Loading [MathJax]/jax/element/mml/optable/GeneralPunctuation.js

Fractional p-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups

  • Received: 01 November 2020 Revised: 01 March 2021 Published: 26 May 2021
  • Primary: 22E30, 35H20; Secondary: 43A80

  • This study examines the existence and multiplicity of non-negative solutions of the following fractional p-sub-Laplacian problem

    {(Δp,g)su=λf(x)|u|α2u+h(x)|u|β2uinΩ,u=0inGΩ,

    where Ω is an open bounded in homogeneous Lie group G with smooth boundary, p>1, s(0,1), (Δp,g)s is the fractional p-sub-Laplacian operator with respect to the quasi-norm g, λ>0, 1<α<p<β<ps, ps:=QpQsp is the fractional critical Sobolev exponents, Q is the homogeneous dimensions of the homogeneous Lie group G with Q>sp, and f, h are sign-changing smooth functions. With the help of the Nehari manifold, we prove that the nonlocal problem on homogeneous group has at least two nontrivial solutions when the parameter λ belong to a center subset of (0,+).

    Citation: Jinguo Zhang, Dengyun Yang. Fractional p-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups[J]. Electronic Research Archive, 2021, 29(5): 3243-3260. doi: 10.3934/era.2021036

    Related Papers:

    [1] Jinguo Zhang, Dengyun Yang . Fractional $ p $-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups. Electronic Research Archive, 2021, 29(5): 3243-3260. doi: 10.3934/era.2021036
    [2] Chungen Liu, Huabo Zhang . Ground state and nodal solutions for fractional Kirchhoff equation with pure critical growth nonlinearity. Electronic Research Archive, 2021, 29(5): 3281-3295. doi: 10.3934/era.2021038
    [3] Heesung Shin, Jiang Zeng . More bijections for Entringer and Arnold families. Electronic Research Archive, 2021, 29(2): 2167-2185. doi: 10.3934/era.2020111
    [4] Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li . Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28(1): 369-381. doi: 10.3934/era.2020021
    [5] Jun Zhou . Initial boundary value problem for a inhomogeneous pseudo-parabolic equation. Electronic Research Archive, 2020, 28(1): 67-90. doi: 10.3934/era.2020005
    [6] Mingqi Xiang, Binlin Zhang, Die Hu . Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping. Electronic Research Archive, 2020, 28(2): 651-669. doi: 10.3934/era.2020034
    [7] Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan . On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29(1): 1709-1734. doi: 10.3934/era.2020088
    [8] Hongyan Guo . Automorphism group and twisted modules of the twisted Heisenberg-Virasoro vertex operator algebra. Electronic Research Archive, 2021, 29(4): 2673-2685. doi: 10.3934/era.2021008
    [9] Nouressadat Touafek, Durhasan Turgut Tollu, Youssouf Akrour . On a general homogeneous three-dimensional system of difference equations. Electronic Research Archive, 2021, 29(5): 2841-2876. doi: 10.3934/era.2021017
    [10] Fabian Ziltener . Note on coisotropic Floer homology and leafwise fixed points. Electronic Research Archive, 2021, 29(4): 2553-2560. doi: 10.3934/era.2021001
  • This study examines the existence and multiplicity of non-negative solutions of the following fractional p-sub-Laplacian problem

    {(Δp,g)su=λf(x)|u|α2u+h(x)|u|β2uinΩ,u=0inGΩ,

    where Ω is an open bounded in homogeneous Lie group G with smooth boundary, p>1, s(0,1), (Δp,g)s is the fractional p-sub-Laplacian operator with respect to the quasi-norm g, λ>0, 1<α<p<β<ps, ps:=QpQsp is the fractional critical Sobolev exponents, Q is the homogeneous dimensions of the homogeneous Lie group G with Q>sp, and f, h are sign-changing smooth functions. With the help of the Nehari manifold, we prove that the nonlocal problem on homogeneous group has at least two nontrivial solutions when the parameter λ belong to a center subset of (0,+).



    We consider the following p-fractional Laplace equation

    {(Δp,g)su=λf(x)|u|α2u+h(x)|u|β2uinΩu=0inGΩ (1.1)

    where Ω is an open bounded domain in homogeneous Lie group G with smooth boundary, p>1, the parameter λ>0, f and h are sign-changing smooth functions, 1<α<p<β<ps:=QpQps, ps is the fractional critical Sobolev exponent in this context and Q>sp is the homogeneous dimension of the homogeneous Lie group G. The operator (Δp,g)s is the fractional p-sub-Laplacian operator on G which is defined by

    (Δp,g)su(x)=2limε0GBg(x,ε)|u(x)u(y)|p2(u(x)u(y))g(y1x)Q+spdy,xG,

    where Bg(x,ε) is the quasi-ball of center xG and radius ε>0 with respect to the homogeneous quasi-norm g. In our work, the homogeneous quasi-norm g:GR+0 is a continuous function satisfying the following properties:

    (ⅰ) g(x)=0 ifandonlyifx=0 for every xG;

    (ⅱ) g(x1)=g(x) for every xG;

    (ⅲ) g(δμ(x))=μg(x) for every μ>0 and for every xG, where δμ is a dilations on homogeneous Lie group G.

    Associated with (1.1), we have the energy functional Iλ:E0gR defined by

    Iλ(u)=1pQ|u(x)u(y)|pg(y1x)Q+spdxdyλαΩf(x)|u|αdx1βΩh(x)|u|βdx.

    By a direct calculation, we have that IλC1(E0g,R) and

    Iλ(u),v=Q|u(x)u(y)|p2(u(x)u(y))(v(x)v(y))g(y1x)Q+spdxdyλΩf(x)|u|α2uvdxΩh(x)|u|β2uvdx,u,vE0g,

    where E0g is a subspace of Eg defined as E0g={uEg:u=0inGΩ} with the norm

    uE0g=(Q|u(x)u(y)|pg(y1x)Q+spdxdy)1p. (1.2)

    Here Q=G2(CΩ×CΩ) and CΩ=GΩ. See Section 2 for more details.

    Recently, a lot of attention is given to the study of fractional operators of elliptic type due to concrete real world applications in finance, thin obstacle problem, optimization, quasi-geostrophic flow etc. Dirichlet boundary value problem in case of fractional Laplacian with polynomial type nonlinearity using variational methods is recently studied in [4,6,21,20,19]. For example, Brändle et. al [4] studied the fractional Laplacian operator (Δ)s equation involving concave-convex nonlinearity for the subcritical case in the Euclidean space RN, they prove that there exists a finite parameter Λ>0 such that for each λ(0,Λ) there exist at least two solutions, for λ=Λ there exists at least one solution and for λ(Λ,+) there is no solution. Barrios et al. [2] studied the non-homogeneous equation involving fractional Laplacian and proved the existence and multiplicity of solutions under suitable conditions of s and q. Zhang, Liu and Jiao [23] studied the fractional equation with critical Sobolev exponent, they proved that the existence and multiplicity of solutions under appropriate conditions on the size of λ. For more other advances on this topic, see [19] for the subcritical, [20] for the critical case, [22] for the supercritical case, and fractional Laplacian equation with Hardy-type potential are shown in [13,14,24,25,26,27]. Moreover, for the fractional p-Laplacian equation, eigenvalue problem related to p-fractional Laplacian is studied in [17,12. Goyal and Sreenadh [15] studied the fractional p-Laplacian equation involving concave-convex nonlinearities. By using the Nehari manifold and the fibering maps methods, they showed that the problem has at least two non-negative solutions.

    In this paper, we present results concerning fractional forms Laplacian operator on homogeneous Lie groups. As usual, the general approach based on homogeneous Lie groups allows one to get insights also in the Abelian case, for example, from the point of view of the possibility of choosing an arbitrary quasi-norm. Moreover, another application of the setting of homogeneous Lie groups is that the results can be equally applied to elliptic and subelliptic problems. We start by discussing fractional Sobolev inequalities on the homogeneous Lie groups. As a consequence of these inequalities, we derive the existence results for the nonlinear problem with fractional p-sub-Laplacian operator and concave-convex nonlinearities and sign-changing weight functions. We also extend this analysis to equations of fractional p-sub-Laplacians and to Riesz type potential operators.

    To the best of our knowledge there is no work for fractional p-sub-Laplacian operator with convex-concave type nonlinearity and sign changing weight functions on the homogeneous Lie groups. We have the following existence result.

    Theorem 1.1. Let G be a homogeneous Lie group with homogeneous dimension Q, and let s(0,1), Q>sp, 1<α<p<β<ps and fLpspsα(Ω), hLpspsβ(Ω), f±=max{±f,0}0, h±=max{±h,0}0. Then there exists Λ>0 such that the equation (1.1) admits at least two non-negative solutions for λ(0,Λ).

    The paper is organized as follows: In Section 2, we study the properties of the Sobolev spaces Ws,pg(G) and E0g on homogeneous groups. In Section 3, we introduce Nehari manifold and study the behavior of Nehari manifold by carefully analyzing the associated fibering maps on homogeneous Lie groups. Section 4 contains the existence of nontrivial solutions in N+λ and Nλ.

    In this section we discuss nilpotent Lie algebras and groups in the spirt of Folland and Stein's book [11] as well as introducing homogeneous Lie groups. For more analyses and details in this direction we refer to the recent open access book [10] and [1,5,3,7,8,9,18] and references therein.

    Let g be a real and finite-dimensional Lie algebra, and let G be the corresponding connected and simply-connected Lie group. The lower central series of g is defined inductively by

    g(1):=g,g(j):=[g,g(j1)].

    If g(s+1)={0} and g(s){0}, then g is said to be nilpotent of step s. A Lie group G is nilpotent of step s whenever its Lie algebra is nilpotent of step s.

    Let exp:gG be a exponential map, and G be a connected and simply-connected nilpotent Lie group with Lie algebra g. Then, exponential map exp is a diffeomorphism from g to G. Let A be a diagonalisable linear operator on g with positive eigenvalues, and μ>0. Define the mappings are of the form

    δμ=exp(Alogμ)=k=0(Alogμ)k.

    Then, let us give a family of dilations of a Lie algebra g as follow

    {δμ:μ>0},

    which satisfies:

    (ⅰ) δμ is a morphism of the Lie algebra g, that is, a linear mapping from g to itself which respects to the Lie bracket:

    [δμX,δμY]=δμ[X,Y],X,Yg,λ>0.

    (ⅱ) δμ1μ2=δμ1δμ2 for all μi>0, i=1,2. If k>0 and {δμ} is a family of dilations on g, then so is {˜δμ}, where ˜δμ=δμk=exp(kAlogμ).

    Remark 2.1. (i) If a Lie algebra g admits a family of dilations, then it is nilpotent, but not all nilpotent Lie algebras admit a dilation structure.

    (ii) Since the exponential mapping exp is a global diffeomorphism from g to G, it induces the corresponding family on G which we may still call the dilations on G and denote by δμ. Thus, for xG we will write δμ(x) or abbreviate it writing simply μx, and the origin of G will be usually denoted by 0.

    Definition 2.1. Let δμ be a dilations on G. We say that a Lie group G is a homogeneous Lie group if:

    (a) It is a connected and simply-connected nilpotent Lie group G whose Lie algebra g is endowed with a family of dilations {δμ}.

    (b) The maps expδμexp1 are group automorphism of G.

    Now, we give some two examples of homogeneous groups.

    Example 2.1. The Euclidean space RN is a homogeneous group with dilation given by the scalar multiplication.

    Example 2.2. If N is a positive integer, the Heisenberg group HN is the group whose underlying manifold is CN×R and whose multiplication is given by

    (z,t)(˜z,˜t)=(z+˜z,t+˜t+2Imz,˜z),

    where (z,t)=(z1,,zN,t)=(x1,y1,,xN,yN,t)HN, xRN yRN and tR. The Heisenberg group HN is a homogeneous group with dilations

    δμ(z1,,zN,t)=(μz1,,μzN,μ2t).

    The mappings {δμ} give the dilation structure to an N-dimensional homogeneous Lie group G with

    δμ(x1,,xN)=(μd1x1,,μdNxN),

    where (x1,,xN) are the exponential coordinates of xG, djN for every j=1,2,,N and 1=d1==dm<dm+1dN for m:=dim(V1). Here the group G and the algebra g are identified through the exponential mapping.

    It is customary to denote with Q:=ki=1idim(Vi) the homogeneous dimension of G which corresponds to the Hausorff dimension of G. From now on Q will always denote the homogneous dimension of G. For example, the homogeneous dimension of Heisenberg group HN is Q:=2N+2.

    Now, we define a homogeneous quasi-norm on a homogeneous Lie group G to be a continuous function g:GR+ with the following properties:

    (ⅰ) g(x)=0 ifandonlyifx=0 for every xG;

    (ⅱ) g(x1)=g(x) for every xG;

    (ⅲ) g(δμ(x))=μg(x) for every μR+ and for every xG.

    For any measurable set EG, we have |δμ(E)|=μQ|E|, d(δμ(x))=μQdx, where |E| denotes the measure of the set E. Let

    Bg(x,r)={yG:g(x1y)<r}

    be the quasi-ball of radius r>0 about x with respect to the homogeneous quasi-norm g. Then, we have that

    |Bg(x,r)|=rQ,xG.

    It can be noticed that Bg(x,r) is the left translate by x of Bg(0,r), which in turn is the image under δr of Bg(0,1). Moreover, let

    Sg(0):={xG:g(x)=1}

    be the unit sphere with respect to the homogeneous quasi-norm g. Then there is a unique positive Radon measure σ on Sg(0) such that for all fL1(G), we have

    Gf(x)dx=0Sg(0)f(ry)rQ1dσ(y)dr.

    Let G be a homogeneous Lie group, with its basis X1, ..., XN, generating its Lie algebra g through their commutators. Then, the sub-Laplacian operator is defined as

    L:=X21++X2N.

    In the sequel, we use the following notations for the horizontal gradient

    G:=(X1,X2,,XN),

    and for the horizontal divergence

    divGv:=Guv.

    Using the Green's first and second formulae, we can define the p-sub-Laplacian on homogeneous groups G as

    Lpu=divG(|Gu|p2Gu),p(1,+).

    Recently, a great deal of attention has been focused on studying of equations or systems involving fractional Laplacian and corresponding nonlocal problems, both for their interesting theoretical structure and for their concrete applications, see [6,17,19,20] and references therein. The fractional p-Laplacian operator (Δp)s, s(0,1), is defined as

    (Δp)su(x)=2limε0+RnB(x,ε)|u(x)u(y)|p2(u(x)u(y))|xy|n+psdy,xRn.

    This type of operator arises in a quite natural way in many different contexts, such as, the thin obstacle problem, finance, phase transitions, anomalous diffusion, flame propagation and many others.

    Let G be a homogeneous Lie group with homogeneous dimension Q, p>1 and s(0,1). Compared to the fractional p-Laplacian problem, the fractional p-sub-Laplacian (Δp,g)s on G can be defined as

    (Δp,g)su(x)=2limε0+GBg(x,ε)|u(x)u(y)|p2(u(x)u(y))g(y1x)Q+spdy,xG,

    where g is a quasi-norm on G and Bg(x,ε) is a quasi-ball with respect to g, with radius ε centered at xG.

    Now we recall the definitions of the fractional Sobolev spaces on homogeneous Lie groups G. For a measurable function u:GR we define the Gagliardo quasi-seminorm by

    [u]s,p,g=(GG|u(x)u(y)|pg(y1x)Q+spdxdy)1p.

    For p>1 and s(0,1), we introduce the the functional Sobolev space on homogeneous Lie groups G by

    Ws,pg(G)={uLp(G)|uismeasurableand[u]s,p,g<+},

    and endowed with the norm

    uWs,pg(G)=uLp(G)+[u]s,p,g.

    Similarly, if ΩG is a Haar measurable set, we define the Sobolev space

    Ws,pg(Ω)={uLp(Ω)|uismeasurableand(ΩΩ|u(x)u(y)|pg(y1x)Q+spdxdy)1p<+},

    endowed with norm

    uWs,pg(Ω)=uLp(Ω)+(ΩΩ|u(x)u(y)|pg(y1x)Q+spdxdy)1p. (2.1)

    In [16,Theorem 2], Kassymov and Suragan given the following analogue of the fractional Sobolev inequality on homogeneous groups G.

    Theorem 2.1. Let G be a homogeneous group with homogeneous dimension Q. Assume that p>1, s(0,1), Q>sp and g denotes a quasi-norm on G. Then, for any measurable and compactly supported function u:GR, there exists a positive constant C=C(Q,p,s)>0 such that

    upLps(G)C[u]ps,p,g=CGG|u(x)u(y)|pg(y1x)Q+spdxdy,

    where ps:=p(Q,s,p)=QpQsp is the fractional critical Sobolev exponents on homogeneous group.

    Let Q=G2(CΩ×CΩ) and CΩ=GΩ. We define the Sobolev space

    Eg={u|u:GRismeasurable,u|ΩLp(Ω)andQ|u(x)u(y)|pg(y1x)Q+spdxdy<+}

    endowed with the norm as following

    uEg=uLp(Ω)+(Q|u(x)u(y)|pg(y1x)Q+spdxdy)1p. (2.2)

    Indeed, if uEg=0, we get that uLp(Ω)=0 and Q|u(x)u(y)|pg(y1x)Q+spdxdy=0. Then, the above equalities imply that u=0 a.e. in Ω and u(x)=u(y)=const. a.e. in Q. So, we can get that u=0 a.e. in G and Eg is a norm on Eg.

    Let E0g={uEg:u=0inGΩ} be a subspace of Eg. Then, for any p>1, E0g is a Banach space and have the following properties.

    Lemma 1. The following hold.

    (i) If uEg, then uWs,pg(Ω) and uWs,pg(Ω)uEg.

    (ii) If uE0g, then uWs,pg(G) and uWs,pg(Ω)uWs,pg(G)=uEg.

    Proof. (i) Let uEg, since Ω×Ω is strictly contained in Q, we have

    ΩΩ|u(x)u(y)|pg(y1x)Q+spdxdyQ|u(x)u(y)|pg(y1x)Q+spdxdy<. (2.3)

    Thus uWs,pg(Ω)uEg and deduced the result (ⅰ).

    (ⅱ) For each uE0g, we get u=0 on GΩ. Hence, uL2(G)=uL2(Ω) and

    GG|u(x)u(y)|pg(y1x)Q+spdxdy=Q|u(x)u(y)|pg(y1x)Q+spdxdy<+, (2.4)

    which and (2.3) yield the result (ⅱ).

    Theorem 2.2. Let G be a homogeneous group with homogeneous dimension Q. Assume that p>1, s(0,1), Q>sp and g be a quasi-norm on G. Then for every uE0g there exists a positive constant c=c(Q,s)>0 depending on Q and s such that

    upLps(Ω)=upLps(G)cGG|u(x)u(y)|pg(y1x)Q+spdxdy.

    Proof. For any uE0g, by Lemma 2.1 (ⅱ) and Theorem 2.1, we know that uWs,pg(G) and Ws,pg(G)Lps(G). Then, we have

    upLps(Ω)=upLps(G)cGG|u(x)u(y)|pg(y1x)Q+spdxdy,

    and completes the proof of Theorem 2.2.

    Lemma 2.2. The space (E0g,E0g) is a reflexive Banach space.

    Proof. Let {uk}kE0g be a Cauchy sequence. By Lemma 2.1 and Theorem 2.2, {uk}k is Cauchy sequence in Lp(Ω) and so {uk}k has a convergent subsequence. We assume uku in Lp(Ω). Since uk=0 in GΩ, we define u=0 in GΩ and then uku strongly in Lp(G) as k. So, there exists a subsequence of {uk}k, still denoted by {uk}k, such that uku a.e. in G. By Fatou's Lemma and using the fact that {uk} is a Cauchy sequence, we get that uE0g and ukuEg0 as k+. Hence E0g is a Banach space. Reflexivity of E0g follows from the fact that E0g is a closed subspace of reflexive Banach space Ws,pg(G).

    From above results, we can defined the following scalar product

    u,vE0g=Q|u(x)u(y)|pg(y1x)Q+spdxdy, (2.5)

    and norm

    uE0g=(Q|u(x)u(y)|pg(y1x)Q+spdxdy)1p (2.6)

    for the reflexive Banach space E0g. Since u=0 a.e. in GΩ, we note that the (2.5) and (2.6) can be extended to all G. Moreover, for any uEg, by Theorem 2.2 and the embedding Lps(Ω)Lp(Ω), there exist C1 and C2>0 such that

    C1Q|u(x)u(y)|pg(y1x)Q+spdxdyupEgC2Q|u(x)u(y)|pg(y1x)Q+spdxdy. (2.7)

    This imply that the norm E0g on E0g is equivalent to the norm Eg on Eg, and the norm E0g involves the interaction between Ω and GΩ. But the norms in (2.1) and (2.2) are not same because Ω×Ω is strictly contained in Q.

    Lemma 2.3. Let {uk}k be a bounded sequence in E0g. Then, there exists uLq(G) such that uku in Lq(G) as k for any q[1,ps).

    Proof. Let {uk}k is bounded in E0g, by Lemmas 2.1 and (2.7), {uk}k is bounded in Ws,pg(Ω) and also in Lp(Ω). Then by assumption on Ω and [4, Corollary 7.2], there exists uLq(Ω) such that up to a subsequence uku in Lq(Ω) as k for any q[1,ps). Since uk=0 on GΩ, we can define u:=0 in GΩ and we get uku in Lq(G).

    From Theorem 2.2, Lemma 2.1 and Lemma 2.3, we have that the embedding E0gLq(Ω) is continuous for any q[1,ps] and compact whenever q[1,ps). Let Sps be the best constant for the embedding of E0gLps(Ω) defined by

    Sps=infuE0g{0}Q|u(x)u(y)|pg(y1x)Q+spdxdy(Ω|u|psdx)pps.

    Since the energy functional Iλ is not bounded below on the space E0g, we consider the Nehari minimization problem: for λ>0,

    cλ=inf{Iλ(u):uNλ},

    where

    Nλ:={uE0g{0}:Iλ(u),u=0}.

    For any uNλ, the following equality hold

    Q|u(x)u(y)|pg(y1x)Q+spdxdy=λΩf(x)|u|αdx+Ωh(x)|u|βdx, (3.1)

    which implies that Nλ contains all nonzero solutions of equation (1.1). Moreover, we have the following result.

    Lemma 3.1. Iλ is coercive and bounded below on Nλ for all λ>0.

    Proof. Let λ>0 and for all uNλ, from (3.1) and Theorem 2.2 there holds

    Iλ(u)=1pQ|u(x)u(y)|pg(y1x)Q+spdxdyλαΩf(x)|u|αdx1βΩh(x)|u|βdx=(1p1β)Q|u(x)u(y)|pg(y1x)Q+spdxdyλ(1α1β)Ωf(x)|u|αdxβppβupE0gλ(βα)αβfpspsαSαppsuαE0g, (3.2)

    which yields that Iλ is bounded below and coercive on Nλ since β>p>α and Sαs>0. This completes the proof of Lemma 3.1.

    Define

    Φλ(u)=Iλ(u),u,uE0g{0}.

    Then, we see that ΦλC1(E0g,R), Nλ=Φ1λ(0){0}, and for all uNλ we get that

    Φλ(u),u=pQ|u(x)u(y)|pg(y1x)Q+spdxdyλαΩf(x)|u|αdxβΩh(x)|u|βdx=(pα)upE0g(βα)Ωh(x)|u|βdx=(pβ)upE0g+(βα)λΩf(x)|u|αdx. (3.3)

    We split Nλ into three parts

    N+λ:={uNλ:Φλ(u),u>0};N0λ:={uNλ:Φλ(u),u=0};Nλ:={uNλ:Φλ(u),u<0}. (3.4)

    On N0λ, the following result hold.

    Lemma 3.2. For each λ>0, let u0 be a local minimizer for Iλ on NλN0λ, then u0 is a critical point of Iλ.

    Proof. Since u0 is a local minimizer for Iλ on Nλ, that is, u0 is a solution of the optimization problem

    min{Iλ(u):uNλ}=min{Iλ(u):Φλ(u)=0}.

    Then, by the theory of Lagrange multipliers, there exists a constant θR such that

    Iλ(u0),u0=θΦλ(u0),u0.

    Since u0N0λ, we have Φλ(u0),u00, thus θ=0, this completes the proof.

    Remark 3.1. Lemmas 3.1 and 3.2 imply that the functional Iλ is bounded below on an appropriate subset of E0g and the minimizers of functional Iλ on subsets N+λ, Nλ giving raise to solutions of (1.1).

    Now, for t>0, define the fibering maps ϕu:tIλ(tu) as

    ϕu(t)=tppQ|u(x)u(y)|pg(y1x)Q+spdxdyλtααΩf(x)|u|αdxtββΩh(x)|u|βdx. (3.5)

    We note that

    ϕu(t)=tp1Q|u(x)u(y)|pg(y1x)Q+spdxdyλtα1Ωf(x)|u|αdxtβ1Ωh(x)|u|βdx=1t(Q|tu(x)tu(y)|pg(y1x)Q+spdxdyλΩf(x)|tu|αdxΩh(x)|tu|βdx)=1tIλ(tu),tu.

    This gives that tuNλ if and only if ϕu(t)=0 and in particular, uNλ if and only if ϕu(1)=0. Moreover, for each uNλ, from (3.1), (3.3) and (3.5) we get that

    ϕu(1)=(p1)Q|u(x)u(y)|pg(y1x)Q+spdxdyλ(α1)Ωf(x)|u|αdx(β1)Ωh(x)|u|βdx=Φλ(u),uIλ(u),u=Φλ(u),u,

    which and (3.4) yield that N+λ, Nλ and N0λ are corresponding to local minima, local maxima and points of inflection of ϕu(t), namely

    N+λ={uNλ:ϕu(1)>0},Nλ={uNλ:ϕu(1)<0},N0λ={uNλ:ϕu(1)=0}.

    In order to understand the Nehari manifold and the fibering maps, we consider the function mu:R+R defined by

    mu(t)=tpαQ|u(x)u(y)|pg(y1x)Q+spdxdytβαΩh(x)|u|βdx.

    Clearly, for any t>0,

    ϕu(t)=tα1(mu(t)λΩf(x)|u|αdx). (3.6)

    Thus, we obtain that

    tuNλϕu(t)=0tisasolutionofequationmu(t)=λΩf(x)|u|αdx. (3.7)

    From the expression (3.5) of ϕu, we see that the behavior of the fibering maps ϕu according to the sign of Ωf(x)|u|αdx and Ωh(x)|u|βdx, then we will study the following four cases.

    Case 1: Ωf(x)|u|αdx<0 and Ωh(x)|u|βdx<0. In this case, we have that mu(t)>0 for all t>0, and the equation

    mu(t)=λΩf(x)|u|αdx (3.8)

    has no solution for all λ>0. See Fig. 1.

    Figure 1.   .

    Case 2: Ωf(x)|u|αdx>0 and Ωh(x)|u|βdx<0. Since Ωh(x)|u|βdx<0, we have mu(t)>0 for all t>0, and limt+mu(t)+. Moreover, for all t>0,

    mu(t)=(pα)tpα1upE0g(βα)tβα1Ωh(x)|u|βdx>0.

    This gives that mu is strictly increasing on (0,+). Then, there exists a unique t1=t1(u)>0 such that mu(t1)=λΩf(x)|u|αdx, see Fig. 2. Moreover, for all t(0,t1), mu(t)<λΩf(x)|u|αdx, and for all t(t1,+), mu(t)>λΩf(x)|u|αdx. So, together with (3.6) we have that ϕu(t) is decreasing on (0,t1), increasing on (t1,) and ϕu(t1)=0. Those imply that the function ϕu has exactly one critical point t1>0 such that t1uN+λ, that is, t1>0 is a global minimum point of function ϕu, see Fig. 3.

    Figure 2.   .
    Figure 3.   .

    Case 3: Ωf(x)|u|αdx<0 and . Since and , we have as . Moreover, from and , we get that the problem (3.8) has a unique solution for each , see Fig. 4. Similarly as Case 2, we obtain that for all , and for all , which yields that is increasing on , decreasing on . So, is the global maximum point of such that , see Fig. 5.

    Figure 4.   .
    Figure 5.   .

    Case 4: and . Similarly as Case 3, we get that , as and there exists

    such that is increasing on , decreasing on and the function attains its maximum value at . See Fig. 6. Moreover,

    Figure 6.   .

    If , there exist , such that and

    Then, for all , for all , and for all , that is, is decreasing on , increasing on , and decreasing on . So, under the condition of , there exist that has a local minimum point and a local maximum point with and . See Fig. 7.

    Figure 7.   .

    From the above proof of Case 4, we have the following result.

    Lemma 3.3. For some with and . Then, there exists a positive constant such that for any , there are unique , such that and and . Moreover,

    Proof. From the proof of Case 4, we only need to prove that there exists such that, for any , the following inequality holds

    (3.9)

    Indeed, since

    Thus, taking , the inequality (3.9) holds for all and completes the proof of Lemma 3.3.

    In this section, we use the results in Section 3 to prove the existence of a nontrivial solution on , as well as on . First, we state the following result. Let

    Lemma 4.1. For each , we have .

    Proof. We consider the following two cases.

    Case Ⅰ: and . We have

    Thus, and so .

    Case Ⅱ: and . Suppose that for all . By (3.3), for each we have

    (4.1)

    and

    (4.2)

    Thus, (4.1), (4.2) and the Sobolev inequality imply that

    (4.3)

    and

    (4.4)

    where is the best constant of Sobolev embedding. So, (4.3) and (4.4) imply that

    contradicting with the assumption.

    Let

    By Lemmas 3.1, 3.2 and 4.1, for each , we get , , and the energy functional is coercive and bounded from below on , and . Then, the Ekeland variational principle implies that has a minimizing sequence on each manifold of , and . Define

    Lemma 4.2. The following facts hold.

    (ⅰ) If , then we have .

    (ⅱ) If ,then we have for some positive constant depending on and .

    Proof. (ⅰ) Let , by (3.3) we have

    Hence

    From this inequality and the definition of , , we deduce that .

    (ⅱ) Let , from (3.3) we have

    (4.5)

    Moreover, by the Hölder inequality and the Sobolev embedding theorem, we obtain

    which and (4.5) imply that

    (4.6)

    Putting together (3.2) and (4.6), we have

    Thus, if , we get that for all for some positive constant , and completes the proof.

    Theorem 4.1. Let be a homogeneous Lie group with homogeneous dimension . Assume that , , and hold. Then has a local minimizer in satisfying and is a nontrivial nonnegative solution of (1.1).

    Proof. Since is coercive and bounded from below on and so on , by the Ekeland variational principle, there exists a minimizing sequence such that

    (4.7)

    From Lemma 3.3 and Case 4, we known that .

    As is coercive on , is a bounded sequence in . Therefore, by Theorem 2.2, there is a subsequence, still denoted by , and such that

    (4.8)

    By the Hölder inequality and Dominated convergence theorem and (4.8), we obtain

    (4.9)

    and

    (4.10)

    First, we claim that , we argue by contradiction, then we have as . Thus

    and

    This contradicts as .

    From (4.8), (4.9) and (4.10), we know is a weak solution of (1.1). We now claim is a nontrivial solution of (1.1). Since , we have

    (4.11)

    which implies that

    (4.12)

    Let in (4.12), by (4.7), (4.8) and , we have that

    Using this inequality, we get that is a nontrivial solution of (1.1).

    Next, we show that in and . Since and (4.11), we obtain

    (4.13)

    Hence, thanks to (4.13) we obtain and , this implies that in .

    Finally, we claim that . Assume by contradiction that , then by Lemma 3.3, there exist unique and such that and . In particular, we have . Since

    there exists such that , from which and Lemma 3.3, we have

    a contraction. Note that and , so by Lemma 3.2, we obtain that is a nontrivial nonnegative solution of (1.1) and conclude the proof.

    Next, we establish the existence of a local minimum for on .

    Theorem 4.2. Let be a homogeneous Lie group with homogeneous dimension , and let , , and . Then has a local minimizer in satisfying and is a nontrivial nonnegative solution of (1.1).

    Proof. From Lemma 4.2, we get for any . Hence and there is a minimizing sequence such that

    (4.14)

    Hence, (4.14) and the coerciveness of yield that is a bounded in . Without loss of generality, we can suppose that such that in , and in for any . Using this results, as the proof of Theorem 4.1, we get that

    and

    Now we claim that . To see this, by contradiction, we suppose that as , from which we have

    and

    This contradicts as .

    Finally we prove in . Other case, we have

    Which contradicts . Hence in as . Similar to Theorem 4.1, the proof can be completed.

    Now, we complete the proof of Theorem 1.1.

    Proof of Theorem 1.1. For , by Theorems 4.1 and 4.2, there are and such that

    So, the equation (1.1) admits at least two nontrivial nonnegative solutions and . Since , it results and are distinct nontrivial nonnegative solutions of problem (1.1) and the thesis is proved.



    [1] Approximations of Sobolev norms in Carnot groups. Commun. Contemp. Math. (2011) 13: 765-794.
    [2] On some critical problems for the fractional Laplacian operator. J. Differential Equations (2012) 252: 6133-6162.
    [3] A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer, 2007.
    [4] A concave-convex elliptic problem involving the fractional Laplacian. Proc. Roy. Soc. Edinburgh Sect. A (2013) 143: 39-71.
    [5] F. Buseghin, N. Garofalo and G. Tralli, On the limiting behavior of some nonlocal semi-norms: A new phenomenon, preprint (2020).
    [6] Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. (2010) 224: 2052-2093.
    [7] M. Capolli, A. Maione, A. M. Salort and E. Vecchi, Asymptotic behaviours in fractional Orlicz-Sobolev spaces on Carnot groups, J. Geom. Anal., 31 (2020), 3196–-3229.. doi: 10.1007/s12220-020-00391-5
    [8] A Pólya-Szegö principle for general fractional Orlicz–Sobolev spaces. Complex Variables and Elliptic Equations (2020) 66: 1-23.
    [9] Fractional Laplacians, perimeters and heat semigroups in Carnot groups. Discrete Contin. Dyn. Syst. (Series S) (2018) 11: 477-491.
    [10] V. Fischer and M. Ruzhansky, Quantization on Nilpotent Lie Groups, volume 314 of Progress in Mathematics, Birkhäuser. (open access book), 2016 doi: 10.1007/978-3-319-29558-9
    [11] (1982) Hardy Spaces on Homogeneous Groups. Princeton, N.J.; University of Tokyo Press, Tokyo: volume 28 of Mathematical Notes. Princeton University Press.
    [12] Fractional -eigenvalues. Riv. Mat. Univ. Parma (2014) 5: 373-386.
    [13] Mass and asymptotics associated to fractional Hardy-Schrödinger operators in critical regimes. Comm Partial Differential Equations (2018) 43: 859-892.
    [14] Borderline variational problems involving fractional Laplacians and critical singularities. Advanced Nonlinear Studies (2015) 15: 527-555.
    [15] Nehari manifold for non-local elliptic operator with concave–convex nonlinearities and sign-changing weight functions. Proc. Indian Acad. Sci. Math. Sci. (2015) 125: 545-558.
    [16] Lyapunov-type inequalities for the fractional p-sub-Laplacian. Advances in Operator Theory (2020) 5: 435-452.
    [17] E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795–-826. doi: 10.1007/s00526-013-0600-1
    [18] M. Ruzhansky, N. Tokmagambetov and N. Yessirkegenov, Best constants in Sobolev and Gagliardo-Nirenberg inequalities on graded groups and ground states for higher order nonlinear subelliptic equations, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 175, 23 pp. doi: 10.1007/s00526-020-01835-0
    [19] Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. (2012) 389: 887-898.
    [20] Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. (2013) 33: 2105-2137.
    [21] Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm. Pure Appl. Math. (2007) 60: 67-112.
    [22] Three solutions for a fractional elliptic problems with critical and supercritical growth. Acta Mathematica Scientia (2016) 36: 1819-1831.
    [23] Multiplicity of positive solutions for a fractional laplacian equations involving critical nonlinearity. Topol. Methods Nonlinear Anal. (2019) 53: 151-182.
    [24] Multiplicity of positive solutions for a nonlocal elliptic problem involving critical Sobolev-Hardy exponents and concave-convex nonlinearities. Acta Math. Sci. (2020) 40B: 679-699.
    [25] Nonlocal elliptic systems involving critical Sobolev-Hardy exponents and concave-convex nonlinearities. Taiwanese J Math. (2019) 23: 1479-1510.
    [26] Multiple solutions for a fractional Laplacian system involving critical Sobolev-Hardy exponents and homogeneous term. Math. Mode. Anal. (2020) 25: 1-20.
    [27] Existence results for a fractional elliptic system with critical Sobolev-Hardy exponents and concave-convex nonlinearities. Math Meth Appl Sci. (2020) 43: 3488-3512.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2714) PDF downloads(209) Cited by(0)

Figures and Tables

Figures(7)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog