Existence and asymptotics of ground states to the nonlinear Dirac equation with Coulomb potential

Abstract

In this paper we study the Dirac equation with Coulomb potential $\begin{array}{rcl} - i α \cdot \nabla u + a β u - \frac{μ}{| x |} u = f (x, | u |) u, x \in R^{3} \end{array}$ where a is a positive constant, μ is a positive parameter, $α = (α_{1}, α_{2}, α_{3})$ , $α_{i}$ and β are $4 \times 4$ Pauli–Dirac matrices. The Dirac operator is unbounded from below and above so the associate energy functional is strongly indefinite. Under some suitable conditions, we prove that the problem possesses a ground state solution which is exponentially decay, and the least energy has continuous dependence about μ. Moreover, we are able to obtain the asymptotic property of ground state solution as $μ \to 0^{+}$ , this result can characterize some relationship of the above problem between $μ > 0$ and $μ = 0$ .

Keywords

Dirac equation Coulomb potential asymptotic property ground state solutions

1. Introduction and main results

It is well known that Dirac proposed a model to the quantum mechanics which, in contrast to the Schrödinger theory, takes into account the relativity theory. More precisely, he proposed a model to describe the evolution of a free relativistic particle in 1928, given by the following Dirac equation $\begin{matrix} i ℏ \frac{\partial ψ}{\partial t} = - i c ℏ α \cdot \nabla ψ + m c^{2} β ψ, \end{matrix}$ where ℏ is Planck’s constant, c the light speed, m the mass, $α = (α_{1}, α_{2}, α_{3})$ and $α_{1}$ , $α_{2}$ , $α_{3}$ and β are the $4 \times 4$ Pauli–Dirac matrices: $\begin{matrix} β = (\begin{matrix} I_{2} & 0 \\ 0 & - I_{2} \end{matrix}), α_{k} = (\begin{matrix} 0 & σ_{k} \\ σ_{k} & 0 \end{matrix}), k = 1, 2, 3, \end{matrix}$ with $\begin{matrix} σ_{1} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), σ_{2} = (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}), σ_{3} = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) . \end{matrix}$ It is not difficult to check that $α_{i}$ and β satisfy the following anticommutation relations $\begin{matrix} α_{k} α_{ℓ} + α_{ℓ} α_{k} = 2 δ_{k ℓ} I_{4}, α_{k} β + β α_{k} = 0 and β^{2} = I_{4}, \end{matrix}$ where $I_{4}$ denotes the identity matrix, $δ_{k ℓ} = 1$ if $k = ℓ$ , $δ_{k ℓ} = 0$ if $k \neq ℓ$ . Physically, nonlinear Dirac equation has been successfully used to describe relativistic particles having a spin $1 / 2$ (see [19]). Moreover, it has been used as effective theories in nonlinear topics, atomic, nuclear, gravitational physics and Bose–Einstein condensates and etc (see [29]). For more details in the physical backgrounds, we refer the readers to [21] and the monograph [40].

Consider the time-dependent nonlinear Dirac equation $\begin{matrix} (1.1) & - i ℏ \frac{\partial ψ}{\partial t} = i c ℏ α \cdot \nabla ψ - m c^{2} β ψ - M (x) ψ + G_{ψ} (x, ψ), \end{matrix}$ where ψ represents the wave function of the state of an electron, M is external potential, The nonlinear coupling G describes a self-interaction among many particles. Typical examples for nonlinear couplings can be found in the self-interacting scalar theories (see [10, 25]). In the present paper, we are mainly interested in studying the standing wave solution of form $ψ (t, x) = e^{\frac{i w t}{ℏ}} u (x)$ under the condition $G (x, e^{i θ} ψ) = G (x, ψ)$ for all $θ \in [0, 2 π]$ . This type of particle-like solution does not change its shape as it evolves in time, hence has a soliton-like behavior. Moreover, we can see that $ψ (t, x)$ satisfies (1.1) if and only if $u : R^{3} \to C^{4}$ satisfies the following stationary equation $\begin{matrix} (1.2) & - i α \cdot \nabla u + a β u + V (x) u = F_{u} (x, u), \end{matrix}$ where $a = m c / ℏ$ , $V (x) = M (x) / c ℏ + w I_{4} / ℏ$ and $F (x, u) = G (x, u) / c ℏ$ .

Mathematical work on nonlinear Dirac equations like the above has been investigated in recent years, and the existence and multiplicity results for such type equations have been discussed in many papers under different assumptions on the potential and nonlinearity by using shooting methods (see [2, 3, 10, 33]) and variational methods (see [4, 16, 17, 20]). We also refer to the monograph [13] and the review [19] for more interesting results for Dirac equations.

The main unusual feature of the Dirac equation is the spectrum of the free Dirac operator $\begin{matrix} A = - i α \cdot \nabla + a β = - i \sum_{k = 1}^{3} α_{k} \partial_{k} + a β \end{matrix}$ is not bounded from below: $σ (A) = (- \infty, a] \cup [a, + \infty)$ which makes the corresponding energy functional strongly indefinite. As we know, the variational method was first used in Esteban and Séré [20] where the authors obtained the existence of infinitely many solutions for problem (1.2) with $V = w$ and the Soler model $\begin{matrix} F (u) = \frac{1}{2} H (\tilde{u} u), H \in C^{2} (R, R), H (0) = 0, where \tilde{u} u : = {(β u, u)}_{C^{4}} \end{matrix}$ by using linking theorem. Thereafter, based on some recently developed critical point theorems from [5] for strongly indefinite functionals, there are some works concerning the existence and multiplicity of solutions to problem (1.2) with general non-autonomous nonlinearity, see [4, 16, 17]. More specifically, Bartsch and Ding [4] studied a class of periodic Dirac equation (1.2) with a scalar potential, and obtained the existence of infinitely many geometrically different solutions. Regarding the non-priodic matrix potentials, Ding and Ruf [16] perform a study about existence and multiplicity of solutions of (1.2) with a non-periodic asymptotically quadratic nonlinearity. The existence results of ground state solution were established in [17, 44] for the non-periodic superquadratic case. Figueiredoa and Pimenta [24] considered the potential is vanishing at infinity but not singular, like the conditions considered by Alves and Souto in [1] which recover the compactness of Sobolev space embedding, they also proved the existence of ground state solutions by applying the generalized Nehari manifold method [38] and deformation lemma. Combining the linking argument [30] and non-Nehari method [39], Zhang et al. [46] investigated the existence and some properties of ground state solutions for the asymptotically periodic case.

Recently, for other related topics involving the semi-classical singular perturbation problem, there are many works devoted to the study of concentration phenomenon of semi-classical states under some different assumptions on the linear and nonlinear potentials, see, for instance [14, 15, 18, 41, 47, 48] and the references therein.

In this paper, we will study a class of singular potential $V (x) = - \frac{μ}{| x |}$ , this type of potential is called Coulomb potential in Physics, and consider the following Dirac equation of form $\begin{matrix} (1.3) & - i α \cdot \nabla u + a β u - \frac{μ}{| x |} u = f (x, | u |) u, x \in R^{3} . \end{matrix}$ Coulomb potential plays a crucial role in quantum electrodynamics, it describes the motion and the interactions between two charged particles, moreover, the physical relevance of the Coulomb potential in quantum mechanics is highlighted in [26]. We also mention that the Coulomb potential has been widely employed in many other areas such as quantum cosmology, atomic physics, nuclear physics and molecular physics, see [40] for further discussion and motivation. From the mathematical point of view, the Coulomb potential has served as mathematical models to illustrate some concepts in elementary particle physics. On the other hand, the main reason of interest in Coulomb potential relies in their criticality: indeed it has the same homogeneity as the free Dirac operator and it do not belong to the Kato’s class, hence it cannot be regarded as a lower order perturbation term. Besides, the singularity of the Coulomb potential influence strongly the structure of spectrum, the decomposition of working apace and the properties of the Dirac–Coulomb operator such as self-adjointness of operator. These new features bring some new mathematical difficulties which make the study of such a problem particularly interesting and challenging.

Compared with the problem with $μ = 0$ , the problem with $μ \neq 0$ becomes more complicated and interesting since the associated energy functional has no more translation invariance, and the classical variational techniques cannot be applied directly. Many attractive and challenging topics on Dirac equation with Coulomb potential remain unsolved. Hence, it makes sense for us to study problem (1.3) thoroughly. Here it is worth pointing out that, regarding the Dirac equation with Coulomb potential, there is only a work studied by Ding and Ruf [16] in which the authors established the existence and multiplicity of solutions for asymptotically linear nonlinearity. However, the other relevant results, such as continuous dependence and asymptotic property of ground state solutions about μ, are not investigated up to now. Due to the special physical importance and motivated by the above facts, in the present paper we will utilize variational methods to study the existence and some asymptotic properties of ground state solutions of problem (1.3). More precisely, the main ingredients of this paper include the following three aspects:

we study the existence, convergence and exponential decay of ground state solutions for sufficiently small parameter μ;

we prove the monotonicity and convergence property of the ground state energy with respect to parameter μ;

we also give the asymptotic behavior of ground state solutions when $μ \to 0^{+}$ , namely, the ground state solutions of problem (1.3) converge to the ones of the problem (1.3) with $μ = 0$ .

Before stating our main results, we assume that the following conditions hold for the nonlinear function f.

(

f_{1}

)

$f \in C^{1} (R^{3} \times R^{+}, R^{+})$ is 1-periodic in $x_{i}$ for $i = 1, 2, 3$ , and there exist $p \in (2, 3)$ and $c > 0$ such that $\begin{matrix} | f (x, s) | ⩽ c (1 + | s |^{p - 2}) for all (x, s) \in R^{3} \times R^{+}; \end{matrix}$

(

f_{2}

)

$f (x, s) = o (1)$ as $s \to 0$ uniformly in x;

(

f_{3}

)

$\frac{F (x, s)}{s^{2}} \to \infty$ as $s \to \infty$ uniformly in x, where $F (x, s) = \int_{0}^{s} f (x, t) t d t$ ;

(

f_{4}

)

$f (x, s)$ is increasing in s on $(0, + \infty)$ ;

(

f_{5}

)

there exist $c_{0} > 0$ and $2 < q ⩽ p$ such that $\begin{matrix} \frac{1}{2} f (x, s) s^{2} - F (x, s) ⩾ c_{0} s^{q} for all (x, s) \in R^{3} \times R^{+} . \end{matrix}$

A typical example is the power function $f (s) = s^{q - 2}$ , $2 < q < 3$ . For the sake of convenience, let E be the working space with an orthogonal decomposition $E = E^{-} \oplus E^{+}$ , and let $Φ_{μ}$ denote the energy functional of problem (1.3), where E and $Φ_{μ}$ will be defined in Section 2. Following from [34, 38], in order to seek for the ground state solutions of problem (1.3), we need introduce the following generalized Nehari manifold $\begin{array}{rcl} N_{μ} : = {u \in E ∖ E^{-} : Φ_{μ}^{'} (u) u = 0 and Φ_{μ}^{'} (u) v = 0 for any v \in E^{-}}, \end{array}$ which has been introduced by Pankov [34], deeply studied by Szulkin and Weth [38]. Moreover, we define the ground state energy level on $N_{μ}$ $\begin{array}{rcl} m_{μ} : = inf_{N_{μ}} Φ_{μ}, μ ⩾ 0 . \end{array}$

Now we are ready to state the main results of the present paper as follows.

Theorem 1.1.
Suppose that $(f_{1})$ – $(f_{4})$ hold and $0 ⩽ μ < \frac{1}{2}$ . Then
problem ( 1.3 ) has at least a ground state solution;

$L$ it is compact in $H^{1} (R^{3})$ , where $L$ denotes the set of all ground state solutions of ( 1.3 );

there exist constants $c, C > 0$ , such that, for $| x | ⩾ 1$ $\begin{matrix} | u (x) | ⩽ C exp (- c | x |) . \end{matrix}$

Theorem 1.2.
Suppose that $(f_{1})$ – $(f_{4})$ hold and $0 ⩽ μ < \frac{1}{2}$ . Let $u_{μ}$ be a ground state solution of ( 1.3 ) and $u_{0}$ be a ground state solution of ( 1.3 ) with $μ = 0$ . Then $m_{μ}$ is decreasing in $μ \in R^{+}$ and $\begin{matrix} lim_{μ \to 0^{+}} m_{μ} = m_{0} . \end{matrix}$
Theorem 1.3.
Suppose that $(f_{1})$ – $(f_{5})$ hold, and let $u_{μ}$ be a ground state solution of problem ( 1.3 ). Then for any sequence ${μ_{n}}$ with $μ_{n} \to 0$ as $n \to \infty$ , up to a subsequence, $u_{μ_{n}} \to u_{0}$ as $n \to \infty$ in $H^{1 / 2} (R^{3}, C^{4})$ , where $u_{0}$ is a ground state solution of problem ( 1.3 ) with $μ = 0$ .
Remark 1.4.
However, we can not say something about the existence of ground state solution of (1.3) with $μ < 0$ and $μ ⩾ \frac{1}{2}$ .

As a motivation we recall that there is a large number of articles concerning the nonlinear Schrödinger equation with the Hardy-type singular potential arising in the nonrelativistic quantum mechanics $\begin{matrix} (1.4) & - Δ u + V (x) u - \frac{μ}{| x |^{2}} u = g (x, u), u \in H_{0}^{1} (Ω) . \end{matrix}$ See [6–9, 11, 12, 22, 23, 28, 35, 36, 45] and the references therein. These authors studied the existence of nontrivial solutions and ground state solutions under suitable assumptions. From a viewpoint of variational methods, the two problems possess different variational structures, the mountain pass and the linking structures, respectively. Contrary to the Schrödinger operator, as we mentioned above, since the Dirac operator has unbounded continuous spectrum and the corresponding energy functional is strongly indefinite, the usual minimax theorems studied the Schrödinger equation can not apply directly for the Dirac equation.

On the other hand, since we work in the whole space, the Palais–Smale or Cerami condition fails to hold, moreover, this combining with the singularity of Coulomb potential at original makes the study via variational methods rather complicated. For these reasons, the problem discussed in the present paper poses a challenge and some difficulties need to be overcame. In order to prove the main results, some arguments are in order. First, we use the non-Nehari manifold method developed by Tang [39], which is completely different from the one of Szulkin and Weth [38], to prove the existence of ground state solutions. The main idea of this method is to construct a special minimizing Cerami sequence for the energy functional outside $N_{μ}$ by using the diagonal method. Second, we establish a global compactness result for bounded $(C)$ -sequences to overcome the lack of compactness of the Sobolev embedding. Third, by using some analysis techniques, we prove some asymptotic properties of ground state solutions.

The paper is organized as follows. In Section 2, we formulate the variational setting and give some useful preliminaries. In Section 3, we study the property of ${(C)}_{c}$ -sequences. The proofs of theorems are completed in Section 4.
2. Variational setting and preliminaries

Throughout this paper, we make use of the following notations. $‖ \cdot ‖_{s}$ denotes the usual norm of the space $L^{s}$ , $1 ⩽ s ⩽ \infty$ ; ${(\cdot, \cdot)}_{2}$ denotes the usual $L^{2}$ inner product; c, $c_{i}$ or $C_{i}$ ( $i = 1, 2, \dots$ ) are some different positive constants.

Recall that the free Dirac operator is defined as $\begin{matrix} A = - i α \cdot \nabla + a β . \end{matrix}$ According to [4], the Dirac operator A is a self-adjoint operator acting on $L^{2} : = L^{2} (R^{3}, C^{4})$ with domain $D (A) = H^{1} : = H^{1} (R^{3}, C^{4})$ , and $σ (A) = σ_{c} (A) = R ∖ (- a, a)$ , where $σ (A)$ and $σ_{c} (A)$ denote the spectrum and the continuous spectrum of A, respectively. Thus the space $L^{2}$ has the following orthogonal decomposition: $\begin{matrix} L^{2} = L^{-} \oplus L^{+}, u = u^{-} + u^{+} \end{matrix}$ such that A is negative definite in $L^{-}$ and positive definite in $L^{+}$ . Denoting by $| A |$ the absolute value of A and by $| A |^{1 / 2}$ its square root, and let $E = D (| A |^{1 / 2})$ be the Hilbert space, endowed with the following inner product $\begin{matrix} (u, v) = Re {(| A |^{1 / 2} u, | A |^{1 / 2} v)}_{2} \end{matrix}$ and the reduced norm $‖ u ‖ = {(u, u)}^{1 / 2}$ , where Re stands for the real part of a complex number.

Following from the conclusions [13, Lemma 7.4], the working apace $E = H^{1 / 2} : = H^{1 / 2} (R^{3}, C^{4})$ and the norm $‖ \cdot ‖$ is equivalent to the usual norm of $H^{1 / 2}$ . Hence E embeds continuously into $L^{q}$ for all $q \in [2, 3]$ and compactly into $L_{loc}^{q}$ for all $q \in [1, 3)$ , then there exists constant $γ_{p} > 0$ such that for all $u \in E$ $\begin{array}{rcl} (2.1) & | u |_{p} ⩽ γ_{p} ‖ u ‖ for all p \in [2, 3] . \end{array}$ Moreover, since $σ (A) = R ∖ (- a, a)$ , then we have $\begin{array}{rcl} (2.2) & a | u |_{2}^{2} ⩽ ‖ u ‖^{2} for all u \in E . \end{array}$ According to the decomposition of $L^{2}$ , we have the decomposition of E $\begin{matrix} E = E^{-} \oplus E^{+}, where E^{\pm} = E \cap L^{\pm}, \end{matrix}$ moreover, it is orthogonal with respect to the inner products ${(\cdot, \cdot)}_{2}$ and $(\cdot, \cdot)$ .

In order to analyze the properties of the energy functional of problem (1.3), Next we need to make a suitable estimate for the Coulomb potential. Setting $V_{μ} : = \frac{μ}{| x |}$ , $μ > 0$ , and note that $\begin{matrix} {(- i α \cdot \nabla)}^{2} = {(- i \sum_{k = 1}^{3} α_{k} \partial_{k})}^{2} = - Δ . \end{matrix}$ Then, using Hardy inequality we get $\begin{array}{rcl} | V_{μ} u |_{2}^{2} ⩽ 4 μ^{2} | \nabla u |_{2}^{2} ⩽ 4 μ^{2} | A u |_{2}^{2} = {| (2 μ A) u |}_{2}^{2}, \end{array}$ thus $\begin{array}{rcl} {| V_{μ}^{1 / 2} u |}_{2}^{2} ⩽ {| {| (2 μ A) |}^{1 / 2} u |}_{2}^{2} = 2 μ {| | A |^{1 / 2} u |}_{2}^{2} = 2 μ ‖ u ‖^{2}, \end{array}$ that is $\begin{array}{rcl} (2.3) & \int_{R^{3}} \frac{μ}{| x |} | u |^{2} ⩽ 2 μ {| | A |^{1 / 2} u |}_{2}^{2} = 2 μ ‖ u ‖^{2} . \end{array}$

It follows from $(f_{1})$ and $(f_{2})$ that for any $ϵ > 0$ , there is $C_{ϵ} > 0$ such that $\begin{matrix} (2.4) & max {f (x, s) s^{2}, F (x, s)} ⩽ ϵ | s |^{2} + C_{ϵ} | s |^{p} for all (x, s), p \in (2, 3), \end{matrix}$ moreover, $(f_{4})$ implies that $\begin{matrix} (2.5) & \frac{1}{2} f (x, s) s^{2} > F (x, s) > 0 for all s \neq 0 . \end{matrix}$

Now on E we define the energy functional of problem (1.3) $\begin{matrix} Φ_{μ} (u) = \frac{1}{2} ({‖ u^{+} ‖}^{2} - {‖ u^{-} ‖}^{2}) - \frac{μ}{2} \int_{R^{3}} \frac{| u |^{2}}{| x |} - \int_{R^{3}} F (x, | u |) . \end{matrix}$ According to the above introduction, the energy functional $Φ_{μ}$ is strongly indefinite. Moreover, by (2.3), (2.4) and a standard argument we can verify that $Φ_{μ} \in C^{1} (E, R)$ under the conditions $(f_{1})$ and $(f_{2})$ . Furthermore, for $u, v \in E$ , there holds $\begin{matrix} Φ_{μ}^{'} (u) v = (u^{+}, v^{+}) - (u^{-}, v^{-}) - μ Re \int_{R^{3}} \frac{u \cdot v}{| x |} - Re \int_{R^{3}} f (x, | u |) u \cdot v . \end{matrix}$ Clearly, this shows that the critical points of $Φ_{μ}$ are solutions of problem (1.3) (see [13, 42]).

For notational simplicity, we denote $\begin{matrix} Ψ_{μ} (u) = \frac{μ}{2} \int_{R^{3}} \frac{| u |^{2}}{| x |} + \int_{R^{3}} F (x, | u |) . \end{matrix}$

Next we introduce some definitions. For a functional $Φ \in C^{1} (E, R)$ , Φ is said to be weakly sequentially lower semi-continuous if for any $u_{n} ⇀ u$ in E one has $Φ (u) ⩽ {lim inf}_{n \to \infty} Φ (u_{n})$ , and $Φ^{'}$ is said to be weakly sequentially continuous if ${lim}_{n \to \infty} Φ^{'} (u_{n}) v = Φ^{'} (u) v$ for each $v \in E$ . We recall that a sequence ${u_{n}} \subset E$ is called ${(C)}_{c}$ -sequence for Φ at the level c if $\begin{matrix} Φ (u_{n}) \to c and (1 + ‖ u_{n} ‖) ‖ Φ^{'} (u_{n}) ‖ \to 0 . \end{matrix}$ We say that Φ satisfy the ${(C)}_{c}$ -condition if any ${(C)}_{c}$ -sequence has a convergent subsequence in E.

To prove the main results, we shall use the following abstract critical point theorem which is taken from [30] and [31].

Lemma 2.1.
Let X be a real Hilbert space with $X = X^{-} \oplus X^{+}$ , and let $Φ \in C^{1} (X, R)$ be of the form $\begin{matrix} Φ (u) = \frac{1}{2} ({‖ u^{+} ‖}^{2} - {‖ u^{-} ‖}^{2}) - Ψ (u), u = u^{-} + u^{+} \in X^{-} \oplus X^{+} . \end{matrix}$ Assume that the following conditions hold: $(Ψ_{1})$
$Ψ \in C^{1} (X, R)$ is bounded from below and weakly sequentially lower semi-continuous;
$(Ψ_{2})$
$Ψ^{'}$ is weakly sequentially continuous;
$(Ψ_{3})$
there exist $R > ρ > 0$ and $e \in X^{+}$ with $‖ e ‖ = 1$ such that $\begin{matrix} κ : = inf Φ (S_{ρ}^{+}) > sup Φ (\partial Q), \end{matrix}$ where $\begin{matrix} S_{ρ}^{+} = {u \in X^{+} : ‖ u ‖ = ρ}, Q = {v + s e : v \in X^{-}, s ⩾ 0, ‖ v + s e ‖ ⩽ R} . \end{matrix}$
Then there exist a constant $c \in [κ, sup Φ (Q)]$ and a sequence ${u_{n}} \subset X$ such that $\begin{array}{rcl} Φ (u_{n}) \to c and (1 + ‖ u_{n} ‖) ‖ Φ^{'} (u_{n}) ‖ \to 0 . \end{array}$
Lemma 2.2.
Assume that $(f_{1})$ – $(f_{4})$ hold and $μ ⩾ 0$ . Then $Ψ_{μ}$ is nonnegative, weakly sequentially lower semi-continuous, and $Ψ_{μ}^{'}$ is weakly sequentially continuous.

Making use of embedding theorem and standard arguments [13], we can check easily Lemma 2.2 holds, here we omit the details. In the following, we give a crucial estimate, which is used repeatedly in the later proof.
Lemma 2.3.
Assume that $(f_{1})$ – $(f_{4})$ hold. Let $μ ⩾ 0$ , $u \in E$ , $v \in E^{-}$ and $t ⩾ 0$ . Then $\begin{matrix} (2.6) & Φ_{μ} (u) ⩾ Φ_{μ} (t u + v) - Φ_{μ}^{'} (u) (\frac{t^{2} - 1}{2} u + t v) . \end{matrix}$
Proof.
Computing directly, we have $\begin{matrix} Φ_{μ} (t u + v) - Φ_{μ} (u) - Φ_{μ}^{'} (u) (\frac{t^{2} - 1}{2} u + t v) & = - \frac{1}{2} ‖ v ‖^{2} - \frac{μ}{2} \int_{R^{3}} \frac{| v |^{2}}{| x |} + \int_{R^{3}} h (x, t) \\ ⩽ - \frac{1}{2} ‖ v ‖^{2} + \int_{R^{3}} h (x, t), \end{matrix}$ where $\begin{matrix} h (x, t) : = Re f (x, | u |) u \cdot (\frac{t^{2} - 1}{2} u + t v) + F (x, | u |) - F (x, | t u + v |) . \end{matrix}$ Next we show that $h (x, t) ⩽ 0$ . Indeed, if $u = 0$ , by $(f_{2})$ and (2.5) we get $h (x, t) = - F (x, | v |) ⩽ 0$ . If $u \neq 0$ . Following the arguments in [38] and [43] we can prove that $h (x, t) ⩽ 0$ . In fact, by $(f_{3})$ we get $h (x, t) \to - \infty$ as $t \to \infty$ . Thus, $h (x, t)$ attains its maximum at some point $t_{0} \in [0, \infty)$ . If $t_{0} = 0$ , it follows from (2.5) that the conclusion holds. If $t_{0} > 0$ , then $\partial_{t} h (x, t_{0}) = 0$ and $\begin{matrix} (2.7) & Re f (x, | u |) u \cdot (t_{0} u + v) = Re f (x, | t_{0} u + v |) u \cdot (t_{0} u + v) . \end{matrix}$ For notational simplicity, let $w = t_{0} u + v$ . Here there are two cases need to discuss: (i) $Re (u \cdot w) \neq 0$ and (ii) $Re (u \cdot w) = 0$ .

Case (i) $Re (u \cdot w) \neq 0$ . Using (2.7) and $(f_{4})$ we have $| u | = | w |$ , $F (x, | u |) = F (x, | w |)$ and $\begin{matrix} (2.8) & Re f (x, | u |) u \cdot w ⩽ Re f (x, | u |) | u |^{2} . \end{matrix}$ Then from (2.5) and (2.8) we deduce that $\begin{matrix} h (x, t_{0}) & = Re f (x, | u |) u \cdot (\frac{t_{0}^{2} - 1}{2} u + t_{0} v) + F (x, | u |) - F (x, | t_{0} u + v |) \\ = Re f (x, | u |) u \cdot (\frac{t_{0}^{2} - 1}{2} u + t_{0} (w - t_{0} u)) + F (x, | u |) - F (x, | w |) \\ = Re f (x, | u |) u \cdot ((\frac{t_{0}^{2}}{2} - t_{0}^{2} + t_{0} - \frac{1}{2}) u + t_{0} w - t_{0} u) \\ = Re [- \frac{1}{2} {(t_{0} - 1)}^{2} f (x, | u |) | u |^{2} + t_{0} (f (x, | u |) u \cdot w - f (x, | u |) | u |^{2})] ⩽ 0 . \end{matrix}$

Case (ii) $Re (u \cdot w) = 0$ . Clearly, $Re f (x, | u |) u \cdot w = 0$ . Using (2.5) we obtain $\begin{matrix} h (x, t_{0}) & = Re f (x, | u |) u \cdot (\frac{t_{0}^{2} - 1}{2} u + t_{0} (w - t_{0} u)) + F (x, | u |) - F (x, | w |) \\ = - \frac{t_{0}^{2}}{2} f (x, | u |) | u |^{2} - \frac{1}{2} f (x, | u |) | u |^{2} + F (x, | u |) + t_{0} Re f (x, | u |) u \cdot w - F (x, | w |) \\ ⩽ - \frac{t_{0}^{2}}{2} f (x, | u |) | u |^{2} - F (x, | w |) ⩽ 0 . \end{matrix}$ By the above discussion we know $h (x, t) ⩽ 0$ for any $t ⩾ 0$ and hence (2.6) holds. □

According to Lemma 2.3, we have the following corollary.
Corollary 2.4.
Assume that $(f_{1})$ – $(f_{4})$ hold and $μ ⩾ 0$ . Then for $u \in N_{μ}$ , $v \in E^{-}$ and $t ⩾ 0$ $\begin{matrix} Φ_{μ} (u) ⩾ Φ_{μ} (t u + v) . \end{matrix}$
Lemma 2.5.
Assume that $(f_{1})$ – $(f_{4})$ hold and $0 ⩽ μ < \frac{1}{2}$ . Then
there exists $ρ > 0$ such that $m_{μ} ⩾ κ : = inf {Φ_{μ} (u) : u \in E^{+}, ‖ u ‖ = ρ} > 0$ .

$‖ u^{+} ‖ ⩾ max {‖ u^{-} ‖, \sqrt{2 m_{μ}}}$ for all $u \in N_{μ}$ .

Proof.
(i) Let $u \in E^{+}$ and $μ ⩾ 0$ , from (2.1), (2.2), (2.3) and (2.4) we deduce that $\begin{matrix} Φ_{μ} (u) & = \frac{1}{2} ‖ u ‖^{2} - \frac{μ}{2} \int_{R^{3}} \frac{| u |^{2}}{| x |} - \int_{R^{3}} F (x, | u |) \\ ⩾ \frac{1}{2} (1 - 2 μ) ‖ u ‖^{2} - ϵ | u |_{2}^{2} - C_{ϵ} | u |_{p}^{p} \\ ⩾ \frac{1}{2} (1 - 2 μ) ‖ u ‖^{2} - ϵ γ_{2}^{2} ‖ u ‖^{2} - γ_{p}^{p} C_{ϵ} ‖ u ‖^{p} . \end{matrix}$ Since $0 ⩽ μ < \frac{1}{2}$ , then we can find $ρ > 0$ such that $\begin{array}{rcl} κ : = inf {Φ_{μ} (u) : u \in E^{+}, ‖ u ‖ = ρ} > 0 . \end{array}$ On the other hand, it is clear that the first inequality holds by Corollary 2.4.

(ii) Let $u \in N_{μ}$ , then by (2.5) we get $\begin{matrix} m_{μ} ⩽ \frac{1}{2} ({‖ u^{+} ‖}^{2} - {‖ u^{-} ‖}^{2}) - \frac{μ}{2} \int_{R^{3}} \frac{| u |^{2}}{| x |} - \int_{R^{3}} F (x, | u |) ⩽ \frac{1}{2} ({‖ u^{+} ‖}^{2} - {‖ u^{-} ‖}^{2}), \end{matrix}$ this shows that $‖ u^{+} ‖ ⩾ max {‖ u^{-} ‖, \sqrt{2 m_{μ}}}$ . □
Remark 2.6.
If $0 ⩽ μ ⩽ μ_{0} < \frac{1}{2}$ , we can choose $ρ > 0$ independent of μ.
Lemma 2.7.
Assume that $(f_{1})$ – $(f_{4})$ hold and $0 ⩽ μ < \frac{1}{2}$ . Then for any $e \in E^{+}$ , $sup Φ_{μ} (E^{-} \oplus R^{+} e) < \infty$ , and there exists $R_{e} > 0$ independent of μ such that $\begin{array}{rcl} Φ_{μ} (u) < 0, \forall u \in E^{-} \oplus R^{+} e, ‖ u ‖ ⩾ R_{e} . \end{array}$ In particular, there is a $R_{0} > ρ$ independent of μ such that $sup Φ (\partial Q_{R}) ⩽ 0$ for $R ⩾ R_{0}$ , where $\begin{matrix} (2.9) & Q_{R} = {s e + v : v \in E^{-}, s ⩾ 0, ‖ s e + v ‖ ⩽ R} . \end{matrix}$
Proof.
Let $e \in E^{+}$ , $t ⩾ 0$ and $u = t e + u^{-} \in E^{-} \oplus R^{+} e$ . Since $μ ⩾ 0$ , then we have $\begin{array}{rcl} Φ_{μ} (u) ⩽ Φ_{0} (u) = \frac{1}{2} ({‖ u^{+} ‖}^{2} - {‖ u^{-} ‖}^{2}) - \int_{R^{3}} F (x, | u |) . \end{array}$ So it suffices to check the conclusion holds for the functional $Φ_{0}$ , and the proof of the functional $Φ_{0}$ is standard, the details can be found in [13, 46]. □

Applying Lemmas 2.1, 2.2, 2.5 and 2.7, we have the following result.
Lemma 2.8.
Assume that $(f_{1})$ – $(f_{4})$ hold and $0 ⩽ μ < \frac{1}{2}$ . Then there exist a constant ${\tilde{c}}_{μ} \in [κ, sup Φ_{μ} (Q)]$ and a sequence ${u_{n}} \subset E$ such that $\begin{matrix} Φ_{μ} (u_{n}) \to {\tilde{c}}_{μ} and ‖ Φ_{μ}^{'} (u_{n}) ‖ (1 + ‖ u_{n} ‖) \to 0 . \end{matrix}$

Next we make use of the non-Nehari method developed by Tang [39] to construct a special ${(C)}_{c_{μ}}$ -sequence for some $c_{μ} \in [κ, m_{μ}]$ , which is very important to seek for the existence of ground state solutions.
Lemma 2.9.
Assume that $(f_{1})$ – $(f_{4})$ hold and $0 ⩽ μ < \frac{1}{2}$ . Then there are a constant $c_{μ} \in [κ, m_{μ}]$ and a sequence ${u_{n}} \subset E$ such that $\begin{matrix} Φ_{μ} (u_{n}) \to c_{μ} and ‖ Φ_{μ}^{'} (u_{n}) ‖ (1 + ‖ u_{n} ‖) \to 0 . \end{matrix}$
Proof.
According to the definition of $m_{μ}$ , we can choose $ξ_{k} \in N_{μ}$ such that $\begin{matrix} (2.10) & m_{μ} ⩽ Φ_{μ} (ξ_{k}) < m_{μ} + \frac{1}{k}, k \in N . \end{matrix}$ Lemma 2.5 shows that $‖ ξ_{k}^{+} ‖ ⩾ \sqrt{2 m_{μ}} > 0$ . Set $e_{k} = ξ_{k}^{+} / ‖ ξ_{k}^{+} ‖$ . Then $e_{k} \in E^{+}$ and $‖ e_{k} ‖ = 1$ . Using Lemma 2.7, there is $R_{k} > max {ρ, ‖ ξ_{k} ‖}$ such that $sup Φ_{μ} (\partial Q_{k}) ⩽ 0$ , where $\begin{matrix} (2.11) & Q_{k} = {s e_{k} + v : v \in E^{-}, s ⩾ 0, ‖ s e_{k} + v ‖ ⩽ R_{k}}, k \in N . \end{matrix}$ Applying Lemma 2.8, there are constants $c_{μ, k} \in [κ, sup Φ_{μ} (Q_{k})]$ and sequences ${u_{k, n}}_{n \in N} \subset E$ satisfying $\begin{matrix} (2.12) & Φ_{μ} (u_{k, n}) \to c_{μ, k} and ‖ Φ_{μ}^{'} (u_{k, n}) ‖ (1 + ‖ u_{k, n} ‖) \to 0, k \in N . \end{matrix}$ By virtue of Corollary 2.4, we have $\begin{matrix} (2.13) & Φ_{μ} (ξ_{k}) ⩾ Φ_{μ} (t ξ_{k} + v), \forall t ⩾ 0, v \in E^{-} . \end{matrix}$ Since $ξ_{k} \in Q_{k}$ , it follows from (2.11)) and (2.13) that $Φ_{μ} (ξ_{k}) = sup Φ_{μ} (Q_{k})$ . Thus, using (2.10) and (2.12) we get $\begin{array}{rcl} Φ_{μ} (u_{k, n}) \to c_{μ, k} < m_{μ} + \frac{1}{k} and ‖ Φ_{μ}^{'} (u_{k, n}) ‖ (1 + ‖ u_{k, n} ‖) \to 0, k \in N . \end{array}$ Up to a subsequence, we choose a sequence ${n_{k}} \subset N$ such that $\begin{array}{rcl} Φ_{μ} (u_{k, n_{k}}) < m_{μ} + \frac{1}{k} and ‖ Φ_{μ}^{'} (u_{k, n_{k}}) ‖ (1 + ‖ u_{k, n_{k}} ‖) < \frac{1}{k}, k \in N . \end{array}$ Let $u_{k} = u_{k, n_{k}}$ , $k \in N$ . Then, after passing to a subsequence, we get $\begin{matrix} Φ_{μ} (u_{k}) \to c_{μ} \in [κ, m_{μ}] and ‖ Φ_{μ}^{'} (u_{k}) ‖ (1 + ‖ u_{k} ‖) \to 0 . \end{matrix}$ □

Taking advantage of a similar way as [38, Lemma 2.6], we have Lemma 2.10.
Assume that $(f_{1})$ – $(f_{4})$ hold and $0 ⩽ μ < \frac{1}{2}$ . Then for any $u \in E ∖ E^{-}$ , we have $N_{μ} \cap (E^{-} \oplus R^{+} u) \neq \emptyset$ , i.e., there exist $t > 0$ and $v \in E^{-}$ such that $t u + v \in N_{μ}$ .

3. Behavior of ${(C)}_{c}$ -sequences

In this section we study the property of ${(C)}_{c}$ -sequences. First, we prove the boundedness of ${(C)}_{c}$ -sequences.

Lemma 3.1.
Assume that ${u_{n}}$ and ${μ_{n}}$ satisfy $0 ⩽ μ_{n} < \frac{1}{2}$ , $(1 + ‖ u_{n} ‖) Φ_{μ_{n}}^{'} (u_{n}) \to 0$ and $Φ_{μ_{n}} (u_{n})$ is bounded from above. Then ${u_{n}}$ is bounded. In particular, the any ${(C)}_{c}$ -sequence of $Φ_{μ}$ at level $c ⩾ 0$ is bounded for $0 ⩽ μ < \frac{1}{2}$ .
Proof.
Assume that ${u_{n}} \subset E$ satisfies $\begin{matrix} (3.1) & (1 + ‖ u_{n} ‖) Φ_{μ_{n}}^{'} (u_{n}) \to 0 and Φ_{μ_{n}} (u_{n}) ⩽ c \end{matrix}$ for some $c > 0$ . Next we show that ${u_{n}}$ is bounded. Arguing by contradiction, we suppose that $‖ u_{n} ‖ \to \infty$ as $n \to \infty$ . Setting $v_{n} = u_{n} / ‖ u_{n} ‖$ , then $‖ v_{n} ‖ = 1$ . Up to a subsequence, we may assume that $v_{n} ⇀ v$ in E and $v_{n} (x) \to v (x)$ a.e. on $R^{3}$ . There are two cases need to discuss for sequence ${v_{n}}$ : vanishing and nonvanishing. Let $\begin{matrix} δ : = lim_{n \to \infty} sup_{y \in R^{3}} \int_{B (y, 1)} {| v_{n}^{+} |}^{2} . \end{matrix}$ If $δ = 0$ , the vanishing lemma (see [32] or [42, Lemma 1.21]) implies that $v_{n}^{+} \to 0$ in $L^{p}$ for any $2 < p < 3$ . Then by (2.4) we know that for any $s > 0$ , there holds $\begin{matrix} (3.2) & \int_{R^{3}} F (x, s | v_{n}^{+} |) \to 0 . \end{matrix}$ Using (2.3), (2.6), (3.1) and (3.2) we obtain $\begin{matrix} (3.3) & \begin{matrix} c & ⩾ Φ_{μ_{n}} (u_{n}) ⩾ Φ_{μ_{n}} (s v_{n}^{+}) + o (1) \\ = \frac{s^{2}}{2} ({‖ v_{n}^{+} ‖}^{2} - μ_{n} \int_{R^{3}} \frac{| v_{n}^{+} |^{2}}{| x |}) + o (1) \\ ⩾ \frac{s^{2}}{2} (1 - 2 μ_{n}) {‖ v_{n}^{+} ‖}^{2} + o (1) . \end{matrix} \end{matrix}$ We deduce from (2.5) that $‖ u_{n}^{+} ‖^{2} - ‖ u_{n}^{-} ‖^{2} ⩾ Φ_{μ_{n}}^{'} (u_{n}) u_{n}$ , moreover, we obtain $\begin{matrix} 2 {‖ u_{n}^{+} ‖}^{2} ⩾ {‖ u_{n}^{+} ‖}^{2} + {‖ u_{n}^{-} ‖}^{2} + Φ_{μ_{n}}^{'} (u_{n}) u_{n} = ‖ u_{n} ‖^{2} + Φ_{μ_{n}}^{'} (u_{n}) u_{n}, \end{matrix}$ this shows that $‖ v_{n}^{+} ‖^{2} ⩾ c_{0}$ for some $c_{0} > 0$ . Clearly, (3.3) yields a contradiction if s is large enough, then the vanishing does not occur. Therefore, $δ > 0$ . Then, passing to a subsequence, we may assume the existence of ${k_{n}} \subset Z^{3}$ such that $\begin{matrix} \int_{B (k_{n}, 1 + \sqrt{3})} {| v_{n}^{+} |}^{2} > \frac{δ}{2} . \end{matrix}$ We define ${\tilde{v}}_{n} (x) = v_{n} (x + k_{n})$ , then there holds $\begin{matrix} \int_{B (0, 1 + \sqrt{3})} {| {\tilde{v}}_{n}^{+} |}^{2} > \frac{δ}{2} . \end{matrix}$ Up to a subsequence, ${\tilde{v}}_{n}^{+} \to {\tilde{v}}^{+}$ in $L_{loc}^{2}$ and ${\tilde{v}}^{+} \neq 0$ . If $\tilde{v} \neq 0$ , then $| u_{n} (x + k_{n}) | = | {\tilde{v}}_{n} (x) | ‖ u_{n} ‖ \to \infty$ . Hence, for $x \in Ω_{0} : = {x \in R^{3} : \tilde{v} \neq 0}$ , using $(F_{3})$ we have $\begin{matrix} lim_{n \to \infty} \frac{F (x, | u_{n} (x + k_{n}) |)}{‖ u_{n} ‖^{2}} = lim_{n \to \infty} \frac{F (x, | u_{n} (x + k_{n}) |)}{| u_{n} (x + k_{n}) |^{2}} | {\tilde{v}}_{n} |^{2} = \infty . \end{matrix}$ According to the periodicity of F in $x \in R^{3}$ and Fatou’s lemma we get $\begin{matrix} 0 & = lim_{n \to \infty} \frac{Φ_{μ_{n}} (u_{n})}{‖ u_{n} ‖^{2}} \\ ⩽ lim_{n \to \infty} (\frac{1}{2} ({‖ v_{n}^{+} ‖}^{2} - {‖ v_{n}^{-} ‖}^{2} - \frac{μ_{n}}{‖ u_{n} ‖^{2}} \int_{R^{3}} \frac{| u_{n} |^{2}}{| x |}) - \int_{Ω_{0}} \frac{F (x, | u_{n} (x + k_{n}) |)}{‖ u_{n} ‖^{2}}) \\ \to - \infty, \end{matrix}$ this is a contradiction. Hence ${u_{n}}$ is bounded in E. □
Lemma 3.2.
If $| x_{n} | \to \infty$ , then for any $u \in E$ , $\begin{matrix} \int_{R^{3}} \frac{| u (x - x_{n}) |^{2}}{| x |} \to 0 as n \to \infty . \end{matrix}$
Proof.
We employ the similar argument as [28, 45] and give the details for completeness. For any $u \in E$ , according to the density of space E, there is ${φ_{m}} \subset C_{0}^{\infty} (R^{3}, C^{4})$ such that $φ_{m} \to u$ in E as $m \to \infty$ . Moreover, we can take suitable $R_{m} > 0$ such that $supp (φ_{m}) \subset B_{R_{m}} (0)$ . For any m we find $n = n (m)$ such that $| x_{n} | - R_{m} ⩾ m$ and ${n (m)}$ is an increasing sequence. Then we obtain $\begin{matrix} (3.4) & \begin{matrix} \int_{R^{3}} \frac{| φ_{m} (x - x_{n}) |^{2}}{| x |} & = \int_{R^{3}} \frac{| φ_{m} |^{2}}{| x + x_{n} |} = \int_{B_{R_{m}} (0)} \frac{| φ_{m} |^{2}}{| x + x_{n} |} \\ ⩽ \frac{1}{| x_{n} | - R_{m}} \int_{B_{R_{m}} (0)} | φ_{m} |^{2} ⩽ \frac{1}{m} | φ_{m} |_{2}^{2} \to 0 . \end{matrix} \end{matrix}$ Using (2.3) and (3.4) we get $\begin{matrix} \int_{R^{3}} \frac{| u (x - x_{n}) |^{2}}{| x |} & = \int_{R^{3}} \frac{1}{| x |} {| u (x - x_{n}) - φ_{m} (x - x_{n}) + φ_{m} (x - x_{n}) |}^{2} \\ ⩽ \int_{R^{3}} \frac{| u (x - x_{n}) - φ_{m} (x - x_{n}) |^{2}}{| x |} + \int_{R^{3}} \frac{| φ_{m} (x - x_{n}) |^{2}}{| x |} \\ ⩽ 2 ‖ φ_{m} - u ‖^{2} + \int_{R^{3}} \frac{| φ_{m} (x - x_{n}) |^{2}}{| x |} \to 0, \end{matrix}$ this shows that the desired conclusion holds. The proof is completed. □

Below we establish a global compactness result for bounded ${(C)}_{c}$ -sequences of $Φ_{μ}$ to overcome the lack of compactness of the Sobolev embedding. Lemma 3.3.
Assume that $0 ⩽ μ < \frac{1}{2}$ and let ${u_{n}}$ be a bounded ${(C)}_{c_{μ}}$ -sequences of $Φ_{μ}$ at level $c_{μ} > 0$ . Then there exists $u_{μ} \in E$ such that $Φ_{μ}^{'} (u_{μ}) = 0$ , moreover, we have either
$u_{n} \to u_{μ}$ in E, or

there exist number $k ⩾ 1$ , nontrivial critical points $u_{1}, \dots, u_{k}$ of $Φ_{0}$ and k sequences of points ${x_{n}^{i}} \subset Z^{3}$ , $1 ⩽ i ⩽ k$ , such that $\begin{array}{l} | x_{n}^{i} | \to + \infty, | x_{n}^{i} - x_{n}^{j} | \to + \infty, if i \neq j, i, j = 1, 2, \dots, k, \\ ‖ u_{n} - u_{μ} - \sum_{i = 1}^{k} u_{i} (\cdot - x_{n}^{i}) ‖ \to 0, \\ c_{μ} = Φ_{μ} (u_{μ}) + \sum_{i = 1}^{k} Φ_{0} (u_{i}) . \end{array}$

Proof.
Let ${u_{n}}$ be a bounded ${(C)}_{c_{μ}}$ -sequences of $Φ_{μ}$ at level $c_{μ} ⩾ 0$ . Then by (2.3) we know that ${u_{n}}$ is bounded in $L^{2} (R^{3}, \frac{1}{| x |})$ . Thus, there exist $u_{μ} \in E$ , up to a subsequence, such that $u_{n} ⇀ u_{μ}$ in E, $u_{n} ⇀ u_{μ}$ in $L^{2} (R^{3}, \frac{1}{| x |})$ , $u_{n} \to u_{μ}$ in $L_{loc}^{2}$ and $u_{n} (x) \to u_{μ} (x)$ a.e. on $R^{3}$ . Moreover, according to Lemma 2.2 we have $Φ_{μ}^{'} (u_{μ}) = 0$ . Setting $v_{n} = u_{n} - u_{μ}$ , then $v_{n}^{+} ⇀ 0$ in $E^{+}$ , $v_{n}^{-} ⇀ 0$ in $E^{-}$ and $v_{n} ⇀ 0$ in $L^{2} (R^{3}, \frac{1}{| x |})$ . By a direct calculation, we have $\begin{matrix} (3.5) & \begin{matrix} {‖ v_{n}^{+} ‖}^{2} = {‖ u_{n}^{+} ‖}^{2} - {‖ u_{μ}^{+} ‖}^{2} + o (1), {‖ v_{n}^{-} ‖}^{2} = {‖ u_{n}^{-} ‖}^{2} - {‖ u_{μ}^{-} ‖}^{2} + o (1), \\ ‖ v_{n} ‖^{2} = ‖ u_{n} ‖^{2} - ‖ u_{μ} ‖^{2} + o (1), \int_{R^{3}} \frac{| v_{n} |^{2}}{| x |} = \int_{R^{3}} \frac{| u_{n} |^{2}}{| x |} - \int_{R^{3}} \frac{| u_{μ} |^{2}}{| x |} + o (1) . \end{matrix} \end{matrix}$ Using Brezis–Lieb lemma and some standard arguments in [13, 42] we can obtain $\begin{matrix} (3.6) & \int_{R^{3}} F (x, | u_{n} |) - F (x, | u_{μ} |) - F (x, | v_{n} |) = o (1), \end{matrix}$ and for any $ϕ \in E$ $\begin{matrix} (3.7) & Re (\int_{R^{3}} f (x, | u_{n} |) u_{n} \cdot ϕ - f (x, | u_{μ} |) u_{μ} \cdot ϕ - f (x, | v_{n} |) v_{n} \cdot ϕ) = o (1) . \end{matrix}$ Then by (3.5), (3.6) and (3.7) we obtain $\begin{matrix} (3.8) & Φ_{μ} (u_{n}) - Φ_{μ} (u_{μ}) - Φ_{μ} (v_{n}) = o (1), \end{matrix}$ and $\begin{matrix} (3.9) & Φ_{μ}^{'} (u_{n}) - Φ_{μ}^{'} (u_{μ}) - Φ_{μ}^{'} (v_{n}) = o (1) . \end{matrix}$

There are two cases need to discuss for sequence ${v_{n}}$ : vanishing and nonvanishing. If ${v_{n}}$ is vanishing, then $\begin{matrix} lim_{n \to \infty} sup_{y \in R^{3}} \int_{B (y, 1)} | v_{n} |^{2} = 0 . \end{matrix}$ By vanishing lemma we have $v_{n} \to 0$ in $L^{p}$ for any $2 < p < 3$ . Since the orthogonal projection of E on $E^{\pm}$ is continuous in $L^{p}$ , then $v_{n}^{+} \to 0$ and $v_{n}^{-} \to 0$ in $L^{p}$ for any $2 < p < 3$ . Thus according to (2.4) we obtain $\begin{matrix} (3.10) & \begin{matrix} Re \int_{R^{3}} (f (x, | u_{n} |) u_{n} - f (x, | u_{μ} |) u_{μ}) \cdot v_{n}^{+} = o (1), \\ Re \int_{R^{3}} (f (x, | u_{μ} |) u_{μ} - f (x, | u_{n} |) u_{n}) \cdot v_{n}^{-} = o (1) . \end{matrix} \end{matrix}$ Since ${u_{n}}$ is bounded ${(C)}_{c_{μ}}$ -sequences, then $Φ_{μ}^{'} (u_{n}) = o (1)$ and $Φ_{μ}^{'} (u_{n}) v_{n}^{\pm} = o (1)$ . Using these facts and $Φ_{μ}^{'} (u_{μ}) = 0$ , we get $\begin{matrix} o (1) & = Φ_{μ}^{'} (u_{n}) v_{n}^{+} \\ = (u_{n}, v_{n}^{+}) - μ Re \int_{R^{3}} \frac{u_{n} \cdot v_{n}^{+}}{| x |} - Re \int_{R^{3}} f (x, | u_{n} |) u_{n} \cdot v_{n}^{+} \\ = {‖ v_{n}^{+} ‖}^{2} - μ Re \int_{R^{3}} \frac{v_{n} \cdot v_{n}^{+}}{| x |} + Φ_{μ}^{'} (u_{μ}) v_{n}^{+} + Re \int_{R^{3}} (f (x, | u_{μ} |) u_{μ} - f (x, | u_{n} |) u_{n}) \cdot v_{n}^{+} \\ = {‖ v_{n}^{+} ‖}^{2} - μ \int_{R^{3}} \frac{| v_{n}^{+} |^{2}}{| x |} - μ Re \int_{R^{3}} \frac{v_{n}^{-} \cdot v_{n}^{+}}{| x |} + Re \int_{R^{3}} (f (x, | u_{μ} |) u_{μ} - f (x, | u_{n} |) u_{n}) \cdot v_{n}^{+}, \end{matrix}$ this combining with (2.4) we get $\begin{matrix} (3.11) & (1 - 2 μ) {‖ v_{n}^{+} ‖}^{2} ⩽ μ Re \int_{R^{3}} \frac{v_{n}^{-} \cdot v_{n}^{+}}{| x |} + Re \int_{R^{3}} (f (x, | u_{n} |) u_{n} - f (x, | u_{μ} |) u_{μ}) \cdot v_{n}^{+} + o (1) . \end{matrix}$ Similarly, there holds $\begin{matrix} o (1) & = Φ_{μ}^{'} (u_{n}) v_{n}^{-} \\ = - {‖ v_{n}^{-} ‖}^{2} - μ \int_{R^{3}} \frac{| v_{n}^{-} |^{2}}{| x |} - μ Re \int_{R^{3}} \frac{v_{n}^{-} \cdot v_{n}^{+}}{| x |} + Re \int_{R^{3}} (f (x, | u_{μ} |) u_{μ} - f (x, | u_{n} |) u_{n}) \cdot v_{n}^{-}, \end{matrix}$ and hence $\begin{matrix} (3.12) & {‖ v_{n}^{-} ‖}^{2} ⩽ - μ Re \int_{R^{3}} \frac{v_{n}^{-} \cdot v_{n}^{+}}{| x |} + Re \int_{R^{3}} (f (x, | u_{μ} |) u_{μ} - f (x, | u_{n} |) u_{n}) \cdot v_{n}^{-} + o (1) . \end{matrix}$ Therefore, from (3.10), (3.11) and (3.12) we deduce that $‖ v_{n} ‖ \to 0$ in E and $u_{n} \to u_{μ}$ in E. So conclusion (i) holds

If ${v_{n}}$ is nonvanishing, then there exist $δ > 0$ , $ϱ > 1$ and ${y_{n}} \subset Z^{3}$ such that $\begin{matrix} (3.13) & \underset{n \to \infty}{lim inf} \int_{B (y_{n}, ϱ)} | v_{n} |^{2} ⩾ δ . \end{matrix}$ Since $v_{n} ⇀ 0$ in E, then ${y_{n}}$ must be unbounded and, up to a subsequence we may assume that $| y_{n} | \to \infty$ . Let us now consider ${\tilde{v}}_{n} = v_{n} (x + y_{n})$ , then by (3.13) we can find $u_{1} \neq 0$ such that up to a subsequence, ${\tilde{v}}_{n} ⇀ u_{1}$ in E, ${\tilde{v}}_{n} \to u_{1}$ in $L_{loc}^{p}$ for $2 ⩽ p < 3$ and ${\tilde{v}}_{n} (x) \to u_{1} (x)$ a.e. on $R^{3}$ . Using the Hölder inequality and Lemma 3.2 we get for any $φ \in E$ $\begin{matrix} | Re \int_{R^{3}} \frac{1}{| x |} v_{n} \cdot φ (\cdot - y_{n}) | ⩽ {(\int_{R^{3}} \frac{| v_{n} |^{2}}{| x |})}^{1 / 2} {(\int_{R^{3}} \frac{| φ (\cdot - y_{n}) |^{2}}{| x |})}^{1 / 2} \to 0 . \end{matrix}$ Then by a direct calculation we have $\begin{matrix} o (1) & = Φ_{μ}^{'} (v_{n}) φ (\cdot - y_{n}) \\ = (v_{n}^{+}, φ^{+} (\cdot - y_{n})) - (v_{n}^{-}, φ^{-} (\cdot - y_{n})) - μ Re \int_{R^{3}} \frac{v_{n} φ (\cdot - y_{n})}{| x |} \\ - Re \int_{R^{3}} f (x, | v_{n} |) v_{n} φ (\cdot - y_{n}) \\ = ({\tilde{v}}_{n}^{+}, φ^{+}) - ({\tilde{v}}_{n}^{-}, φ^{-}) - Re \int_{R^{3}} f (x, | {\tilde{v}}_{n} |) {\tilde{v}}_{n} \cdot φ + o (1) \\ = Φ_{0}^{'} ({\tilde{v}}_{n}) φ + o (1), \end{matrix}$ combining Lemma 2.2 we deduce that $Φ_{0}^{'} (u_{1}) φ = 0$ , which implies that $u_{1}$ is a nontrivial critical point of $Φ_{0}$ . Now we consider $v_{n}^{1} = u_{n} - u_{μ} - u_{1} (\cdot - y_{n})$ , similar to (3.5) there holds $\begin{matrix} (3.14) & \begin{matrix} {‖ v_{n}^{1 +} ‖}^{2} = {‖ u_{n}^{+} ‖}^{2} - {‖ u_{μ}^{+} ‖}^{2} - {‖ u_{1}^{+} ‖}^{2} + o (1), \\ {‖ v_{n}^{1 -} ‖}^{2} = {‖ u_{n}^{-} ‖}^{2} - {‖ u_{μ}^{-} ‖}^{2} - {‖ u_{1}^{-} ‖}^{2} + o (1), \\ {‖ v_{n}^{1} ‖}^{2} = ‖ u_{n} ‖^{2} - ‖ u_{μ} ‖^{2} - ‖ u_{1} ‖^{2} + o (1) . \end{matrix} \end{matrix}$ Observe that since $u_{n} ⇀ u_{μ}$ in $L^{2} (R^{3}, \frac{1}{| x |})$ we obtain $\begin{matrix} Re \int_{R^{3}} \frac{u_{n} \cdot u_{μ}}{| x |} \to Re \int_{R^{3}} \frac{| u_{μ} |^{2}}{| x |}, \end{matrix}$ moreover, using Lemma 3.2 we have $\begin{matrix} (3.15) & \begin{matrix} \int_{R^{3}} \frac{| v_{n}^{1} |^{2}}{| x |} & = \int_{R^{3}} \frac{| u_{n} - u_{μ} |^{2}}{| x |} - \int_{R^{3}} \frac{| u_{1} (\cdot - y_{n}) |^{2}}{| x |} + o (1) \\ = \int_{R^{3}} \frac{| u_{n} |^{2}}{| x |} - \int_{R^{3}} \frac{| u_{μ} |^{2}}{| x |} + o (1) . \end{matrix} \end{matrix}$ Similar to (3.6), there holds $\begin{matrix} (3.16) & \int_{R^{3}} F (x, | u_{n} |) - F (x, | u_{μ} |) - F (x, | u_{1} |) - F (x, | v_{n}^{1} |) = o (1) . \end{matrix}$ Therefore, from (3.14), (3.15) and (3.16) we deduce that $\begin{matrix} (3.17) & Φ_{μ} (v_{n}^{1}) = Φ_{μ} (u_{n}) - Φ_{μ} (u_{μ}) - Φ_{0} (u_{1}) + o (1) \end{matrix}$ and we take $x_{n}^{1} : = y_{n}$ . Now we replace $v_{n}$ by $v_{n}^{1}$ and repeat the above argument for the vanishing case and nonvanishing case, that is, if $\begin{matrix} lim_{n \to \infty} sup_{y \in R^{3}} \int_{B (y, 1)} {| v_{n}^{1} |}^{2} = 0, \end{matrix}$ then $v_{n}^{1} \to 0$ in E and by (3.14) and (3.17) we take $k = 1$ . Otherwise as in nonvanishing case we find ${y_{n}} \subset Z^{3}$ such that (3.13) holds for ${v_{n}^{1}}$ . Then passing to a subsequence $| y_{n} | \to \infty$ and $| y_{n} - x_{n}^{1} | \to \infty$ as $n \to \infty$ . Similar to the above argument, let ${\tilde{v}}_{n}^{1} (x) = v_{n} (x + y_{n})$ , then we can find $u_{2} \neq 0$ such that up to a subsequence, ${\tilde{v}}_{n}^{1} ⇀ u_{2}$ in E, ${\tilde{v}}_{n}^{1} \to u_{2}$ in $L_{loc}^{p}$ for $2 ⩽ p < 3$ and ${\tilde{v}}_{n}^{1} (x) \to u_{2} (x)$ a.e. on $R^{3}$ . Moreover, $u_{2}$ is a nontrivial critical point of $Φ_{0}$ by Lemma 2.2. Denote $v_{n}^{2} = u_{n} - u_{μ} - u_{1} (\cdot - x_{n}^{1}) - u_{2} (\cdot - y_{n})$ , and similar to (3.14) and (3.17), we obtain $\begin{matrix} {‖ v_{n}^{2 +} ‖}^{2} = {‖ u_{n}^{+} ‖}^{2} - {‖ u_{μ}^{+} ‖}^{2} - {‖ u_{1}^{+} ‖}^{2} - {‖ u_{2}^{+} ‖}^{2} + o (1), \\ {‖ v_{n}^{2 -} ‖}^{2} = {‖ u_{n}^{-} ‖}^{2} - {‖ u_{μ}^{-} ‖}^{2} - {‖ u_{1}^{-} ‖}^{2} - {‖ u_{2}^{-} ‖}^{2} + o (1), \\ {‖ v_{n}^{2} ‖}^{2} = ‖ u_{n} ‖^{2} - ‖ u_{μ} ‖^{2} - ‖ u_{1} ‖^{2} - ‖ u_{2} ‖^{2} + o (1), \\ Φ_{μ} (v_{n}^{2}) = Φ_{μ} (u_{n}) - Φ_{μ} (u_{μ}) - Φ_{0} (u_{1}) - Φ_{0} (u_{2}) + o (1), \end{matrix}$ and $x_{n}^{2} : = y_{n}$ . Again we repeat the above arguments to finish the proof of conclusion (ii) and the iterations must stop after finite steps, this is because there is a constant $σ > 0$ such that $\begin{matrix} (3.18) & ‖ u ‖ ⩾ σ for any u \neq 0 with Φ_{0}^{'} (u) = 0 . \end{matrix}$ Indeed, according to the fact $Φ_{0}^{'} (u) u^{\pm} = 0$ and (2.4) we have $\begin{matrix} {‖ u^{+} ‖}^{2} ⩽ \int_{R^{3}} | f (x, | u |) u u^{+} | ⩽ ϵ ‖ u^{+} ‖ ‖ u ‖ + C_{ϵ} ‖ u^{+} ‖ ‖ u ‖^{p - 1} \end{matrix}$ and $\begin{matrix} {‖ u^{-} ‖}^{2} ⩽ \int_{R^{3}} | f (x, | u |) u u^{-} | ⩽ ϵ ‖ u^{-} ‖ ‖ u ‖ + C_{ϵ} ‖ u^{-} ‖ ‖ u ‖^{p - 1} . \end{matrix}$ Hence $\begin{matrix} ‖ u ‖^{2} ⩽ 2 ϵ ‖ u ‖^{2} + 2 C_{ϵ} ‖ u ‖^{p}, \end{matrix}$ which implies that (3.18) holds. The proof is completed. □

4. Proof of the main results

In this section we give the proof of the main results.

Proof of Theorem 1.1.

We will take three steps to complete the proof.

Step 1. Existence of ground state solutions. According to Lemma 2.9, there exists a ${(C)}_{c_{μ}}$ -sequence ${u_{n}}$ of $Φ_{μ}$ such that $\begin{matrix} Φ_{μ} (u_{n}) \to c_{μ} ⩽ m_{μ} and ‖ Φ_{μ}^{'} (u_{n}) ‖ (1 + ‖ u_{n} ‖) \to 0 . \end{matrix}$ Moreover, Lemma 3.1 shows that ${u_{n}}$ is bounded in E, then passing to a subsequence, $u_{n} ⇀ u_{μ}$ in E, $u_{n} (x) \to u_{μ} (x)$ a.e. on $R^{3}$ , and Lemma 2.2 shows that $Φ_{μ}^{'} (u_{μ}) = 0$ .

Next we verify that $u_{μ} \neq 0$ . To do this, we need to prove $u_{n} \to u_{μ}$ in E. Indeed, for $μ = 0$ , combining with concentration compactness principle [32] and the invariance of $Φ_{0}$ under translation, by a standard method we can prove that $Φ_{0}$ has a nontrivial ground state solution $u_{0} \in N_{0}$ satisfying $Φ_{0} (u_{0}) = m_{0}$ , the result and details of proof can be found in [46]. Now let us assume that $0 < μ < \frac{1}{2}$ and consider $\begin{matrix} Q (u_{0}) : = {u = t u_{0} + v : v \in E^{-}, t ⩾ 0, ‖ t u_{0} + v ‖ ⩽ R} . \end{matrix}$ Observe that, let $t_{n} u_{0} + v_{n} \in Q (u_{0})$ , then passing to a subsequence we may assume that $t_{n} \to t_{0}$ , $v_{n} ⇀ v_{0}$ in $E^{-}$ and $L^{2} (R^{3}, \frac{1}{| x |})$ and $v_{n} (x) \to v_{0} (x)$ a.e. on $R^{3}$ . Making use of the weak lower semi-continuity of norm, we see that $t_{n} u_{0} + v_{n} ⇀ t_{0} u_{0} + v_{0} \in Q (u_{0})$ , so $Q (u_{0})$ is weakly sequentially closed. Again using the weak lower semi-continuity of norm and Fatou’s lemma we can easily obtain $\begin{matrix} \underset{n \to \infty}{lim sup} Φ_{μ} (t_{n} u_{0} + v_{n}) ⩽ Φ_{μ} (t_{0} u_{0} + v_{0}), \end{matrix}$ this shows that $Φ_{μ}$ is weakly sequentially upper semi-continuous. Hence $Φ_{μ}$ attains its maximum in $Q (u_{0})$ . Assume that $t_{0} u_{0} + v_{0} \in Q (u_{0})$ such that $\begin{matrix} Φ_{μ} (t_{0} u_{0} + v_{0}) = max_{u \in Q (u_{0})} Φ_{μ} (u) > 0, \end{matrix}$ then $t_{0} u_{0} + v_{0} \in N_{μ}$ . Therefore by Corollary 2.4, we have $\begin{matrix} (4.1) & m_{0} = Φ_{0} (u_{0}) ⩾ Φ_{0} (t_{0} u_{0} + v_{0}) > Φ_{μ} (t_{0} u_{0} + v_{0}) ⩾ m_{μ} ⩾ c_{μ} . \end{matrix}$ If ${u_{n}}$ does not converge strongly to $u_{μ}$ in E ( $u_{n} ↛ u_{μ}$ in E), then by Lemma 3.3 we can find number $k ⩾ 1$ and nontrivial solutions $u_{1}, \dots, u_{k}$ of problem (1.3) with $μ = 0$ such that $\begin{matrix} lim_{n \to \infty} Φ_{μ} (u_{n}) = c_{μ} = Φ_{μ} (u_{μ}) + \sum_{i = 1}^{k} Φ_{0} (u_{i}) ⩾ k m_{0} ⩾ m_{0} . \end{matrix}$ Obviously, this is a contradiction with (4.1). So $u_{n} \to u_{μ}$ in E and $u_{μ} \neq 0$ .

Following from the above discussion we know that $u_{μ}$ is a nontrivial critical point of $Φ_{μ}$ . Furthermore, using (2.5) and Fatou’s lemma we get $\begin{matrix} m_{μ} & ⩾ c_{μ} = lim_{n \to \infty} (Φ_{μ} (u_{n}) - \frac{1}{2} Φ_{μ}^{'} (u_{n}) u_{n}) \\ = lim_{n \to \infty} \int_{R^{3}} (\frac{1}{2} f (x, | u_{n} |) | u_{n} |^{2} - F (x, | u_{n} |)) \\ ⩾ \int_{R^{3}} lim_{n \to \infty} (\frac{1}{2} f (x, | u_{n} |) | u_{n} |^{2} - F (x, | u_{n} |)) \\ = Φ_{μ} (u_{μ}) - \frac{1}{2} Φ_{μ}^{'} (u_{μ}) u_{μ} = Φ_{μ} (u_{μ}), \end{matrix}$ which implies that $Φ_{μ} (u_{μ}) ⩽ m_{μ}$ , then $Φ_{μ} (u_{μ}) = m_{μ}$ by the definition of $m_{μ}$ . Therefore, $u_{μ}$ is a ground state solution of problem (1.3).

Step 2. Compactness of the set of all ground state solutions. Let $L$ be the set of all ground state solutions of $Φ_{μ}$ , following from Step 1, $L \neq \emptyset$ . Let ${u_{n}} \subset L$ , then $u_{n} \in N_{μ}$ , $Φ_{μ} (u_{n}) = m_{μ}$ and $Φ_{μ}^{'} (u_{n}) = 0$ . Clearly, ${u_{n}}$ is a ${(C)}_{m_{μ}}$ -sequence, and ${u_{n}}$ is bounded by Lemma 3.1. Similar to the proof of Step 1, up to a subsequence, we can deduce that there exists u such that $u_{n} \to u$ in E and $u \in L$ . Observe that $\begin{matrix} A u = \frac{μ}{| x |} u + f (x, | u |) u . \end{matrix}$ Setting $\begin{matrix} A_{0} = - i \nabla \cdot α = - i \sum_{i = 1}^{3} α_{k} \partial_{k} and A = A_{0} + a β . \end{matrix}$ By the Minkowski inequality and the Hardy inequality, we have $\begin{matrix} {| A (u_{n} - u) |}_{2} & = {(\int_{R^{3}} {| \frac{μ}{| x |} (u_{n} - u) + (f (x, | u_{n} |) u_{n} - f (x, | u |) u) |}^{2})}^{1 / 2} \\ ⩽ μ {(\int_{R^{3}} \frac{| u_{n} - u |^{2}}{| x |^{2}})}^{1 / 2} + {(\int_{R^{3}} {| f (x, | u_{n} |) u_{n} - f (x, | u |) u |}^{2})}^{1 / 2} \\ ⩽ μ {(\int_{R^{3}} 4 {| \nabla (u_{n} - u) |}^{2})}^{1 / 2} + {(\int_{R^{3}} {| f (x, | u_{n} |) u_{n} - f (x, | u |) u |}^{2})}^{1 / 2} \\ ⩽ 2 μ {| A (u_{n} - u) |}_{2} + {(\int_{R^{3}} {| f (x, | u_{n} |) u_{n} - f (x, | u |) u |}^{2})}^{1 / 2} . \end{matrix}$ Moreover, since $0 ⩽ μ < \frac{1}{2}$ , by (2.4) and $u_{n} \to u$ in E we get $\begin{matrix} {| A (u_{n} - u) |}_{2} = o (1), \end{matrix}$ which implies that $u_{n} \to u$ in $H^{1}$ .

Step 3. Exponential decay of ground state solutions. Observe that $\begin{matrix} A_{0}^{2} = - Δ and A_{0} u = - a β u + \frac{μ}{| x |} u + f (x, | u |) u, \end{matrix}$ computing directly, we have $\begin{matrix} - Δ u & = A_{0} (A_{0} u) = A_{0} (- a β u + \frac{μ}{| x |} u + f (x, | u |) u) \\ = - a^{2} u + {(\frac{μ}{| x |} + f (x, | u |))}^{2} u + μ A_{0} (\frac{1}{| x |}) u + A_{0} f (x, | u |) u, \end{matrix}$ that is, $\begin{matrix} (4.2) & Δ u = a^{2} u - {(\frac{μ}{| x |} + f (x, | u |))}^{2} u - μ A_{0} (\frac{1}{| x |}) u - A_{0} f (x, | u |) u . \end{matrix}$ Letting $\begin{matrix} sgn u = \{\begin{matrix} \frac{\bar{u}}{| u |} & if u \neq 0, \\ 0 & if u = 0 . \end{matrix} \end{matrix}$ Using Kato’s inequality and (4.2), there exist $R > 0$ and $Λ > 0$ such that $\begin{matrix} (4.3) & \begin{matrix} Δ | u | & ⩾ Re [Δ u (sgn u)] \\ = Re [a^{2} u \frac{\bar{u}}{| u |} - {(\frac{μ}{| x |} + f (x, | u |))}^{2} u \frac{\bar{u}}{| u |} - μ A_{0} (\frac{1}{| x |}) u \frac{\bar{u}}{| u |} - A_{0} f (x, | u |) u \frac{\bar{u}}{| u |}] \\ ⩾ - Λ | u | - Re [μ A_{0} (\frac{1}{| x |}) u \frac{\bar{u}}{| u |} + A_{0} f (x, | u |) u \frac{\bar{u}}{| u |}] \end{matrix} \end{matrix}$ for all $| x | ⩾ R$ .

Observe that, for $ξ_{k} : R^{3} \to R$ , $k = 1, 2, 3$ , $\begin{matrix} M_{ξ} : = \sum_{i = 1}^{3} α_{k} ξ_{k} = (\begin{matrix} 0 & 0 & ξ_{3} & ξ_{1} - i ξ_{2} \\ 0 & 0 & ξ_{1} + i ξ_{2} & - ξ_{3} \\ ξ_{3} & ξ_{1} - i ξ_{2} & 0 & 0 \\ ξ_{1} + i ξ_{2} & - ξ_{3} & 0 & 0 \end{matrix}) \end{matrix}$ is a Hermitian matrix. Applying to $ξ_{k} = \partial (\frac{1}{| x |}) / \partial x_{k}$ and $\partial f / \partial x_{k}$ , we see plainly that $\begin{matrix} (4.4) & Re [μ A_{0} (\frac{1}{| x |}) u \frac{\bar{u}}{| u |}] = 0 and Re [A_{0} f (x, | u |) u \frac{\bar{u}}{| u |}] = 0 . \end{matrix}$ It then follows from (4.3) and (4.4) that $\begin{matrix} Δ | u | ⩾ - Λ | u |, for | x | ⩾ R . \end{matrix}$ According to the sub-solution estimate [27, 37] we have $\begin{matrix} (4.5) & | u (x) | ⩽ C_{0} \int_{B_{1} (x)} | u (y) | d y, for | x | ⩾ R \end{matrix}$ with $C_{0}$ independent of x and $u \in L$ .

Since $L$ is compact in $H^{1}$ , $| u (x) | \to 0$ as $| x | \to \infty$ uniformly in $u \in L$ . In fact, if not, then by (4.5) there exist $\tilde{c} > 0$ , $u_{j} \in L$ and $x_{j} \in R^{3}$ with $| x_{j} | \to \infty$ such that $\begin{matrix} \tilde{c} ⩽ | u_{j} (x_{j}) | ⩽ C_{0} \int_{B_{1} (x_{j})} | u_{j} (y) | d y . \end{matrix}$ We may assume that $u_{j} \to u \in L$ in $H^{1}$ by the compactness of $L$ , then we get $\begin{matrix} \tilde{c} & ⩽ | u_{j} (x_{j}) | ⩽ C_{0} \int_{B_{1} (x_{j})} | u_{j} | ⩽ C_{0} \int_{B_{1} (x_{j})} | u_{j} - u | + C_{0} \int_{B_{1} (x_{j})} | u | \\ ⩽ \tilde{C} {(\int_{R^{3}} | u_{j} - u |^{2})}^{1 / 2} + C_{0} \int_{B_{1} (x_{j})} | u | \to 0, \end{matrix}$ which implies a contradiction. Therefore, it follows from $(f_{2})$ that there exists $R > 0$ such that $\begin{matrix} | \frac{μ}{| x |} + f (x, | u |) | ⩽ \frac{a^{2}}{2} \end{matrix}$ for all $| x | ⩾ R$ and $u \in L$ . This, together with (4.3) and (4.4), implies $\begin{matrix} Δ | u | ⩾ \frac{a^{2}}{2} | u |, for | x | ⩾ R, u \in L . \end{matrix}$ Let $Γ (x)$ be a fundamental solution of equation $- Δ Γ + \frac{a^{2}}{2} Γ = 0$ . According to the uniform boundedness, we may choose $Γ (x)$ such that $| u (x) | ⩽ \frac{a^{2}}{2} Γ (x)$ holds on $| x | = R$ for all $u \in L$ . Let $z = | u | - \frac{a^{2}}{2} Γ$ , then $\begin{matrix} Δ z = Δ | u | - Δ Γ ⩾ \frac{a^{2}}{2} (| u | - \frac{a^{2}}{2} Γ) = \frac{a^{2}}{2} z, for | x | ⩾ R . \end{matrix}$ Using the maximum principle we can conclude that $z (x) ⩽ 0$ for $| x | ⩾ R$ , i.e., $| u (x) | ⩽ \frac{a^{2}}{2} Γ (x)$ for $| x | ⩾ R$ . It is well known that there are $C > 0$ such that $\begin{matrix} Γ (x) ⩽ C exp (- \sqrt{a^{2} / 2} | x |) \end{matrix}$ for $| x | ⩾ 1$ . Hence, we get $\begin{matrix} | u (x) | ⩽ C exp (- \sqrt{a^{2} / 2} | x |) \end{matrix}$ for $| x | ⩾ 1$ and $u \in L$ . The proof is completed. □

Proof of Theorem 1.2.

Plainly, $Φ_{μ_{1}} (u) ⩽ Φ_{μ_{2}} (u)$ if $μ_{1} ⩾ μ_{2}$ , hence $m_{μ_{1}} ⩽ m_{μ_{2}}$ and $m_{μ}$ is decreasing. Next we verify the convergence of $m_{μ}$ when μ tends to 0. Indeed, let $u_{μ} \in N_{μ}$ be a ground state solution of problem (1.3) and $0 ⩽ μ < \frac{1}{2}$ . From Lemma 2.10 we deduce that there exist $t_{μ} > 0$ and $v_{μ} \in E^{-}$ such that $t_{μ} u_{μ} + v_{μ} \in N_{0}$ . Then, by Corollary 2.4 we get $\begin{matrix} (4.6) & \begin{matrix} m_{μ} & = Φ_{μ} (u_{μ}) ⩾ Φ_{μ} (t_{μ} u_{μ} + v_{μ}) \\ = Φ_{0} (t_{μ} u_{μ} + v_{μ}) - \frac{μ}{2} \int_{R^{3}} \frac{| t_{μ} u_{μ} + v_{μ} |^{2}}{| x |} \\ ⩾ m_{0} - \frac{μ}{2} \int_{R^{3}} \frac{| t_{μ} u_{μ} + v_{μ} |^{2}}{| x |} . \end{matrix} \end{matrix}$ Similarly, let $u_{0} \in N_{0}$ be a ground state solution of problem (1.3) with $μ = 0$ . Also using Lemma 2.10 there exist $t_{0} > 0$ and $v_{0} \in E^{-}$ such that $t_{0} u_{0} + v_{0} \in N_{μ}$ . Then, by Corollary 2.4 we get $\begin{matrix} (4.7) & \begin{matrix} m_{0} & = Φ_{0} (u_{0}) ⩾ Φ_{0} (t_{0} u_{0} + v_{0}) \\ = Φ_{μ} (t_{0} u_{0} + v_{0}) + \frac{μ}{2} \int_{R^{3}} \frac{| t_{0} u_{0} + v_{0} |^{2}}{| x |} \\ ⩾ m_{μ} + \frac{μ}{2} \int_{R^{3}} \frac{| t_{0} u_{0} + v_{0} |^{2}}{| x |} . \end{matrix} \end{matrix}$ Since $μ ⩾ 0$ , then we have $\begin{matrix} (4.8) & m_{0} ⩾ m_{μ} = Φ_{μ} (u_{μ}) . \end{matrix}$ To prove the convergence property of $m_{μ}$ , we take sequence $μ_{n} \to 0^{+}$ as $n \to \infty$ . Let $u_{μ_{n}} \in N_{μ_{n}}$ , then $u_{μ_{n}}$ satisfies $Φ_{μ_{n}} (u_{μ_{n}}) = m_{μ_{n}} ⩽ m_{0}$ and $Φ_{μ_{n}}^{'} (u_{μ_{n}}) u_{μ_{n}} = 0$ . For notational simplicity, we denote $u_{μ_{n}}$ by $u_{n}$ . According to the proof of Lemma 3.1, we can see that ${u_{n}}$ is bounded. If $\begin{matrix} lim_{n \to \infty} sup_{y \in R^{3}} \int_{B (y, 1)} {| u_{n}^{+} |}^{2} = 0, \end{matrix}$ then by the vanishing lemma we get $u_{n}^{+} \to 0$ in $L^{p}$ for all $p \in (2, 3)$ . According to this fact, from $u_{n} \in N_{μ_{n}}$ , it follows that $\begin{matrix} {‖ u_{n}^{+} ‖}^{2} = μ_{n} Re \int_{R^{3}} \frac{u_{n} u_{n}^{+}}{| x |} + Re \int_{R^{3}} f (x, | u_{n} |) u_{n} u_{n}^{+} \to 0 . \end{matrix}$ Then it is easy to get ${lim sup}_{n \to \infty} Φ_{μ_{n}} (u_{n}) ⩽ 0$ , which contradicts to Lemma 2.5-(i). Therefore, there exist $δ > 0$ , $ϱ > 1$ and ${y_{n}} \subset Z^{3}$ such that $\begin{matrix} \underset{n \to \infty}{lim inf} \int_{B (y_{n}, ϱ)} {| u_{n}^{+} |}^{2} ⩾ δ, \end{matrix}$ then passing to a subsequence we find $u \in E$ such that $u_{n}^{+} (\cdot + y_{n}) \to u^{+}$ in $L_{loc}^{2}$ and $u^{+} \neq 0$ . Moreover, we may assume that $u_{n} (\cdot + y_{n}) ⇀ u$ in E, $u_{n} (x + y_{n}) \to u (x)$ , $u_{n}^{+} (x + y_{n}) \to u^{+} (x)$ a.e. on $R^{3}$ . Let $t_{n} u_{n} + v_{n} \in N_{0}$ and $t_{n} > 0$ , $v_{n} \in E^{-}$ . Since $F ⩾ 0$ and the boundedness of ${u_{n}}$ , by (2.5) we have $\begin{matrix} (4.9) & \begin{matrix} {‖ u_{n}^{+} ‖}^{2} & = {‖ u_{n}^{-} + \frac{v_{n}}{t_{n}} ‖}^{2} + \frac{1}{t_{n}^{2}} \int_{R^{3}} f (x, | t_{n} u_{n} + v_{n} |) | t_{n} u_{n} + v_{n} |^{2} \\ ⩾ {‖ u_{n}^{-} + \frac{v_{n}}{t_{n}} ‖}^{2} + 2 \int_{R^{3}} \frac{F (x, t_{n} | u_{n} + \frac{v_{n}}{t_{n}} |)}{t_{n}^{2}} ⩾ {‖ u_{n}^{-} + \frac{v_{n}}{t_{n}} ‖}^{2}, \end{matrix} \end{matrix}$ this shows that ${u_{n} + \frac{v_{n}}{t_{n}}}$ is also bounded. Hence we may assume that $(u_{n}^{-} + \frac{v_{n}}{t_{n}}) (x) \to v (x)$ a.e. on $R^{3}$ for some $v \in E^{-}$ . Now we claim that ${t_{n}}$ is bounded. If not, then $| t_{n} u_{n} (x) + v_{n} (x) | = t_{n} | u_{n} + \frac{v_{n}}{t_{n}} | \to \infty$ provided that $(u^{+} + v) (x) \neq 0$ . Using $(f_{3})$ and Fatou’s lemma we get $\begin{matrix} \int_{R^{3}} \frac{F (x, t_{n} | u_{n} + \frac{v_{n}}{t_{n}} |)}{t_{n}^{2}} \to \infty, \end{matrix}$ which contradicts (4.9), and so ${t_{n}}$ is bounded. Then ${t_{n} u_{n}^{+}}$ and ${t_{n} u_{n}^{-} + v_{n}}$ are both bounded, and by (2.3) we get $\begin{matrix} (4.10) & \frac{μ_{n}}{2} \int_{R^{3}} \frac{| t_{n} u_{n} + v_{n} |^{2}}{| x |} ⩽ μ_{n} ‖ t_{n} u_{n} + v_{n} ‖^{2} \to 0 . \end{matrix}$ Therefore, using (4.6), (4.8) and (4.10) one has $m_{μ_{n}} \to m_{0}$ as $μ_{n} \to 0^{+}$ . The proof is completed. □

Proof of Theorem 1.3.

Let ${μ_{n}}$ be a sequence with $μ_{n} \to 0^{+}$ as $n \to \infty$ and ${u_{μ_{n}}}$ be a sequence of ground state solutions of problem (1.3) with $μ = μ_{n}$ and denote $u_{n} : = u_{μ_{n}}$ . As in the proof of Theorem 1.2, ${u_{n}}$ is bounded, then passing to a subsequence, we may assume that $u_{n} ⇀ u_{0}$ in E, $u_{n} \to u_{0}$ in $L_{loc}^{p}$ for $2 ⩽ p < 3$ and $u_{n} (x) \to u_{0} (x)$ a.e. on $R^{3}$ . Observe that for any $φ \in E$ , by the Hölder inequality and (2.3) we have $\begin{matrix} Φ_{0}^{'} (u_{n}) φ = Φ_{μ_{n}}^{'} (u_{n}) φ + μ_{n} Re \int_{R^{3}} \frac{u_{n} φ}{| x |} \to 0 . \end{matrix}$ This shows that $Φ_{0}^{'} (u_{0}) = 0$ , and so $u_{0}$ is a nontrivial critical point of $Φ_{0}$ . Setting $\begin{matrix} G (x, | u |) : = \frac{1}{2} f (x, | u |) | u |^{2} - F (x, | u |) . \end{matrix}$ Using Fatou’s lemma and the conclusion of Theorem 1.2 we obtain $\begin{matrix} m_{0} & = lim_{n \to \infty} Φ_{μ_{n}} (u_{n}) = lim_{n \to \infty} (Φ_{μ_{n}} (u_{n}) - \frac{1}{2} Φ_{μ_{n}}^{'} (u_{n}) u_{n}) \\ = lim_{n \to \infty} \int_{R^{3}} G (x, | u_{n} |) ⩾ \int_{R^{3}} G (x, | u_{0} |) \\ = Φ_{0} (u_{0}) - \frac{1}{2} Φ_{0}^{'} (u_{0}) u_{0} = Φ_{0} (u_{0}) ⩾ m_{0}, \end{matrix}$ which implies that $u_{0}$ is a ground state solution of problem (1.3) with $μ = 0$ . Moreover, according to the above fact we also obtain $\begin{matrix} (4.11) & lim_{n \to \infty} \int_{R^{3}} G (x, | u_{n} |) = \int_{R^{3}} G (x, | u_{0} |) . \end{matrix}$ Next we prove $u_{n} \to u_{0}$ in E. It follows from (3.6) and (3.7) we can obtain $\begin{matrix} lim_{n \to \infty} \int_{R^{3}} (G (x, | u_{n} |) - G (x, | u_{n} - u_{0} |) - G (x, | u_{0} |)) = 0 . \end{matrix}$ It follows from (4.11) and $(f_{5})$ that $\begin{matrix} c_{0} \int_{R^{3}} | u_{n} - u_{0} |^{q} ⩽ \int_{R^{3}} G (x, | u_{n} - u_{0} |) \to 0, \end{matrix}$ which implies that $u_{n} \to u_{0}$ in $L^{q}$ . Since ${u_{n}}$ is bounded in $L^{2}$ and $L^{3}$ , then by the interpolation inequalities we get $u_{n} \to u_{0}$ in $L^{τ}$ for any $2 < τ < 3$ . Hence, using (2.3), (2.4) and the continuity of orthogonal projection of E on $E^{+}$ we have $\begin{matrix} {‖ u_{n}^{+} - u_{0}^{+} ‖}^{2} & = Φ_{μ_{n}}^{'} (u_{n}) (u_{n}^{+} - u_{0}^{+}) - (u_{0}^{+}, u_{n}^{+} - u_{0}^{+}) \\ + μ_{n} Re \int_{R^{3}} \frac{u_{n} \cdot (u_{n}^{+} - u_{0}^{+})}{| x |} + Re \int_{R^{3}} f (x, | u_{n} |) u_{n} \cdot (u_{n}^{+} - u_{0}^{+}) \\ \to 0, \end{matrix}$ this implies that $u_{n}^{+} \to u_{0}^{+}$ in E. Similarly, we can prove that $u_{n}^{-} \to u_{0}^{-}$ in E. So $u_{n} \to u_{0}$ in E. This completes the proof. □

Footnotes

Acknowledgements

This work was supported by the Natural Science Foundation of Hunan Province (2021JJ30189) and the China Scholarship Council (Nos. 201908430218, 201908430219) for visiting the University of Craiova (Romania). W. Zhang and J. Zhang would like to thank the China Scholarship Council and the Embassy of the People’s Republic of China in Romania.

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