Existence and asymptotic behavior of solutions for Choquard equations with singular potential under Berestycki

Abstract

We prove the existence and asymptotic behavior of solutions to the following problem: $\begin{matrix} \{\begin{matrix} - Δ u + V (x) u - g (x) u = (I_{α} * F (u)) f (u), x \in R^{N}; \\ u \in H^{1} (R^{N}), \end{matrix} \end{matrix}$ where $g (x) : = \frac{μ}{| x |}$ is called the Coulomb potential, $g (x) : = \frac{β}{| x |^{2}}$ is called the Hardy potential (the inverse-square potential). $μ, β > 0$ are parameters, $I_{α} : R^{N} ⟶ R$ is the Riesz potential. Moreover, the nonlinearity f satisfies Berestycki–Lions type conditions which are introduced by Moroz and Van Schaftingen (Trans. Amer. Math. Soc. 367 (2015) 6557–6579). When $μ \in (0, α (N - 2) / 2 (α + 1))$ and $β \in (0, α {(N - 2)}^{2} / 4 (2 + α))$ , under some mild assumptions on V, we establish the existence and asymptotic behavior of solutions. Particularly, our results extend some relate ones in the literature.

Keywords

Choquard equation ground state solution Berestycki–Lions type conditions Coulomb potential Hardy potential

1. Introduction and main results

We consider the following Choquard equation with Coulomb or Hardy potential: $\begin{matrix} (1.1) & \{\begin{matrix} - Δ u + V (x) u - g (x) u = (I_{α} * F (u)) f (u), x \in R^{N}; \\ u \in H^{1} (R^{N}), \end{matrix} \end{matrix}$ where $N ⩾ 3$ , $α \in (0, N)$ , $g (x) : = \frac{μ}{| x |}$ or $g (x) : = \frac{β}{| x |^{2}}$ , μ and $β > 0$ are parameters, F is the primitive function of f and $I_{α} : R^{N} ⟶ R$ is the Riesz potential defined by $\begin{matrix} I_{α} (x) = \frac{Γ (\frac{N - 2}{2})}{Γ (\frac{α}{N}) 2^{α} π^{\frac{N}{2}} | x |^{N - α}}, x \in R^{N} ∖ {0} . \end{matrix}$ $V : R^{N} \to R$ and $f : R \to R$ satisfy the following basic assumptions:

$V \in C (R^{N}, (0, \infty))$ , ${inf}_{R^{N}} V (x) > 1$ and $V_{\infty} : = {lim}_{| x | \to \infty} V (x) < + \infty$ ;

$f \in C (R, R)$ and there exists a constant $C_{0} > 0$ such that $\begin{matrix} | f (t) t | ⩽ C_{0} (| t |^{\frac{N + α}{N - 2}} + | t |^{\frac{N + α}{N - 2}}), \forall t \in R; \end{matrix}$

$F (t) = o (t^{\frac{N + α}{N}})$ as $t \to 0$ and $F (t) = o (t^{\frac{N + α}{N - 2}})$ as $| t | \to \infty$ ;

there exists $s_{0} > 0$ such that $F (s_{0}) \neq 0$ .

Note that (F1)–(F3) were almost necessary and sufficient conditions and regarded as the Berestycki–Lions type conditions to Choquard equations which introduced by Lions and Berestycki in [3] firstly. Later, Moroz and Van Schaftingen investigated the Choquard equation (1.1) with $g (x) = 0$ in [18].

The main reason of interest in Coulomb and Hardy potential rely in their criticality: indeed they have the same homogeneity as the Laplacian and do not belong to the Kato class [21], hence they cannot be regarded as a lower order perturbation term. And the additional singular potential breaks the compactness of embedding $H^{1} (R^{N}) ↪ L^{2} (R^{N}, | x |^{- 2} d x)$ , $H^{1} (R^{N}) ↪ L^{2} (R^{N}, | x |^{- 1} d x)$ which leads some new difficulties. Besides, Hardy potential with the rate of decay are critical also in nonrelativistic quantum mechanics, as they represent an intermediate threshold between regular potentials (for which there are ordinary stationary states) and singular potentials (for which the energy is not lower-bounded and the particle falls to the center), for more details see [6]. We also mention that inverse square singular potentials arise in many other physical contexts: molecular physics, quantum cosmology [2], linearization of combustion models [1,7]. We present that the conditions on V, f give rise to difficulties in proving the boundedness of (PS)-sequence and the lack of compactness of the embedding of $H^{1} (R^{N})$ in the Lebesgue spaces $L^{p} (R^{N})$ , $p \in (2, 2^{*})$ , which prevents us from using the variational techniques in a standard way. The first difficulty can be overcome by the work of Jeanjean–Toland [11] and Pohožaev identity. In order to solve the last difficulty, our main tool is the bounded (PS)-sequence decomposition in [10].

If $g (x) = 0$ , (1.1) reduces to the nonlinear Choquard equation of the form: $\begin{matrix} (1.2) & \{\begin{matrix} - Δ u + V (x) u = (I_{α} * F (u)) f (u), x \in R^{N}; \\ u \in H^{1} (R^{N}), \end{matrix} \end{matrix}$ which has been extensively studied by using variational methods, see [17–19,22], and references therein. In view of (F1), (F2) and Hardy–Littlewood–Sobolev inequality, for some $p \in (2, 2^{*})$ and any $ε > 0$ , one has $\begin{array}{rcl} \int_{R^{N}} (I_{α} * F (u)) F (u) d x & = & \frac{Γ (\frac{N - α}{2})}{Γ (\frac{α}{2}) 2^{α} π^{\frac{N}{2}}} \int_{R^{N}} \int_{R^{N}} \frac{F (u (x)) F (u (y))}{| x - y |^{N - α}} d x d y \\ ⩽ & C_{1} {‖ F (u) ‖}_{\frac{2 N}{N + α}}^{2} \\ (1.3) & ⩽ & ε (‖ u ‖_{2}^{\frac{2 (N + α)}{N - 2}} + ‖ u ‖_{2^{*}}^{\frac{2 (N + α)}{N - 2}}) + C_{ε} ‖ u ‖_{p}^{\frac{(N + α) p}{N}}, \forall u \in H^{1} (R^{N}) . \end{array}$

Using (1.3), it is standard to check that under (V1), (F1) and (F2), the energy functional of (1.2) defined in $H^{1} (R^{N})$ by $\begin{matrix} (1.4) & J (u) = \frac{1}{2} \int_{R^{N}} (| \nabla u |^{2} + V (x) u^{2}) d x - \frac{1}{2} \int_{R^{N}} (I_{α} * F (u)) F (u) d x \end{matrix}$ is continuously differentiable and its critical points correspond to the weak solutions of (1.2).

The limit equation of (1.2) is the following autonomous form: $\begin{matrix} (1.5) & \{\begin{matrix} - Δ u + V_{\infty} u = (I_{α} * F (u)) f (u), x \in R^{N}; \\ u \in H^{1} (R^{N}), \end{matrix} \end{matrix}$ whose energy functional is as follows: $\begin{matrix} (1.6) & J_{\infty} (u) = \frac{1}{2} \int_{R^{N}} (| \nabla u |^{2} + V_{\infty} u^{2}) d x - \frac{1}{2} \int_{R^{N}} (I_{α} * F (u)) F (u) d x . \end{matrix}$ For $V_{\infty} = 1$ , Moroz and Van Schafingen [18] have proved that the problem has a ground state solution under (F1)–(F3); And if $f \in C (R, R)$ satisfies (F1) and $u \in H^{1} (R^{N})$ is the solution of (1.5), one has $\begin{matrix} (1.7) & P_{\infty} (u) = \frac{N - 2}{2} ‖ \nabla u ‖_{2}^{2} + \frac{N V_{\infty}}{2} ‖ u ‖_{2}^{2} - \frac{(N + α)}{2} \int_{R^{N}} (I_{α} * F (u)) F (u) d x = 0 . \end{matrix}$ Tang and Chen [22] extended the above results on (1.5) obtained by Moroz and Van Schafingen [18] to (1.2), where V satisfies (V1) and the following assumptions:

$V \in C (R^{N}, R)$ , $V (x) ⩽ V_{\infty}$ but $V (x) ≢ V_{\infty}$ ;

$V \in C^{1} (R^{N}, R)$ , ${inf}_{x \in R^{N}} (\nabla V (x) \cdot x) > - \infty$ , and there exists $θ^{'} \in (0, \frac{1}{2})$ and $\overline{R} ⩾ 0$ such that $\begin{matrix} \nabla V (x) \cdot x ⩽ \{\begin{matrix} \frac{{(N - 2)}^{2}}{2 | x |^{2}}, & 0 < | x | < \overline{R}; \\ θ^{'} α V (x), & | x | ⩾ \overline{R} . \end{matrix} \end{matrix}$

For $N = 3$ , $α = 2$ and $F (s) = \frac{s^{2}}{2}$ , (1.5) covers in particular the Choquard–Pekar equation: $\begin{matrix} (1.8) & \{\begin{matrix} - Δ u + u = (I_{2} * | u |^{2}) u, x \in R^{3}; \\ u \in H^{1} (R^{3}), \end{matrix} \end{matrix}$ introduced at least in 1954, in a work by Peaker describing the quantum mechanics of a polaron at rest. In 1976, Choquard used (1.8) to describe an electron trapped in its own hole, in a certain approximation to Hartree–Fock theory of one component plasma [13]. In 1996, Penrose proposed (1.8) as a model of self-gravitating matter [16]. In this context, equation (1.8) is usually called the nonlinear Schrödinger–Newton equation. Note that if u solves (1.8), then the function Ψ defined by $Ψ (t, x) = e^{i t} u (x)$ is a solitary wave of the focusing time-dependent Hartree equation $\begin{matrix} i Ψ_{t} + Δ Ψ = - (I_{2} * | Ψ |^{2}) Ψ, Ψ \in R_{+} \times R^{N} . \end{matrix}$ In this context, (1.8) is also known as the stationary nonlinear Hartree equation. The existence of solutions for stationary equation (1.8) was proved by variational methods by Lieb, Lions and Menzala [13–15] and also by ordinary differential equations techniques [4,16,23].

Define $\begin{matrix} (1.9) & \bar{μ} = \frac{α (N - 2)}{2 (α + 1)} and \bar{β} = \frac{α {(N - 2)}^{2}}{4 (2 + α)} . \end{matrix}$ Using (1.3), it is standard to check that under $μ \in (0, \bar{μ})$ , (V1) and (F1), the energy functional defined in $H^{1} (R^{N})$ by $\begin{matrix} (1.10) & J_{μ} (u) = \frac{1}{2} \int_{R^{N}} (| \nabla u |^{2} + V (x) u^{2}) d x - \frac{μ}{2} \int_{R^{N}} \frac{u^{2}}{| x |} d x - \frac{1}{2} \int_{R^{N}} (I_{α} * F (u)) F (u) d x \end{matrix}$ is continuously differentiable and its critical points correspond to the weak solutions of (1.1) with $g (x) = \frac{μ}{| x |}$ . Similarly, under $β \in (0, \bar{β})$ , (V1) and (F1), the energy functional defined in $H^{1} (R^{N})$ by $\begin{matrix} (1.11) & J_{β} (u) = \frac{1}{2} \int_{R^{N}} (| \nabla u |^{2} + V (x) u^{2}) d x - \frac{β}{2} \int_{R^{N}} \frac{u^{2}}{| x |^{2}} d x - \frac{1}{2} \int_{R^{N}} (I_{α} * F (u)) F (u) d x \end{matrix}$ is continuously differentiable and its critical points correspond to the weak solutions of (1.1) with $g (x) = \frac{β}{| x |^{2}}$ .

In the second part of this paper, motivated by [10,11,22], we shall obtain a ground state solution for (1.1), which has minimal “energy” $J_{μ}$ in the set of all nontrivial solutions, under $μ \in (0, \bar{μ})$ , (V1)–(V3) and (F1)–(F3). The same results can be obtained for $J_{β}$ .

Our first two results are as follows.

Theorem 1.1.
Assume that $μ \in (0, \bar{μ})$ , V and f satisfy (V1)–(V3) and (F1)–(F3). Then problem (1.1) with $g (x) = \frac{μ}{| x |}$ has a solution $\bar{u} \in H^{1} (R^{N}) ∖ {0}$ such that $J_{μ} (\bar{u}) = {inf}_{K_{μ}} J_{μ}$ , where $\begin{matrix} (1.12) & K_{μ} : = {\bar{u} \in H^{1} (R^{N}) ∖ {0} : {(J_{μ})}^{'} (u) = 0} . \end{matrix}$
Theorem 1.2.
Assume that $β \in (0, \bar{β})$ , V and f satisfy (V1)–(V3) and (F1)–(F3). Then problem (1.1) with $g (x) = \frac{β}{| x |^{2}}$ has a solution $\bar{u} \in H^{1} (R^{N}) ∖ {0}$ such that $J_{β} (\bar{u}) = {inf}_{K_{β}} J_{β}$ , where $\begin{matrix} (1.13) & K_{β} : = {\bar{u} \in H^{1} (R^{N}) ∖ {0} : {(J_{β})}^{'} (u) = 0} . \end{matrix}$

To prove Theorem 1.1, as in Jeanjean–Tanaka [9], for $λ \in [\frac{1}{2}, 1]$ and $μ \in (0, \bar{μ})$ , we consider the family of functionals $J_{μ}^{λ} (u) : H^{1} (R^{N}) \to R$ defined by $\begin{matrix} (1.14) & J_{μ}^{λ} (u) = \frac{1}{2} \int_{R^{N}} (| \nabla u |^{2} + V (x) u^{2}) d x - \frac{μ}{2} \int_{R^{N}} \frac{u^{2}}{| x |} d x - \frac{λ}{2} \int_{R^{N}} (I_{α} * F (u)) F (u) d x . \end{matrix}$ We can prove that these functionals all have a Mountain Pass geometry, and we denote $c_{μ}^{λ}$ the corresponding Mountain Pass level. Specially, $c_{μ}^{1} : = c_{μ}$ , corresponding to (1.6), we also let $\begin{matrix} (1.15) & J_{\infty}^{λ} (u) = \frac{1}{2} \int_{R^{N}} (| \nabla u |^{2} + V_{\infty} u^{2}) d x - \frac{λ}{2} \int_{R^{N}} (I_{α} * F (u)) F (u) d x . \end{matrix}$ By [22, Corollary 1.2], for any $λ \in [\frac{1}{2}, 1]$ , there exists a minimizer $u_{\infty}^{λ}$ of $J_{\infty}^{λ}$ on $M_{\infty}^{λ}$ , where $\begin{matrix} (1.16) & M_{\infty}^{λ} = {u \in H^{1} (R^{N}) ∖ {0} : P_{\infty}^{λ} (u) = 0}, \end{matrix}$ and $\begin{matrix} (1.17) & P_{\infty}^{λ} (u) = \frac{N - 2}{2} ‖ \nabla u ‖_{2}^{2} + \frac{N V_{\infty}}{2} ‖ u ‖_{2}^{2} - \frac{(N + α) λ}{2} \int_{R^{N}} (I_{α} * F (u)) F (u) d x . \end{matrix}$ Specially, $M_{\infty}^{1} = M_{\infty}$ , $P_{\infty}^{1} = P_{\infty}$ , $J_{\infty}^{1} = J_{\infty}$ . Let $\begin{matrix} A_{μ} (u) = \frac{1}{2} \int_{R^{N}} (| \nabla u |^{2} + V (x) u^{2}) d x - \frac{μ}{2} \int_{R^{N}} \frac{u^{2}}{| x |} d x, B (u) = \frac{1}{2} \int_{R^{N}} (I_{α} * F (u)) F (u) d x \end{matrix}$ then $J_{μ}^{λ} (u) = A_{μ} (u) - λ B (u)$ . Since $B (u)$ is not sign definite, it prevents us from employing Jeanjean’s monotonicity trick [8]. Thanks to the work of Jeanjean–Toland [11], $J_{μ}^{λ}$ still has a bounded (PS)-sequence ${u_{n} (λ)}_{n \in N_{+}} \subseteq H^{1} (R^{N})$ , at level $c_{μ}^{λ}$ for almost every $λ \in [\frac{1}{2}, 1]$ . By [22], we can find a constant $\bar{λ} \in [\frac{1}{2}, 1)$ and prove the following inequality directly $\begin{matrix} (1.18) & c_{μ}^{λ} < m_{\infty}^{λ}, \forall λ \in (\bar{λ}, 1], \end{matrix}$ where $m_{\infty}^{λ} : = {inf}_{u \in M_{\infty}^{λ}} J_{\infty}^{λ} (u)$ . Specially, $m_{\infty}^{1} = m_{\infty}$ . Applying (1.18) and a precise decomposition of bounded (PS)-sequence in [10], we can get a nontrivial critical point $u_{λ}$ of $J_{μ}^{λ}$ which possesses energy $c_{μ}^{λ}$ for almost every $λ \in [\bar{λ}, 1]$ . Take ${λ_{j}}_{j \in N_{+}} \subseteq (\bar{λ}, 1]$ , satisfying $λ_{j} \to 1$ as $j \to \infty$ , then for every $λ_{j}$ , we can get a nontrivial critical point $u_{λ_{j}}$ of $J_{μ}^{λ_{j}}$ which possesses energy $c_{μ}^{λ_{j}}$ . Moreover with a Pohožaev identity, we can prove that ${u_{λ_{j}}}_{j \in N_{+}}$ is bounded in $H^{1} (R^{N})$ , then applying (1.18) and a precise decomposition of bounded (PS)-sequence [10], we can get a nontrivial point $\bar{u}$ of $J_{μ}$ . Finally, by taking the minimizing sequence, we can prove that (1.1) with $g (x) = \frac{μ}{| x |}$ admits a least energy solution under $μ \in (0, \bar{μ})$ , (V1)–(V3) and (F1)–(F3). The same steps above are also applicable for $J_{β}$ , and similar results can be obtained. Here we make parallel definitions of $J_{β}$ .

For $λ \in [\frac{1}{2}, 1]$ , $β \in (0, \bar{β})$ , we define $J_{β}^{λ} (u) : H^{1} (R^{N}) \to R$ by $\begin{matrix} (1.19) & J_{β}^{λ} (u) = \frac{1}{2} \int_{R^{N}} (| \nabla u |^{2} + V (x) u^{2}) d x - \frac{β}{2} \int_{R^{N}} \frac{u^{2}}{| x |^{2}} d x - \frac{λ}{2} \int_{R^{N}} (I_{α} * F (u)) F (u) d x \end{matrix}$ denoting $C_{β}^{λ}$ the corresponding Mountain Pass level. Specially, $C_{β}^{1} : = C_{β}$ .

In the third part of this paper, motivated by [12,18], we are interested in the asymptotic behavior of solutions for $J_{μ}$ and $J_{β}$ with respect to the μ and β. We construct a family of nontrivial solutions of $J_{μ}$ which converge to a nontrivial solution of (1.2) as $μ \to 0^{+}$ and a family of ground state solutions of $J_{β}$ which converge to a nontrivial solution of (1.2) as $β \to 0^{+}$ , under (V1)–(V3), (F1)–(F3) and following additional condition on V:

$V \in C^{1} (R^{N}, R)$ and there exists $θ \in [0, 1)$ such that $t \to \frac{N V (t x) + \nabla V (t x) \cdot (t x)}{t^{α}} + \frac{{(N - 2)}^{3} θ}{4 t^{α + 2} | x |^{2}}$ is nonincreasing on $(0, \infty)$ for all $x \in R^{N} ∖ {0}$ .

In this direction, we have the following two theorems.
Theorem 1.3.
Assume that V and f satisfy (V1), (V2), (V4) and (F1)–(F3). Let ${u_{n}}_{n \in N_{+}}$ be a sequence of nontrivial solutions from the proof of Theorem 1.1 with $μ_{n} \to 0^{+}$ as $n \to \infty$ . Then $u_{n} \to u$ along a subsequence strongly in $H^{1} (R^{N})$ . Moreover, u is a nontrivial solution of (1.2).
Theorem 1.4.
Assume that V and f satisfy (V1), (V2), (V4) and (F1)–(F3). Let ${u_{n}}_{n \in N_{+}}$ be a sequence of ground state from Theorem 1.2 with $β_{n} \to 0^{+}$ as $n \to \infty$ . Then $u_{n} \to u$ along a subsequence strongly in $H^{1} (R^{N})$ . Moreover, u is a ground state solution of (1.2).

To prove Theorem 1.3, we can take a sequence ${μ_{n}}_{n \in N_{+}} \subseteq (0, \bar{μ})$ satisfying $μ_{n} \to 0^{+}$ as $n \to \infty$ . From the proof of Theorem 1.1, we know that there exists $u_{n} \in H^{1} (R^{N}) ∖ {0}$ which satisfies $\begin{matrix} (1.20) & J_{μ_{n}} (u_{n}) ⩽ C_{μ_{n}}, {(J_{μ_{n}})}^{'} (u_{n}) = 0, \forall n \in N_{+} . \end{matrix}$ Applying Pohožaev identity, the boundness of ${u_{n}}_{n \in N_{+}}$ can be proved, then through bounded (PS)-sequence decomposition [10], we can reach the conclusion of Theorem 1.3 under (V1), (V2), (V4) and (F1)–(F3).

To prove Theorem 1.4, we can divide it into the following steps:

showing $C_{β} = {\hat{m}}_{β}$ for sufficient small $β \in (0, \bar{β})$ , where ${\hat{m}}_{β} : = {inf}_{K_{β}} J_{β} (u)$ ;

choosing ${β_{n}}_{n \in N_{+}} \subseteq (0, \bar{β})$ satisfying $β_{n} \to 0^{+}$ as $n \to \infty$ , then applying Theorem 1.2, there exists $u_{n} \in H^{1} (R^{N}) ∖ {0}$ which satisfies $\begin{matrix} J_{β_{n}} (u_{n}) = {\hat{m}}_{β_{n}} = C_{β_{n}}, {(J_{β_{n}})}^{'} (u_{n}) = 0, \forall n \in N_{+}, \end{matrix}$ and we can prove that ${u_{n}}_{n \in N_{+}}$ is bounded in $H^{1} (R^{N})$ ;

showing that $C_{β_{n}} \to C_{0}$ as $n \to \infty$ , where $C_{0}$ is the Mountain Pass level of (1.2);

with a precise decomposition of bounded (PS)-sequence in [10], we can verify the conclusion of Theorem 1.4.

Throughout the paper, we make use of the following notations:

$H^{1} (R^{N})$ denotes the usual Sobolev space equipped with the inner product and norm $\begin{matrix} (u, v) = \int_{R^{N}} (\nabla u \cdot \nabla v + u v) d x, ‖ u ‖ = {(u, u)}^{\frac{1}{2}}, \forall u, v \in H^{1} (R^{N}); \end{matrix}$

$L^{s} (R^{N})$ ( $1 ⩽ s < \infty$ ) denotes the Lebesgue space with the norm $‖ u ‖_{s} = {(\int_{R^{N}} | u |^{s} d x)}^{\frac{1}{s}}$ ;

$C_{0}^{\infty} (R^{N})$ denotes the space of the functions in finitely differentiable with compact support in $R^{N}$ ;

$D^{1, 2} (R^{N})$ is the completion of $C_{0}^{\infty} (R^{N})$ with respect to the norm $\begin{matrix} ‖ u ‖_{D^{1, 2} (R^{N})}^{2} = \int_{R^{N}} | \nabla u |^{2} d x; \end{matrix}$

$⟨ \cdot, \cdot ⟩$ denote action of dual;

For any $u \in H^{1} (R^{N}) ∖ {0}$ , $u_{t} (x) : = u (t^{- 1} x)$ for $t > 0$ ;

For any $x \in H^{1} (R^{N})$ and $r > 0$ , $B_{r} (x) : = {y \in R^{N}; | y - x | < r}$ ;

$C, C_{1}, C_{2}, \dots$ denote positive constants possibly different in different places.

The rest of the paper is organized as follows. In Section 2, we give the proofs of Theorem 1.1 and Theorem 1.2. Section 3 is devoted to analyzing the asymptotic properties of solutions of problem (1.1).
2. Ground state solution for (1.1)

In this section, we give the proofs of Theorem 1.1 and Theorem 1.2. The existence of ground state solutions for (1.1) were proved by variational methods which were introduced by Jeanjean, Toland and Tanaka [10,11]. Firstly, we prove the existence of nontrivial solutions for the equations, and then we prove the existence of ground state solutions by taking a minimization sequence. By a simple calculation, we can verify Lemma 2.1 and Lemma 2.2.

Lemma 2.1.
Assume that $μ \in (0, \bar{μ})$ , (V1) and (F1)–(F2) hold. If ${u_{n}}_{n \in N_{+}} \subseteq H^{1} (R^{N})$ is bounded, then there exists $α_{0}, C_{1} > 0$ such that $\begin{array}{l} (2.1) & \int_{R^{N}} (| \nabla u |^{2} + V (x) u^{2}) d x - μ \int_{R^{N}} \frac{u^{2}}{| x |} d x ⩾ α_{0} ‖ u ‖^{2}, \forall u \in H^{1} (R^{N}) . \\ (2.2) & \int_{R^{N}} (I_{α} * F (u_{n})) F (u_{n}) d x ⩾ - C_{1} . \end{array}$
Proof.
Due to $μ \in (0, \bar{μ})$ , one has $\begin{matrix} (2.3) & μ ⩽ \frac{α}{2 (α + 1)} (N - 2) . \end{matrix}$ According to Hardy inequality, we have $\begin{matrix} (2.4) & ‖ \nabla u ‖_{2}^{2} ⩾ \frac{{(N - 2)}^{2}}{4} \int_{R^{N}} \frac{u^{2}}{| x |^{2}} d x, \forall u \in H^{1} (R^{N}) . \end{matrix}$ Thus, by Hölder inequality, Young inequality, (V1), (2.3) and (2.4), one has $\begin{array}{l} \int_{R^{N}} (| \nabla u |^{2} + V (x) u^{2}) d x - μ \int_{R^{N}} \frac{u^{2}}{| x |} d x \\ ⩾ \int_{R^{N}} (| \nabla u |^{2} + V (x) u^{2}) d x - μ ‖ u ‖_{2} {(\int_{R^{N}} \frac{u^{2}}{| x |^{2}} d x)}^{\frac{1}{2}} \\ ⩾ ‖ u ‖^{2} - \frac{2 μ}{N - 2} ‖ u ‖_{2} ‖ \nabla u ‖_{2} \\ = (1 - \frac{α}{2 (α + 1)}) ‖ u ‖^{2} + \frac{α}{2 (α + 1)} ‖ u ‖^{2} - \frac{2 μ}{N - 2} ‖ u ‖_{2} ‖ \nabla u ‖_{2} \\ ⩾ (1 - \frac{α}{2 (α + 1)}) ‖ u ‖^{2} + (\frac{α}{2 (α + 1)} - \frac{μ}{N - 2}) ‖ u ‖^{2} \\ ⩾ (1 - \frac{α}{2 (α + 1)}) ‖ u ‖^{2} \\ = α_{0} ‖ u ‖^{2}, \end{array}$ where $α_{0} : = 1 - \frac{α}{2 (α + 1)} > 0$ , for any $u \in H^{1} (R^{N})$ .

By (1.4) and Sobolev inequality, for any $ε > 0$ , one has $\begin{array}{rcl} | \int_{R^{N}} (I_{α} * F (u_{n})) F (u_{n}) d x | & ⩽ & \frac{Γ (\frac{N - α}{2})}{Γ (\frac{α}{2}) 2^{α} π^{\frac{N}{2}}} \int_{R^{N}} \int_{R^{N}} \frac{| F (u_{n} (x)) | | F (u_{n} (y)) |}{| x - y |^{N - α}} d x d y \\ ⩽ & ε (‖ u_{n} ‖_{2}^{\frac{2 (N + α)}{N}} + ‖ u_{n} ‖_{2^{}}^{\frac{2 (N + α)}{N - 2}}) + C_{ε} ‖ u_{n} ‖_{P}^{\frac{(N + α) P}{N}} \\ ⩽ & C_{1} . \end{array}$ □
Lemma 2.2.
Assume that* $μ \in (0, \bar{μ})$ , (V1) and (F1)–(F3) hold. Then for any $λ \in [\frac{1}{2}, 1]$ , $J_{μ}^{λ} (u)$ has Mountain Pass geometry, where $J_{μ}^{λ} (u)$ is defined by (1.14).
Proof.
By (1.3), (2.1) and Sobolev continuous embedding, for any $ε > 0$ , one has $\begin{array}{rcl} J_{μ}^{λ} (u) & ⩾ & \frac{1}{2} α_{0} ‖ u ‖^{2} - \frac{λ}{2} \int_{R^{N}} (I_{α} * F (u)) F (u) d x \\ ⩾ & \frac{1}{2} α_{0} ‖ u ‖^{2} - ε (‖ u ‖_{2}^{\frac{2 (N + α)}{N}} + ‖ u ‖_{2^{}}^{\frac{2 (N + α)}{N - 2}}) + C_{ε} ‖ u ‖_{p}^{\frac{(N + α) p}{N}} \\ ⩾ & \frac{1}{2} α_{0} ‖ u ‖^{2} - ε C_{2} (‖ u ‖^{\frac{2 (N + α)}{N}} + ‖ u ‖^{\frac{2 (N + α)}{N - 2}}) + C_{3} C_{ε} ‖ u ‖^{\frac{(N + α) p}{N}}, \end{array}$ which implies that there exists $ρ > 0$ , $r > 0$ such that $\begin{matrix} (2.5) & J_{μ}^{λ} (u) ⩾ r > 0, \forall u \in \partial B_{\partial ρ} (0) . \end{matrix}$

In view of [22, Theorem 1.3], $J_{\infty}$ has a minimizer $u_{1}^{\infty} \neq 0$ on $M_{\infty}$ , where $M_{\infty}$ is defined by (1.16) taking $λ = 1$ , one has $\begin{matrix} P_{\infty} (u_{1}^{\infty}) = \frac{N - 2}{2} {‖ \nabla u_{1}^{\infty} ‖}_{2}^{2} + \frac{N V_{\infty}}{2} {‖ u_{1}^{\infty} ‖}_{2}^{2} - \frac{(N + α)}{2} \int_{R^{N}} (I_{α} F (u_{1}^{\infty})) F (u_{1}^{\infty}) d x = 0 . \end{matrix}$ Due to $u_{1}^{\infty} \neq 0$ , one has $\begin{matrix} (2.6) & \int_{R^{N}} (I_{α} * F (u_{1}^{\infty})) F (u_{1}^{\infty}) d x > 0 . \end{matrix}$ By (V1), we have $V_{sup} : = {sup}_{R^{N}} V (x) \in (1, \infty)$ , let $\begin{matrix} J_{}^{\frac{1}{2}} (u) : = \frac{1}{2} \int_{R^{N}} (| \nabla u |^{2} + V_{sup} u^{2}) d x - \frac{1}{4} \int_{R^{N}} (I_{α} F (u)) F (u) d x . \end{matrix}$ Then, we have $\begin{matrix} J_{}^{\frac{1}{2}} ({(u_{1}^{\infty})}_{t}) = \frac{t^{N - 2}}{2} {‖ \nabla u_{1}^{\infty} ‖}_{2}^{2} + \frac{t^{N}}{2} V_{sup} {‖ u_{1}^{\infty} ‖}_{2}^{2} - \frac{t^{N + α}}{4} \int_{R^{N}} (I_{α} F (u_{1}^{\infty})) F (u_{1}^{\infty}) d x . \end{matrix}$ Using (2.6) and $α \in (0, N)$ , it is easy to verify, that $\begin{matrix} lim_{t \to + \infty} J_{}^{\frac{1}{2}} ({(u_{1}^{\infty})}_{t}) = - \infty < 0, \end{matrix}$ then there exists $T > 0$ such that $\begin{matrix} J_{}^{\frac{1}{2}} ({(u_{1}^{\infty})}_{T}) < 0, ‖ {(u_{1}^{\infty})}_{T} ‖ > ρ . \end{matrix}$ In view of $J_{μ}^{λ} ({(u_{1}^{\infty})}_{T}) ⩽ J_{*}^{\frac{1}{2}} ({(u_{1}^{\infty})}_{T})$ for any $λ \in [\frac{1}{2}, 1]$ , we have $\begin{matrix} (2.7) & J_{μ}^{λ} ({(u_{1}^{\infty})}_{T}) < 0, ‖ {(u_{1}^{\infty})}_{T} ‖ > ρ, \forall λ \in [\frac{1}{2}, 1] . \end{matrix}$ □
Proposition 2.3 ([11]).

Let X be a Banach space and let $J \subseteq R^{+}$ be an interval, and $\begin{matrix} Φ_{λ} (u) = A (u) - λ B (u), \forall λ \in J \end{matrix}$ be a family of $C^{1}$ -functional on X such that

either $A (u) \to \infty$ or $B (u) \to \infty$ , as $‖ u ‖ \to \infty$ ;

B maps every bounded set of X into a set of $R$ bounded below;

there are two points $v_{1}$ , $v_{2}$ in X such that $\begin{matrix} {\tilde{C}}_{λ} : = inf_{y \in \tilde{Γ}} max_{t \in [0, 1]} Φ_{λ} (y (t)) > max {Φ_{λ} (v_{1}), Φ_{λ} (v_{2})}, \end{matrix}$ where $\tilde{Γ} = {y \in C ([0, 1], X) : y (0) = v_{1}, y (1) = v_{2}}$ ;

then, for almost every

λ \in J

, there exists a sequence

{u_{n} (λ)}_{n \in N_{+}}

such that

${u_{n} (λ)}_{n \in N_{+}}$ is bounded in X;

$Φ_{λ} (u_{n} (λ)) \to {\tilde{C}}_{λ}$ ;

$Φ_{λ}^{'} (u_{n} (λ)) \to 0$ in $X^{*}$ , where $X^{*}$ is the dual of X.

Employing Lemma 2.1 and Lemma 2.2, it is obvious that $J_{μ}^{λ}$ satisfies the conditions of Proposition 2.3. Then we have the following lemma.

Lemma 2.4.
Assume that $μ \in (0, \bar{μ})$ , (V1) and (F1)–(F3) hold. Then
$J_{μ}^{λ} ({(u_{1}^{\infty})}_{T}) < 0$ for all $λ \in [\frac{1}{2}, 1]$ ;

there exists a positive constant $K_{μ}$ independent of λ such that for all $λ \in [\frac{1}{2}, 1]$ $\begin{matrix} C_{μ}^{λ} : = inf_{y \in Γ} max_{t \in [0, 1]} J_{μ}^{λ} (y (t)) ⩾ K_{μ} > max {J_{μ}^{λ} (0), J_{μ}^{λ} ({(u_{1}^{\infty})}_{T})}, \end{matrix}$ where $\begin{matrix} Γ = {y \in C ([0, 1], H^{1} (R^{N})) : y (0) = 0, y (1) = {(u_{1}^{\infty})}_{T}}; \end{matrix}$

$c_{μ}^{λ}$ is bounded for $λ \in [\frac{1}{2}, 1]$ ;

${lim sup}_{λ \to λ_{0}} c_{μ}^{λ} ⩽ c_{μ}^{λ_{0}}$ for $λ_{0} \in (\frac{1}{2}, 1]$ .
The proof is the same as [22, Lemma 3.4].

The following lemma is crucial and plays a key role in the proof of Theorem 1.1.
Lemma 2.5.
Assume that $μ \in (0, \bar{μ})$ , (V1)–(V2) and (F1)–(F3) hold. Then there exists $\bar{λ} \in [\frac{1}{2}, 1]$ such that $\begin{matrix} (2.8) & c_{μ}^{λ} < m_{\infty}^{λ}, \forall λ \in [\bar{λ}, 1] . \end{matrix}$
Proof.
Since the limit equation of (1.1) is equal to (1.5), in view of [22, Lemma 3.5] we know that $C_{0}^{λ} < m_{\infty}^{λ}$ , $\forall λ \in (\bar{λ}, 1]$ for some $\bar{λ} \in [\frac{1}{2}, 1)$ , where $C_{0}^{λ}$ is the Mountain Pass level of $J_{μ}^{λ}$ with $μ = 0$ . Specially, $C_{0}^{1} : = C_{0}$ . It is easy to verify that $c_{μ}^{λ} ⩽ C_{0}^{λ}$ , for any $μ \in (0, \bar{μ})$ . □

Under the hypothesis $μ \in (0, \bar{μ})$ , (V1) and (F1)–(F3), by Lemma 2.1, Lemma 2.2, Proposition 2.3 and Lemma 2.4, for almost every $λ \in [\bar{λ}, 1]$ , there exists a bounded sequence ${u_{n}^{λ}}_{n \in N_{+}} \subseteq H^{1} (R^{N})$ such that $\begin{matrix} (2.9) & J_{μ}^{λ} (u_{n}^{λ}) \to c_{μ}^{λ}, {(J_{μ}^{λ})}^{'} (u_{n}^{λ}) \to 0, as n \to \infty . \end{matrix}$

The proof of Theorem 1.1 is based on a Pohožaev identity for (1.1) with $g (x) = \frac{μ}{| x |}$ , which is needed to verify the boundedness of (PS)-sequence.
Lemma 2.6.
Assume that $μ \in (0, \bar{μ})$ , (V1)–(V3) and (F1)–(F2) hold. If $u \in H^{1} (R^{N})$ is a weak solution for (1.1) with $g (x) = \frac{μ}{| x |}$ , then we have the following identity: $\begin{array}{rcl} P_{μ} (u) & = & \frac{N - 2}{2} ‖ \nabla u ‖_{2}^{2} + \frac{1}{2} \int_{R^{N}} (N V (x) + \nabla V (x) \cdot x) u^{2} d x \\ (2.10) & - \frac{μ (N - 1)}{2} \int_{R^{N}} \frac{u^{2}}{| x |} d x - \frac{N + α}{2} \int_{R^{N}} (I_{α} * F (u)) F (u) d x = 0 . \end{array}$
Proof.
To get the Pohožaev type identity for (1.1) with $g (x) = \frac{μ}{| x |}$ , we use a truncation argument due to [18, Proposition 3.5]. Let $φ \in C_{0}^{1} (R^{N})$ such that $φ (x) = 1$ for $| x | ⩽ 1$ , $φ (x) = 0$ for $| x | ⩾ 2$ and $| \nabla φ | ⩽ C_{1}$ in $R^{N}$ . Since $u \in H^{1} (R^{N})$ is a weak solution of (1.1) with $g (x) = \frac{μ}{| x |}$ , by a standard regularity argument (see [5, Proposition 2.3] and [20, Theorem 2]), we can show that $u \in H_{loc}^{2} (R^{N})$ . Let $φ_{ε} (x) : = φ (ε x)$ . Then one has $\begin{matrix} (2.11) & \begin{matrix} 0 ⩽ φ_{ε} (x) ⩽ 1, x \cdot \nabla φ_{ε} (x) \equiv 0, \forall ε | x | \in [0, 1] \cup [2, + \infty), \\ | x \cdot \nabla φ_{ε} (x) | ⩽ C_{2}, \forall x \in R^{N} . \end{matrix} \end{matrix}$ We choose the function $v_{ε} \in H^{1} (R^{N})$ defined for $ε \in (0, \infty)$ and $x \in R^{N}$ by $\begin{matrix} v_{ε} (x) : = φ_{ε} (x) x \cdot \nabla u (x) \end{matrix}$ as a test function in Eq. (1.1), then we obtain $\begin{matrix} (2.12) & \int_{R^{N}} (\nabla u \cdot \nabla v_{ε} + V (x) u v_{ε}) d x - μ \int_{R^{N}} \frac{u}{| x |} v_{ε} d x = \int_{R^{N}} (I_{α} * F (u)) f (u) v_{ε} d x . \end{matrix}$ For every $ε > 0$ , we can compute by integration by parts $\begin{matrix} \int_{R^{N}} V (x) u v_{ε} d x & = \int_{R^{N}} φ_{ε} (x) V (x) u (x) (x \cdot \nabla u) d x \\ = \frac{1}{2} \int_{R^{N}} φ_{ε} (x) V (x) (x \cdot \nabla u^{2}) d x \\ = - \frac{1}{2} \int_{R^{N}} φ_{ε} (x) [x \cdot \nabla V (x) + N V (x)] u^{2} d x - \frac{1}{2} \int_{R^{N}} V (x) (x \cdot \nabla φ_{ε}) u^{2} d x . \end{matrix}$ (V1)–(V3) imply that $x \cdot \nabla V (x) + N V (x)$ is bounded, and it follows from the Lebesgue’s domination convergence theorem and the absolute continuity of integrals that $\begin{array}{rcl} (2.13) & lim_{ε \to 0} \int_{R^{N}} V (x) u v_{ε} d x = - \frac{1}{2} \int_{R^{N}} [\nabla V (x) \cdot x + N V (x)] u^{2} d x . \end{array}$ Since $u \in W_{loc}^{2, 2} (R^{N})$ , one has $\begin{matrix} div {[(x \cdot \nabla u) \nabla u - \frac{| \nabla u |^{2}}{2} x] φ_{ε}} \\ = \sum_{i = 1}^{N} \frac{\partial}{\partial x_{i}} [φ_{ε} (x \cdot \nabla u) \frac{\partial u}{\partial x_{i}}] - \frac{1}{2} \sum_{i = 1}^{N} \frac{\partial}{\partial x_{i}} (φ_{ε} | \nabla u |^{2} x_{i}) \\ = φ_{ε} (x \cdot \nabla u) Δ u + (x \cdot \nabla u) (\nabla u \cdot \nabla φ_{ε}) - \frac{| \nabla u |^{2}}{2} (x \cdot \nabla φ_{ε}) - \frac{N - 2}{2} φ_{ε} | \nabla u |^{2} . \end{matrix}$ It follows that $\begin{array}{rcl} \int_{R^{N}} \nabla u \cdot \nabla v_{ε} d x & = & \int_{R^{N}} \nabla u \cdot \nabla [φ_{ε} (x \cdot \nabla u)] d x \\ = & - \int_{R^{N}} φ_{ε} (x \cdot \nabla u) Δ u d x \\ = & \int_{R^{N}} [(x \cdot \nabla u) (\nabla u \cdot \nabla φ_{ε}) - \frac{| \nabla u |^{2}}{2} (x \cdot \nabla φ_{ε}) - \frac{N - 2}{2} φ_{ε} | \nabla u |^{2}] d x \\ = & \int_{R^{N}} (x \cdot \nabla u) (\nabla u \cdot \nabla φ_{ε}) d x - \frac{1}{2} \int_{R^{N}} (x \cdot \nabla φ_{ε}) | \nabla u |^{2} d x \\ - \frac{N - 2}{2} \int_{R^{N}} φ_{ε} | \nabla u |^{2} d x, \end{array}$ Lebesgue’s domination convergence again is applicable since $\nabla u \in L^{2} (R^{N})$ and we obtain $\begin{matrix} (2.14) & lim_{ε \to 0} \int_{R^{N}} \nabla u \cdot \nabla v_{ε} d x = - \frac{N - 2}{2} \int_{R^{N}} | \nabla u |^{2} d x . \end{matrix}$ Define $B_{r} = {x \in R^{N} : | x | < r}$ and $B_{r}^{c} : = R^{N} ∖ B_{r}$ for some sufficiently small $r > 0$ . Since $u \in W_{loc}^{2, 2} (B_{r}^{c})$ , we obtain $\begin{array}{rcl} \int_{B_{r}^{c}} \frac{u}{| x |} v_{ε} d x & = & \int_{B_{r}^{c}} \frac{u}{| x |} φ_{ε} (x) (x \cdot \nabla u) d x \\ = & - (N - 1) \int_{B_{r}^{c}} \frac{u^{2}}{| x |} φ_{ε} (x) d x - \int_{B_{r}^{c}} \frac{u}{| x |} φ_{ε} (x) (x \cdot \nabla u) d x \\ - \int_{B_{r}^{c}} \frac{u^{2}}{| x |} (x \cdot \nabla φ_{ε}) d x - \int_{\partial B_{r}} \frac{u^{2}}{| x |} φ_{ε} (x) (x \cdot ν) d S . \end{array}$ Hence, we have $\begin{array}{rcl} \int_{B_{r}^{c}} \frac{u}{| x |} v_{ε} d x & = & - \frac{N - 1}{2} \int_{B_{r}^{c}} \frac{u^{2}}{| x |} φ_{ε} (x) d x - \frac{ε}{2} \int_{B_{r}^{c}} \frac{u^{2}}{| x |} (x \cdot \nabla φ_{ε}) d x \\ - \frac{1}{2} \int_{\partial B_{r}} \frac{u^{2}}{| x |} φ_{ε} (x) (x \cdot ν) d S . \end{array}$ By Lebesgue’s dominated convergence and the absolute continuity of integrals, we obtain $\begin{array}{l} lim_{ε \to 0^{+}} \int_{B_{r}^{c}} \frac{u}{| x |} v_{ε} d x = - \frac{N - 1}{2} \int_{B_{r}^{c}} \frac{u^{2}}{| x |} d x - \int_{\partial B_{r}} \frac{u^{2}}{| x |} x \cdot ν d S, \\ lim_{r \to 0^{+}} [- \frac{N - 1}{2} \int_{B_{r}^{c}} \frac{u^{2}}{| x |} d x - \int_{\partial B_{r}} \frac{u^{2}}{| x |} x \cdot ν d S] = - \frac{N - 1}{2} \int_{R^{N}} \frac{u^{2}}{| x |} d x, \end{array}$ then, we have $\begin{matrix} (2.15) & lim_{ε \to 0^{+}} \int_{R^{N}} \frac{u}{| x |} v_{ε} d x = lim_{ε \to 0^{+}, r \to 0^{+}} \int_{B_{r}^{c}} \frac{u}{| x |} v_{ε} d x = - \frac{N - 1}{2} \int_{R^{N}} \frac{u^{2}}{| x |} d x . \end{matrix}$ The last term in (2.12) can be rewritten by integration by parts for every $ε > 0$ as $\begin{array}{l} \int_{R^{N}} (I_{α} * F (u)) f (u) v_{ε} d x \\ = \int_{R^{N}} \int_{R^{N}} (F \circ u) (y) I_{α} (x - y) φ_{ε} (x) (x \cdot \nabla (F \circ u) (x)) d x d y \\ = \frac{1}{2} \int_{R^{N}} \int_{R^{N}} I_{α} (x - y) [(F \circ u) (y) φ_{ε} (x) (x \cdot \nabla (F \circ u) (x)) \\ + (F \circ u) (x) φ_{ε} (y) (y \cdot \nabla (F \circ u) (y))] d x d y \\ = - \int_{R^{N}} \int_{R^{N}} F (u (y)) I_{α} (x - y) [N φ_{ε} (x) + x \cdot \nabla φ_{ε} (x)] F (u (x)) d x d y \\ (2.16) & + \frac{N - α}{2} \int_{R^{N}} \int_{R^{N}} F (u (y)) I_{α} (x - y) \frac{(x - y) \cdot [x φ_{ε} (x) - y φ_{ε} (y)]}{| x - y |^{2}} F (u (x)) d x d y . \end{array}$ Note that $\begin{array}{l} | (x - y) \cdot [x φ_{ε} (x) - y φ_{ε} (y)] | \\ = | | x - y |^{2} φ_{ε} (x) + (x - y) \cdot y [φ_{ε} (x) - φ_{ε} (y)] | \\ ⩽ | x - y |^{2} + | x - y |^{2} | y | | \nabla φ_{ε} (y + ξ (x - y)) | \\ ⩽ | x - y |^{2} [1 + (| x - y | + | y + ξ (x - y) |) | \nabla φ_{ε} (y + ξ (x - y)) |] \\ ⩽ | x - y |^{2} [1 + | φ_{ε} (x) - φ_{ε} (y) | + | y + ξ (x - y) | | \nabla φ_{ε} (y + ξ (x - y)) |] \\ (2.17) & ⩽ (2 + C_{2}) | x - y |^{2}, \forall x, y \in R^{N} . \end{array}$ From (2.16), (2.17) and the Lebesgue’s dominated convergence theorem, we can thus conclude $\begin{array}{rcl} (2.18) & lim_{ε \to 0^{+}} \int_{R^{N}} (I_{α} * F (u)) f (u) v_{ε} d x = - \frac{N + α}{2} \int_{R^{N}} (I_{α} * F (u)) F (u) d x . \end{array}$ (2.12), (2.13), (2.14), (2.15) and (2.18) imply that (2.10) holds. □
Lemma 2.7.
Assume that (V1) and (V3) hold. Then $\begin{matrix} (2.19) & (2 + α) ‖ \nabla u ‖_{2}^{2} + \int_{R^{N}} (α V (x) - \nabla V (x) \cdot x) u^{2} d x ⩾ α (1 - θ^{'}) ‖ u ‖^{2}, \forall u \in H^{1} (R^{N}) . \end{matrix}$
Proof.
From (2.4) and (V1), (V3), for any $u \in H^{1} (R^{N})$ , we have $\begin{array}{l} (2 + α) ‖ \nabla u ‖_{2}^{2} + \int_{R^{N}} (α V (x) - \nabla V (x) \cdot x) u^{2} d x \\ = (2 + α) ‖ \nabla u ‖_{2}^{2} - \frac{{(N - 2)}^{2}}{2} \int_{R^{N}} \frac{u^{2}}{| x |^{2}} d x \\ + \int_{R^{N}} [α V (x) - \nabla V (x) \cdot x + \frac{{(N - 2)}^{2}}{2 | x |^{2}}] u^{2} d x \\ ⩾ α ‖ \nabla u ‖_{2}^{2} + \int_{R^{N}} [α V (x) - \nabla V (x) \cdot x + \frac{{(N - 2)}^{2}}{2 | x |^{2}}] u^{2} d x \\ ⩾ α ‖ \nabla u ‖_{2}^{2} + α (1 - θ^{'}) \int_{R^{N}} V (x) u^{2} d x \\ ⩾ α (1 - θ^{'}) ‖ u ‖^{2} . \end{array}$ □

For any bounded (PS)-sequence of $J_{μ}^{λ}$ we have the following decomposition.
Lemma 2.8 ([10]).

Assume that $μ \in (0, \bar{μ})$ , (V1) and (F1)–(F3) hold, let ${u_{n}^{λ}}_{n \in N_{+}}$ be a bounded (PS)-sequence for $J_{μ}^{λ}$ , for $λ \in [\frac{1}{2}, 1]$ . Then there exists a subsequence of ${u_{n}^{λ}}_{n \in N_{+}}$ , still denoted by ${u_{n}^{λ}}_{n \in N_{+}}$ , an integer $l \in N \cup {0}$ , a sequence ${y_{n}}_{n \in N_{+}}$ and $w^{k} \in H^{1} (R^{N})$ for $1 ⩽ k ⩽ l$ , such that

$u_{n}^{λ} ⇀ u_{λ}$ with ${(J_{μ}^{λ})}^{'} (u_{λ}) = 0$ ;

$w^{k} \neq 0$ and $({(J_{\infty}^{λ})}^{'} (w^{k}) = 0$ for $1 ⩽ k ⩽ l$ ;

$‖ u_{n}^{λ} - u_{λ} - \sum_{k = 1}^{l} w^{k} (\cdot + y_{n}^{k}) ‖ \to 0$ ;

$J_{μ}^{λ} (u_{n}^{λ}) \to J_{\infty}^{λ} (u_{λ}) + \sum_{i = 1}^{l} J_{\infty}^{λ} (w^{i})$ .

where we agree that in the case

l = 0

the above holds without

w^{k}

By Lemma 2.8, we have the following lemma.

Lemma 2.9.
Assume that $μ \in (0, \bar{μ})$ , (V1)–(V3) and (F1)–(F3) hold. Then for almost every $λ \in (\bar{λ}, 1]$ , there exists $u_{λ} \in H^{1} (R^{N}) ∖ {0}$ such that $\begin{matrix} (2.20) & {(J_{μ}^{λ})}^{'} (u_{λ}) = 0, J_{μ}^{λ} (u_{λ}) = c_{μ}^{λ} . \end{matrix}$
Proof.
Under (V1) and (F1)–(F3), Proposition 2.3 implies that for almost every $λ \in (\bar{λ}, 1]$ , there exists a bounded sequence ${u_{n}^{λ}}_{n \in N_{+}} \subseteq H^{1} (R^{N})$ satisfying (2.9). By Lemma 2.8, there exists a subsequence of ${u_{n}^{λ}}_{n \in N_{+}}$ , still denoted by ${u_{n}^{λ}}_{n \in N_{+}}, u_{λ} \in H^{1} (R^{N})$ , an integer $l \in N \cup {0}$ and $w^{1}, w^{2}, \dots, w^{l} \in H^{1} (R^{N}) ∖ {0}$ such that $\begin{array}{l} u_{n}^{λ} ⇀ u_{λ} \in H^{1} (R^{N}), J_{λ}^{'} (u_{λ}) = 0; \\ {(J_{\infty}^{λ})}^{'} (w^{k}) = 0, J_{\infty}^{λ} (w^{k}) ⩾ m_{\infty}^{λ}, 1 ⩽ k ⩽ l; \end{array}$ and $\begin{matrix} c_{μ}^{λ} = J_{μ}^{λ} (u_{λ}) + \sum_{k = 1}^{l} J_{\infty}^{λ} (w^{k}) . \end{matrix}$ Since ${(J_{μ}^{λ})}^{'} (u_{λ}) = 0$ , then by Lemma 2.6, we have $\begin{matrix} (2.21) & P_{μ}^{λ} (u_{λ}) = 0, \end{matrix}$ where $\begin{array}{l} P_{μ}^{λ} (u_{λ}) & = \frac{N - 2}{2} ‖ \nabla u_{λ} ‖_{2}^{2} + \frac{1}{2} \int_{R^{N}} (N V (x) + \nabla V (x) \cdot x) u_{λ}^{2} d x \\ (2.22) & - \frac{μ (N - 1)}{2} \int_{R^{N}} \frac{u_{λ}^{2}}{| x |} d x - \frac{(N + α) λ}{2} \int_{R^{N}} (I_{α} * F (u_{λ})) F (u_{λ}) d x . \end{array}$ From (2.4), (2.19), (2.21) and Hölder inequality, one has $\begin{array}{rcl} J_{μ}^{λ} (u_{λ}) & = & J_{μ}^{λ} (u_{λ}) - \frac{1}{N + α} P_{μ}^{λ} (u_{λ}) \\ = & \frac{2 + α}{2 (N + α)} ‖ \nabla u_{λ} ‖_{2}^{2} + \frac{1}{2 (N + α)} \int_{R^{N}} (α V (x) - \nabla V (x) \cdot x) u_{λ}^{2} d x \\ - \frac{(α + 1) μ}{2 (N + α)} \int_{R^{N}} \frac{u_{λ}^{2}}{| x |} d x \\ ⩾ & \frac{α (1 - θ^{'})}{2 (N + α)} ‖ u_{λ} ‖^{2} - \frac{μ (α + 1)}{2 (N + α)} \int_{R^{N}} \frac{u_{λ}^{2}}{| x |} d x \\ ⩾ & \frac{α}{4 (N + α)} ‖ u_{λ} ‖^{2} - \frac{μ (α + 1)}{(N + α) (N - 2)} ‖ \nabla u_{λ} ‖_{2} ‖ u_{λ} ‖_{2} . \end{array}$ Due to $μ < \bar{μ}$ , there exists $λ_{μ}^{'} \in (0, 1)$ such that $\begin{matrix} μ ⩽ λ_{μ}^{'} \cdot \frac{α (N - 2)}{2 (α + 1)}, \end{matrix}$ thus, by Young inequality, one has $\begin{array}{l} \frac{α}{4 (N + α)} ‖ u_{λ} ‖^{2} - \frac{μ (α + 1)}{(N + α) (N - 2)} ‖ \nabla u_{λ} ‖_{2} ‖ u_{λ} ‖_{2} \\ = \frac{(1 - λ_{μ}^{'}) α}{4 (N + α)} ‖ u_{λ} ‖^{2} + \frac{λ_{μ}^{'} α}{4 (N + α)} ‖ u_{λ} ‖^{2} - \frac{μ (α + 1)}{(N + α) (N - 2)} ‖ \nabla u_{λ} ‖_{2} ‖ u_{λ} ‖_{2} \\ ⩾ \frac{(1 - λ_{μ}^{'}) α}{4 (N + α)} ‖ u_{λ} ‖^{2} + \frac{λ_{μ}^{'} α (N - 2) - 2 μ (α + 1)}{4 (N + α) (N - 2)} ‖ u_{λ} ‖^{2} \\ (2.23) & ⩾ \frac{(1 - λ_{μ}^{'}) α}{4 (N + α)} ‖ u_{λ} ‖^{2} . \end{array}$ Which shows $J_{μ}^{λ} (u_{λ}) ⩾ 0$ , in view of $\begin{matrix} c_{μ}^{λ} = J_{μ}^{λ} (u_{λ}) + \sum_{k = 1}^{l} J_{\infty}^{λ} (w^{k}) . \end{matrix}$ If $u_{λ} = 0$ , then $l ⩾ 1$ and $\begin{matrix} c_{μ}^{λ} = \sum_{k = 1}^{l} J_{\infty}^{λ} (w^{k}) ⩾ m_{\infty}^{λ}, \end{matrix}$ which contradicts to Lemma 2.5, thus $u_{λ} \neq 0$ and by (2.23) we deduce $J_{μ}^{λ} (u_{λ}) > 0$ . By Lemma 2.5, we have $c_{μ}^{λ} < m_{\infty}^{λ}$ for $λ \in (\bar{λ}, 1]$ , which implies $l = 0$ and $J_{μ}^{λ} (u_{λ}) = C_{μ}^{λ}$ . □
Proof of Theorem 1.1.
In view of Lemma 2.9, for $μ \in (0, \bar{μ})$ we know that for almost every $λ \in (\bar{λ}, 1]$ , there exists $u_{λ} \in H^{1} (R^{N}) ∖ {0}$ satisfying (2.20). Then there exists two sequence ${λ_{n}}_{n \in N_{+}} \subseteq (\bar{λ}, 1]$ and ${u_{λ_{n}}}_{n \in N_{+}} \subseteq H^{1} (R^{N}) ∖ {0}$ such that $\begin{matrix} (2.24) & λ_{n} \to 1, c_{μ}^{λ_{n}} \to c_{μ}^{}, {(J_{μ}^{λ_{n}})}^{'} (u_{λ_{n}}) = 0, J_{μ}^{λ_{n}} (u_{λ_{n}}) = c_{μ}^{λ_{n}}, as n \to \infty . \end{matrix}$ It is obvious that $\begin{array}{l} J_{μ} (u_{λ_{n}}) = J_{μ}^{λ_{n}} (u_{λ_{n}}) + \frac{λ_{n} - 1}{2} \int_{R^{N}} (I_{α} F (u_{λ_{n}})) F (u_{λ_{n}}) d x, \\ {(J_{μ})}^{'} (u_{λ_{n}}) φ = {(J_{μ}^{λ_{n}})}^{'} (u_{λ_{n}}) φ + (λ_{n} - 1) \int_{R^{N}} (I_{α} * F (u_{λ_{n}})) f (u_{λ_{n}}) φ d x, \end{array}$ where $φ \in C_{0}^{\infty} (R^{N})$ and $C_{0}^{\infty} (R^{N})$ is dense in $H^{1} (R^{N})$ . If ${u_{λ_{n}}}_{n \in N_{+}}$ is bounded in $H^{1} (R^{N})$ , we have $\begin{array}{l} (2.25) & J_{μ} (u_{λ_{n}}) \to lim_{n \to \infty} c_{μ}^{λ_{n}} = c_{μ}^{}, as n \to \infty, \\ (2.26) & J_{μ}^{'} (u_{λ_{n}}) \to 0, as n \to \infty . \end{array}$ From (2.21), (2.23) and (2.24), one has $\begin{array}{rcl} C_{4} & ⩾ & c_{μ}^{λ_{n}} = J_{μ}^{λ_{n}} (u_{λ_{n}}) \\ = & J_{μ}^{λ_{n}} (u_{λ_{n}}) - P_{μ}^{λ_{n}} (u_{λ_{n}}) \\ ⩾ & \frac{(1 - λ_{μ}^{'}) α}{4 (N + α)} ‖ u_{λ_{n}} ‖^{2}, \end{array}$ which shows that ${u_{λ_{n}}}_{n \in N_{+}}$ is bounded in $H^{1} (R^{N})$ . In view of Lemma 2.4, we have $c_{μ}^{} ⩽ c_{μ}$ . Hence, in view of the proof of Lemma 2.9, we can show that there exists $\bar{u} \in H^{1} (R^{N}) ∖ {0}$ such that $\begin{matrix} (2.27) & J_{μ} (\bar{u}) = c_{μ}^{}, {(J_{μ})}^{'} (\bar{u}) = 0 . \end{matrix}$ Let ${\hat{m}}_{μ} : = {inf}_{u \in K_{μ}} J_{μ} (u)$ , where $K_{μ}$ is defined in (1.12). By the evidence mentioned above, we know that $K_{μ} \neq \emptyset$ and ${\hat{m}}_{μ} ⩽ c_{μ}$ . For any $u \in K_{μ}$ , Lemma 2.6 implies $P_{μ} (u) = 0$ . Hence, it follows from (2.23) that $J_{μ} (u) > 0$ for all $u \in K_{μ}$ , and so ${\hat{m}}_{μ} ⩾ 0$ . Let ${u_{n}}_{n \in N_{+}} \subseteq K_{μ}$ such that $\begin{matrix} {(J_{μ})}^{'} (u_{n}) = 0, J_{μ} (u_{n}) \to {\hat{m}}_{μ}, as n \to \infty . \end{matrix}$ In view of Lemma 2.5, ${\hat{m}}_{μ} ⩽ c_{μ} < m_{\infty}$ , by a similar argument as the proof of Lemma 2.9, we can prove that there exists $\bar{u} \in H^{1} (R^{N}) ∖ {0}$ such that $\begin{matrix} (2.28) & {(J_{μ})}^{'} (\bar{u}) = 0, J_{μ} (\bar{u}) = {\hat{m}}_{μ} . \end{matrix}$ This shows that $\bar{u}$ is the least energy solution of (1.1) with $g (x) = \frac{μ}{| x |}$ . □

In the following lemma, we use a similar method to show that there are parallel results for (1.1) with $g (x) = \frac{β}{| x |^{2}}$ .
Lemma 2.10.
Assume that* $β \in (0, \bar{β})$ and (V1) hold. Then exists $η_{0} > 0$ such that $\begin{matrix} (2.29) & \int_{R^{N}} (| \nabla u |^{2} + V (x) u^{2}) d x - β \int_{R^{N}} \frac{u^{2}}{| x |^{2}} d x ⩾ η_{0} ‖ u ‖^{2}, \forall u \in H^{1} (R^{N}) . \end{matrix}$
Proof.
In view of (V1) and (2.4), for any $u \in H^{1} (R^{N})$ , one has $\begin{array}{rcl} \int_{R^{N}} (| \nabla u |^{2} + V (x) u^{2}) d x - β \int_{R^{N}} \frac{u^{2}}{| x |^{2}} d x & ⩾ & \int_{R^{N}} | \nabla u |^{2} + V (x) u^{2} d x - \frac{4 β}{{(N - 2)}^{2}} ‖ \nabla u ‖_{2}^{2} \\ ⩾ & (1 - \frac{4 β}{{(N - 2)}^{2}}) ‖ \nabla u ‖_{2}^{2} + \int_{R^{N}} u^{2} d x \\ ⩾ & (1 - \frac{α}{2 + α}) ‖ u ‖^{2} \\ = & η_{0} ‖ u ‖^{2}, \end{array}$ where $η_{0} : = 1 - \frac{α}{2 + α}$ . □
Lemma 2.11.
Assume that $β \in (0, \bar{β})$ , (V1) and (F1)–(F3) hold. Then for any $λ \in [\frac{1}{2}, 1]$ , $J_{β}^{λ} (u)$ has Mountain Pass geometry, where $J_{β}^{λ} (u)$ is defined by (1.20).
Proof.
In view of $J_{β}^{λ} ({(u_{1}^{\infty})}_{T}) ⩽ J_{}^{\frac{1}{2}} ({(u_{1}^{\infty})}_{T}) < 0$ , the proof of Lemma 2.11 is the same as Lemma 2.2. □

By Lemma 2.10 and Lemma 2.11, we know that $J_{β}^{λ}$ satisfies the conditions of Proposition 2.3, then for almost every $λ \in [\frac{1}{2}, 1]$ , $J_{β}^{λ}$ has a nontrivial solution at $C_{β}^{λ}$ which is defined in the following lemma.
Lemma 2.12.
Assume* $β \in (0, \bar{β})$ , (V1) and (F1)–(F3) hold. Then
$J_{β}^{λ} ({(u_{1}^{\infty})}_{T}) < 0$ for all $λ \in [\frac{1}{2}, 1]$ ;

there exists a positive constant $K_{β}$ independent of λ such that for all $λ \in [\frac{1}{2}, 1]$ $\begin{matrix} C_{β}^{λ} : = inf_{y \in Γ} max_{t \in [0, 1]} J_{β}^{λ} (y (t)) ⩾ K_{β} > max {J_{β}^{λ} (0), J_{β}^{λ} ({(u_{1}^{\infty})}_{T})}, \end{matrix}$ where $\begin{matrix} Γ = {y \in C ([0, 1], H^{1} (R^{N}) : y (0) = 0, y (1) = {(u_{1}^{\infty})}_{T}}; \end{matrix}$

$C_{β}^{λ}$ is bounded for $λ \in [\frac{1}{2}, 1]$ ;

${lim sup}_{λ \to λ_{0}} C_{β}^{λ} ⩽ C_{β}^{λ_{0}}$ for $λ_{0} \in (\frac{1}{2}, 1]$ .
The proof is the same as [22, Lemma 3.4].

The following lemma is crucial and plays a key role in the proof of Theorem 1.2.
Lemma 2.13.
Assume that $β \in (0, \bar{β})$ , (V1)–(V2) and (F1)–(F3) hold. Then there exists $\bar{λ} \in [\frac{1}{2}, 1]$ such that $\begin{matrix} (2.30) & C_{β}^{λ} < m_{\infty}^{λ} for λ \in [\bar{λ}, 1] . \end{matrix}$
Proof.
The proof is the same as Lemma 2.5. □

Under hypothesis $β \in (0, \bar{β})$ , (V1) and (F1)–(F3), by Proposition 2.3 and Lemma 2.12, for almost every $λ \in (\bar{λ}, 1]$ , there exists a bounded sequence ${u_{n}^{λ}}_{n \in N_{+}} \subseteq H^{1} (R^{N})$ such that $\begin{matrix} (2.31) & J_{β}^{λ} (u_{n}^{λ}) \to C_{β}^{λ}, {(J_{β}^{λ})}^{'} (u_{n}^{λ}) \to 0, as n \to \infty . \end{matrix}$

Similar to the proof of Lemma 2.6, we can prove the following lemma.
Lemma 2.14.
Assume that $β \in (0, \bar{β})$ , (V1)–(V3) and (F1)–(F2) hold. If $u \in H^{1} (R^{N})$ is a weak solution for (1.1) with $g (x) = \frac{β}{| x |^{2}}$ , then we have the following identity: $\begin{array}{rcl} P_{β} (u) & = & \frac{N - 2}{2} ‖ \nabla u ‖_{2}^{2} + \frac{1}{2} \int_{R^{N}} (N V (x) + \nabla V (x) \cdot x) u^{2} d x \\ (2.32) & - \frac{β (N - 2)}{2} \int_{R^{N}} \frac{u^{2}}{| x |^{2}} d x - \frac{N + α}{2} \int_{R^{N}} (I_{α} * F (u)) F (u) d x = 0 . \end{array}$

For any bounded (PS)-sequence of $J_{β}^{λ}$ we have the following decomposition.
Lemma 2.15 ([10]).

Assume that $β \in (0, \bar{β})$ , (V1) and (F1)–(F3) hold, let ${u_{n}^{λ}}_{n \in N_{+}}$ be a bounded (PS)-sequence for $J_{β}^{λ}$ , for $λ \in [\frac{1}{2}, 1]$ . Then there exists a subsequence of ${u_{n}^{λ}}_{n \in N_{+}}$ , still denoted by ${u_{n}^{λ}}_{n \in N_{+}}$ , an integer $l \in N \cup {0}$ , a sequence ${y_{n}}_{n \in N_{+}}$ and $w^{k} \in H^{1} (R^{N})$ for $1 ⩽ k ⩽ l$ , such that

$u_{n}^{λ} ⇀ u_{λ}$ with ${(J_{β}^{λ})}^{'} (u_{λ}) = 0$ ;

$w^{k} \neq 0$ and $({(J_{\infty}^{λ})}^{'} (w^{k}) = 0$ for $1 ⩽ k ⩽ l$ ;

$‖ u_{n}^{λ} - u_{λ} - \sum_{k = 1}^{l} w^{k} (\cdot + y_{n}^{k}) ‖ \to 0$ ;

$J_{β}^{λ} (u_{n}^{λ}) \to J_{\infty}^{λ} (u_{λ}) + \sum_{i = 1}^{l} J_{\infty}^{λ} (w^{i})$ ,

where we agree that in the case

l = 0

the above holds without

w^{k}

By Lemma 2.15, we have the following lemma.

Lemma 2.16.
Assume that $β \in (0, \bar{β})$ , (V1)–(V3) and (F1)–(F3) hold. Then for almost every $λ \in (\bar{λ}, 1]$ , there exists $u_{λ} \in H^{1} (R^{N}) ∖ {0}$ such that $\begin{matrix} (2.33) & J_{β}^{λ} (u_{λ}) = C_{β}^{λ}, {(J_{β}^{λ})}^{'} (u_{λ}) = 0 . \end{matrix}$
Proof.
The proof of this lemma is similar to Lemma 2.9. □

The proof of Theorem 1.2 is similar to Theorem 1.1
3. Asymptotic behavior of solutions for (1.1)

In this section, motivate by [12,18], we prove the asymptotic properties of nontrivial solutions of (1.1) with $g (x) = \frac{μ}{| x |}$ and ground state solutions of (1.1) with $g (x) = \frac{β}{| x |^{2}}$ .

Proof of Theorem 1.3.

In view of Theorem 1.1, we know that there exists $u_{μ} \in H^{1} (R^{N}) ∖ {0}$ satisfying $\begin{matrix} J_{μ} (u_{μ}) = {c_{μ}}^{*} ⩽ c_{μ}, {(J_{μ})}^{'} (u_{μ}) = 0 \end{matrix}$ for any $μ \in (0, \bar{μ})$ . Hence, taking ${μ_{n}}_{n \in N_{+}} \subseteq (0, \bar{μ})$ , and satisfying $μ_{n} \to 0^{+}$ as $n \to \infty$ , then there exists ${u_{n}}_{n \in N_{+}} \subseteq H^{1} (R^{N}) ∖ {0}$ such that $\begin{matrix} J_{μ_{n}} (u_{n}) = C_{μ_{n}}^{*} ⩽ C_{μ_{n}}, {(J_{μ_{n}})}^{'} (u_{n}) = 0, \forall n \in N_{+} . \end{matrix}$ Notice that $C_{μ_{n}} ⩽ C_{0}$ , where $C_{0}$ is the Mountain pass level of (1.4). Thus, it follows Lemma 2.6, (2.19), (2.4), Hölder inequality and Young inequality that $\begin{array}{rcl} C_{0} & ⩾ & C_{μ_{n}} ⩾ C_{μ_{n}}^{*} = J_{μ_{n}} (u_{n}) \\ = & J_{μ_{n}} (u_{n}) - \frac{1}{N + α} P_{μ_{n}} (u_{n}) \\ = & \frac{2 + α}{2 (N + α)} ‖ \nabla u_{n} ‖_{2}^{2} + \frac{1}{2 (N + α)} \int_{R^{N}} (α V (x) - \nabla V (x) \cdot x) u_{n}^{2} d x \\ - \frac{μ_{n} (α + 1)}{2 (N + α)} \int_{R^{N}} \frac{u_{n}^{2}}{| x |} d x \\ ⩾ & \frac{α (1 - θ^{'})}{2 (N + α)} ‖ u_{n} ‖^{2} - \frac{μ_{n} (α + 1)}{2 (N + α)} \int_{R^{N}} \frac{u_{n}^{2}}{| x |} d x \\ ⩾ & \frac{α}{4 (N + α)} ‖ u_{n} ‖^{2} - \frac{μ_{n} (α + 1)}{(N + α) (N - 2)} ‖ \nabla u_{n} ‖_{2} ‖ u_{n} ‖_{2} \\ (3.1) & ⩾ & \frac{α}{4 (N + α)} ‖ u_{n} ‖^{2} - \frac{μ_{n} (α + 1)}{2 (N + α) (N - 2)} ‖ u_{n} ‖^{2} . \end{array}$ Since $μ_{n} \to 0^{+}$ as $n \to + \infty$ , for sufficient small $μ_{n}$ , we have $\frac{α}{4 (N + α)} - \frac{μ_{n} (α + 1)}{4 (N + α) (N - 2)} > 0$ . Then (3.1) implies that ${u_{n}}_{n \in N_{+}}$ is bounded in $H^{1} (R^{N})$ . In view of the boundedness of ${u_{n}}_{n \in N_{+}}$ and $μ_{n} \to 0^{+}$ , one has $\begin{array}{l} J_{μ_{n}} (u_{n}) = J (u_{n}) + o (1), as n \to + \infty, \\ ⟨ {(J_{μ_{n}})}^{'} (u_{n}), φ ⟩ = ⟨ {(J)}^{'} (u_{n}), φ ⟩ + o (1), as n \to + \infty, \end{array}$ where $φ \in C_{0}^{\infty} (R^{N})$ . Since ${C_{μ_{n}}^{*}}_{n \in N_{+}}$ is a bounded sequence in the field of real numbers and has positive upper and lower bounds, then there must at least a convergent subsequence. So passing to a subsequence, one has $\begin{array}{l} J_{μ_{n}} (u_{n}) \to C_{*}^{'} > 0, as n \to + \infty, \\ {(J_{μ_{n}})}^{'} (u_{n}) \to 0, as n \to + \infty, \end{array}$ thus, we have $\begin{array}{l} J (u_{n}) \to C_{*}^{'} ⩽ C_{0}, as n \to + \infty, \\ {(J)}^{'} (u_{n}) \to 0, as n \to + \infty . \end{array}$ By Lemma 2.5 and Lemma 2.8, passing to a subsequence, we obtain that there exists $\bar{u} \in H^{1} (R^{N}) ∖ {0}$ such that $\begin{matrix} u_{n} \to \bar{u} in H^{1} (R^{N}), as n \to \infty, \end{matrix}$ and $\begin{matrix} J (\bar{u}) = C_{*} ⩽ C_{0}, {(J)}^{'} (\bar{u}) = 0 . \end{matrix}$ □

Proof of Theorem 1.4.

We will take 5 steps to complete the proof.

Step 1: We claim that for any $u \in K_{β}$ , where $β > 0$ sufficient small, there holds $\begin{matrix} \int_{R^{N}} (I_{α} * F (u)) F (u) d x > 0 . \end{matrix}$ In fact, we recall $P_{β} (u) = 0$ from Lemma 2.14. By [22, Lemma 2.4], there exists $y_{1} > 0$ such that $\begin{matrix} (N - 2) ‖ \nabla u ‖_{2}^{2} + \int_{R^{N}} (N V (x) + \nabla V (x) \cdot x) u^{2} d x ⩾ y_{1} ‖ u ‖^{2}, \forall u \in H^{1} (R^{N}) . \end{matrix}$ Thus, by (V1), (V2) and (V4) we have $\begin{array}{rcl} (N + α) \int_{R^{N}} (I_{α} * F (u)) F (u) d x & = & (N - 2) ‖ \nabla u ‖_{2}^{2} + \int_{R^{N}} (N V (x) + \nabla V (x) \cdot x) u^{2} d x \\ - β (N - 2) \int_{R^{N}} \frac{u^{2}}{| x |^{2}} d x \\ ⩾ & y_{1} ‖ u ‖^{2} - \frac{4 β}{N - 2} ‖ \nabla u ‖_{2}^{2} \\ (3.2) & ⩾ & (y_{1} - \frac{4 β}{N - 2}) ‖ u ‖^{2}, \end{array}$ with $β > 0$ sufficient small, then there exists $β_{0} \in (0, \bar{β})$ such that $y_{1} - \frac{4 β_{0}}{N - 2} > 0$ .

Step 2: Showing $C_{β} = {\hat{m}}_{β}$ , for $β > 0$ sufficient small, where ${\hat{m}}_{β} : = {inf}_{K_{β}}$ . On the one hand, from Theorem 1.2, we know that there exists $w \in H^{1} (R^{N}) ∖ {0}$ such that $J_{β} (w) = C_{β}^{*} ⩽ C_{β}$ and ${(J_{β})}^{'} (w) = 0$ , thus $w \in K_{β}$ , then we obtain $C_{β} ⩾ J_{β} (w) ⩾ {\hat{m}}_{β}$ . On the other hand, by (V4) and (2.4), for any $u \in H^{1} (R^{N})$ and $t \in (0, \infty)$ one has $\begin{array}{l} J_{β} (u) - J_{β} (u_{t}) - \frac{1 - t^{N + α}}{N + α} P_{β} (u) \\ = \frac{(N + α) (1 - t^{N - 2}) - (N - 2) (1 - t^{N + α})}{2 (N + α)} ‖ \nabla u ‖_{2}^{2} \\ + \int_{R^{N}} {(\frac{1}{2} - \frac{N (1 - t^{N + α})}{2 (N + α)}) V (x) - \frac{t^{N}}{2} V (t x) - \frac{1 - t^{N + α}}{2 (N + α)} \nabla V (x) \cdot x} u^{2} d x \\ - \frac{β [2 + α - (N + α) t^{N - 2} + (N - 2) t^{N + α}]}{2 (N + α)} \int_{R^{N}} \frac{u^{2}}{| x |^{2}} d x \\ ⩾ \frac{(1 - θ) g (t)}{2 (N + α)} ‖ \nabla u ‖_{2}^{2} - \frac{β g (t)}{2 (N + α)} \int_{R^{N}} \frac{u^{2}}{| x |^{2}} d x \\ ⩾ \frac{(1 - θ) g (t)}{2 (N + α)} ‖ \nabla u ‖_{2}^{2} - \frac{2 β g (t)}{(N + α) {(N - 2)}^{2}} ‖ \nabla u ‖_{2}^{2} \\ (3.3) & = (\frac{1 - θ}{2} - \frac{2 β}{{(N - 2)}^{2}}) \frac{g (t)}{N + α} ‖ \nabla u ‖_{2}^{2}, \end{array}$ where $\begin{matrix} g (t) = 2 + α - (N + α) t^{N - 2} + (N - 2) t^{N + α} . \end{matrix}$ By taking the derivative, we know that $g (t) > 0$ for $t \neq 1$ . Hence, for any $0 < β < \frac{(1 - θ) {(N - 2)}^{2}}{4}$ and $t \in (0, \infty)$ , one has $\begin{matrix} (3.4) & J_{β} (u) ⩾ J_{β} (u_{t}), \forall u \in K_{β}, \end{matrix}$ where the equation holds if and only if $t = 1$ . Define $\bar{β_{0}} = min {β_{0}, \frac{(1 - θ) {(N - 2)}^{2}}{4}}$ , if $β \in (0, \bar{β_{0}})$ and $u \in K_{β}$ , in view of ${lim}_{t \to \infty} J_{β} (u_{t}) = - \infty$ , we have ${max}_{t > 0} J_{β} (u (\frac{x}{t})) = J_{β} (u) ⩾ C_{β}$ . Thus, we obtain ${\hat{m}}_{β} ⩾ C_{β}$ for any $β \in (0, \bar{β_{0}})$ .

Step 3: In view of Theorem 1.2, we know that there exists $u_{β} \in H^{1} (R^{N}) ∖ {0}$ satisfying $\begin{matrix} J_{β} (u_{β}) = {\hat{m}}_{β}, {(J_{β})}^{'} (u_{β}) = 0, \end{matrix}$ for any $β \in (0, \bar{β_{0}})$ . Hence, taking ${β_{n}}_{n \in N_{+}} \subseteq (0, \bar{β_{0}})$ , which satisfies $β_{n} \to 0^{+}$ as $n \to \infty$ , then there exists ${u_{n}}_{n \in N_{+}} \subseteq H^{1} (R^{N}) ∖ {0}$ such that $\begin{matrix} J_{β_{n}} (u_{n}) = {\hat{m}}_{β_{n}} = C_{β_{n}}, {(J_{β_{n}})}^{'} (u_{n}) = 0 . \end{matrix}$ Notice that $C_{β_{n}} ⩽ C_{0}$ , then by $β_{n} \in (0, \bar{β_{0}})$ , (V3), Lemma 2.14 and (2.4), one has $\begin{array}{rcl} (N + α) J_{β_{n}} (u_{n}) & = & (N + α) J_{β_{n}} (u_{n}) - P_{β_{n}} (u_{n}) \\ = & \frac{α + 2}{2} ‖ \nabla u_{n} ‖_{2}^{2} + \frac{1}{2} \int_{R^{N}} (α V (x) - \nabla V (x) \cdot x) u_{n}^{2} d x \\ - \frac{β_{n} (2 + α)}{2} \int_{R^{N}} \frac{u_{n}^{2}}{| x |^{2}} d x \\ ⩾ & \frac{α + 2}{2} ‖ \nabla u_{n} ‖^{2} + \frac{1}{2} \int_{R^{N}} α V (x) u_{n}^{2} d x - \frac{1}{2} \int_{B_{\bar{R}}} \frac{{(N - 2)}^{2}}{2 | x |^{2}} u_{n}^{2} d x \\ - \frac{1}{2} \int_{B_{\bar{R}}^{c}} θ^{'} α V (x) u_{n}^{2} d x - \frac{β_{n} (2 + α)}{2} \int_{R^{N}} \frac{u_{n}^{2}}{| x |^{2}} d x \\ ⩾ & \frac{α + 2}{2} ‖ \nabla u_{n} ‖^{2} + \frac{1}{2} \int_{R^{N}} α (1 - θ^{'}) V (x) u_{n}^{2} d x \\ - \frac{{(N - 2)}^{2}}{4} \int_{R^{N}} \frac{u_{n}^{2}}{| x |^{2}} d x - \frac{2 β_{n} (2 + α)}{{(N - 2)}^{2}} ‖ \nabla u_{n} ‖_{2}^{2} \\ ⩾ & (\frac{α}{2} - \frac{2 β_{n} (2 + α)}{{(N - 2)}^{2}}) ‖ \nabla u_{n} ‖_{2}^{2} \\ (3.5) & ⩾ & \frac{α}{4} ‖ \nabla u_{n} ‖_{2}^{2}, \end{array}$ which implies that ${u_{n}}_{n \in N_{+}}$ is bounded in $D^{1, 2} (R^{N})$ . By (3.5) and Lemma 2.7, one has $\begin{array}{rcl} 2 (N + α) J_{β_{n}} (u_{n}) & = & (2 + α) ‖ \nabla u_{n} ‖_{2}^{2} + \int_{R^{N}} (α V (x) - \nabla V (x) \cdot x) u_{n}^{2} d x \\ - (2 + α) β_{n} \int_{R^{N}} \frac{u_{n}^{2}}{| x |^{2}} d x \\ (3.6) & ⩾ & α (1 - θ^{'}) ‖ u_{n} ‖^{2} - (2 + α) β_{n} \int_{R^{N}} \frac{u_{n}^{2}}{| x |^{2}} d x, \end{array}$ then by (3.6), one has $\begin{matrix} (3.7) & 2 (N + α) J_{β_{n}} (u_{n}) + \frac{4 β_{n} (N + α)}{{(N - 2)}^{2}} ‖ \nabla u_{n} ‖_{2}^{2} ⩾ α (1 - θ^{'}) ‖ u_{n} ‖^{2} . \end{matrix}$ This implies that ${u_{n}}_{n \in N_{+}}$ is bounded in $H^{1} (R^{N})$ .

Step 4: Showing $C_{β_{n}} \to C_{0}$ as $n \to + \infty$ . Notice that $C_{β_{n}} ⩽ C_{0}$ , then we only seek out that ${lim inf}_{n \to \infty} C_{β_{n}} ⩾ C_{0}$ . Since ${u_{n}}_{n \in N_{+}}$ is bounded in $H^{1} (R^{N})$ , we have $\begin{matrix} C_{β_{n}} = J_{β_{n}} (u_{n}) = max_{t > 0} J_{β_{n}} (u_{n} (\frac{x}{t})) = max_{t > 0} J (u_{n} (\frac{x}{t})) + o (1) ⩾ C_{0} + o (1) \end{matrix}$ as $n \to \infty$ . This implies that $lim {inf}_{n \to \infty} C_{β_{n}} ⩾ C_{0}$ .

Step 5: By the above steps, we know that $\begin{array}{l} J_{β_{n}} (u_{n}) = J (u_{n}) + o (1), as n \to \infty, \\ ⟨ {(J_{β_{n}})}^{'} (u_{n}), φ ⟩ = ⟨ J^{'} (u_{n}), φ ⟩ + o (1), as n \to \infty, \end{array}$ where $φ \in C_{0}^{\infty} (R^{N})$ . Then, one has $\begin{matrix} J (u_{n}) \to C_{0}, J^{'} (u_{n}) \to 0, as n \to \infty . \end{matrix}$ we recall $C_{0} < m_{\infty}$ from [22, Lemma 3.5]. In view of Lemma 2.8, passing to a subsequence, there exists $u \in H^{1} (R^{N}) ∖ {0}$ such that $\begin{matrix} u_{n} \to u in H^{1} (R^{N}), as n \to \infty, \end{matrix}$ and $\begin{matrix} J (u) = C_{0}, J^{'} (u) = 0 . \end{matrix}$ The proof is now complete. □

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Existence and asymptotic behavior of solutions for Choquard equations with singular potential under Berestycki–Lions type conditions

Abstract

Keywords

1. Introduction and main results

References