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Review Topical Sections

Reactive oxygen species, health and longevity

  • Reactive oxygen species (ROS) are considered responsible of ageing in animal and humans. Mitochondria are both source and target of ROS. Various strategies to reduce ROS production have been considered to extend lifespan. Caloric restriction, exercise, and antioxidants are thought to be able to protect cells from structural and functional damage. However, there is evidence that ROS production has a detrimental effect on health, but at physiological levels are necessary to stimulate longevity. They play an important effect on secondary signal transduction stimulating innate immunology and mitochondriogenesis. During exercise at moderate intensity, skeletal muscles generate ROS that are necessary for the remodelling of the muscular cells. Physical inactivity determines excessive ROS production and muscle atrophy. Caloric restriction (CR) can reduce ROS generation and improve longevity while antioxidant supplementation has shown a negative effect on longevity reducing the muscle adaptation to exercise and increasing mortality risk in patients with chronic diseases. The role of ROS in chronic diseases in also influenced by sex steroids that decrease in aging. The physiology of longevity is the result of integrated biological mechanisms that influence mitochondrial function and activity. The main objective of this review is to evaluate the effects of ROS on mitochondriogenesis and lifespan extension.

    Citation: Vittorio Emanuele Bianchi, Giancarlo Falcioni. Reactive oxygen species, health and longevity[J]. AIMS Molecular Science, 2016, 3(4): 479-504. doi: 10.3934/molsci.2016.4.479

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  • Reactive oxygen species (ROS) are considered responsible of ageing in animal and humans. Mitochondria are both source and target of ROS. Various strategies to reduce ROS production have been considered to extend lifespan. Caloric restriction, exercise, and antioxidants are thought to be able to protect cells from structural and functional damage. However, there is evidence that ROS production has a detrimental effect on health, but at physiological levels are necessary to stimulate longevity. They play an important effect on secondary signal transduction stimulating innate immunology and mitochondriogenesis. During exercise at moderate intensity, skeletal muscles generate ROS that are necessary for the remodelling of the muscular cells. Physical inactivity determines excessive ROS production and muscle atrophy. Caloric restriction (CR) can reduce ROS generation and improve longevity while antioxidant supplementation has shown a negative effect on longevity reducing the muscle adaptation to exercise and increasing mortality risk in patients with chronic diseases. The role of ROS in chronic diseases in also influenced by sex steroids that decrease in aging. The physiology of longevity is the result of integrated biological mechanisms that influence mitochondrial function and activity. The main objective of this review is to evaluate the effects of ROS on mitochondriogenesis and lifespan extension.


    In this paper, we study a class of Schrödinger-Poisson system with the following version

    {Δu+u+K(x)ϕu=|u|p1u+μh(x)u in R3,Δϕ=K(x)u2 in R3, (1)

    where p(3,5), μ>0, K(x) and h(x) are nonnegative functions. System (1) can be looked on as a non-autonomous version of the system

    {Δu+u+ϕu=f(u) in R3,Δϕ=u2 in R3, (2)

    which has been derived from finding standing waves of the Schrödinger-Poisson system

    {iψtΔψ+ϕψ=f(ψ) in R3,Δϕ=|ψ|2 in R3.

    A starting point of studying system (1) is the following fact. For any uH1(R3) and KL(R3), there is a unique ϕuD1,2(R3) with

    ϕu(x)=14πR3K(y)|u(y)|2|xy|dy

    such that Δϕu=K(x)u2, see e.g. [11,20]. Inserting this ϕu into the first equation of the system (1), we get that

    Δu+u+K(x)ϕuu=|u|p1u+μh(x)u,uH1(R3). (3)

    Problem (3) can be also looked on as a usual semilinear elliptic equation with an additional nonlocal perturbation K(x)ϕuu. Our aim here is to prove some new phenomenon of (3) due to the presence of the term K(x)ϕuu. Before giving the main results, we state the following assumptions.

    (A1) h(x)0, h(x)0 in R3 and h(x)L32(R3)L(R3).

    (A2) There exist b>0 and H0>0 such that h(x)H0eb|x| for all xR3.

    (A3) K(x)0 and K(x)L2(R3)L(R3).

    (A4) There exist a>0 and K0>0 such that K(x)K0ea|x| for all xR3.

    From Lemma 2.1, we know that under the condition (A1), the following eigenvalue problem

    Δu+u=μh(x)u,u H1(R3)

    has a first eigenvalue μ1>0 and μ1 is simple. Denote

    F(u):=R3K(x)ϕu(x)|u(x)|2dx

    and introduce the energy functional Iμ:H1(R3)R associated with (3)

    Iμ(u)=12u2+14F(u)R3(1p+1|u|p+1+μ2h(x)u2)dx,

    where u2=R3(|u|2+u2)dx. From [11] and the Sobolev inequality, Iμ is well defined and IμC1(H1(R3),R). Moreover, for any vH1(R3),

    Iμ(u),v=R3(uv+uv+K(x)ϕuuv|u|p1uv+μh(x)uv)dx.

    It is known that there is a one to one correspondence between solutions of (3) and critical points of Iμ in H1(R3). Note that if uH1(R3) is a solution of (3), then (u,ϕu) is a solution of the system (1). If u0 and u is a solution of (3), then (u,ϕu) is a nonnegative solution of (1) since ϕu is always nonnegative. We call uH1(R3){0} a bound state of (3) if Iμ(u)=0. At this time (u,ϕu) is called a bound state of (1). A bound state u is called a ground state of (3) if Iμ(u)=0 and Iμ(u)Iμ(w) for any bound state w. In this case, we call (u,ϕu) a ground state of (1). The first result is about μ less than μ1.

    Theorem 1.1. Suppose that the assumptions of (A1) - (A4) hold and 0<b<a<2. If 0<μμ1, then problem (3) has at least one nonnegative bound state.

    The second result is about μ in a small right neighborhood of μ1.

    Theorem 1.2. Under the assumptions of (A1) - (A4), if 0<b<a<1, then there exists δ>0 such that, for any μ(μ1,μ1+δ),

    (1) problem (3) has at least one nonnegative ground state u0,μ with Iμ(u0,μ)<0. Moreover, u0,μ(n)0 strongly in H1(R3) for any sequence μ(n)>μ1 and μ(n)μ1;

    (2) problem (3) has another nonnegative bound state u2,μ with Iμ(u2,μ)>0. Moreover, u2,μ(n)uμ1 strongly in H1(R3) for any sequence μ(n)>μ1 and μ(n)μ1, where uμ1 satisfies Iμ1(uμ1)=0 and Iμ1(uμ1)>0.

    The proofs of Theorem 1.1 and Theorem 1.2 are based on critical point theory. There are several difficulties in the road of getting critical points of Iμ in H1(R3) since we are dealing with the problem in the whole space R3, the embedding from H1(R3) into Lq(R3) (2<q<6) is not compact, the appearance of a nonlocal term K(x)ϕuu and the non coercive linear part. To explain our strategy, we review some related known results. For the system (2), under various conditions of f, there are a lot of papers dealing with the existence and nonexistence of positive solutions (u,ϕu)H1(R3)×D1,2(R3), see for example [2,23] and the references therein. The lack of compactness from H1(R3)Lq(R3) (2<q<6) was overcome by restricting the problem in H1r(R3) which is a subspace of H1(R3) containing only radial functions. The existence of multiple radial solutions and non-radial solutions have been obtained in [2,13]. See also [6,15,16,17,18,19,24,29,30] for some other results related to the system (2).

    While for nonautonomous version of Schrödinger-Poisson system, only a few results are known in the literature. Jiang et.al.[21] have studied the following Schrödinger-Poisson system with non constant coefficient

    {Δu+(1+λg(x))u+θϕ(x)u=|u|p2u in R3,Δϕ=u2in R3,lim|x|ϕ(x)=0,

    in which the authors prove the existence of ground state solution and its asymptotic behavior depending on θ and λ. The lack of compactness was overcome by suitable assumptions on g(x) and λ large enough. The Schrödinger-Poisson system with critical nonlinearity of the form

    {Δu+u+ϕu=V(x)|u|4u+μP(x)|u|q2uin R3,Δϕ=u2in R3,2<q<6,μ>0

    has been studied by Zhao et al. [31]. Besides some other conditions, Zhao et. al. [31] assume that V(x)C(R3,R), lim|x|V(x)=V(0,) and V(x)V for xR3 and prove the existence of one positive solution for 4<q<6 and each μ>0. It is also proven the existence of one positive solution for q=4 and μ large enough. Cerami et. al. [11] study the following type of Schrödinger-Poisson system

    {Δu+u+L(x)ϕu=g(x,u) in R3,Δϕ=L(x)u2 in R3. (4)

    Besides some other conditions and the assumption L(x)L2(R3), they prove the existence and nonexistence of ground state solutions. We emphasize that L(x)L2(R3) will imply suitable compactness property of the coupled term L(x)ϕu. Huang et. al. [20] have used this property to prove the existence of multiple solutions of (4) when g(x,u)=a(x)|u|p2u+μh(x)u and μ stays in a right neighborhood of μ1. The lack of compactness was overcome by suitable assumptions on the sign changing function a(x). While for (3), none of the aboved mentioned properties can be used. We have to analyze the energy level of the functional Iμ such that the Palais-Smale ((PS) for short) condition may hold at suitable interval. Also for (3), another difficulty is to find mountain pass geometry for the functional Iμ in the case of μμ1. We point out that for the semilinear elliptic equation

    Δu=a(x)|u|p2u+˜μk(x)u, in RN, (5)

    Costa et.al.[14] have proven the mountain pass geometry for the related functional of (5) when ˜μ˜μ1, where ˜μ1 is the first eigenvalue of Δu=˜μk(x)u in D1,2(RN). Costa et. al. have managed to do these with the help of the condition RNa(x)˜ep1dx<0, where ˜e1 is a positive eigenfunction corresponding to ˜μ1. In the present paper, it is not possible to use such kind of condition. We will develop further the techniques in [20] to prove the mountain pass geometry. A third difficulty is to look for a ground state of (3). A usual method of getting a ground state is by minimizing the functional Iμ over the Nehari set {uH1(R3){0} : Iμ(u),u=0}. But in the case of μ>μ1, one can not do like this because we do not know if 0 belongs to the boundary of this Nehari set. To overcome this trouble, we will minimize the functional over the set {uH1(R3){0} : Iμ(u)=0}.

    This paper is organized as follows. In Section 2, we give some preliminaries. Special attentions are focused on several lemmas analyzing the Palais-Smale conditions of the functional Iμ, which will play an important role in the proofs of Theorem 1.1 and Theorem 1.2. In Section 3, we prove Theorem 1.1. And Section 4 is devoted to the proof of Theorem 1.2.

    Notations. Throughout this paper, o(1) is a generic infinitesimal. The H1(R3) denotes dual space of H1(R3). Lq(R3) (1q+) is a Lebesgue space with the norm denoted by uLq. The Sp+1 is defined by

    Sp+1=infuH1(R3){0}R3(|u|2+|u|2)dx(R3|u|p+1dx)2p+1.

    For any ρ>0 and xR3, Bρ(x) denotes the ball of radius ρ centered at x. C or Cj (j=1, 2, ) denotes various positive constants, whose exact value is not important.

    In this section, we give some preliminary lemmas, which will be helpful to analyze the (PS) conditions for the functional Iμ. Firstly, for any uH1(R3) and KL(R3), defining the linear functional

    Lu(v)=R3K(x)u2vdx,vD1,2(R3),

    one may deduce from the Hölder and the Sobolev inequalities that

    |Lu(v)|Cu2L125vL6Cu2L125vD1,2. (6)

    Hence, for any uH1(R3), the Lax-Milgram theorem implies that there exists a unique ϕuD1,2(R3) such that Δϕ=K(x)u2 in D1,2(R3). Moreover it holds that

    ϕu(x)=14πR3K(y)u2(y)|xy|dy.

    Clearly ϕu(x)0 for any xR3. We also have that

    ϕu2D1,2=R3|ϕu|2dx=R3K(x)ϕuu2dx. (7)

    Using (6) and (7), we obtain that

    ϕuL6CϕuD1,2Cu2L125Cu2. (8)

    Then we deduce that

    R3K(x)ϕu(x)u2(x)dxCu4. (9)

    Hence on H1(R3), both the functional

    F(u)=R3K(x)ϕu(x)u2(x)dx (10)

    and

    Iμ(u)=12u2+14F(u)R3(1p+1|u|p+1+μ2h(x)u2)dx (11)

    are well defined and C1. Moreover, for any vH1(R3),

    Iμ(u),v=R3(uv+uv+K(x)ϕuuv|u|p1uvμh(x)uv)dx.

    The following Lemma 2.1 is a direct consequence of [28,Lemma 2.13].

    Lemma 2.1. Assume that the hypothesis (A1) holds. Then the functional uH1(R3)R3h(x)u2dx is weakly continuous and for each vH1(R3), the functional uH1(R3)R3h(x)uvdx is weakly continuous.

    Using the spectral theory of compact symmetric operators on Hilbert space, the above lemma implies the existence of a sequence of eigenvalues (μn)nN of

    Δu+u=μh(x)u,inH1(R3)

    with μ1<μ2 and each eigenvalue being of finite multiplicity. The associated normalized eigenfunctions are denoted by e1,e2, with ei=1, i=1,2,. Moreover, one has μ1>0 with an eigenfunction e1>0 in R3. In addition, we have the following variational characterization of μn:

    μ1=infuH1(R3){0}u2R3h(x)u2dx,μn=infuSn1{0}u2R3h(x)u2dx,

    where Sn1={span{e1,e2,,en1}}.

    Next we analyze the (PS) condition of the functional Iμ in H1(R3). The following definition is standard.

    Definition 2.2. For dR, the functional Iμ is said to satisfy (PS)d condition if for any (un)nNH1(R3) with Iμ(un)d and Iμ(un)0, the (un)nN contains a convergent subsequence in H1(R3). The functional Iμ is said to satisfy (PS) conditions if Iμ satisfies (PS)d condition for any dR.

    Lemma 2.3. Let (un)nNH1(R3) be such that Iμ(un)dR and Iμ(un)0, then (un)nN is bounded in H1(R3).

    Proof. For n large enough, we have that

    d+1+o(1)un=Iμ(un)14Iμ(un),un=14un2μ4R3h(x)u2ndx+p34(p+1)R3|un|p+1dx. (12)

    Note that p+1p1>32 for p(3,5). Then for any ϑ>0, we obtain from hL32(R3)L(R3) that

    R3h(x)u2ndx(R3|un|p+1dx)2p+1(R3|h(x)|p+1p1dx)p1p+12ϑp+1R3|un|p+1dx+p1p+1ϑ2p1R3|h(x)|p+1p1dx.

    Choosing ϑ=p32μ, we get

    d+1+o(1)un14un2D(p,h)μp+1p1, (13)

    where D(p,h)=p14(p+1)(p32)2p1R3|h(x)|p+1p1dx. Hence (un)nN is bounded in H1(R3).

    The following lemma is a variant of Brezis-Lieb lemma. One may find the proof in [20].

    Lemma 2.4. [20] If a sequence (un)nNH1(R3) and unu0 weakly in H1(R3), then

    limnF(un)=F(u0)+limnF(unu0).

    Lemma 2.5. There is a δ1>0 such that for any μ[μ1,μ1+δ1), any solution u of (3) satisfies

    Iμ(u)>p12(p+1)Sp+1p1p+1.

    Proof. Since u is a solution of (3), we get that

    Iμ(u)=12(u2μR3h(x)u2dx)+14F(u)1p+1R3|u|p+1dx=p12(p+1)(u2μR3h(x)u2dx)+p34(p+1)F(u).

    Noticing that u2μ1R3h(x)u2dx for any uH1(R3), we deduce that for any u0,

    Iμ1(u)p34(p+1)F(u)>0.

    Next, we claim: there is a δ1>0 such that for any μ[μ1,μ1+δ1), any solution u of (3) satisfies

    Iμ(u)>p12(p+1)Sp+1p1p+1.

    Suppose this claim is not true, then there is a sequence μ(n)>μ1 with μ(n)μ1 and solutions uμ(n) of (3) such that

    Iμ(n)(uμ(n))p12(p+1)Sp+1p1p+1.

    Note that Iμ(n)(uμ(n))=0. Then we deduce that for n large enough,

    Iμ(n)(uμ(n))+o(1)uμ(n)Iμ(n)(uμ(n))14Iμ(n)(uμ(n)),uμ(n)14uμ(n)2D(p,h)(μ(n))p+1p1.

    This implies that (uμ(n))nN is bounded in H1(R3). Since for any nN, uμ(n)2μ1R3h(x)(uμ(n))2dx, we obtain that as μ(n)μ1

    uμ(n)2μ(n)R3h(x)(uμ(n))2dx(1μ(n)μ1)uμ(n)20

    because (uμ(n))nN is bounded in H1(R3). Noting that

    Iμ(n)(uμ(n))=p12(p+1)(uμ(n)2μ(n)R3h(x)(uμ(n))2dx)+p34(p+1)F(uμ(n)),

    we deduce that

    lim infnIμ(n)(uμ(n))p34(p+1)lim infnF(uμ(n))0,

    which contradicts to the

    Iμ(n)(uμ(n))p12(p+1)Sp+1p1p+1.

    This proves the claim and the proof of Lemma 2.5 is complete.

    Lemma 2.6. If μ[μ1,μ1+δ1), then Iμ satisfies (PS)d condition for any d<0.

    Proof. Let (un)nNH1(R3) be a (PS)d sequence of Iμ with d<0. Then for n large enough,

    d+o(1)=12un2μ2R3h(x)u2ndx+14F(un)1p+1R3|un|p+1dx

    and

    Iμ(un),un=un2μR3h(x)u2ndx+F(un)R3|un|p+1dx.

    Then we can prove that (un)nN is bounded in H1(R3). Without loss of generality, we may assume that unu0 weakly in H1(R3) and unu0 a. e. in R3. Denoting wn:=unu0, we obtain from Brezis-Lieb lemma and Lemma 2.4 that for n large enough,

    un2=u02+wn2+o(1),
    F(un)=F(u0)+F(wn)+o(1)

    and

    unp+1Lp+1=u0p+1Lp+1+wnp+1Lp+1+o(1).

    Using Lemma 2.1, we also have that R3h(x)u2ndxR3h(x)u20dx as n. Therefore

    d+o(1)=Iμ(un)=Iμ(u0)+12wn2+14F(wn)1p+1R3|wn|p+1dx. (14)

    Noticing Iμ(un),ψ0 for any ψH1(R3), we obtain that Iμ(u0)=0. From which we deduce that

    u02μR3h(x)u20dx+F(u0)=R3|u0|p+1dx. (15)

    Since (un)nN is bounded in H1(R3), we obtain from Iμ(un)0 that

    o(1)=un2μR3h(x)u2ndx+F(un)R3|un|p+1dx.

    Combining this with (15) as well as Lemma 2.1, we obtain that

    o(1)=wn2+F(wn)R3|wn|p+1dx. (16)

    Recalling the definition of Sp+1, we have that u2Sp+1u2Lp+1 for any u H1(R3). Now we distinguish two cases:

    (i) R3|wn|p+1dx as n\to\infty ;

    \bf (ii) \int_{\mathbb{R}^3}|w_n|^{p+1}dx \to 0 as n\to\infty .

    Suppose that the case (ⅰ) occurs. We may obtain from (16) that

    \|w_n\|^2 \geq S_{p+1} \left(\|w_n\|^2 + F(w_n) - o(1)\right)^{\frac2{p+1}}.

    Hence we get that for n large enough,

    \begin{equation} \|w_n\|^2 \geq S_{p+1}^{\frac{p+1}{p-1}} + o(1). \end{equation} (17)

    Therefore using (14), (16) and (17), we deduce that for n large enough,

    \begin{eqnarray} & d& + o(1) = I_\mu(u_n) \\ & = & I_\mu(u_0) + \frac{1}{2}\|w_n\|^2 + \frac{1}{4}F(w_n) -\frac{1}{p+1}\int_{\mathbb{R}^3}|w_n|^{p+1}dx. \\ & = & I_\mu(u_0) + \frac{p-1}{2(p+1)}\|w_n\|^2 + \frac{p-3}{4(p+1)}F(w_n) \\ & > & - \frac{p-1}{2(p+1)}S_{p+1}^{\frac{p+1}{p-1}} + \frac{p-1}{2(p+1)}\|w_n\|^2 + \frac{p-3}{4(p+1)}F(w_n) \\ & > & 0, \end{eqnarray} (18)

    which contradicts to the condition d < 0 . This means that the case (ⅰ) does not occur. Therefore the case (ⅱ) occurs. Using (16), we deduce that \|w_n\|^2 \to 0 as n\to\infty . Hence we have proven that u_n\to u_0 strongly in H^1(\mathbb{R}^3) .

    Next we give a mountain pass geometry for the functional I_\mu .

    Lemma 2.7. There exist \delta_2 > 0 with \delta_2 \leq \delta_1 , \rho > 0 and \alpha > 0 , such that for any \mu\in [\mu_1, \mu_1 + \delta_2) , I_{\mu}|_{\partial B_{\rho}}\geq \alpha > 0.

    Proof. For any u\in H^1(\mathbb{R}^3) , there exist t\in \mathbb{R} and v\in S_1^\perp such that

    \begin{equation} u = te_1+v, \; \hbox{where}\; \int_{\mathbb{R}^3} \left(\nabla v \nabla e_1 + ve_1\right)dx = 0. \end{equation} (19)

    Hence we deduce that

    \begin{equation} \|u\|^2 = \|\nabla (te_1+v)\|_{L^2}^2 + \|te_1+v\|_{L^2}^2 = t^2+\|v\|^2, \end{equation} (20)
    \begin{equation} \mu_2\int_{\mathbb{R}^3} h(x)v^2dx\leq \|v\|^2, \; \; \mu_1\int_{\mathbb{R}^3} h(x)e_1^2dx = \|e_1\|^2 = 1 \end{equation} (21)

    and

    \begin{equation} \mu_1\int_{\mathbb{R}^3} h(x)e_1 v dx = \int_{\mathbb{R}^3} \left(\nabla v \nabla e_1+ve_1\right)dx = 0. \end{equation} (22)

    We first consider the case of \mu = \mu_1 . Denoting \theta_1: = (\mu_2-\mu_1)/2\mu_2 > 0 , then by the relations from (19) to (22), we obtain that

    \begin{array}{rl} I_{\mu_1}(u)& = \frac{1}{2} \|u\|^2+\frac{1}{4}F(u)- \frac{\mu_1}{2}\int_{\mathbb{R}^3}h(x)u^2dx - \frac{1}{p+1}\int_{\mathbb{R}^3} |u|^{p+1}dx\\ & = \frac{1}{2} \|t e_1 + v\|^2 +\frac{1}{4} F(te_1+v)\\ & \quad -\frac{\mu_1}{2}\int_{\mathbb{R}^3}h(x)(te_1+v)^2dx-\frac{1}{p+1}\int_{\mathbb{R}^3} |te_1 + v|^{p+1}dx\\ & \geq\frac{1}{2}\left(1-\frac{\mu_1}{\mu_2}\right) \|v\|^2 +\frac{1}{4} F(te_1+v)-\frac{1}{p+1}\int_{\mathbb{R}^3} |te_1 + v|^{p+1}dx\\ & \geq \theta_1\|v\|^2 +\frac{1}{4} F(te_1+v) - C_1|t|^{p+1} - C_2\|v\|^{p+1}. \end{array}

    Next we estimate the term F(te_1+v) . Using the expression of F(u) , we have that

    F(te_1+v) = \frac{1}{4\pi}\int_{\mathbb{R}^3\times \mathbb{R}^3}\frac{K(x)K(y)(te_1(y)+v(y))^2(te_1(x)+v(x))^2}{|x-y|}dydx.

    Since

    \begin{array}{rl} & \left(te_1(y)+v(y)\right)^2\left(te_1(x)+v(x)\right)^2 = t^4 (e_1(y))^2(e_1(x))^2 + (v(y))^2(v(x))^2\\ & \qquad + 2t^3\left(e_1(y)(e_1(x))^2v(y) + e_1(x)(e_1(y))^2v(x)\right)\\ & \qquad + 2t\left(e_1(x)v(x)(v(y))^2 + e_1(y)v(y)(v(x))^2\right)\\ & \qquad +t^2\left((e_1(x))^2(v(y))^2 + 4e_1(y)e_1(x)v(y)v(x) + (e_1(y))^2(v(x))^2\right), \end{array}

    we know that

    \begin{equation} \left|\int_{\mathbb{R}^3\times \mathbb{R}^3}\frac{K(x)K(y)\left(e_1(y)(e_1(x))^2v(y) + e_1(x)(e_1(y))^2v(x)\right)}{|x-y|}dydx\right|\leq C\|v\|; \end{equation} (23)
    \begin{equation} \left|\int_{\mathbb{R}^3\times \mathbb{R}^3}\frac{K(x)K(y)\left(2(e_1(x))^2(v(y))^2 + 4e_1(y)e_1(x)v(y)v(x)\right)}{|x-y|}dydx\right|\leq C\|v\|^2 \end{equation} (24)

    and

    \begin{equation} \left|\int_{\mathbb{R}^3\times \mathbb{R}^3}\frac{K(x)K(y)\left(e_1(x)v(x)(v(y))^2 + e_1(y)v(y)(v(x))^2\right)}{|x-y|}dydx\right|\leq C\|v\|^3. \end{equation} (25)

    Hence

    \begin{array}{rl} I_{\mu_1}(u)& \geq \theta_1\|v\|^2 + \theta_2|t|^4 - C_1|t|^{p+1} - C_2\|v\|^{p+1} \\ & \qquad -C_3|t|^3\|v\| - C_4|t|^2\|v\|^2 - C_5|t|\|v\|^3+\frac{1}{4}F(v), \end{array}

    where \theta_2 = \frac14 \int_{\mathbb{R}^3}K(x)\phi_{e_1} e_1^2 dx . Note that

    t^2\|v\|^2 \leq \frac2{p+1} |t|^{p+1} + \frac{p-1}{p+1} \|v\|^{\frac{2(p+1)}{p-1}},
    |t|\|v\|^3 \leq \frac1{p+1} |t|^{p+1} + \frac{p}{p+1}\|v\|^{\frac{3(p+1)}{p}}

    and for some q_0 with 2 < q_0 < 4 , we also have that

    |t|^3\|v\| \leq \frac{1}{q_0} \|v\|^{q_0} + \frac{q_0 - 1}{q_0} |t|^{\frac{3q_0}{q_0 - 1}}.

    Therefore we deduce that

    \begin{equation} \begin{array}{rl} & I_{\mu_1}(u) \geq \theta_1 \|v\|^2 + \theta_2 |t|^4 - \frac{C_3}{q_0} \|v\|^{q_0} - \frac{C_3(q_0 - 1)}{q_0} |t|^{\frac{3q_0}{q_0 - 1}}\\ & \quad -\frac{2C_4}{p+1} |t|^{p+1} - \frac{(p-1)C_4}{p+1}\|v\|^{\frac{2(p+1)}{p-1}} - \frac{C_5}{p+1} |t|^{p+1} \\ & \quad - \frac{pC_5}{p+1}\|v\|^{\frac{3(p+1)}{p}} - C|t|^{p+1} - C\|v\|^{p+1}. \end{array} \end{equation} (26)

    From q_0 > 2 and \frac{3q_0}{q_0 - 1} > 4 (since q_0 < 4 ), we know that there are positive constants \theta_3 , \theta_4 and \tilde{\theta}_3 , \tilde{\theta}_4 such that

    I_{\mu_1}(u) \geq \theta_3 \|v\|^2 + \theta_4 |t|^4

    provided that \|v\| \leq \tilde{\theta}_3 and |t|\leq \tilde{\theta}_4 . Hence there are positive constants \theta_5 and \tilde{\theta}_5 such that

    \begin{equation} I_{\mu_1}(u) \geq \theta_5 \|u\|^4\qquad \hbox{for}\quad \|u\|^2 \leq \tilde{\theta}_5^2. \end{equation} (27)

    Set \bar{\delta} : = \min\{ \frac{\mu_1}{2}\theta_5\tilde{\theta}^2_5,\ \ \mu_2 - \mu_1\} > 0 and \delta_2 : = \min \{\bar{\delta}, \delta_1\} . Then for any \mu\in [\mu_1, \mu_1 + \delta_2) , we deduce from (27) that

    \begin{array}{rl} I_\mu (u) & = I_{\mu_1}(u) + \frac{1}{2}(\mu_1 - \mu)\int_{\mathbb{R}}h(x)u^2dx\\ & \geq \theta_5 \|u\|^4 - \frac{\mu - \mu_1}{2\mu_1}\|u\|^2\\ & = \|u\|^2 \left(\theta_5 \|u\|^2 - \frac{\mu - \mu_1}{2\mu_1}\right)\\ & \geq \|u\|^2\left(\frac{1}{2} \theta_5 \tilde{\theta}^2_5 - \frac{1}{4}\theta_5\tilde{\theta}^2_5\right) = \frac{1}{4}\theta_5\tilde{\theta}^2_5\|u\|^2 \end{array}

    for \frac{1}{2}\tilde{\theta}^2_5 \leq \|u\|^2 \leq \tilde{\theta}^2_5 . Choosing \rho^2 = \frac{1}{2}\tilde{\theta}^2_5 and \alpha = \frac{1}{4}\theta_5\tilde{\theta}^2_5\rho^2, we finish the proof of Lemma 2.7.

    In this section, our aim is to prove Theorem 1.1. For 0 < \mu < \mu_1 , it is standard to prove that the functional I_{\mu} contains mountain pass geometry. For \mu = \mu_1 , as we have seen in Lemma 2.7, with the help of the competing between the Poisson term K(x)\phi_u u and the nonlinear term, the 0 is a local minimizer of the functional I_{\mu_1} and I_{\mu_1} contains mountain pass geometry. To get a mountain pass type critical point of the functional I_\mu , it suffices to prove the (PS)_{d} condition by the mountain pass theorem of [3]. In the following we will focus our attention to the case of \mu = \mu_1 , since the case of 0 < \mu < \mu_1 is similar.

    Proposition 3.1. Let the assumptions (A1)-(A4) hold and 0 < b <a < 2 . Define

    d_{\mu_1} = \inf\limits_{\gamma \in \Gamma_1}\sup\limits_{t\in [0,1]}I_{\mu_1}(\gamma(t))

    with

    \Gamma_1 = \left\{ \gamma \in C([0,1], H^1(\mathbb{R}^3))\ : \ \gamma(0) = 0,\ I_{\mu_1}(\gamma(1)) < 0 \right\}.

    Then d_{\mu_1} is a critical value of I_{\mu_1} .

    Before proving Proposition 3.1, we analyze the (PS)_{d_{\mu_1}} condition of I_{\mu_1} . Let U(x) be the unique positive solution of -\Delta u + u = |u|^{p-1}u in H^1(\mathbb{R}^3) . We know that for any \varepsilon \in (0,1) , there is a C \equiv C(\varepsilon)>0 such that U(x) \leq C e^{-(1-\varepsilon)|x|}.

    Lemma 3.2. If the assumptions (A1)-(A4) hold and 0 < b <a < 2 , then the d_{\mu_1} defined in Proposition 3.1 satisfies d_{\mu_1} < \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}}.

    Proof. It suffices to find a path \gamma(t) starting from 0 such that

    \sup\limits_{t\in [0,1]}I_{\mu_1}(\gamma(t)) < \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}}.

    Define U_R(x) = U(x-R\theta) with \theta = (0,0,1) . Note that for the U_R defined as above, the I_{\mu_1}(tU_R) \to -\infty as t\to +\infty and I_{\mu_1}(tU_R) \to 0 as t\to 0 . We know that there is a unique T_R > 0 such that \frac{\partial}{\partial t}I_{\mu_1}(t U_R)|_{t = T_R} = 0 , which is

    \|U_R\|^2 -\mu_1\int h(x)U_R^2 dx+ T_R^2F(U_R) - T_R^{p-1}\int U_R^{p+1} dx = 0.

    If T_R\to 0 as R\to\infty , then \|U_R\|^2 -\mu_1\int h(x)U_R^2dx \to 0 as R\to\infty , which is impossible. If T_R\to\infty as R\to\infty , then as R\to\infty ,

    \frac{1}{T_R^2}\left(\|U_R\|^2 -\mu_1\int h(x)U_R^2dx\right)+F(U_R) = T_R^{p-3}\int U_R^{p+1}dx \to \infty,

    which is impossible either. Hence we only need to estimate I_{\mu_1}(tU_R) for t in a finite interval and we may write

    I_{\mu_1}(tU_R) \leq g(t) + CF(U_R),

    where

    g(t) = \frac{t^2}{2}\left(\|U_R\|^2 - \mu_1 \int_{\mathbb{R}^3} h(x) U_R^2 dx\right) - \frac{|t|^{p+1}}{p+1}\int_{\mathbb{R}^3} U_R^{p+1} dx.

    Noting that under the assumptions (A1)-(A4) , we obtain that for R large enough,

    \begin{equation} \begin{array}{rl} F(U_R) & \leq \left(\int_{\mathbb{R}^3}K(x)^{\frac65}U_R^{\frac{12}{5}}dx\right)^{\frac56} \left(\int_{\mathbb{R}^3} \phi_{U_R}^6dx\right)^{\frac16}\\ & \leq C\left(\int_{\mathbb{R}^3} e^{-\frac65 a|x+R\theta|}(U(x))^{\frac{12}{5}} dx\right)^{\frac56}\\ & \leq C\left(\int_{\mathbb{R}^3} e^{-{\frac65}aR}e^{({\frac65}a-{\frac{12}5}(1-\varepsilon))|x|} dx\right)^{\frac56} \leq C e^{-aR} \end{array} \end{equation} (28)

    since 0 < a < 2 . We can also prove that

    \begin{equation} \begin{array}{rl} & \int_{\mathbb{R}^3}h(x)U_R^2dx = \int_{\mathbb{R}^3}h(x+R\theta)U^2(x)dx\\ & \geq C\int_{\mathbb{R}^3} e^{-b|x+R\theta|}U^2(x) dx \geq C\int_{\mathbb{R}^3} e^{-b|x| -bR } U^2(x) dx\\ & \geq C e^{-bR} \int_{\mathbb{R}^3} e^{-b|x|} U^2(x)dx \geq C e^{-bR}. \end{array} \end{equation} (29)

    It is now deduced from (28) and (29) that

    \begin{array}{rl} & \sup\limits_{t > 0} I_{\mu_1}(tU_R) \leq \sup\limits_{t > 0} g(t) + C e^{-aR} \\ &\leq \frac{p-1}{2(p+1)} \left(\|U_R\|^2 - \mu_1 \int_{\mathbb{R}^3} h(x) U_R^2 dx\right)^{\frac{p+1}{p-1}}(\|U_R\|_{L^{p+1}}^{-2})^{\frac{p+1}{p-1}} + C e^{-aR} \\ &\leq \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}} - Ce^{-bR} + o(e^{-bR}) + C e^{-aR} \\ & < \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}} \end{array}

    for R large enough since 0 < b < a . The proof is complete.

    Lemma 3.3. Under the assumptions (A1)-(A4) , I_{\mu_1} satisfies (PS)_d condition for any d < \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}} .

    Proof. Let (u_n)_{n\in\mathbb{N}}\subset H^1(\mathbb{R}^3) be a (PS)_d sequence of I_{\mu_1} with d < \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}} . Then we have that for n large enough,

    d + o(1) = \frac{1}{2}\|u_n\|^2 - \frac{\mu_1}{2}\int_{\mathbb{R}^3}h(x)u_n^2dx + \frac{1}{4}F(u_n) -\frac{1}{p+1}\int_{\mathbb{R}^3}|u_n|^{p+1}dx

    and

    \langle I'_{\mu_1} (u_n), u_n\rangle = \|u_n\|^2 - \mu_1\int_{\mathbb{R}^3}h(x)u_n^2dx + F(u_n) - \int_{\mathbb{R}^3}|u_n|^{p+1}dx.

    Hence we can deduce that (u_n)_{n\in\mathbb{N}} is bounded in H^1(\mathbb{R}^3) . Going if necessary to a subsequence, we may assume that u_n \rightharpoonup u_0 weakly in H^1(\mathbb{R}^3) and u_n \to u_0 a. e. in \mathbb{R}^3 . Denote w_n : = u_n - u_0 . We then obtain from Brezis-Lieb lemma and Lemma 2.4 that for n large enough,

    \|u_n\|^2 = \|u_0\|^2 + \|w_n\|^2 +o(1),\quad F(u_n) = F(u_0) + F(w_n) +o(1)

    and

    \|u_n\|^{p+1}_{L^{p+1}} = \|u_0\|^{p+1}_{L^{p+1}} + \|w_n\|^{p+1}_{L^{p+1}} +o(1).

    Since \int_{\mathbb{R}^3}h(x)u_n^2dx\to \int_{\mathbb{R}^3}h(x)u_0^2dx as n\to\infty , we deduce that

    \begin{eqnarray} & & d + o(1) = I_{\mu_1}(u_n) = I_{\mu_1}(u_0) + \frac{1}{2}\| w_n\|^2\\ & & \qquad \quad + \frac{1}{4}F(w_n) -\frac{1}{p+1}\int_{\mathbb{R}^3}|w_n|^{p+1}dx. \end{eqnarray} (30)

    From \langle I'_{\mu_1} (u_n), \psi\rangle \to 0 for any \psi\in H^1(\mathbb{R}^3) , one may deduce that I'_{\mu_1} (u_0) = 0 . Therefore

    \|u_0\|^2 - \mu_1\int_{\mathbb{R}^3}h(x)u_0^2dx + F(u_0) = \int_{\mathbb{R}^3}|u_0|^{p+1}dx

    and then

    I_{\mu_1}(u_0) \geq \frac{p-1}{2(p+1)}\left(\|u_0\|^2 - \mu_1 \int_{\mathbb{R}^3}h(x)u_0^2dx\right) + \frac{p-3}{4(p+1)}F(u_0) \geq 0.

    Now using an argument similar to the proof of (16), we obtain that

    \begin{equation} o(1) = \|w_n\|^2 + F(w_n) - \int_{\mathbb{R}^3}|w_n|^{P+1}dx. \end{equation} (31)

    By the relation \|u\|^2 \geq S_{p+1} \|u\|_{L^{p+1}}^2 for any u\in H^1(\mathbb{R}^3) , we proceed our discussion according to the following two cases:

    \bf (I) \int_{\mathbb{R}^3}|w_n|^{p+1}dx\not\to 0 as n\to\infty ;

    \bf (II) \int_{\mathbb{R}^3}|w_n|^{p+1}dx \to 0 as n\to\infty .

    Suppose that the case (I) occurs. Then up to a sbusequence, we may obtain from (31) that

    \|w_n\|^2 \geq S_{p+1} \left(\| w_n\|^2 + F(w_n) - o(1)\right)^{\frac2{p+1}},

    which implies that for n large enough,

    \|w_n\|^2 \geq S_{p+1}^{\frac{p+1}{p-1}} + o(1).

    It is deduced from this and (30) that d \geq \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}}, which is a contradiction. Therefore the case (II) must occur. This and (31) imply that \|w_n\| \to 0 . Hence we have proven that I_{\mu_1} satisfies (PS)_d condition for any d < \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}} .

    Proof of Proposition 3.1. Since 0 is a local minimizer of I_{\mu_1} and for v\neq 0 , I_{\mu_1}(s v) \to -\infty as s\to +\infty , Lemma 3.2, Lemma 3.3 and the mountain pass theorem [3] imply that d_{\mu_1} is a critical value of I_{\mu_1} .

    Proof of Theorem 1.1. By Proposition 3.1, the d_{\mu_1} is a critical value of I_{\mu_1} and d_{\mu_1} > 0 . The proof of nonnegativity for at least one of the corresponding critical point is inspired by the idea of [1]. In fact, since I_{\mu_1}(u) = I_{\mu_1}(|u|) for any u \in H^1(\mathbb{R}^3) , for every n\in \mathbb{N} , there exists \gamma_n\in \Gamma_1 with \gamma_n(t)\geq 0 (a.e. in \mathbb{R}^3 ) for all t\in [0, 1] such that

    \begin{equation} d_{\mu_1} \leq \max\limits_{t\in [0, 1]} I_{\mu_1}(\gamma_n(t)) < d_{\mu_1} + \frac{1}{n}. \end{equation} (32)

    By Ekeland's variational principle [5], there exists \gamma_n^*\in \Gamma_1 satisfying

    \begin{equation} \left\{ \begin{array}{ll} d_{\mu_1} \leq \max\limits_{t\in [0, 1]} I_{\mu_1} (\gamma_n^*(t))\leq \max\limits_{t\in [0, 1]} I_{\mu_1} (\gamma_n(t)) < d_{\mu_1} +\frac{1}{n};\\ \max\limits_{t\in [0, 1]}\|\gamma_n(t)-\gamma_n^*(t)\| < \frac{1}{\sqrt{n}}; \\ \hbox{ there exists} \; t_n\in [0, 1]\; \hbox{such that}\; z_n = \gamma_n^*(t_n) \hbox{ satisfies}: \\ I_{\mu_1}(z_n) = \max\limits_{t\in [0, 1]} I_{\mu_1}(\gamma_n^*(t)), \;\hbox{and}\; \|I'_{\mu_1}(z_n)\|\leq\frac{1}{\sqrt{n}}. \end{array} \right. \end{equation} (33)

    By Lemma 3.2 and Lemma 3.3 we get a convergent subsequence (still denoted by (z_n)_{n\in \mathbb{N}} ). We may assume that z_n\rightarrow z in H^1(\mathbb{R}^3) as n\rightarrow \infty . On the other hand, by (33), we also arrive at \gamma_n(t_n)\rightarrow z in H^1(\mathbb{R}^3) as n\rightarrow \infty . Since \gamma_n(t)\geq 0 , we conclude that z\geq 0 , z\not\equiv 0 in \mathbb{R}^3 with I_{\mu_1}(z)>0 and it is a nonnegative bound state of (3) in the case of \mu = \mu_1 .

    In this section, we always assume the conditions (A1)-(A4) . We will prove the existence of ground state and bound states of (3) as well as their asymptotical behavior with respect to \mu . We emphasize that if 0 < \mu < \mu_1 , then one may consider a minimization problem like

    \inf\{ I_\mu(u)\ : \ u\in \mathcal{M}\},\quad \mathcal{M} = \{u\in H^1(\mathbb{R}^3)\ :\ \langle I'_\mu(u), u\rangle = 0\}

    to get a ground state solution. But for \mu \geq \mu_1 , we can not do like this because for \mu > \mu_1 , we do not know if 0 \not\in \partial\mathcal{M} . To overcome this difficulty, we define the set of all nontrivial critical points of I_\mu in H^1(\mathbb{R}^3) :

    \mathcal{N} = \{u\in H^1(\mathbb{R}^3)\backslash\{0\}:I_\mu'(u) = 0\}.

    And then we consider the following minimization problem

    \begin{equation} c_{0,\mu} = \inf\{I_\mu(u):u\in\mathcal{N}\}. \end{equation} (34)

    Lemma 4.1. Let \delta_2 and \rho be as in Lemma 2.7 and \mu\in (\mu_1, \mu_1 + \delta_2) . Define the following minimization problem

    d_{0, \mu} = \inf\limits_{\|u\| < \rho} I_\mu (u).

    Then the d_{0, \mu} is achieved by a nonnegative function w_{0,\mu}\in H^1(\mathbb{R}^3) . Moreover this w_{0,\mu} is a nonnegative solution of (3).

    Proof. Firstly, we prove that -\infty < d_{0, \mu} < 0 for \mu\in (\mu_1, \mu_1 + \delta_2) . Keeping the expression of I_\mu(u) in mind, we obtain from the Sobolev inequality that

    \begin{eqnarray} I_\mu(u) & = &\frac12 \|u\|^2-\frac{\mu}{2}\int_{\mathbb{R}^3} h(x)u^2dx + \frac14 F(u) -\frac{1}{p+1}\int_{\mathbb{R}^3}|u|^{p+1}dx \\ &\geq& \frac12 \|u\|^2-\frac{\mu}{2\mu_1}\|u\|^2 - C\|u\|^{p+1} > -\infty \end{eqnarray}

    as \|u\| < \rho . Next, for any t > 0 , we have that

    I_\mu(te_1) = \frac{t^2}2 \|e_1\|^2-\frac{\mu t^2}{2}\int_{\mathbb{R}^3} h(x)e_1^2dx + \frac{t^4}4 F(e_1) - \frac{t^{p+1}}{p+1}\int_{\mathbb{R}^3}|e_1|^{p+1}dx.

    It is now deduced from \mu_1\int_{\mathbb{R}^3} h(x)e_1^2dx = \|e_1\|^2 that

    I_\mu(te_1) = \frac{t^2}2\left(1-\frac{\mu}{\mu_1}\right)\|e_1\|^2 + \frac{t^4}4 F(e_1) -\frac{t^{p+1}}{p+1}\int_{\mathbb{R}^3}|e_1|^{p+1}dx.

    Since \mu > \mu_1 , we obtain that for t small enough, the I_\mu(t e_1) < 0 . Thus we have proven that -\infty < d_{0, \mu} < 0 for \mu\in (\mu_1, \mu_1 + \delta_2) .

    Secondly, let (v_n)_{n\in\mathbb{N}} be a minimizing sequence, that is, \|v_n\| < \rho and I_\mu(v_n)\to d_{0, \mu} as n\to\infty . By the Ekeland's variational principle, we can obtain that there is a sequence (u_n)_{n\in\mathbb{N}}\subset H^1(\mathbb{R}^3) with \|u_n\| < \rho such that as n\to\infty ,

    I_\mu(u_n) \to d_{0, \mu}\qquad \hbox{and}\qquad I'_\mu(u_n) \to 0.

    Then we can prove that (u_n)_{n\in\mathbb{N}} is bounded in H^1(\mathbb{R}^3) . Using Lemma 2.6, we obtain that (u_n)_{n\in\mathbb{N}} contains a convergent subsequence, still denoted by (u_n)_{n\in\mathbb{N}} , such that u_n\to u_0 strongly in H^1(\mathbb{R}^3) . Noticing the fact that if (v_n)_{n\in\mathbb{N}} is a minimizing sequence, then (|v_n|)_{n\in\mathbb{N}} is also a minimizing sequence, we may assume that for each n\in \mathbb{N} , the u_n\geq 0 in \mathbb{R}^3 . Therefore we may assume that u_0 \geq 0 in \mathbb{R}^3 . The I'_\mu(u_n) \to 0 and u_n\to u_0 strongly in H^1(\mathbb{R}^3) imply that I_\mu'(u_0) = 0 . Hence choosing w_{0,\mu} \equiv u_0 , we know that w_{0,\mu} is a nonnegative solution of the (3).

    We emphasize that the above lemma does NOT mean that w_{0,\mu} is a ground state of (3). But it does imply that \mathcal{N} \neq\emptyset for any \mu\in (\mu_1, \mu_1 + \delta_2) . Now we are in a position to prove that the c_{0,\mu} defined in (34) can be achieved.

    Lemma 4.2. For \mu\in (\mu_1, \mu_1 + \delta_2) , the c_{0,\mu} is achieved by a nontrivial v_{0, \mu}\in H^1(\mathbb{R}^3) , which is a nontrivial critical point of I_\mu and hence a solution of the (3).

    Proof. By Lemma 4.1, we know that \mathcal{N} \neq\emptyset for \mu\in (\mu_1, \mu_1 + \delta_2) . Hence we have that c_{0,\mu} < 0 . Next we prove that the c_{0,\mu} > -\infty .

    For any u\in \mathcal{N} , since I'_\mu(u) = 0 , then \langle I'_\mu(u), u\rangle = 0 . Then we can deduce that

    I_\mu(u) = I_\mu(u) - \frac{1}{4}\langle I'_\mu(u), u\rangle \geq \frac{1}{4}\|u\|^2 - D(p,h) \mu^{\frac{p+1}{p-1}}.

    Therefore the c_{0,\mu} > -\infty.

    Now let (u_n)_{n\in\mathbb{N}}\subset \mathcal{N} be a sequence such that

    I_\mu(u_n) \to c_{0,\mu}\qquad \hbox{and}\qquad I'_\mu(u_n) = 0.

    Since -\infty < c_{0,\mu} < 0 , we know from Lemma 2.6 that (u_n)_{n\in\mathbb{N}} contains a convergent subsequence in H^1(\mathbb{R}^3) and then we may assume without loss of generality that u_n \to v_0 strongly in H^1(\mathbb{R}^3) . Therefore we have that I_\mu(v_0) = c_{0,\mu} and I'_\mu(v_0) = 0 . Choosing v_{0,\mu} \equiv v_0 and we finish the proof of the Lemma 4.2.

    Next, to analyze further the (PS)_d condition of the functional I_\mu , we have to prove a relation between the minimizer w_{0,\mu} obtained in Lemma 4.1 and the minimizer v_{0,\mu} obtained in Lemma 4.2.

    Lemma 4.3. There exists \delta_3 \in (0, \delta_2] such that for any \mu\in (\mu_1, \mu_1 + \delta_3) , the v_{0,\mu} obtained in Lemma 4.2 can be chosen to coincide the w_{0,\mu} obtained in Lemma 4.1.

    Proof. The proof is divided into two steps. In the first place, for u\neq 0 and I_{\mu_1}'(u) = 0 , we have that

    \|u\|^2 - \mu_1\int_{\mathbb{R}^3} h(x)u^2dx + F(u) = \int_{\mathbb{R}^3} |u|^{p+1}dx

    and hence

    I_{\mu_1}(u) = \frac{p-1}{2(p+1)} \left(\|u\|^2 - \mu_1\int_{\mathbb{R}^3} h(x)u^2dx\right) + \frac{p-3}{4(p+1)}F(u).

    Since \|u\|^2 \geq \mu_1\int_{\mathbb{R}^3} h(x)u^2dx for any u\in H^1(\mathbb{R}^3) , we obtain that

    I_{\mu_1}(u) \geq \frac{p-3}{4(p+1)}F(u) > 0.

    In the second place, denoted by u_{0,\mu} a ground state obtained in Lemma 4.2. For any sequence \mu^{(n)} > \mu_1 and \mu^{(n)} \to \mu_1 as n\to\infty , we have that u_{0,\mu^{(n)}} satisfies

    I'_{\mu^{(n)}}(u_{0,\mu^{(n)}}) = 0

    and we also have that

    c_{0, \mu^{(n)}} = I_{\mu^{(n)}}(u_{0,\mu^{(n)}}) < 0.

    Hence we deduce that (u_{0,\mu^{(n)}})_{n\in \mathbb{N}} is bounded in H^1(\mathbb{R}^3) . Since I'_{\mu^{(n)}}(u_{0,\mu^{(n)}}) = 0 , one also has that

    \begin{array}{rl} I_{\mu^{(n)}}(u_{0,\mu^{(n)}}) & = \frac{p-1}{2(p+1)} \left(\|u_{0,\mu^{(n)}}\|^2 - \mu^{(n)}\int_{\mathbb{R}^3} h(x)(u_{0,\mu^{(n)}})^2dx\right)\\ & \qquad \quad + \frac{p-3}{4(p+1)}F(u_{0,\mu^{(n)}}). \end{array}

    Using the definition of \mu_1 , we obtain that, as n\to \infty ,

    \|u_{0,\mu^{(n)}}\|^2 - \mu^{(n)}\int_{\mathbb{R}^3} h(x)(u_{0,\mu^{(n)}})^2dx \geq \left(1-\frac{\mu^{(n)}}{\mu_1}\right)\|u_{0,\mu^{(n)}}\|^2\to 0

    because (u_{0,\mu^{(n)}})_{n\in \mathbb{N}} is bounded in H^1(\mathbb{R}^3) . Next since (u_{0,\mu^{(n)}})_{n\in\mathbb{N}} is bounded in H^1(\mathbb{R}^3) , we may assume without loss of generality that u_{0,\mu^{(n)}}\rightharpoonup \tilde{u}_0 weakly in H^1(\mathbb{R}^3) .

    Claim. As n\to \infty , the u_{0,\mu^{(n)}} \to \tilde{u}_0 strongly in H^1(\mathbb{R}^3) and \tilde{u}_0 = 0 .

    Proof of the Claim. From u_{0,\mu^{(n)}}\rightharpoonup \tilde{u}_0 weakly in H^1(\mathbb{R}^3) , we may assume that u_{0,\mu^{(n)}}\to \tilde{u}_0 a. e. in \mathbb{R}^3 . Using these and the fact of I'_{\mu^{(n)}}(u_{0,\mu^{(n)}}) = 0 , we deduce that I'_{\mu_1}(\tilde{u}_0) = 0 . Then similar to the proof in Lemma 2.6, we obtain that

    \begin{eqnarray} o(1) + I_{\mu^{(n)}}(u_{0,\mu^{(n)}}) & = & I_{\mu^{(n)}}(\tilde{u}_0) + \frac{1}{2}\|\tilde{w}_n\|^2 \\ & & + \frac{1}{4}F(\tilde{w}_n) - \frac{1}{p+1}\int_{\mathbb{R}^3}|\tilde{w}_n|^{p+1}dx, \end{eqnarray} (35)

    where \tilde{w}_n : = u_{0,\mu^{(n)}} - \tilde{u}_0 .

    Now we distinguish two cases:

    \bf (i) \int_{\mathbb{R}^3}|\tilde{w}_n|^{p+1}dx\not\to 0 as n\to\infty ;

    \bf (ii) \int_{\mathbb{R}^3}|\tilde{w}_n|^{p+1}dx \to 0 as n\to\infty .

    Suppose that the case (ⅰ) occurs. We may deduce from a proof similar to Lemma 2.6 that

    I_{\mu^{(n)}}(u_{0,\mu^{(n)}}) + o(1) \geq I_{\mu_1}(\tilde{u}_0) + \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}},

    which is a contradiction because I_{\mu_1}(\tilde{u}_0) > - \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}} by Lemma 2.5 and the fact of I_{\mu^{(n)}}(u_{0,\mu^{(n)}}) < 0 . Therefore the case (ii) occurs, which implies that u_{0,\mu^{(n)}} \to \tilde{u}_0 strongly in H^1(\mathbb{R}^3) (the proof is similar to those in Lemma 2.6). From this we also deduce that F(\tilde{w}_n) \to F(\tilde{u}_0).

    Next we prove that \tilde{u}_0 = 0 . Arguing by a contradiction, if \tilde{u}_0 \neq 0 , then we know from I'_{\mu^{(n)}}(u_{0,\mu^{(n)}}) = 0 that

    \liminf\limits_{n\to\infty} I_{\mu^{(n)}}(u_{0,\mu^{(n)}}) \geq \frac{p-3}{4(p+1)} F(\tilde{u}_0) > 0,

    which is also a contradiction since I_{\mu^{(n)}}(u_{0,\mu^{(n)}}) < 0 . Therefore \tilde{u}_0 = 0 .

    Hence there is \delta_3 \in (0, \delta_2] such that for any \mu\in (\mu_1, \mu_1 + \delta_3) , \|u_{0,\mu}\| < \rho , which implies that c_{0,\mu} = d_{0,\mu} . Using Lemma 4.1, we can get a nonnegative ground state of (3), called w_{0,\mu} and c_{0,\mu} = d_{0,\mu} = I_\mu(w_{0,\mu}) . The proof is complete.

    Remark 4.4. The proof of Lemma 4.3 implies that (1) of Theorem 1.2 holds.

    In the following, we are going to prove the existence of another nonnegative bound state solution of (3). To obtain this goal, we have to analyze further the (PS)_d condition of the functional I_\mu .

    Lemma 4.5. Under the assumptions of (A1)-(A4) , if \mu\in (\mu_1, \mu_1 + \delta_3) , then I_\mu satisfies (PS)_d condition for any d < c_{0,\mu} + \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}} .

    Proof. Let (u_n)_{n\in\mathbb{N}}\subset H^1(\mathbb{R}^3) be a (PS)_d sequence of I_\mu with d < c_{0,\mu} + \frac{p-1}{2(p+1)}S_{p+1}^{\frac{p+1}{p-1}} . Then we have that for n large enough,

    d + o(1) = \frac{1}{2}\|u_n\|^2 - \frac{\mu}{2}\int_{\mathbb{R}^3}h(x)u_n^2dx + \frac{1}{4}F(u_n) -\frac{1}{p+1}\int_{\mathbb{R}^3}|u_n|^{p+1}dx

    and

    \langle I'_\mu (u_n), u_n\rangle = \|u_n\|^2 - \mu\int_{\mathbb{R}^3}h(x)u_n^2dx + F(u_n) - \int_{\mathbb{R}^3}|u_n|^{p+1}dx.

    Similar to the proof in Lemma 2.3, we can deduce that (u_n)_{n\in\mathbb{N}} is bounded in H^1(\mathbb{R}^3) . Going if necessary to a subsequence, we may assume that u_n \rightharpoonup u_0 weakly in H^1(\mathbb{R}^3) and u_n \to u_0 a. e. in \mathbb{R}^3 . Denote w_n : = u_n - u_0 . We then obtain from Brezis-Lieb lemma and Lemma 2.4 that for n large enough,

    \|u_n\|^2 = \|u_0\|^2 + \|w_n\|^2 +o(1),
    F(u_n) = F(u_0) + F(w_n) +o(1)

    and

    \|u_n\|^{p+1}_{L^{p+1}} = \|u_0\|^{p+1}_{L^{p+1}} + \|w_n\|^{p+1}_{L^{p+1}} +o(1).

    Using Lemma 2.1, we also have that \int_{\mathbb{R}^3}h(x)u_n^2dx\to \int_{\mathbb{R}^3}h(x)u_0^2dx as n\to\infty . Therefore we deduce that

    \begin{eqnarray} & & d + o(1) = I_\mu(u_n) = I_\mu(u_0) + \frac{1}{2}\| w_n\|^2\\ & & \qquad \quad + \frac{1}{4}F(w_n) -\frac{1}{p+1}\int_{\mathbb{R}^3}|w_n|^{p+1}dx. \end{eqnarray} (36)

    Since \langle I'_\mu (u_n), \psi\rangle \to 0 for any \psi\in H^1(\mathbb{R}^3) , we know that I'_\mu (u_0) = 0 . Moreover we have that

    I_\mu(u_0) \geq c_{0,\mu}

    and

    \|u_0\|^2 - \mu\int_{\mathbb{R}^3}h(x)u_0^2dx + \int_{\mathbb{R}^3}\phi_{u_0}u_0^2 = \int_{\mathbb{R}^3}|u_0|^{p+1}dx.

    Note that (u_n)_{n\in\mathbb{N}} is bounded in H^1(\mathbb{R}^3) . The Brezis-Lieb lemma, Lemma 2.4 and

    o(1) = \|u_n\|^2 - \mu\int_{\mathbb{R}^3}h(x)u_n^2dx + F(u_n) - \int_{\mathbb{R}^3}|u_n|^{p+1}dx

    imply that

    \begin{equation} o(1) = \| w_n\|^2 + F(w_n) -\int_{\mathbb{R}^3}|w_n|^{p+1}dx. \end{equation} (37)

    Using \|u\|^2 \geq S_{p+1} \|u\|^2_{L^{p+1}} for any u\in H^1(\mathbb{R}^3) , we distinguish two cases:

    \bf (I) \int_{\mathbb{R}^3}|w_n|^{p+1}dx\not\to 0 as n\to\infty ;

    \bf (II) \int_{\mathbb{R}^3}|w_n|^{p+1}dx \to 0 as n\to\infty .

    Suppose (I) occurs. Up to a subsequence, we may obtain from (37) that

    \|w_n\|^2 \geq S_{p+1} \left(\|w_n\|^2 + F(w_n) - o(1)\right)^{\frac2{p+1}}.

    Hence we get that for n large enough,

    \begin{equation} \|w_n\|^2 \geq S_{p+1}^{\frac{p+1}{p-1}} + o(1). \end{equation} (38)

    Therefore using (36) and (38), we deduce that for n large enough,

    \begin{eqnarray} & d & + o(1) = I_\mu(u_n) \\ & = & I_\mu(u_0) + \frac{1}{2}\|w_n\|^2 + \frac{1}{4}F(w_n) -\frac{1}{p+1}\int_{\mathbb{R}^3}|w_n|^{p+1}dx \\ & = & I_\mu(u_0) + \frac{p-1}{2(p+1)}\|w_n\|^2 + \frac{p-3}{4(p+1)}F(w_n) \\ & \geq & c_{0,\mu} + \frac{p-1}{2(p+1)}\|w_n\|^2 + \frac{p-3}{4(p+1)}F(w_n) \\ & > & c_{0,\mu} + \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}}, \end{eqnarray} (39)

    which contradicts to the assumption d < c_{0,\mu} + \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}} . Therefore the case (Ⅱ) must occur, i.e., \int_{\mathbb{R}^3}|w_n|^{p+1}dx\to 0 as n\to\infty . This and (37) imply that \|w_n\| \to 0 . Hence we have proven that I_\mu satisfies (PS)_d condition for any d < c_{0,\mu} + \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}} .

    Next, for the w_{0,\mu} obtained in Lemma 4.3, we define

    d_{2, \mu} = \inf\limits_{\gamma \in \Gamma_2}\sup\limits_{t\in [0,1]}I_\mu(\gamma(t))

    with

    \Gamma_2 = \{ \gamma \in C([0,1], H^1(\mathbb{R}^3))\ : \ \gamma(0) = w_{0, \mu},\ I_\mu(\gamma(1)) < c_{0,\mu} \}.

    Lemma 4.6. Suppose that the conditions (A1)-(A4) hold and 0<b<a<1 . If \mu\in (\mu_1, \mu_1 + \delta_3) , then

    d_{2, \mu} < c_{0,\mu} + \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}}.

    Proof. It suffices to find a path starting from w_{0, \mu} and the maximum of the energy functional over this path is strictly less than c_{0,\mu} + \frac{p-1}{2(p+1)} S_{p+1}^{\frac{p+1}{p-1}}. To simplify the notation, we denote w_0 : = w_{0,\mu} , which corresponds to the critical value c_{0,\mu} . We will prove that there is a T_0 such that the path \gamma(t) = w_0 + t T_0U_R is what we need, here U_R(x) \equiv U(x-R\theta) is defined as before. Similar to the discussion in the proof of Lemma 3.2, we only need to estimate I_\mu(w_0 + tU_R) for positive t in a finite interval. By direct calculation, we have that

    \begin{array}{rl} I_\mu(w_0 + tU_R) & = \frac{1}{2}\left(\left\|w_0 + tU_R\right\|^2 - \mu \int_{\mathbb{R}^3} h(x)|w_0 + t U_R|^2dx\right) \\ & \quad + \frac{1}{4}F(w_0 + tU_R) - \frac1{p+1} \int_{\mathbb{R}^3}|w_0 + t U_R|^{p+1}dx \\ & = I_\mu(w_0) + A_1 + A_2 + A_3 +\frac{t^2}{2}\|U_R\|^2 - \frac{\mu}{2}\int_{\mathbb{R}^3}h(x)U_R^2dx, \end{array}

    where

    A_1 = \langle w_0, tU_R\rangle - \mu t\int_{\mathbb{R}^3} h(x)w_0 U_R dx,
    A_2 = \frac{1}{4}\left(F(w_0 + tU_R) - F(w_0)\right)

    and

    A_3 = \frac1{p+1} \int_{\mathbb{R}^3}\left(|w_0|^{p+1} - |w_0 + t U_R|^{p+1}\right)dx.

    Since w_0 is a solution of (3), we have that

    A_1 = \int_{\mathbb{R}^3}(w_0)^p t U_R dx - \int_{\mathbb{R}^3} K(x)\phi_{w_0}w_0 t U_R dx.

    From an elementary inequality:

    (a+b)^q - a^q \geq b^q + q a^{q-1}b,\qquad q > 1,\ \ a > 0, b > 0,

    we deduce that

    |A_3| \leq - \frac1{p+1}\int_{\mathbb{R}^3} |t U_R|^{p+1}dx - \int_{\mathbb{R}^3} |w_0|^p tU_Rdx.

    For the estimate of A_2 , using the expression of F(u) = \int_{\mathbb{R}^3} K(x)\phi_u u^2 dx and the symmetry property of the integral with respect to x and y , we can obtain that

    \begin{array}{rl} |A_2| &\leq t\int_{\mathbb{R}^3}K(x)\phi_{w_0}w_0 U_R dx + \frac{t^2}{2}\int_{\mathbb{R}^3}K(x)\phi_{w_0} (U_R)^2 dx \\ & \qquad \quad + \frac{t^4}{4}\int_{\mathbb{R}^3} K(x)\phi_{U_R}(U_R)^2 dx + t^3\int_{\mathbb{R}^3} K(x)\phi_{U_R}w_0 U_R dx \\ & \qquad \qquad \quad + t^2\int_{\mathbb{R}^3\times\mathbb{R}^3}\frac{K(x)K(y)w_0(x)w_0(y)U_R(x)U_R(y)}{|x-y|}dxdy. \end{array}

    Since w_0 is a nonnegative solution of (3) and w_0\in L^\infty(\mathbb{R}^3) , we obtain from the assumption on K(x) that

    \begin{array}{rl} & \int_{\mathbb{R}^3\times\mathbb{R}^3}\frac{K(x)K(y)w_0(x)w_0(y)U_R(x)U_R(y)}{|x-y|}dxdy \\ & = \int_{\mathbb{R}^3}K(x)\phi_{\sqrt{w_0 U_R}}w_0 U_R dx\\ & \leq \|\phi_{\sqrt{w_0 U_R}}\|_{L^6} \left(\int_{\mathbb{R}^3} K(x)^{\frac65}(w_0U_R)^{\frac65} dx\right)^{\frac56} \\ & \leq C \left(\int_{\mathbb{R}^3} e^{-\frac65 aR} e^{\left(\frac65 a - \frac65(1-\delta)\right)|x|} dx\right)^{\frac56}\\ & \leq C e^{-a R}\quad \hbox{since}\quad 0 < a < 1. \end{array}

    Similarly we can deduce that for R large enough,

    \int_{\mathbb{R}^3}K(x)\phi_{w_0}w_0 U_R dx \leq C e^{-a R}, \ \ \int_{\mathbb{R}^3}K(x)\phi_{w_0} (U_R)^2 dx \leq C e^{-a R},
    \int_{\mathbb{R}^3} K(x)\phi_{U_R}(U_R)^2 dx \leq C e^{-a R}\ \ \hbox{and}\ \ \int_{\mathbb{R}^3} K(x)\phi_{U_R}w_0 U_R dx \leq C e^{-a R}.

    Since \int_{\mathbb{R}^3} h(x) (U_R)^2 dx \geq C e^{-b R} for R large enough, we obtain that

    \begin{array}{rl} & I_\mu(w_0 + tU_R) \leq I_\mu(w_0) +\frac{t^2}{2}\|U_R\|^2dx - \frac{\mu}{2}\int_{\mathbb{R}^3}h(x)U_R^2dx\\ & \qquad \qquad - \frac1{p+1}\int_{\mathbb{R}^3} |t U_R|^{p+1}dx+C e^{-a R}\\ & \leq I_\mu(w_0) + \frac{p-1}{2(p+1)}S_{p+1}^{\frac{p+1}{p-1}} +C e^{-a R} - C e^{-b R} + o(e^{-b R})\\ & < c_{0,\mu} + \frac{p-1}{2(p+1)}S_{p+1}^{\frac{p+1}{p-1}} \end{array}

    for R large enough since 0 < b < a < 1 . The proof is complete.

    Proposition 4.7. Under the conditions (A1)-(A4), if \mu\in (\mu_1, \mu_1 + \delta_3) and w_{0,\mu} be the minimizer obtained in Lemma 4.3, then the d_{2, \mu} is a critical value of I_\mu .

    Proof. Since for \mu\in (\mu_1, \mu_1 + \delta_3) , we know from Lemma 4.1 and Lemma 4.3 that the w_{0, \mu} is a local minimizer of I_\mu . Moreover, one has that I_\mu(w_{0, \mu} + s U_R) \to -\infty as s\to +\infty . Therefore Lemma 4.5, Lemma 4.7 and the mountain pass theorem of [3] imply that d_{2, \mu} is a critical value of I_\mu .

    Proof of Theorem 1.2. The conclusion (1) of Theorem 1.2 follows from Lemma 4.3 and Remark 4.4. It remains to prove (2) of Theorem 1.2. By Proposition 4.7, the d_{2, \mu} is a critical value of I_{\mu} and d_{2, \mu} > 0 . The proof of nonnegativity for at least one of the corresponding critical point is inspired by the idea of [1]. In fact, since I_{\mu}(u) = I_{\mu}(|u|) for any u\in H^1(\mathbb{R}^3) , for every n\in \mathbb{N} , there exists \gamma_n\in \Gamma_2 with \gamma_n(t)\geq 0 (a.e. in \mathbb{R}^3 ) for all t\in [0, 1] such that

    \begin{equation} d_{2, \mu} \leq \max\limits_{t\in [0, 1]} I_{\mu} (\gamma_n(t)) < d_{2, \mu} + \frac{1}{n}. \end{equation} (40)

    By Ekeland's variational principle, there exists \gamma_n^*\in \Gamma_2 satisfying

    \begin{equation} \left\{ \begin{array}{ll} d_{2, \mu} \leq \max\limits_{t\in [0, 1]} I_{\mu} (\gamma_n^*(t))\leq \max\limits_{t\in [0, 1]} I_{\mu} (\gamma_n(t)) < d_{2, \mu} +\frac{1}{n};\\ \max\limits_{t\in [0, 1]}\|\gamma_n(t)-\gamma_n^*(t)\| < \frac{1}{\sqrt{n}}; \\ \hbox{ there exists} \; t_n\in [0, 1]\; \hbox{such that}\; z_n : = \gamma_n^*(t_n) \hbox{ satisfies}: \\ I_{\mu}(z_n) = \max\limits_{t\in [0, 1]} I_{\mu} (\gamma_n^*(t)), \;\hbox{and}\; \|I'_{\mu}(z_n)\|\leq\frac{1}{\sqrt{n}}. \end{array} \right. \end{equation} (41)

    By Lemma 4.6 we get a convergent subsequence (still denoted by (z_n)_{n\in \mathbb{N}} ). We may assume that z_n\rightarrow z strongly in H^1(\mathbb{R}^3) as n\rightarrow \infty . On the other hand, by (41), we also arrive at \gamma_n(t_n)\rightarrow z strongly in H^1(\mathbb{R}^3) as n\rightarrow \infty . Since \gamma_n(t)\geq 0 , we conclude that z\geq 0 , z\not\equiv 0 in \mathbb{R}^3 with I_{\mu}(z)>0 and it is a nonnegative solution of problem (3).

    Next, let u_{2,\mu} be the nonnegative solution given by the above proof, that is, I_{\mu}'(u_{2,\mu}) = 0 and I_{\mu}(u_{2,\mu}) = d_{2, \mu} . We claim that for any sequence \mu^{(n)}> \mu_1 and \mu^{(n)}\to \mu_1 , there exist a sequence of solution u_{2,\mu^{(n)}} of (3) with \mu = \mu^{(n)} and a u_{\mu_1} with I_{\mu_1}'(u_{\mu_1}) = 0 such that u_{2,\mu^{(n)}}\rightarrow u_{\mu_1} strongly in H^1(\mathbb{R}^3) . In fact, denoted by w_{0,\mu^{(n)}} the minimizer corresponding to d_{0, \mu^{(n)}} , according to the definition of d_{2, \mu} and the proof of Lemma 4.6, we deduce that for n large enough,

    0 < \alpha \leq d_{2, \mu^{(n)}}\leq \max\limits_{s > 0}I_{\mu^{(n)}}(w_{0,\mu^{(n)}} + sU_R)

    and

    I_{\mu^{(n)}}(w_{0,\mu^{(n)}} + sU_R) \leq \frac{p-1}{2(p+1)}S_{p+1}^{\frac{p+1}{p-1}} +C e^{-a R} - C e^{-b R} + o(e^{-b R}),
    \begin{equation} \limsup\limits_{n\to\infty} d_{2, \mu^{(n)}} \leq \frac{p-1}{2(p+1)}S_{p+1}^{\frac{p+1}{p-1}}. \end{equation} (42)

    Next, similar to the proof in Lemma 2.3, we can deduce that (u_{2,\mu^{(n)}})_{n\in \mathbb{N}} is bounded in H^1(\mathbb{R}^3) . Going if necessary to a subsequence, we may assume that u_{2,\mu^{(n)}} \rightharpoonup \tilde{u}_2 weakly in H^1(\mathbb{R}^3) and u_{2,\mu^{(n)}} \to \tilde{u}_2 a. e. in \mathbb{R}^3 . Then we have that I'_{\mu_1}(\tilde{u}_2) = 0 . Moreover I_{\mu_1}(\tilde{u}_2) \geq 0 . If (u_{2,\mu^{(n)}})_{n\in \mathbb{N}} does not converge strongly to \tilde{u}_2 in H^1(\mathbb{R}^3) , then using an argument similar to the proof of Lemma 4.5, we may deduce that

    I_{\mu^{(n)}}(u_{2,\mu^{(n)}}) \geq I_{\mu_1}(\tilde{u}_2) + \frac{p-1}{2(p+1)}S_{p+1}^{\frac{p+1}{p-1}},

    which contradicts to (42). Hence u_{2,\mu^{(n)}}\to \tilde{u}_2 strongly in H^1(\mathbb{R}^3) and hence I_{\mu_1}(\tilde{u}_2) > 0 . The proof is complete by choosing u_{\mu_1} = \tilde{u}_2 .

    The author thanks the unknown referee for helpful comments.

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