Special Issues

Normalized solutions for Choquard equations with general nonlinearities

  • In this paper, we prove the existence of positive solutions with prescribed L2-norm to the following Choquard equation:

    Δuλu=(IαF(u))f(u),    xR3,

    where λR,α(0,3) and Iα:R3R is the Riesz potential. Under the weaker conditions, by using a minimax procedure and some new analytical techniques, we show that for any c>0, the above equation possesses at least a couple of weak solution (ˉuc,ˉλc)Sc×R such that ˉuc22=c.

    Citation: Shuai Yuan, Sitong Chen, Xianhua Tang. Normalized solutions for Choquard equations with general nonlinearities[J]. Electronic Research Archive, 2020, 28(1): 291-309. doi: 10.3934/era.2020017

    Related Papers:

    [1] Shuai Yuan, Sitong Chen, Xianhua Tang . Normalized solutions for Choquard equations with general nonlinearities. Electronic Research Archive, 2020, 28(1): 291-309. doi: 10.3934/era.2020017
    [2] Xudong Shang . Normalized ground states to the nonlinear Choquard equations with local perturbations. Electronic Research Archive, 2024, 32(3): 1551-1573. doi: 10.3934/era.2024071
    [3] Lingzheng Kong, Haibo Chen . Normalized solutions for nonlinear Kirchhoff type equations in high dimensions. Electronic Research Archive, 2022, 30(4): 1282-1295. doi: 10.3934/era.2022067
    [4] Jun Wang, Yanni Zhu, Kun Wang . Existence and asymptotical behavior of the ground state solution for the Choquard equation on lattice graphs. Electronic Research Archive, 2023, 31(2): 812-839. doi: 10.3934/era.2023041
    [5] Quanqing Li, Zhipeng Yang . Existence of normalized solutions for a Sobolev supercritical Schrödinger equation. Electronic Research Archive, 2024, 32(12): 6761-6771. doi: 10.3934/era.2024316
    [6] Haijun Luo, Zhitao Zhang . Existence and stability of normalized solutions to the mixed dispersion nonlinear Schrödinger equations. Electronic Research Archive, 2022, 30(8): 2871-2898. doi: 10.3934/era.2022146
    [7] Cheng He, Changzheng Qu . Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28(4): 1545-1562. doi: 10.3934/era.2020081
    [8] Yony Raúl Santaria Leuyacc . Elliptic equations in $ \mathbb{R}^2 $ involving supercritical exponential growth. Electronic Research Archive, 2024, 32(9): 5341-5356. doi: 10.3934/era.2024247
    [9] Hami Gündoğdu . RETRACTED ARTICLE: Impact of damping coefficients on nonlinear wave dynamics in shallow water with dual damping mechanisms. Electronic Research Archive, 2025, 33(4): 2567-2576. doi: 10.3934/era.2025114
    [10] Lin Shen, Shu Wang, Yongxin Wang . The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28(2): 691-719. doi: 10.3934/era.2020036
  • In this paper, we prove the existence of positive solutions with prescribed L2-norm to the following Choquard equation:

    Δuλu=(IαF(u))f(u),    xR3,

    where λR,α(0,3) and Iα:R3R is the Riesz potential. Under the weaker conditions, by using a minimax procedure and some new analytical techniques, we show that for any c>0, the above equation possesses at least a couple of weak solution (ˉuc,ˉλc)Sc×R such that ˉuc22=c.



    This paper is dedicated to deal with the existence of normalized solutions to the generalized Choquard equation as follows:

    Δuλu=(IαF(u))f(u),    xR3, (1.1)

    where λR, α(0,3), Iα:R3R is the Riesz potential. Problem (1.1) is a nonlocal one due to the existence of the nonlocal nonlinearity. When λR is a fixed and assigned a parameter or even with an additional external, the existence of (1.1) has been studied during the last decade.

    For example, when λ=1, α=2 and F(u)=u2, (1.1) comes back to the description of the quantum theory of a polaron at rest by Pekar [22] and the modeling of an electron trapped in its own hole (in the work of Choquard in 1976), in a certain approximation to Hartree-Fock theory of one-component plasma [17]. The equation is also known as the Schrödinger-Newton equation, which was proposed by Penrose [23] in 1996 as a model of self-gravitating matter. Under this condition, the existence of nontrivial solutions was investigated by various variational methods by Lieb and Menzala [17,19] and also by ordinary differential equations methods [11,21,27]. There are also many papers investigating the Choquard equation under the general pure nonlinearity condition,

    Δu+u=(Iα|u|p)|u|p2u,    xRN, (1.2)

    where N3 and α(0,N), We can refer to [5,15,17,20]. In [20], Moroz and Van Schaftingen obtained that problem (1.2) has a nontrivial solution when N+αN<p<N+αN2.

    Nowadays, since physicist are more and more interested in the normalized solutions, like [1,2,3,13,14,30], mathematical researchers are committed to investigate the solutions with prescribed L2-norm, that is, solutions which satisfy u22=c>0 for a priori given constant. Such prescribed L2-norm solutions of (1.2) can be obtained by looking for critical points of the following functional

    IN(u)=12RN|u|2dx12RN(IαF(u))F(u)dx (1.3)

    on the constraint

    Sc={uH1(RN):u22=c}, (1.4)

    where F(u)=u0f(t)dt. In this sense, the parameter λR cannot be fixed but regarded as a Lagrange multiplier, and each critical point ucSc of IN|Sc, corresponds a Lagrange multiplier λcR such that (uc,λc) solves (weakly) (1.2). In particular, if ucSc is a minimizer of problem

    σ(c):=infuScIN(u), (1.5)

    then there exists λcR such that IN(uc)=λcuc, hence, (uc,λc) is a solution of (1.2).

    In [12], Jeanjean proved the existence of normalized solutions of the following Schrödinger equation

    Δuf(u)=λu in RN, (1.6)

    where N1, f:RR satisfies the following cases:

    (f1) fC(R,R) and f is odd;

    (f2)  α,βR with 2N+4N<αβ<2 such that

    αG(s)g(s)sβG(s),

    where G(s)=s0g(t)dt and 2=2NN2 if N3 and 2=+ if N=1,2;

    (f3) let ˉG(s)=g(s)s2G(s). Then ˉG exists and

    ˉG(s)s>2N+4NˉG(s).

    Jeanjean deduced the existence of normalized solutions by dealing with the minimization problem

    infuH1(RN),u2=cRN[12|u|2F(u)]dx,

    and the author verified the existence of the mountain pass structure on the constraint defined by Sc. Moreover, one of the highlights in the proof is that the auxiliary functional ˜I:H1(RN)×RR is introduced, defined by:

    ˜I(u,t)=e2t2RN|u|2dx1eNtRNG(eNt2u)dx.

    By applying the new functional ˜I, Jeanjean proved that for any fixed c>0, problem (1.6) has a couple of weak solution (uc,λc)H1(RN)×R such that uc2=c under the conditions (f1), (f2) and (f3).

    Bellazzini, Jeanjean and Luo [4] verified the existence of standing waves with prescribed L2-norm for the following Schrödinger-Poisson equation:

    Δu+(|x|1|u|2)u|u|q2u=λu,    xR3, (1.7)

    where q(103,6), and which different from [12] is that the function defined by

    ˆI(u)=12R3|u|2dx+14R3R3|u(x)|2|u(y)|2|xy|dxdy1qR3|u|qdx, (1.8)

    is no more bounded from below on the constraint:

    Sc={uH1(R3):u22=c}. (1.9)

    To overcome this difficulty, they first investigated the mountain-pass structure of ˆI on the constraint Sc, and then they show the existence of special bounded Palais-Smale sequence {un} at the level γ(c) which surrounds around the constraint set

    Mc={uSc:ˆJ(u):=ddtˆI(ut)|t=1=0}, (1.10)

    that is ˆJ(un)=o(1), where ut(x)=t3/2u(tx). In particular, Mc used in [4] acts as a natural restriction and γ(c) equals numerically to

    ˆm(c)=infuMcˆI(u). (1.11)

    As far as we know, there seems to be only one paper [16] dealing with the Choquard equation in the sense of prescribed L2-norm, Li and Ye considered the Choquard equation (1.1) in the Ndimension space under the following conditions:

    (F1') f(s)=0 for s0 and there exists r(N+α+2N,N+αN2) such that

    lim|s|0f(s)|s|r2s=0  and  lim|s|+F(s)|s|r=+;

    (F2') lim|s|+F(s)|s|N+αN2=0;

    (F3') there exists θ11 such that θ1˜F(s)˜F(ts) for sR and t[0,1], where ˜F(s)=f(s)sN+α+2NF(s);

    (F4') f(s)s<N+αN2F(s) for all s>0;

    (F5') let ˉF(s):=f(s)sN+αNF(s). ˉF(s) exists and

    ˉF(s)s>N+α+2NˉF(s);

    (F6') there exists 0<θ2<1 and t0 such that for all sR and |t|t0,

    F(ts)θ2|t|N+α+2NF(s).

    In fact, the nonlinearity term in paper [16] needs the assumption fC1(R,R). A natural question is whether the above result in [16] on the existence of normalized solutions to (1.1) can be generalized to more general fC(R,R). The purpose of the present paper is to address this question. To this end, we introduce the following assumptions:

    (F1) f(s)=0 for s0 and there exists r(3+α+23,3+α) such that

    lim|s|0f(s)|s|r2s=0,  and  lim|s|+F(s)|s|r=+;

    (F2) lim|s|+F(s)|s|3+α=0 and lim|s|0F(s)|s|2=0;

    (F3) f(s)s<(3+α)F(s) for all s>0;

    (F4) there exists θ11 such that θ1˜F(s)˜F(ts) for sR and t[0,1], where ˜F(s)=f(s)s9+2α6F(s);

    (F5) [f(t)t3+α3F(t)]/|t|6+2α3t is nondecreasing on (,0) and (0,+).

    In this paper, we define

    I(u)=12R3|u|2dx12R3(IαF(u))F(u)dx (1.12)

    and

    Mc={uSc:J(u):=ddtI(ut)|t=1=0},

    where the definition of Sc is given by (1.9). Our main result is as follows:

    Theorem 1.1. Assume that (F1)-(F5) hold. Then for any constant c>0, (1.1) has a couple of solutions (ˉvc,ˉλc)Sc×R such that ˉvc>0 and

    I(ˉvc)=infvMcI(v)=infvScmaxt>0I(vt)>0.

    Notice that, we proved the existence of normalized solution of problem (1.1) under the assumptions (F1)-(F5). Compared to [16], case (F5) plays an important role to overcome the difficulty caused by the absence of condition (F5), that is, we generalized the problem (1.1) concerning the prescribe L2-norm solutions to fit on more general nonlinearity term. But also, the absence of (F5) in [16] causes new difficulties. In the proof, we present a new and more general approach to overcome this difficulty.

    Now, we give our main idea for the proof of Theorem 1.1.

    By (F1) and (F2), there exists some C>0 such that

    |F(s)|C(|s|r+|s|3+α). (1.13)

    By Hardy-Littlewood-Sobolev inequality: fLp(R3), gLq(R3), if 0<α<3, 1<p,q<+, and 1p+1q+3α3=2, then

    R3(Iαf)gdxCfpgq,

    we see that F(u)L63+α(R3) for each uH1(R3) and IC1(H1(R3),R).

    Inspired by [16], (F4) implies that

    f(s)s9+2α6F(s)0 for any sR. (1.14)

    Then for any s>0, F(s)s9+2α6 is nondecreasing in s>0. By (F1), we conclude that:

    F(s)0 for any sR. (1.15)

    Then by (1.14), we see that f(s)0 for all sR and F(s) is nondecreasing in sR.

    As in [4,12], I is no more bounded from below on Sc by (F1), similarly we shall seek for a critical point satisfying a minimax characterization, i.e., we try to prove, I possesses a mountain pass geometry on the constrain Sc.

    Definition 1.2. For given c>0, we say that I(u) possesses a mountain pass geometry on Sc if there exists ρc>0 such that

    γ(c)=infgΓcmaxτ[0,1]I(g(τ))>maxgΓcmax{I(g(0)),I(g(1))}, (1.16)

    where Γc={gC([0,1],Sc):g(0)22ρc,I(g(1))<0}.

    Let us recall this, to obtain this conclusion, the authors in [4] constructed some sequence of paths {gn}Γc which have nice `shape' properties, and by Taylor's formula which is relies on IC2(H1(R3),R), the author deduced a localization lemma concerning the specific (PS) sequence. Different from his work, in the present paper we shall investigate the following auxiliary functional:

    ˜I(v,t)=I(β(v,t))=e2t2v2212e(3+α)tR3(IαF(e3t2v))F(e3t2v)dx,

    and also we shall know the fact that ˜I possesses the same mountain pass structure on Sc×R as the functional I|Sc. Based on this fact, in Lemma 2.2, we find a (PS)γ(c) sequence {un} with the additional property J(un)0, and then prove the convergence of {un}, this idea comes from [12] in which the classical Schrödinger equation (1.6) was studied.

    Since we have obtained the boundness of {un}, next we using scaling tramsform to verify the convergence of {un}. Because the nonlocal term and the gradient term in I scale differently, we overcome this difficulty by verifying whether γ(c) is nonincreasing. As in [8], we first prove that γ(c) is nonincreasing and then combining with the fact γ(c)=m(c) which is verified in Lemma 2.10, then we can prove the convergence of {un}.

    In [4] the fact I may be not C2 prevents us using the Implicit Function Theorem which influence above approach, then there needs new techniques and more subtle analyses to apply to more general fC1. To deduce the convergence of (PS)γ(c) sequence {un}, we shall establish a new key inequality with the help of (F5), see Lemma 2.4, which is also inspired by [6,7,9,10,24,26,28]. In particular, we present a new and more general approach to recover the compactness of minimizing sequence.

    Throughout the paper we use the following notations:

    H1(R3) denotes the usual Sobolev space equipped with the inner product and norm

    (u,v)=R3(uv+uv)dx,  u=(u,u)1/2,   u,vH1(R3);

    Ls(R3) (1s<) denotes the Lebesgue space with the norm us=(R3|u|sdx)1/s;

    for any uH1(R3), ut(x):=t3/2u(tx);

    for any xR3 and r>0, Br(x):={yR3:|yx|<r};

    S=infuD1,2(R3){0}u22/u26;

    C,C1,C2, denote positive constants possibly different in different places.

    To prove Theorem 1.1, recalling the Gagliardo-Nirenberg inequality, that is, let p[2,6),

    uLpCpuβL2u1βL2,

    where β=3(121p).

    In the following lemma, we show that I possesses the mountain pass geometry on the constraint Sc.

    Lemma 2.1. Assume that (F1),(F2) and (F4) hold. Then for any c>0, there exist 0<k1<k2 and u1,u2Sc such that u1Ak1 and u2Ak2, where

    Ak1={uSc:u22k1,I(u)>0} (2.1)

    and

    Ak2={uSc:u22>k2,I(u)<0}. (2.2)

    Moreover, I has a mountain pass geometry on the constraint Sc.

    Proof. Given any k>0, let

    Bk={uSc:u22k}. (2.3)

    We shall check that there exist 0<k1<k2 such that

    I(u)>0,   uBk2 and supuBk1I(u)<infuBk2I(u). (2.4)

    We have known that F(u)L63+α(R3), then by the Hardy-Littlewood-Sobolev inequality, Sobolev embedding inequality and the Gagliardo-Nirenberg inequality, we can find

    |R3(IαF(u))F(u)dx|C(R3|F(u)|63+αdx)3+α3C(R3|u|6r3+αdx+R3|u|2dx)3+α3C[u2r6r3+α+(R3|u|2dx)3+α3]C(u3r(3+α)2+u6+2α2). (2.5)

    Hence, we have that

    I(u)12u22C2u3r(3+α)2C2u6+2α2. (2.6)

    Since 3r3α>2 and 6+2α>2, it follows from (2.6) that there exist k2>0 small and ρ>0 such that

    infuBk2I(u)ρ>0 and I(u)>0 for uBk2. (2.7)

    On the other hand, use (2.5) again, we have

    |I(u)|12u22+12|R3(IαF(u))F(u)dx|12u22+C2u3r(3+α)2+C2u6+2α2. (2.8)

    which implies

    supuBk|I(u)|0 as k0. (2.9)

    Combining (2.7) with (2.9), there exists k1(0,k2) small such that

    supuBk1I(u)<ρinfuBk2I(u).

    and (2.4) follows.

    Let

    ut(x)=t3/2u(tx),     t>0,uH1(R3). (2.10)

    Then ut2=u2, and so utSc for any uSc and t>0. Note that

    I(ut)=t22u2212t3+αR3(IαF(t3/2u))F(t3/2u)dx. (2.11)

    Using (F1), (2.10) and Fatou's Lemma, which is inspired by [16], we can see that

        lim inftR3(IαF(t3/2u)|t3/2u||u|r)F(t3/2u)|t3/2u||u|rR3lim inft[(F(t3/2u)|t3/2u||u|r)(x)F(t3/2u)|t3/2u||u|r]+. (2.12)

    Hence, we have

    I(ut)t3r3α=u222t3r(3+α+2)12R3(IαF(t3/2u)|t3/2u||u|r)F(t3/2u)|t3/2u||u|r   as   t+. (2.13)

    So I(ut) as t+. For any uSc, there exist t1>0 small and t2>1 large such that

    ut122=t21u22k1,  ut222=t22u22>k2 and I(ut2)<0. (2.14)

    Set u1=ut1 and u2=ut2. Then (2.14) yields

    u122k1,    u222>k2.

    This fact indicates that u1Ak1 and u2Ak2.

    We next claim that I possesses a mountain pass geometry on Sc. For

    Γc:={gC([0,1],Sc):g(0)22k1,I(g(1))<0},

    if Γc, then for any gΓc, (2.4) implies g(0)22k1<k2<g(1)22. Thus, by the intermediate value theorem, there exists τ0(0,1) such that g(τ0)22=k2, i.e., g(τ0)Bk2. It follows from (2.4) that

    maxt[0,1]I(g(t))I(g(τ0))infuBk2I(u)>supuBk1I(u),     gΓc,

    which, together with the arbitrariness of gΓc, implies

    γ(c)=infgΓcmaxt[0,1]I(g(t))>maxgΓcmax{I(g(0)),I(g(1))}. (2.15)

    Indeed, to obtain the desired conclusion, it suffices to check that Γc. For any uSc, set

    g0(τ)=u(1τ)t1+τt2,     τ[0,1].

    It follows from (2.14) that g0γ(c). Hence, Γc and the proof is completed.

    Next, inspired by [6,12], we will show the existence of a (PS) sequence for the functional I on the constraint Sc attaching the property J(un)0, where

    J(u)=u2232R3(IαF(u))[f(u)u3+α3F(u)]dx,     uH1(R3). (2.16)

    To achieve this, we define a continuous map β:H:=H1(R3)×RH1(R3) by

    β(v,t)(x)=e3t2v(etx)  for vH1(RN), tR, and xR3, (2.17)

    where H is a Banach space equipped with the product norm (v,t)H:=(v2+|t|2)1/2. We introduce the following auxiliary functional:

    ˜I(v,t)=I(β(v,t))=e2t2v2212e(3+α)tR3(IαF(e3t2v))F(e3t2v)dx. (2.18)

    It is easy to see that ˜IC1(H,R), and for any (w,s)H,

    ˜I(v,t),(w,s)=e2tR3vwdx12e(3+α)tR3(IαF(e3t2v))f(e3t2v)e3t2wdx    +e2tsv22+(3+α)s2e(3+α)tR3(IαF(e3t2v))F(e3t2v)dx    3s2e(3+α)tR3(IαF(e3t2v))f(e3t2v)e3t2vdx. (2.19)

    Set

    ˜γ(c):=inf˜g˜Γcmaxτ[0,1]˜I(˜g(τ)), (2.20)

    where

    ˜Γc={˜gC([0,1],Sc×R):˜g(0)Ak1×{0},˜g(1)Ak2×{0}}.

    and the sets Ak1 and Ak2 are defined in Lemma 2.1. For any gΓc, let ˜g0(τ)=(g(τ),0) for τ[0,1]. It is easy to see that ˜g0˜Γc, and then ˜Γc. Since Γc={β˜g:˜g˜Γc}, then we know the minimax value of I agrees to ˜I, i.e. γ(c)=˜γ(c), moreover, (2.15) leads to

    ˜γ(c)=γ(c)>maxgΓcmax{I(g(0)),I(g(1))}=max˜g˜Γcmax{˜I(˜g(0)),˜I(˜g(1))}. (2.21)

    Following by [29], we recall that for any c>0, Sc is a submanifold of H1(R3) with codimension 1 and the tangent space at Sc is given

    Tu={vH1(R3):R3uvdx=0}. (2.22)

    The norm of the C1 restriction functional I|Sc is defined by

    I|Sc(u)=supvTu,v=1I(u),v. (2.23)

    And the tangent space at (u,t)Sc×R is given as

    ˜Tu,t={(v,s)H:R3uvdx=0}. (2.24)

    The norm of the derivative of the C1 restriction functional ˜I|Sc×R is defined by

    ˜I|Sc×R(u,t)=sup(v,s)˜Tu,t,(v,s)H=1˜I|Sc×R(u,t),(v,s). (2.25)

    Learning from [12,Proposition 2.2], we have the following proposition.

    Proposition 1. Assume that ˜I has a mountain pass geometry on the constraint Sc×R. Let ˜gn˜Γc be such that

    maxτ[0,1]˜I(˜gn(τ))˜γ(c)+1n. (2.26)

    Then there exists a sequence (un,tn)Sc×R such that

    (ⅰ) ˜I(un,tn)[˜γ(c)1n,˜γ(c)1n];

    (ⅰ) minτ[0,1](un,tn)˜gn(τ)H1n;

    (ⅰ) ˜I|Sc×R(un,tn)2n, i.e.,

    |˜I(un,tn),(v,s)|2n(v,s)H,     (v,s)˜Tun,tn.

    Applying proposition 1 to ˜I and also by [8], we conclude the following key lemma.

    Lemma 2.2. Assume that (F1),(F2) and (F4) hold. Then for any c>0, there exists a sequence {vn}Sc such that

    I(vn)γ(c)>0,  I|Sc(vn)0  and  J(vn)0. (2.27)

    Proof. Given {gn}Γc satisfy

    maxτ[0,1]I(gn(τ))γ(c)+1n. (2.28)

    In order to obtain the desired sequence, we first apply proposition 1 to ˜I. We define

    ˜gn(τ)=(gn(τ),0),     τ[0,1].

    It is easy to know that ˜gn˜Γc and ˜I(˜gn(τ))=I(gn(τ)). Since ˜γ(c)=γ(c), it follows from (2.28) that

    maxτ[0,1]˜I(˜gn(τ))˜γ(c)+1n. (2.29)

    From the preceding proposition 1, there exists a sequence {(un,tn)}Sc×R such that

    (ⅰ) ˜I(un,tn)˜γ(c);

    (ⅰ) minτ[0,1](un,tn)(gn(τ),0)H0;

    (ⅰ) ˜I|Sc×R(un,tn)2n.

    Set vn:=β(un,tn), and the definition of β is given in (2.17). Since vnSc and ˜γ(c)=γ(c), it follows from (i) that

    I(vn)γ(c). (2.30)

    Accoring to (2.19) and (ⅱ), we derive

    I(vn),w=˜I(un,tn),(β(w,tn),0)2n(β(w,tn),0)H,     wTvn. (2.31)

    To prove I|Sc(vn)0, by (2.31), it suffices to prove that {(β(w,tn),0)} is uniformly bounded in H and {(β(w,tn),0)}˜Γun,tn. For any wTvn, i.e.,

    R3vnwdx=R3e3tn2un(etnx)w(x)dx=0,

    we can see that

    R3un(x)β(w,tn)dx=R3un(x)e3tn2w(etnx)dx=R3e3tn2un(etnx)w(x)dx=0,

    follows

    (β(w,tn),0)˜Γun,tn. (2.32)

    Then by (ⅱ), we have

    |tn|minτ[0,1](un,tn)˜gn(τ)H1 for large nN,

    which leads to

    (β(w,tn),0)2H=β(w,tn)2=e2tnw22+w22 e2w2 for large nN. (2.33)

    This shows that {(β(w,tn),0)}˜Γun,tn is uniformly bounded in H, and so I|Sc(vn)0. In the end, by (ⅲ), we obtain

    |˜I(un,tn),(0,1)|=J(β(un,tn))=J(vn)=o(1). (2.34)

    Hence, {vn} satisfies (2.27).

    In connection with the additional minimax characterization of γ(c), we have the following Lemma 2.10. To achieve this goal, we have to establish some new inequalities, which is the crucial procedure for our convenience to obtain our final conclusion of this paper.

    Lemma 2.3. Assume that (F1),(F2),(F4) and (F5) hold. Then

    h(t):=(3+α)t32NR3(IαF(u))F(u)dx+12(t3+α)R3(IαF(t32u))F(t32u)dx    t32R3(IαF(u))f(u)udx0,     t>0. (2.35)

    Proof. For any tR, we have

    ddth(t)=(3+α)t122R3(IαF(u))F(u)dx3+α2t3+α+1R3(IαF(t32u))F(t32u)dx   +32t3+α+1R3(IαF(t32u))f(t32u)(t32u)dx3t122R3(IαF(u))f(u)udx (2.36)

    Now we only need to study q(t,τ1,τ2) which is defined by the following form:

    q(t,τ1,τ2)=F(t32τ1)[32t3+α+1f(t32τ2)t32τ23+α2t3+α+1F(t32τ2)]    F(τ1)[3t2f(τ2)τ2(3+α)t122F(τ2)]
    =32|τ2|9+2α3t12F(t32τ1)[f(t32τ2)t32τ23+α3F(t32τ2)|t32τ2|9+2α3]    32|τ2|9+2α3t12F(τ1)[f(τ2)τ23+α3F(τ2)|τ2|9+2α3]{0, t1,0, 0<t<1,

    By (F5) and (1.15), we can easily get the above conclusion, which implies that h(t)h(1)=0 for all t>0, This shows that (2.35) holds.

    By the preceding scaling (2.10), we have

    I(ut)=t22u2212t3+αR3(IαF(t3/2u))F(t3/2u)dx. (2.37)

    It can be easily checked that J(u)=ddtI(ut)|t=1, where the definition of J is given in (2.16). Set

    h1(t):=4t323t21,  t0. (2.38)

    After basic calculations, we can see

    h1(1)=0,    h1(t)>0,     t[0,1)(1,+). (2.39)

    Inspired by [7,25], we obtain the following key inequality.

    Lemma 2.4. Assume that (F1),(F2),(F4) and (F5) hold. Then

    I(u)I(ut)+2(1t32)3J(u)+h1(t)6u22,    uH1(R3), t>0 (2.40)

    and

    I(u)23J(u)16u22,       uH1(R3). (2.41)

    Proof. By (1.12), (2.16), (2.35), (2.37), (2.38) and (2.39), we have

    I(u)I(ut)=1t22R3|u|2dx12R3(IαF(u))F(u)dx+12t3+αR3(IαF(t32u))F(t32u)dx=2(1t32)3J(u)+(1t32)R3(IαF(u))f(u)udx3+2(3+α)(1t32)6R3(IαF(u))F(u)dx+4t323t216u22+12t3+αR3(IαF(t32u))F(t32u)dx
    =2(1t32)3J(u)+h1(t)6u22+h(t)+R3(IαF(u))[f(u)u9+2α6F(u)]2(1t32)3J(u)+h1(t)6u22,       uH1(R3),  t>0. (2.42)

    This shows that (2.40) holds. Letting t0 in (2.40), we derive that (2.41) holds.

    Following the Lemma 2.4 naturally, we obtain the following corollary.

    Corollary 1. Assume that (F1),(F2),(F4) and (F5) hold. Then

    I(u)=maxt>0I(ut),     uMc. (2.43)

    Lemma 2.5. Assume that (F1),(F2),(F4) and (F5) hold. Then for any uH1(R3){0}, there exists a unique tu>0 such that utuMc.

    Proof. Let uH1(R3){0} be fixed and define a function ζ(t):=I(ut) on (0,). Clearly, by (1.12) and (2.16), we have

    ζ(t)=0  tu22=32t3+α+1R3IαF(t3/2u))[f(t3/2u)t3/2uN+α3F(t3/2u)]dx   1tJ(ut)=0    utMc. (2.44)

    Note that (F1) leads to

    |F(s)||s|r  for    |s|δ, (2.45)

    From (2.37) and (2.45), we infer that

    I(ut)t22u2212t3r3αR3(IαF(u))F(u)dx, (2.46)

    which, together with 2<3r3α implies that ζ(t)>0 for t>0 small enough. Moreover, by (2.37) and (2.13), it is easy to verify that limt0ζ(t)=0 and ζ(t)<0 for t large enough. We conclude that maxt(0,)ζ(t) is achieved at tu>0 so that ζ(tu)=0 and utuMc.

    In order to finish this proof, it is suffices to show that tu is unique for any uH1(R3){0}. Otherwise, for any given uH1(R3){0}, there exist positive constants t1t2 such that ut1,ut2Mc, i.e. J(ut1)=J(ut2)=0, then (2.39) and (2.40) lead to

    I(ut1)>I(ut2)+2[t321t322]3t321J(ut1)=I(ut2)>I(ut1)+2[t322t321]3t322J(ut2)=I(ut1). (2.47)

    This contradiction shows us that tu>0 is unique for any uH1(R3){0}.

    Combining the Corollary 1 and Lemma 2.5, we can easily obtain the following facts.

    Lemma 2.6. Assume that (F1),(F2),(F4) and (F5) hold. Then

    infuMcI(u)=m(c)=infuScmaxt>0I(ut).

    Lemma 2.7. Assume that (F1),(F2),(F4) and (F5) hold. The function cm(c) is nonincreasing on (0,).

    Proof. To achieve this purpose, it is sufficient to verify whether in the condition that for any c1<c2 and ε>0 arbitrary, we have

    m(c2)m(c1)+ε (2.48)

    By the definition of m(c1), there exists uMc1 such that I(u)m(c1)+ε/4. Let ηC0(RN) satisfies

    η(x)={1, |x|1,[0,1], 1|x|<2,0, |x|2.

    For any small δ(0,1], let

    uδ(x)=η(δx)u(x). (2.49)

    It is easy to obtain that uδu in H1(R3) as δ0. Then we have

    I(uδ)I(u)m(c1)+ε4,    J(uδ)J(u)=0. (2.50)

    From Lemma 2.5, for any δ>0, there exists tδ>0 such that uδtδMc for some c>0. Next we show that {tδ} is bounded. Actually, if tδ as δ0, since uδu0 in H1(R3) as δ0, in view of (F1), we infer that

    0=limδ0I(utδδ)t2δ=12u22+limδ0R3|IαF(t3/2δu)t5+α2δ|F(t3/2δ)ut5+α2δdx=,

    which is a contradiction. So we may assume that up to a subsequence, tδˉt as δ0, and so J(uδtδ)J(uˉt), which jointly with J(u)=0 implies that ˉt=1. By (2.40), we have

    I(utδδ)I(uδ)2(1t32δ)3J(uδ)+h1(tδ)6utδδ22,

    which, together with (2.50), implies that there exists δ0(0,1) small enough such that

    I(utδ0δ0)I(uδ0)+ε8I(u)+ε4m(c1)+ε2. (2.51)

    Let vC0(RN) be such that suppvB2Rδ0BRδ0 with Rδ0=2/δ0. Set

    v0=c2uδ022v22v,

    for which we have v022=c2uδ022. For λ(0,1), we define wλ=uδ0+vλ0 with vλ02=v02. Observing that

    dist{suppuδ0,suppvλ0}2Rδ0λRδ0=2δ0(2λ1)>0, (2.52)

    following which we can easily obtain

    |wλ(x)|2=|uδ0(x)+vλ0(x)|2=|uδ0(x)|2+|vλ0(x)|2, (2.53)
    wλ22=uδ0+vλ022=uδ02+vλ022=uδ02+v022, (2.54)
    wλ22=uδ0+vλ022=uδ02+vλ022=uδ02+λ2v022, (2.55)
    R3F(wλ)dx=R3F(uδ0+vλ0)dx=R3F(uδ0)dx+R3F(vλ0)dx=R3F(uδ0)dx+λ3R3F(λ32v0)dx (2.56)

    Then (2.55), (2.56) and (F2) imply that as λ0,

    wλ22uδ02,    R3F(wλ)dxR3F(uδ0)dx (2.57)

    and by (2.57), we have

    R3R3F(wλ(x))F(wλ(y))|xy|3αdxdyR3R3F(uδ0(x)F(uδ0(y))|xy|3αdxdy, (2.58)

    which lead to

    I(wλ)I(uδ0)  and  J(wλ)J(uδ0). (2.59)

    By (2.54), we have wλSc2. Using Lemma 2.5, there always exists tλ>0 such that wλtλMc2. As the preceding proof, the sequence {tλ} is bounded. Then assume that up to a subsequence, tλˆt as λ0. Combining the convergence (2.57) with the Hardy-Littlewood-Sobolev inequality, a standard argument can be used to show that as λ0,

    R3F(wλtλ)dxR3F(uδ0ˆt)dx. (2.60)

    and

    R3R3F(wtλλ(x))F(wtλλ(y))|xy|3αdxdyR3R3F(uˆtδ0(x)F(uˆtδ0(y))|xy|3αdxdy (2.61)

    Deduced by (2.60) and (2.61), there exists λ0(0,1) small enough such I(wλtλ)I(uδ0ˆt)+ε/2. And then it follows from (2.43) and (2.51) that

    m(c2)I(wλtλ)I(uδ0ˆt)+ε2maxt>0I(uδ0t)+ε2=I(uδ0tδ0)+ε2m(c1)+ε. (2.62)

    The proof is completed.

    Inspired by the above works, we have established the additional minimax characterization of γ(c), which can be summarized as the following lemma.

    Lemma 2.8. Assume that (F1),(F2),(F4) and (F5) hold. Then γ(c)=m(c) for any c>0.

    Proof. By (2.14), for any uMc, there exist t1<0 small and t2>1 large such that ut1Ak1 and ut2Ak2. Set

    ˉg(τ)=u(1τ)t1+τt2,     τ[0,1],

    we have ˉgΓc. By (2.43), we have

    γ(c)maxτ[0,1]I(ˉg(τ))=I(u),

    and so γ(c)infuMcI(u)=m(c) for any c>0.

    On the other hand, by (2.41), we have

    J(u)32I(u)+14u22,     uSc.

    which implies

    J(g(1))32I(g(1))<0,     gΓc.

    Moreover, it is easy to verify that there exists u0Bk1 such that J(u0)>0. Hence, any path in Γc has to go through Mc. We deduce that

    maxτ[0,1]I(g(τ))infuMcI(u)=m(c),     gΓc,

    and so γ(c)m(c) for any c>0. Therefore, γ(c)=m(c) for any c>0.

    Let H be a real Hilbert space, we define its norm and scalar products as H and (,)H respectively. Let (X,X) be a real Banach space, and devoted its dual space by X satisfying XHX and M={xXxH=1} be a submanifold of X of codimension 1.

    Lemma 2.9. Let J:XR be a C1 functional and J|M be a C1 functional restricted to M, assume that {xn}M is a bounded sequence in X. Then the following are equivalent:

    (1) J|M(xn)0 as n+;

    (1) J(xn)J(xn),xnxn0 in X as n+.

    Lemma 2.10. Let {vn}Sc be a bounded (PS)γc sequence of I|S(c). Then there exists a sequence {λn}R and λcR, vcH1(R3) such that

    (1) vnvc in H1(R3);

    (1) λnλc in R;

    (1) I(vn)λcvn0 in H1(R3).

    Proof. (1) since {vn} is bounded in H1(R3), we have vnvc in H1(R3).

    (2) Since I|Sc(vn)0 in H1(R3), by preceding Lemma 2.11, we obtain that

    I(vn)I(vn),vnvn0  in  H1(R3).

    It means that for any ωH1(R3),

    I(vn)I(vn),vnvn,ω=R3vnωR3(IαF(vn))f(vn)vnωλnR3vnw0,

    where

    λn=vn22R3(IαF(vn))f(vn)vnvn22,

    Then

    I(vn)λnvn0  in  H1(R3) (2.63)

    and λn is bounded which is deduced by the boundedness of {vn} and the Hardy-Littewood-Sobolev inequality. Finally, there exists λcR such that λnλc.

    (3) follows immediately (1), (2) and (2.63).

    Lemma 2.11. Assume that fC(R,R) satisfies the following condition:

     C>0 such that for every sR,|sf(s)|C(|s|N+αN+|s|N+αN2).

    If (u.λ)H1(R3)×R solves problem (1.1), then

    12u2232λu223+α2R3(IαF(u))F(u)=0. (2.64)

    Next Lemma can also be found in [16], for the sake of completeness and convenience for reading, we show it here again.

    Lemma 2.12. Assume that (F1)(F3) hold and (ˉvc,ˉλc)S(c)×R is a weak solution of problem (1.1), then ˉvcMc and λ<0.

    Proof. Since (ˉvc,ˉλc)S(c)×R is a weak solution of (1.1), by Lemma 2.13, we infer that

    12ˉvc22=32ˉλcˉvc22+3+α2R3(IαF(ˉvc))F(ˉvc)=32ˉvc2232R3(IαF(ˉvc))f(ˉvc)ˉvc+3+α2R3(IαF(ˉvc))F(ˉvc), (2.65)

    where we use that

    ˉλc=ˉvc22R3(IαF(ˉvc))f(ˉvc)ˉvcˉvc22. (2.66)

    Then, we have

    ˉvc22+3+α2R3(IαF(ˉvc))F(ˉvc)32R3(IαF(ˉvc))f(ˉvc)ˉvc=0. (2.67)

    i.e. ˉvcMc.

    By (2.66),

    ˉλcc=ˉvc22R3(IαF(ˉvc))f(ˉvc)ˉvc=12R3(IαF(ˉvc))f(ˉvc)ˉvc3+α2R3(IαF(ˉvc))F(ˉvc)0. (2.68)

    hence ˉλc<0. Just suppose ˉλc=0, then (F3) and (2.68) imply F(ˉλc)=0. If F(ˉλc)=0, then by (3.1) again we have that ˉvc2=0, hence I(ˉvc)=0, which is a contradiction. Then question (1.1) has no nontrivial solution in H1(R3), hence ˉλc must be negative for that ˉvc is a nontrivial solution of (1.1).

    In view of Lemmas 2.8 and 2.11, for each c>0, there exists a sequence {vn}Sc such that

    I(vn)m(c)>0,  I|Sc(vn)0  and  J(vn)0. (3.1)

    By (2.41) and (3.1), we have

    m(c)+o(1)=I(vn)23J(vn)16vn22, (3.2)

    which, combining with vn22=c, implies {vn} is bounded in H1(R3). Then there exists vH1(R3) such that up to a subsequence, vnv in H1(R3), vnv in Lsloc(R3) for 2s<6 and vnv a.e. in R3. Since m(c)=γ(c)>0, by Lions' concentration compactness principle [29,Lemma 1.21] and a standard procedure, we can obtain that {vn} is non-vanishing, and so there exist δ>0 and {yn}R3 such that B1(yn)|vn|2dx>δ. Let ˉvn(x)=vn(x+yn). Then we have ˉvn=vn and

    I(ˉvn)m(c),    J(ˉvn)=o(1),    B1(0)|ˉvn|2dx>δ. (3.3)

    Therefore, there exists ˉvH1(R3){0} such that, passing to a subsequence,

    {ˉvnˉv,in H1(R3);ˉvnˉv,in Lsloc(R3),  s[1,6);ˉvnˉv,a.e. on R3. (3.4)

    Let wn=ˉvnˉv. Then (3.4) and the Brezis-Lieb type Lemma yield

    ˉv22:=ˉcc,    wn22:=ˉcnc for large nN (3.5)

    and

    I(ˉvn)=I(ˉv)+I(wn)+o(1)  and  J(ˉvn)=J(ˉv)+J(wn)+o(1). (3.6)

    Let

    Ψ(u):=I(u)23J(u)=16u22+R3[f(u)u9+2α6F(u)]dx,     uH1(R3). (3.7)

    Then Ψ(u)>0 for all uH1(R3){0}. Moreover, it follows from (3.3), (3.6) and (3.7) that

    Ψ(wn)=m(c)Ψ(ˉv)+o(1),    J(wn)=J(ˉv)+o(1). (3.8)

    If there exists a subsequence of such that , by (F4), (3.7), (3.8), the Fatou's lemma and the weak lower semicontinuity of norm, we can deduce that . Next, we verify that this still be true for . Assume that . We claim that . Otherwise, if , then (3.8) implies for large . According to the Lemma 2.5, there exists such that . Then we can know from (1.12), (2.16), (2.40), (3.7), (3.8), Lemmas 2.7 and 2.8 that

    which contradicts . This indicates that . In view of Lemma 2.5, there exists such that . Then we can know from (2.40), (3.7), the weak semicontinuity of norm, Fatou's lemma and Lemma 2.7 that

    which implies for . In the end, we prove that . Using Lemma 2.12, there exists such that

    (3.9)

    Since , a standard procedure can be used to show that

    (3.10)

    Combining (3.9) with (3.10), we have . Hence, for any , (1.1) has a couple of solutions such that

    And by condition (F1) and the strong maximum principle, we conclude that for all . This completes the proof.

    The authors express their sincere gratitude to the anonymous referee for all insightful comments and valuable suggestions.



    [1] Normalized solutions for a system of coupled cubic Schrödinger equations on . J. Math. Pures Appl. (2016) 106: 583-614.
    [2] A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems. J. Funct. Anal. (2017) 272: 4998-5037.
    [3] Scaling properties of functionals and existence of constrained minimizers. J. Funct. Anal. (2011) 261: 2486-2507.
    [4] Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations. Proc. Lond. Math. Soc. (2013) 107: 303-339.
    [5] High energy solutions of the Choquard equation. Discrete Contin. Dyn. Syst. (2018) 38: 3023-3032.
    [6] Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete Contin. Dyn. Syst. (2019) 39: 5867-5889.
    [7] Berestycki-Lions conditions on ground state solutions for a Nonlinear Schrödinger equation with variable potentials. Adv. Nonlinear Anal. (2020) 9: 496-515.
    [8] S. Chen, X. Tang and S. Yuan, Normalized solutions for Schrödinger-Poisson equations with general nonlinearities, J. Math. Anal. Appl., 481 (2020), 123447, 24 pp. doi: 10.1016/j.jmaa.2019.123447
    [9] On the planar Schrödinger-Poisson system with the axially symmetric potential. J. Differential Equations (2020) 268: 945-976.
    [10] Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations. J. Differential Equations (2020) 268: 2672-2716.
    [11] Stationary solutions of the Schrödinger-Newton model–an ODE approach. Differential Integral Equations (2008) 21: 665-679.
    [12] Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal. (1997) 28: 1633-1659.
    [13] Sharp nonexistence results of prescribed -norm solutions for some class of Schrödinger-Poisson and quasi-linear equations. Z. Angew. Math. Phys. (2013) 64: 937-954.
    [14] Multiple normalized solutions for quasi-linear Schrödinger equations. J. Differential Equations (2015) 259: 3894-3928.
    [15] On finite energy solutions of fractional order equations of the Choquard type. Discrete Contin. Dyn. Syst. (2019) 39: 1497-1515.
    [16] G.-B. Li and H.-Y. Ye, The existence of positive solutions with prescribed -norm for nonlinear Choquard equations, J. Math. Phys., 55 (2014), 19 pp. doi: 10.1063/1.4902386
    [17] Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Studies in Appl. Math. (1977) 57: 93-105.
    [18] Solutions of Hartree-Fock equations for Coulomb systems. Comm. Math. Phys. (1987) 109: 33-97.
    [19] On regular solutions of a nonlinear equation of Choquard's type. Proc. Roy. Soc. Edinburgh Sect. A (1980) 86: 291-301.
    [20] Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics. J. Funct. Anal. (2013) 265: 153-184.
    [21] Spherically-symmetric solutions of the Schrödinger-Newton equations. Classical Quantum Gravity (1998) 15: 2733-2742.
    [22] S. I. Pekar, Üntersuchung über die Elektronentheorie der Kristalle, Akademie-Verlag, 1954.
    [23] On gravity's role in quantum state reduction. Gen. Relativity Gravitation (1996) 28: 581-600.
    [24] Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete Contin. Dyn. Syst. (2017) 37: 4973-5002.
    [25] Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions. Adv. Nonlinear Anal. (2020) 9: 413-437.
    [26] X. Tang, S. Chen, X. Lin and J. Yu, Ground state solutions of Nehari-Pankov type for Schrödinger equations with local super-quadratic conditions, J. Differential Equations, 268 (2019). doi: 10.1016/j.jde.2019.10.041
    [27] An analytical approach to the Schrödinger-Newton equations. Nonlinearity (1999) 12: 201-216.
    [28] Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four. Adv. Nonlinear Anal. (2019) 8: 715-724.
    [29] M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1
    [30] H. Ye, The mass concentration phenomenon for -critical constrained problems related to Kirchhoff equations, Z. Angew. Math. Phys., 67 (2016), 16 pp. doi: 10.1007/s00033-016-0624-4
  • This article has been cited by:

    1. Xudong Shang, Pei Ma, Normalized solutions to the nonlinear Choquard equations with Hardy-Littlewood-Sobolev upper critical exponent, 2023, 521, 0022247X, 126916, 10.1016/j.jmaa.2022.126916
    2. Xinfu Li, Standing waves to upper critical Choquard equation with a local perturbation: Multiplicity, qualitative properties and stability, 2022, 11, 2191-950X, 1134, 10.1515/anona-2022-0230
    3. Chunyu Lei, Miaomiao Yang, Binlin Zhang, Sufficient and Necessary Conditions for Normalized Solutions to a Choquard Equation, 2023, 33, 1050-6926, 10.1007/s12220-022-01151-3
    4. Xinfu Li, Nonexistence, existence and symmetry of normalized ground states to Choquard equations with a local perturbation, 2023, 68, 1747-6933, 578, 10.1080/17476933.2021.2007378
    5. Jiankang Xia, Xu Zhang, Normalized saddle solutions for a mass supercritical Choquard equation, 2023, 364, 00220396, 471, 10.1016/j.jde.2023.03.049
    6. Shengbing Deng, Junwei Yu, Normalized solutions for a Choquard equation with exponential growth in $$\mathbb {R}^{2}$$, 2023, 74, 0044-2275, 10.1007/s00033-023-01994-y
    7. Yuxi Meng, Xiaoming He, Normalized Solutions for the Fractional Choquard Equations with Hardy–Littlewood–Sobolev Upper Critical Exponent, 2024, 23, 1575-5460, 10.1007/s12346-023-00875-z
    8. Xudong Shang, Normalized ground states to the nonlinear Choquard equations with local perturbations, 2024, 32, 2688-1594, 1551, 10.3934/era.2024071
    9. Wenjing Chen, Zexi Wang, Normalized solutions for a biharmonic Choquard equation with exponential critical growth in $$\mathbb {R}^4$$, 2024, 75, 0044-2275, 10.1007/s00033-024-02200-3
    10. Zhenyu Guo, Wenyan Jin, Normalized solutions of linear and nonlinear coupled Choquard systems with potentials, 2024, 15, 2639-7390, 10.1007/s43034-024-00348-7
    11. Wenjing Chen, Zexi Wang, Normalized Ground States for a Fractional Choquard System in $$\mathbb {R}$$, 2024, 34, 1050-6926, 10.1007/s12220-024-01629-2
    12. Zi-Heng Zhang, Jian-Lun Liu, Hong-Rui Sun, Existence and Asymptotical Behavior of $$L^2$$-Normalized Standing Wave Solutions to HLS Lower Critical Choquard Equation with a Nonlocal Perturbation, 2024, 23, 1575-5460, 10.1007/s12346-024-01060-6
    13. Jinxia Wu, Xiaoming He, Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations, 2024, 13, 2191-950X, 10.1515/anona-2024-0038
    14. Yuxi Meng, Bo Wang, Multiplicity of Normalized Solutions to a Class of Non-autonomous Choquard Equations, 2025, 35, 1050-6926, 10.1007/s12220-024-01844-x
    15. Meng Li, Jinchun He, Haoyuan Xu, Meihua Yang, Normalized solutions of linearly coupled Choquard system with potentials, 2024, 0170-4214, 10.1002/mma.10556
    16. Ziheng Zhang, Jianlun Liu, Hong-Rui Sun, Normalized solutions to HLS upper critical focusing Choquard equation with a non-autonomous nonlocal perturbation, 2024, 14, 1664-2368, 10.1007/s13324-024-00979-y
    17. Meiling Zhu, Xinfu Li, Existence of normalized solutions to Choquard equation with general mixed nonlinearities, 2024, 1747-6933, 1, 10.1080/17476933.2024.2429102
    18. Shuai Yuan, Yuning Gao, Normalized solutions of Kirchhoff equations with Hartree-type nonlinearity, 2023, 31, 1844-0835, 271, 10.2478/auom-2023-0015
    19. Yuxuan Tong, Thin Van Nguyen, Sihua Liang, Concentration phenomena of normalized solutions for a fractional p-Laplacian Schrödinger–Choquard system in RN, 2025, 144, 10075704, 108665, 10.1016/j.cnsns.2025.108665
    20. Yinbin Deng, Yulin Shi, Xiaolong Yang, Normalized solutions to Choquard equation including the critical exponents and a logarithmic perturbation, 2025, 429, 00220396, 204, 10.1016/j.jde.2025.02.038
    21. Ling Huang, Giulio Romani, Positive solutions with prescribed mass for a planar Choquard equation with critical growth, 2025, 76, 0044-2275, 10.1007/s00033-025-02478-x
    22. Tao Wang, Jing Lai, Hui Guo, Multiple nodal solutions for Choquard type equations with an asymptotically linear term, 2025, 0003-6811, 1, 10.1080/00036811.2025.2496284
  • Reader Comments
  • © 2020 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(4752) PDF downloads(558) Cited by(22)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog