 and 1.5 be given in section 3.

    In order to prove our main results, we need to introduce a basic definition and some lemmas.

    Definition 2.1. For all k0, the truncation function defined by

    Tk(s)=max{k,min{k,s}},Gk(s)=sTk(s).

    Let 0<ε<1, we approximate problem (1.1) by the following non-singular problem

    {div(M(x,T1ε(uε))uε)+uε|uε|2(|uε|+ε)θ+1=fε,xΩ,uε=0,xΩ, (2.1)

    where fε=T1ε(f). Problem (2.1) admits at least a solution uεH10(Ω)L(Ω) by Theorem 2 of [8]. Due to the fact that fε0 and quadratic lower order term has the same sign of the solution, it is easy to prove that uε0 by taking uε as a test function in (2.1).

    Lemma 2.2. Let uε be the solutions to problem (2.1). Then

    Ωuε|uε|2(uε+ε)θ+1Ωf. (2.2)

    Proof. For fixed h>0, taking Th(uε)h as a test function in (2.1). Dropping the first term, we obtain

    Ωuε|uε|2(uε+ε)θ+1Th(uε)hΩfεTh(uε)h.

    Using the fact that fεf and Th(uε)h1, then

    Ωuε|uε|2(uε+ε)θ+1Th(uε)hΩf.

    Letting h0, we deduce (2.2) by the Fatou Lemma.

    Lemma 2.3. Let δ>0 and 0<ε<1. Then there exists C>0, such that

    αδ(t+ε)θ2(ρ+t)γ+tt+εC.

    for every t0.

    Proof. Clearly, if tε, we have tt+ε12, while if t<ε, we have

    αδ(t+ε)θ2(ρ+t)γαδ(ρ+t)γ(2ε)2θαδ22θ(ρ+1)γ,

    since ε<1. Therefore, Lemma 2.3 is proved.

    In this section, C denotes a generic constant whose value might change from line to line. We prove the existence results of Theorems 1.2 and 1.5 by considering the following approximate problem

    {div(M(x,T1ε(uε))uε)+uε|uε|2(uε+ε)θ+1=fε,xΩ,uε=0,xΩ. (3.1)

    Proof of Theorem 1.2. Step 1: Let δ=θNN1, then δ>0 by (1.6). Choosing (uε+ε)δ(uε+ε)δ1 as a test function in the approximate problem (3.1), we find

    ΩM(x,T1ε(uε))uεuε[δ(uε+ε)δ1+(1δ)(uε+ε)δ2]+Ωuε(uε+ε)δ|uε|2(uε+ε)θ+1=Ωuε|uε|2(uε+ε)θ+1(uε+ε)δ1+Ωfε[(uε+ε)δ(uε+ε)δ1].

    Combining (1.2)-(1.3) and dropping the positive term, we obtain

    Ω|uε|2(uε+ε)δθ[α(1δ)(uε+ε)θ2(ρ+uε)γ+uεuε+ε]Ωuε|uε|2(uε+ε)θ+1(uε+ε)δ1+Ωfε(uε+ε)δ.

    Since 1δ>0, according to Lemma 2.3, we have

    CΩ|uε|2(uε+ε)δθΩuε|uε|2(uε+ε)θ+1(uε+ε)δ1+Ωfε(uε+ε)δ.

    Using the fact that uε0,fεf and (2.2), we obtain

    CΩ|uε|2(uε+ε)δθεδ1Ωuε|uε|2(uε+ε)θ+1+Ωf(uε+ε)δεδ1Ωf+Ωf(uε+ε)δ. (3.2)

    Observe that the left hand side of (3.2) can be rewritten as

    CΩ|[(uε+ε)δθ+22εδθ+22]|2. (3.3)

    Then, (3.2) and (3.3) imply

    CΩ|[(uε+ε)δθ+22εδθ+22]|2εδ1Ωf+Ωf(uε+ε)δ. (3.4)

    By the Sobolev inequality, satisfy

    [Ω|(uε+ε)δθ+22εδθ+22|2]22CΩ|[(uε+ε)δθ+22εδθ+22]|2. (3.5)

    Using the Hölder inequality and (3.4)-(3.5), we get

    [Ω|(uε+ε)δθ+22εδθ+22|2]22CfLm(Ω)+CfLm(Ω)[Ω(uε+ε)δm]1m.

    Since |(t+ε)sεs|2C[(t+ε)2s1] for every t0 and for suitable constant C independent on ε, then we find

    (Ω[(uε+ε)2(δθ+2)21])22CfLm(Ω)+CfLm(Ω)[Ω(uε+ε)δm]1m. (3.6)

    Thanks to the choice of δ, we have

    2(δθ+2)2=δm=NN1.

    Moreover 22>1m since m<N2. Then (3.6) implies that

    ΩuNN1εC. (3.7)

    Observe that δθ=NN1, then, (3.2), (3.7) follow

    Ω|uε|2(ε+uε)NN1C. (3.8)

    Combining (3.7)-(3.8) with the Hölder inequality, we obtain

    Ω|uε|=Ωuε(ε+uε)N2N2(ε+uε)N2N2[Ω|uε|2(ε+uε)NN1]12[Ω(ε+uε)NN1]12C.

    Then we get that {uε} is bounded in W1,10(Ω). Hence, there exists a subsequence {uε}, which converges to a measurable function u a.e. in Lr(Ω) with 1r<NN1.

    Step 2: First, we are going to estimate {uεk}|uε|. Choosing [(uε+ε)δ(k+ε)δ]+ as a test function in (3.1). By (1.2)-(1.3) and Lemma 2.3, we have

    {uεk}|uε|2(ε+uε)NN1({uεk}|f|m)1m({uεk}(ε+uε)NN1)1mC({uεk}|f|m)1m.

    Using the Hölder inequality and (3.7), we find

    {uεk}|uε|={uεk}uε(ε+uε)N2N2(ε+uε)N2N2C({uεk}|f|m)12m. (3.9)

    Choosing Tk(uε) as a test function in (3.1). Dropping the nonnegative lower order term, by (1.2)-(1.3) and the boundedness of uε in LNN1(Ω), we get

    Ω|Tk(uε)|2k(ρ+k)γαfL1(Ω). (3.10)

    This implies that Tk(uε)Tk(u) weakly in W1,20(Ω).

    Let E be a measurable subset of Ω, and i=1,,N. By the Hölder inequality and (3.9)-(3.10), we obtain

    E|uεxi|E|uε|E|Tk(uε)|+{uεk}|uε|meas(E)12(E|Tk(uε)|2)12+C({uεk}|f|m)12m. (3.11)

    The estimates (3.7) and (3.11) shows that the sequence {uεxi} is equi-integrable. Thus, by the Dunford–Pettis theorem, there exists a subsequence {uε} and Vi in L1(Ω), such that unxiVi in L1(Ω). Since uεxi is the distributional partial derivative of uε, then we have

    Ωuεxiφ=Ωuεφxi,φC0(Ω),

    for every ε>0.

    Since uεxiVi in L1(Ω) and uεu in L1(Ω), we find

    ΩViφ=Ωuφxi,φC0(Ω).

    This implies that Vi=uxi for every i.

    Step 3: We prove that uε|uε|2(uε+ε)θ+1 is equi-integrable. Let E⊂⊂Ω, then

    Euε|uε|2(uε+ε)θ+1E{uεk}uε|uε|2(uε+ε)θ+1+E{uεk}uε|uε|2(uε+ε)θ+1.

    For every subset E⊂⊂Ω,

    E{uεk}uε|uε|2(uε+ε)θ+1E{uεk}1uθε|Tk(uε)|2CE{uεk}|Tk(uε)|2,

    since uεC>0 in E by Proposition 2 of [9]. Moreover, since Tk(uε)Tk(u) weakly in W1,20(Ω), then there exists εn,δ>0, such that

    E{uεk}|Tk(uε)|dxϵ2,εεn, (3.12)

    for every ϵ>0 if μ(E)<δ.

    Choosing T1(uεTk1(uε)) as a test function in the approximate problem (3.1), dropping the nonnegative term, we have

    {uεk}uε|uε|2(uε+ε)θ+1{uεk1}f. (3.13)

    Observe there exists a constant C>0, such that μ(uεk1)Ck1. As uε are uniformly bounded in LNN1(Ω). This implies the right hand side of (3.13) converges to 0 as k. Thus, we deduce there exists k0>1, such that

    {uεk}uε|uε|2(uε+ε)θ+1ϵ2,k>k0, (3.14)

    for every ϵ>0. The (3.12), (3.14) imply that uε|uε|2(uε+ε)θ+1 is equi-integrable and converges a.e. to |u|2uθ.

    Let u the weak limit of the sequence of approximated solutions uε. Thanks to (2.2), we have

    Ωuε|uε|2(uε+ε)θ+1Ωf.

    Using the Fatou lemma, that uε convergence to u a.e, uε convergence to u a.e and the strict positivity of uε imply

    Ω|u|2uθΩfC.

    This show that |u|2uθL1(Ω).

    Since uε is bounded and uεu a.e, it follow M(x,T1ε(uε)uεM(x,u)u a.e. Hence, we can pass to the limit in (3.1). Thus prove that uW1,10(Ω) is a distributional solution of (1.1) and yields the conclusion of the proof of Theorem 1.1.

    Proof of Theorem 1.5. Step 1: For 0<ε<1, according to Lemma 2.2, we have

    12θ+1{uε1}|uε|2uθε{uε1}uε|uε|2(uε+ε)θ+1fL1(Ω). (3.15)

    By the Sobolev inequality, (3.15) lead to

    [Ω|u2θ2ε1|2]22CfL1(Ω), (3.16)

    which implies that

    [Ωu(2θ)22ε]22C+CfL1(Ω). (3.17)

    Observe that θ=(2θ)22=NN1. Then (3.17) shows that

    ΩuNN1εC. (3.18)

    Using the Hölder inequality and (3.15), (3.18), we obtain

    Ω|G1(uε)|={uε1}|G1(uε)|uθ2εuθ2ε[{uε1}|uε|2uθε]12[{uε1}uθε]12CfL1(Ω).

    This fact show that G1(uε) is bounded in W1,10(Ω).

    Choosing T1(uε) as a test function in (3.1), it is easy to prove that T1(uε) is bounded in H10(Ω)), hence in W1,10(Ω). Since uε=G1(uε)+T1(uε), we deduce that uε is bounded in W1,10(Ω).

    Moreover, due to (3.15) and the Hölder inequality, we have

    {uεk}|uε|={uεk}|uε|uθ2εuθ2εCf12L1(Ω). (3.19)

    That (3.10), (3.19) implies, for every measurable subset E, we have

    E|uεxi|E|uε|E|Tk(uε)|+{uεk}|uε|meas(E)12[k(ρ+k)γαfL1(Ω)]12+Cf12L1(Ω).

    Thus, we prove that uεu in W1,10(Ω). Then pass to the limit in problem (3.1), as in the proof of Theorem 1.2, it is sufficient to observe that uW1,10(Ω) is a distributional solution of (1.1). This concludes the proof the Theorem 1.5.

    In this paper, we main consider the existence of W1,10(Ω) solutions to a elliptic equation with principal part having noncoercivity. The main results show that the singular quadratic term has an important impact on this existence.

    This research was partially supported by the National Natural Science Foundation of China (No. 11761059), Program for Yong Talent of State Ethnic Affairs Commission of China (No. XBMU-2019-AB-34), Fundamental Research Funds for the Central Universities (No.31920200036) and First-rate Discipline of Northwest Minzu University.

    The authors declare that there is no conflict of interests regarding the publication of this article.



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    3. Hocine Ayadi, Rezak Souilah, The impact of a singular first-order term in some degenerate elliptic equations involving Hardy potential, 2024, 9, 2662-2009, 10.1007/s43036-024-00324-x
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