1 Introduction

In this paper, we consider the existence of ground state solution for the following equation:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u=(I_{\alpha }*F(u))f(u)+g(u), &{} \quad x\in \mathbb {R}^N; \\ u\in H^1(\mathbb {R}^N), \end{array} \right. \end{aligned}$$
(1.1)

where \(N\ge 3\), \(\alpha \in (0, N)\) and \(I_{\alpha }: \mathbb {R}^N\rightarrow \mathbb {R}\) is the Riesz potential defined by

$$\begin{aligned} I_{\alpha }(x)=\frac{\Gamma \left( \frac{N-\alpha }{2}\right) }{\Gamma \left( \frac{\alpha }{2}\right) 2^{\alpha }\pi ^{N/2}|x|^{N-\alpha }}, \ \ \ \ x \in \mathbb {R}^N{\setminus } \{0\}, \end{aligned}$$

\(F(t)=\int _{0}^{t}f(s)\mathrm {d}s\), \(V: \mathbb {R}^N\rightarrow \mathbb {R}\) and \(f,g: \mathbb {R}\rightarrow \mathbb {R}\) satisfy the following assumptions:

  1. (V1)

    \(V\in {\mathcal {C}}(\mathbb {R}^N, [0, \infty ))\) and \(V(x)\le V_{\infty }:=\lim _{|y|\rightarrow \infty }V(y)\) for all \(x\in \mathbb {R}^N\);

  2. (V2)

    \(V\in {\mathcal {C}}^1(\mathbb {R}^N, \mathbb {R})\) and there exists \(\theta \in [0, 1)\) such that either of the following cases holds:

    1. (i)

      \(\nabla V(x)\cdot x\le \frac{\theta (N-2)^2}{2|x|^2}\) for all \(x\in \mathbb {R}^N{\setminus } \{0\}\),

    2. (ii)

      \(\Vert \max \{\nabla V(x)\cdot x,0\}\Vert _{N/2}\le 2\theta S\), where \(S=\inf _{u\in H^1(\mathbb {R}^N){\setminus } \{0\}}\frac{\Vert \nabla u\Vert _2^2}{\Vert u\Vert _{2^*}^2}\);

  3. (F1)

    \(f\in {\mathcal {C}}(\mathbb {R}, \mathbb {R})\), \(f(t)=o(t^{\alpha /N})\) as \(t\rightarrow 0\) and \(f(t)=o\left( t^{(\alpha +2)/(N-2)}\right) \) as \(|t|\rightarrow \infty \);

  4. (F2)

    \(F(t)\ge 0\) for all \(t\in \mathbb {R}\) and \(\mathrm {meas}\{t\in \mathbb {R}: F(t)=0\}=0\);

  5. (G1)

    \(g\in {\mathcal {C}}(\mathbb {R}, \mathbb {R})\), \(g(t)=o(|t|)\) as \(t\rightarrow 0\) and \(g(t)=o(|t|^{(N-2)/2N})\) as \(|t|\rightarrow \infty \).

By (V1), (F1), (G1), the Hardy–Littlewood–Sobolev and the Sobolev embedding theorem, the energy functional \({\mathcal {I}}: H^1(\mathbb {R}^N)\rightarrow \mathbb {R}\) associated with (1.1) is continuously differentiable defined by

$$\begin{aligned} {\mathcal {I}}(u) = \frac{1}{2}\int _{\mathbb {R}^N}\left[ |\nabla u|^2+V(x)u^2\right] \mathrm {d}x -\frac{1}{2}\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x -\int _{\mathbb {R}^N}G(u)\mathrm {d}x,\nonumber \\ \end{aligned}$$
(1.2)

and its critical points correspond to the weak solutions of (1.1). A solution is called a ground state solution if its energy is minimal among all nontrivial solutions.

Equation (1.1) can be viewed as a local nonlinear perturbation of the following Choquard equation

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u=(I_{\alpha }*F(u))f(u), &{}\quad x\in \mathbb {R}^N; \\ u\in H^1(\mathbb {R}^N), \end{array} \right. \end{aligned}$$
(1.3)

which has a strong physical meaning, and appears in several physical contexts, for example, for \(N = 3, \alpha = 2\), \(V(x)=1\) and \(f(u) = u\), it was used to study the quantum theory of a polaron at rest by Pekar [24]; to describe an electron trapped in its own hole by Choquard [17]; to model a self-gravitating matter by Penrose [20]. In a few decade, Eq. (1.3) has been studied by variational methods. For the special form of (1.3):

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+u=(I_{\alpha }*|u|^{q})|u|^{q-2}u, &{}\quad x\in \mathbb {R}^N; \\ u\in H^1(\mathbb {R}^N), \end{array} \right. \end{aligned}$$
(1.4)

Moroz and Van Schaftingen [21] obtained the existence of ground state solutions and qualitative properties of solutions within an optimal range exponents q satisfying by the intercriticality condition: \(1+\alpha /N<q<(N+\alpha )/(N-2)\), and showed that it has no nontrivial solution when either \(q\le 1+\alpha /N\) or \(q\ge (N+\alpha )/(N-2)\), where endpoints of the above interval are lower and upper critical exponents for the Choquard equation. Later, in another paper [22], they proved the existence of a ground state solution for (1.3) with \(V=1\) under Berestycki–Lions assumptions on f, by using a scaling technique introduced by Jeanjean [11] whose key is to construct a Palais–Smale sequence ((PS) sequence in short) that satisfies asymptotically the Pohoz̆aev identity (a Pohoz̆aev–Palais–Smale sequence in short). For more existence results on (1.3) or (1.4), we refer to [1,2,3, 8, 10, 18, 19, 23, 26, 33].

Recently, many researchers began to focus on the existence of ground state solutions for the Choquard equation with a local nonlinear perturbation like (1.1). It seems that the first result is due to Van Schaftingen and Xia [31]. By the mountain pass lemma and a concentration compactness argument, they proved the existence and symmetry of ground state solutions for (1.1) where \(V=1\), \(f(u)=|u|^{\frac{\alpha }{N}-1}u\) and g satisfies (G1) and the following super-linear conditions:

  1. (G2)

    there exists \(\mu >2\) such that \(0<\mu G(t)\le g(t)t\) for all \(t\ne 0\);

  2. (G3)

    there exists \(\Lambda _0>0\) such that \(\liminf _{|t|\rightarrow 0}\frac{G(t)}{|t|^{N/4+2}}\ge \Lambda _0\).

Note that (G2) is a well-known assumption of Ambrosetti–Rabinowitz type, which can help verify the mountain pass geometry and the boundedness of (PS) sequence for the corresponding functional. Inspired by [31], Li et al. [16] considered the following equation:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+u=(I_{\alpha }*|u|^{q})|u|^{q-2}u+|u|^{p-2}u, &{}\quad x\in \mathbb {R}^N; \\ u\in H^1(\mathbb {R}^N) \end{array} \right. \end{aligned}$$
(1.5)

with \(1+\alpha /N<q<(N+\alpha )/(N-2)\) and \(2<p<2^*\), and obtained a ground state solution of mountain pass type under some additional assumptions on p and q. Regarding existence results for (1.1) with \(V=1\) and \(f(u)=|u|^{\frac{(N+\alpha )}{(N-2)}-2}u\), we quote Ao [4], Li and Tang [14] and Li and Ma [15]. It is worth pointing out that the nonlinearity f is always a power function and the local nonlinear perturbation g is super-linear at infinity in the above-mentioned papers. In fact, the approaches used in these papers rely heavily on the homogeneous of degree s (\(s=1+\alpha /N, q, (N+\alpha )/(N-2)\)), the constant potential V and the super-linear growth of g. It is difficult to generalize the results on existence of ground state solutions for (1.5) to (1.1) with a variable potential V and general interaction functions f and g.

Motivated by the above works, especially [16], in this paper, we shall establish the existence of ground state solutions for (1.1), and improve and generalize the results on (1.5) obtained in [16] to (1.1). In particular, different from the existing literature, in our argument, f and g only need to satisfy (F1), (F2) and (G1). Compared with the related results, we must overcome the difficulties due to the following unpleasant facts.

  1. (a)

    Since f and g satisfy neither the Ambrosetti–Rabinowitz growth condition nor monotonicity properties, the usual Nehari manifold to obtain existence of nontrivial solutions does not work anymore.

  2. (b)

    No condition is imposed on the nonlinear terms f and g near infinity, except \(f(t)=o(t^{(\alpha +2)/(N-2)})\) and \(g(t)=o(|t|^{(N-2)/2N})\) as \(|t|\rightarrow \infty \). So we have to take care of the combined effects and the interaction of the nonlocal nonlinear term and the local nonlinear term to verify the mountain pass geometry.

  3. (c)

    The fact that \(V\not \equiv \) constant in (1.1) prevents us from constructing a Pohoz̆aev–Palais–Smale sequence as in [22]. Moreover, we have to introduce other skills to recover the compactness since our work space is \(H^1(\mathbb {R}^N)\) not \(H_{r}^1(\mathbb {R}^N)\).

These difficulties enforce the implementation of new ideas and techniques. To the best of our knowledge, there seem to be no results for (1.1) on this topic until now.

Now, we are in a position to state our first result.

Theorem 1.1

Assume that V, f and g satisfy (V1), (V2), (F1), (F2) and (G1). Then (1.1) has a ground state solution \({\bar{u}}\in H^1(\mathbb {R}^N)\) such that \({\mathcal {I}}({\bar{u}}) =\inf _{{\mathcal {K}}}{\mathcal {I}}\), where

$$\begin{aligned} {\mathcal {K}}:=\{u\in H^1(\mathbb {R}^N){\setminus } \{0\} : {\mathcal {I}}'(u)=0\}. \end{aligned}$$

Applying Theorem 1.1 to the following perturbed problem:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+[V_{\infty }-\varepsilon h(x)]u=(I_{\alpha }*F(u))f(u)+G(u), &{}\quad x\in \mathbb {R}^N; \\ u\in H^1(\mathbb {R}^N), \end{array} \right. \end{aligned}$$
(1.6)

where \(V_{\infty }\) is a positive constant and the function \(h \in {\mathcal {C}}^1(\mathbb {R}^N, \mathbb {R})\) verifies:

  1. (H1)

    \(h(x) \ge 0\) for all \(x\in \mathbb {R}^N\) and \(\lim _{|x|\rightarrow \infty }h(x)=0\);

  2. (H2)

    \(\sup _{x\in \mathbb {R}^N}[-|x|^2\nabla h(x)\cdot x]<\infty \),

we have the following corollary.

Corollary 1.2

Assume that h, f and g satisfy (H1), (H2), (F1), (F2) and (G1). Then there exists a constant \({\hat{\varepsilon }}>0\) such that (1.6) has a ground state solution \({\bar{u}}_{\varepsilon }\in H^1(\mathbb {R}^N){\setminus } \{0\}\) for all \(0<\varepsilon \le {\hat{\varepsilon }}\).

Next, we further provide a minimax characterization of the ground state energy. Inspired by [6, 7, 9], we introduce a monotonicity condition on V as follows:

  1. (V3)

    \(V\in {\mathcal {C}}^1(\mathbb {R}^N, \mathbb {R})\) and \(t\mapsto NV(tx)+\nabla V(tx)\cdot (tx)+\frac{(N-2)^3}{4t^{2}|x|^2}\) is nonincreasing on \((0,\infty )\) for all \(x\in \mathbb {R}^N{\setminus } \{0\}\).

And we define the following Pohoz̆aev functional on \(H^1(\mathbb {R}^N)\):

$$\begin{aligned} \begin{aligned} {\mathcal {P}}(u)&:= \frac{N-2}{2}\Vert \nabla u\Vert _2^2+\frac{1}{2}\int _{{\mathbb {R}}^N}[NV(x)+\nabla V(x)\cdot x]u^2\mathrm {d}x\\&\ \ \ \ -\frac{N+\alpha }{2}\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x-N\int _{\mathbb {R}^N}G(u)\mathrm {d}x. \end{aligned} \end{aligned}$$
(1.7)

In view of [16, Proposition 3.1], if \({\bar{u}}\) is a solution of (1.1), then it satisfies Pohoz̆aev identity \({\mathcal {P}}({\bar{u}})=0\). Let

$$\begin{aligned} {\mathcal {M}}:= \{u\in H^1(\mathbb {R}^N){\setminus } \{0\} : {\mathcal {P}}(u)=0\}. \end{aligned}$$
(1.8)

Then every solution of (1.1) is contained in \({\mathcal {M}}\), we call \({\mathcal {M}}\) the Pohoz̆aev manifold of \({\mathcal {I}}\). Our second main result is as follows.

Theorem 1.3

Assume that V, f and g satisfy (V1)–(V3), (F1), (F2) and (G1). Then (1.1) has a solution \({\bar{u}}\in H^1(\mathbb {R}^N)\) such that \({\mathcal {I}}({\bar{u}})=\inf _{{\mathcal {M}}}{\mathcal {I}} =\inf _{u\in H^1(\mathbb {R}^N){\setminus }\{0\}}\max _{t> 0}{\mathcal {I}}(u_t)>0\), where \(u_t(x):=u(x/t)\).

Applying Theorem 1.3 to the limiting problem:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V_{\infty }u=(I_{\alpha }*F(u))f(u)+g(u) , &{}\quad x\in \mathbb {R}^N; \\ u\in H^1(\mathbb {R}^N), \end{array} \right. \end{aligned}$$
(1.9)

we have the following corollary.

Corollary 1.4

Assume that f and g satisfy (F1), (F2) and (G1). Then (1.9) has a solution \({\bar{u}}\in H^1(\mathbb {R}^N)\) such that \({\mathcal {I}}^{\infty }({\bar{u}})=\inf _{{\mathcal {M}}^{\infty }}{\mathcal {I}}^{\infty } =\inf _{u\in H^1(\mathbb {R}^N){\setminus }\{0\}}\max _{t> 0}{\mathcal {I}}^{\infty }(u_t)>0\), where

$$\begin{aligned} {\mathcal {I}}^{\infty }(u)= & {} \frac{1}{2}\int _{\mathbb {R}^N}\left[ |\nabla u|^2+V_{\infty }u^2\right] \mathrm {d}x\nonumber \\&-\frac{1}{2}\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x -\int _{\mathbb {R}^N}G(u)\mathrm {d}x, \end{aligned}$$
(1.10)
$$\begin{aligned} {\mathcal {P}}^{\infty }(u)= & {} \frac{N-2}{2}\Vert \nabla u\Vert _2^2+\frac{NV_{\infty }}{2}\Vert u\Vert _2^2\nonumber \\&-\frac{N+\alpha }{2}\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x -N\int _{\mathbb {R}^N}G(u)\mathrm {d}x=0 \end{aligned}$$
(1.11)

and

$$\begin{aligned} {\mathcal {M}}^{\infty }:= \{u\in H^1(\mathbb {R}^N){\setminus } \{0\} : {\mathcal {P}}^{\infty }(u)=0\}. \end{aligned}$$
(1.12)

Remark 1.5

Applying Corollary 1.4 to Eq. (1.5) considered in [16], (1.5) admits a ground state solution provided \(1+\alpha /N<q<(N+\alpha )/(N-2)\) and \(2<p<2^*\), while the extra conditions on p and q used in [16] are removed. Our results generalize and improve the main results in [16], and also extend the previous related ones in the literature.

To prove Theorem 1.1, following an approximation procedure developed by Jeanjean and Toland [13], we construct a sequence \(\{u_n\}\) of exact critical points of nearby functionals which satisfies \(\lambda _n\uparrow 1\), \({\mathcal {I}}_{\lambda _n}'(u_n)=0\) and \({\mathcal {I}}_{\lambda _n}(u_n)\rightarrow c_{*}>0\), where

$$\begin{aligned}&{\mathcal {I}}_{\lambda }(u)={\mathcal {I}}(u)+(1-\lambda )\int _{\mathbb {R}^N} \left[ \frac{1}{2}(I_{\alpha }*F(u))F(u)+G(u)\right] \mathrm {d}x, \\&\ \ \forall \ u\in H^1(\mathbb {R}^N),\ \lambda \in [1/2,1]. \end{aligned}$$

Note that the variable potential V(x) in (1.1) breaks down the invariance under translations in \(\mathbb {R}^N\). To circumvent this obstacle, we borrow the idea used in [25] which rely on a comparison of the mountain pass level with the ground state energy for the corresponding limit problem (1.9). But, in our assumptions the function \(\frac{1}{2}(I_{\alpha }*F(u))F(u)+G(u)\) may be sign-changing and the ground state solutions of the limit problem (1.9) are not positive definite. These facts, together with the appearance of the nonlocal nonlinear term would require our extra efforts. More precisely, as in [29], we give a new minimax characterization of the ground state energy for the limit functional \({\mathcal {I}}_{\lambda }^{\infty }\) (see (3.3) below), and establish the key inequality:

$$\begin{aligned} c_{\lambda }<m_{\lambda }^{\infty } :=\inf _{u\in {\mathcal {M}}_{\lambda }^{\infty }}{\mathcal {I}}_{\lambda }^{\infty }(u) =\inf _{u\in H^1(\mathbb {R}^N){\setminus }\{0\}}\max _{t > 0}{\mathcal {I}}_{\lambda }^{\infty }(u_t) \end{aligned}$$

for \(\lambda \in ({\bar{\lambda }}, 1]\) by using some new analytical skills and finer calculations (see Lemma 3.5), and then prove the strong convergence of critical points \(\{u_n\}\) based on the above inequality and the global compactness lemma.

To prove Theorem 1.3, inspired by the works in [5, 29], we look for a minimizer for the minimization problem \(m:=\inf _{{\mathcal {M}}}{\mathcal {I}}\) and then prove that the minimizer is a ground state solution of (1.1). More precisely, we first choose a minimizing sequence \(\{u_n\}\) of \({\mathcal {I}}\) on \({\mathcal {M}}\) satisfying

$$\begin{aligned} {\mathcal {I}}(u_n)\rightarrow m=\inf _{{\mathcal {M}}}{\mathcal {I}}, \ \ \ \ \ {\mathcal {P}}(u_n)= 0. \end{aligned}$$
(1.13)

Then we show, with a concentration-compactness argument and “the least energy squeeze approach”, that there exist \({\hat{u}}\in H^1(\mathbb {R}^N){\setminus }\{0\}\) and \({\hat{t}}>0\) such that, after a translation and extraction of a subsequence, \(u_n\rightharpoonup {\hat{u}}\) in \(H^1(\mathbb {R}^N)\), and \(({\hat{u}})_{{\hat{t}}}\in {\mathcal {M}}\) is a minimizer of m (see Lemma 2.11), since the lack of information on \({\mathcal {I}}'(u_n)\) prevents us from using the global compactness lemma in \(H^1(\mathbb {R}^N)\). In the final, we prove that \({\bar{u}}\) is a critical point of \({\mathcal {I}}\) by combining the deformation lemma and intermediary theorem for continuous functions (see Lemma 2.12).

Throughout the paper we make use of the following notations:

  • \(H^1(\mathbb {R}^N)\) denotes the usual Sobolev space equipped with the inner product and norm

    $$\begin{aligned} (u, v)=\int _{\mathbb {R}^N}(\nabla u\cdot \nabla v+uv)\mathrm {d}x, \ \ \Vert u\Vert =(u, u)^{1/2}, \quad \ \forall \ u,v\in H^1(\mathbb {R}^N). \end{aligned}$$

  • \(L^s(\mathbb {R}^N) (1\le s< \infty )\) denotes the Lebesgue space with the norm \(\Vert u\Vert _s =\left( \int _{\mathbb {R}^N}|u|^s\mathrm {d}x\right) ^{1/s}\).

  • For any \(u\in H^1(\mathbb {R}^N){\setminus } \{0\}\), \(u_t(x):=u(t^{-1}x)\) for \(t>0\).

  • For any \(x\in \mathbb {R}^N\) and \(r>0\), \(B_r(x):=\{y\in \mathbb {R}^N: |y-x|<r \}\).

  • \(C_1, C_2,\cdots \) denote positive constants possibly different in different places.

Under (V1), there exists a constant \(\gamma _0>0\) such that

$$\begin{aligned} \gamma _0\Vert u\Vert ^2\le \int _{\mathbb {R}^N}\left[ |\nabla u|^2+V(x)u^2\right] \mathrm {d}x \le \max \{1, V_{\infty }\}\Vert u\Vert ^2, \ \ \quad \ \forall \ u\in H^1(\mathbb {R}^N).\qquad \end{aligned}$$
(1.14)

By (F1) and Hardy–Littlewood–Sobolev inequality, for some \(\kappa \in (2, 2^*)\) and any \(\epsilon >0\), one has

$$\begin{aligned}&\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x\nonumber \\&\quad = \frac{\Gamma \left( \frac{N-\alpha }{2}\right) }{\Gamma \left( \frac{\alpha }{2}\right) 2^{\alpha }\pi ^{N/2}} \int _{\mathbb {R}^N}\int _{\mathbb {R}^N}\frac{F(u(x))F(u(y))}{|x-y|^{N-\alpha }}\mathrm {d}x\mathrm {d}y \le {\mathcal {C}}_1\Vert F(u)\Vert _{2N/(N+\alpha )}^2\nonumber \\&\quad \le \epsilon \left( \Vert u\Vert _2^{2(N+\alpha )/N}+\Vert u\Vert _{2^*}^{2(N+\alpha )/(N-2)}\right) +C_{\epsilon }\Vert u\Vert _{\kappa }^{(N+\alpha )\kappa /N}, \ \ \ \ \forall \ u\in H^1(\mathbb {R}^N).\nonumber \\ \end{aligned}$$
(1.15)

The rest of the paper is organized as follows. As mentioned above, since the proof of Theorem 1.1 require the helps of a ground state solution for the limiting problem (1.9) and a minimax characterization of its energy, for the sake of convenience, the proof of Theorem 1.3 is provided in Sect. 2. Section 3 is devoted to the proof of Theorem 1.1.

2 Proof of Theorem 1.3

In this section, we give the proof of Theorem 1.3. Since \(V(x)\equiv V_{\infty }\) satisfies (V1)–(V3), thus all conclusions on \({\mathcal {I}}\) are also true for \({\mathcal {I}}^{\infty }\). For (1.9), we always assume that \(V_{\infty }>0\).

First, by a simple calculation, we can verify the following lemma.

Lemma 2.1

The following two inequalities hold:

$$\begin{aligned}&2-Nt^{N-2}+(N-2)t^N > 0, \ \ \ \ \forall \ t\in [0, 1)\cup (1, +\infty ), \end{aligned}$$
(2.1)
$$\begin{aligned}&\beta (t):=\alpha -(N+\alpha )t^{N}+Nt^{N+\alpha } > \beta (1)= 0, \ \ \ \ \forall \ t\in [0, 1)\cup (1, +\infty ). \end{aligned}$$
(2.2)

Moreover, (V3) implies the following inequality holds:

$$\begin{aligned}&Nt^N\left[ V(x)-V(tx)\right] +\left( t^{N}-1\right) \nabla V(x)\cdot x \nonumber \\&\quad \ge -\frac{(N-2)^2\left[ 2-Nt^{N-2}+(N-2)t^{N}\right] }{4|x|^2}, \ \ \forall \ t\ge 0, \ x\in \mathbb {R}^N{\setminus } \{0\}. \end{aligned}$$
(2.3)

Inspired by [30], we establish a key functional inequality as follows.

Lemma 2.2

Assume that (V1), (V3), (F1), (F2) and (G1) hold. Then

$$\begin{aligned} {\mathcal {I}}(u)\ge {\mathcal {I}}(u_t)+\frac{1-t^{N}}{N}{\mathcal {P}}(u)+\frac{\beta (t)}{2N} \int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x, \ \ \ \ \forall \ u\in H^1(\mathbb {R}^N), \ \ t > 0.\nonumber \\ \end{aligned}$$
(2.4)

Proof

According to Hardy inequality, we have

$$\begin{aligned} \Vert \nabla u\Vert _2^2 \ge \frac{(N-2)^2}{4}\int _{\mathbb {R}^N}\frac{u^2}{|x|^2}\mathrm {d}x, \ \ \forall \ u\in H^1(\mathbb {R}^N). \end{aligned}$$
(2.5)

Note that

$$\begin{aligned} {\mathcal {I}}(u_t)= & {} \frac{t^{N-2}}{2}\Vert \nabla u\Vert _2^2+\frac{t^N}{2}\int _{{\mathbb {R}}^N}V(tx)u^2\mathrm {d}x\nonumber \\&-\frac{t^{N+\alpha }}{2}\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x-t^N\int _{\mathbb {R}^N}G(u)\mathrm {d}x. \end{aligned}$$
(2.6)

Thus, by (1.2), (1.7), (2.1), (2.3), (2.5) and (2.6), one has

$$\begin{aligned}&{\mathcal {I}}(u)-{\mathcal {I}}(u_t)\\&\quad = \frac{1-t^{N-2}}{2}\Vert \nabla u\Vert _2^2+\frac{1}{2}\int _{{\mathbb {R}}^N}\left[ V(x)-t^{N}V(tx)\right] u^2\mathrm {d}x\\&\qquad -\frac{1-t^{N+\alpha }}{2}\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x -(1-t^{N})\int _{\mathbb {R}^N}G(u)\mathrm {d}x\\&\quad = \frac{1-t^{N}}{N}\left\{ \frac{N-2}{2}\Vert \nabla u\Vert _2^2\right. \\&\qquad \left. +\frac{1}{2}\int _{\mathbb {R}^N} [NV(x)+\nabla V(x)\cdot x]u^2\mathrm {d}x -\frac{N+\alpha }{2}\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x\right. \\&\qquad \left. -N\int _{{\mathbb {R}}^N}G(u)\mathrm {d}x\right\} +\frac{2-Nt^{N-2}+(N-2)t^N}{2N}\Vert \nabla u\Vert _2^2\\&\qquad +\frac{1}{2}\int _{\mathbb {R}^N}\left\{ t^N[V(x)-V(tx)] -\frac{1-t^N}{N}\nabla V(x)\cdot x\right\} u^2\mathrm {d}x\\&\qquad +\frac{\alpha -(N+\alpha )t^{N}+Nt^{N+\alpha }}{2N}\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x\\&\quad \ge \frac{1-t^N}{N}{\mathcal {P}}(u)+\frac{1}{2N}\int _{\mathbb {R}^N}\bigg \{\frac{(N-2)^2 \left[ 2-Nt^{N-2}+(N-2)t^{N}\right] }{4|x|^2}\bigg .\\&\qquad +\bigg .Nt^N[V(x)-V(tx)]-(1-t^N)\nabla V(x)\cdot x\bigg \}u^2\mathrm {d}x +\frac{\beta (t)}{2N}\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x\\&\quad \ge \frac{1-t^N}{N}{\mathcal {P}}(u)+\frac{\beta (t)}{2N} \int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x. \end{aligned}$$

This shows that (2.4) holds. \(\square \)

From Lemma 2.2, we have the following two corollaries.

Corollary 2.3

Assume that (F1), (F2) and (G1) hold. Then

$$\begin{aligned} {\mathcal {I}}^{\infty }(u)\ge & {} {\mathcal {I}}^{\infty }(u_t)+\frac{1-t^N}{N}{\mathcal {P}}^{\infty }(u) +\frac{2-Nt^{N-2}+(N-2)t^N}{2N}\Vert \nabla u\Vert _2^2,\nonumber \\&\ \ \ \ \ \ \ \ \forall \ u\in H^1(\mathbb {R}^N), \quad \ t > 0. \end{aligned}$$
(2.7)

Corollary 2.4

Assume that (V1), (V3), (F1), (F2) and (G1) hold. Then for \(u\in {\mathcal {M}}\)

$$\begin{aligned} {\mathcal {I}}(u) = \max _{t> 0}{\mathcal {I}}(u_t). \end{aligned}$$
(2.8)

Next, we shall construct a saddle point structure with respect to the fibre \(\{u_t : t > 0\}\subset H^1(\mathbb {R}^N)\) for \(u\in H^1(\mathbb {R}^3){\setminus }\{0\}\). For this purpose, we need the following inequality.

Lemma 2.5

Assume that (V1) and (V3) hold. Then

  1. (i)

    \(|\nabla V(x)\cdot x|\rightarrow 0\) as \(|x|\rightarrow \infty \);

  2. (ii)

    there exist two constants \(\gamma _1, \gamma _2>0\) such that for all

    $$\begin{aligned}&\gamma _1\Vert u\Vert ^2\le (N-2)\Vert \nabla u\Vert _2^2 +\int _{\mathbb {R}^N}\left[ NV(x)+\nabla V(x)\cdot x\right] u^2\mathrm {d}x\nonumber \\&\quad \le \gamma _2\Vert u\Vert ^2, \ \ \forall \ u\in H^1(\mathbb {R}^N). \end{aligned}$$
    (2.9)

Proof

(i) Arguing by contradiction, we assume that there exist \(\{x_n\}\subset \mathbb {R}^N\) and \(\delta >0\) such that

$$\begin{aligned} |x_n|\rightarrow \infty , \ \ \text{ and } \ \ \nabla V(x_n)\cdot x_n\ge \delta \ \text{ or }\ \nabla V(x_n)\cdot x_n\le -\delta , \ \ \ \ \forall \ n\in \mathbb {N}. \end{aligned}$$
(2.10)

Now, we only distinguish two cases: (1) \(\nabla V(x_n)\cdot x_n\ge \delta , \forall \ n\in \mathbb {N}\) and (2) \(\nabla V(x_n)\cdot x_n\le -\delta , \forall \ n\in \mathbb {N}\).

Case (1) \(\nabla V(x_n)\cdot x_n\ge \delta , \forall \ n\in \mathbb {N}\). Note that (2.3) with \(t=0\) gives

$$\begin{aligned} \nabla V(x)\cdot x\le \frac{(N-2)^2}{2|x|^2}, \ \ \ \ \forall \ x\in \mathbb {R}^N{\setminus } \{0\}. \end{aligned}$$
(2.11)

Then (2.11) implies that

$$\begin{aligned} \delta \le \nabla V(x_n)\cdot x_n\le \frac{(N-2)^2}{2|x_n|^2}=o(1), \end{aligned}$$
(2.12)

which is a obvious contradiction.

Case (2) \(\nabla V(x_n)\cdot x_n\le -\delta \) for all \(n\in \mathbb {N}\). Clearly, (2.1) yields

$$\begin{aligned} 2-2^{N-2}N +2^N(N-2)>0. \end{aligned}$$
(2.13)

From (2.3), with \(t=2\), and (2.13), we derive

$$\begin{aligned} -\delta\ge & {} \nabla V(x_n)\cdot x_n\nonumber \\\ge & {} \frac{N2^{N}[V(2x_n)-V(x_n)]}{2^{N}-1} -\frac{(N-2)^2[2-2^{N-2}N+2^N(N-2)]}{4(2^N-1)|x_n|^2}=o(1). \end{aligned}$$

Again this contradiction proves that (i) holds.

(ii) Note the item (i) implies that \(\nabla V(x)\cdot x\) is bounded for all \(x\in \mathbb {R}^N\). From (V1), (2.3), with \(t \rightarrow \infty \), and (2.11), we deduce

$$\begin{aligned} -\frac{(N-2)^3}{4|x|^2}+NV_{\infty } \le NV(x)+\nabla V(x)\cdot x\le NV_{\infty }+\frac{(N-2)^2}{2|x|^2}, \ \ \ \ \forall \ x\in \mathbb {R}^N{\setminus } \{0\}.\nonumber \\ \end{aligned}$$
(2.14)

Thus it follows from (V1) and (2.14) that

$$\begin{aligned}&(N-2)\Vert \nabla u\Vert _2^2+\int _{\mathbb {R}^N}\left[ NV(x)+\nabla V(x)\cdot x\right] u^2\mathrm {d}x\nonumber \\\le & {} (N-2+2)\Vert \nabla u\Vert _2^2+NV_{\infty }\Vert u\Vert _2^2\nonumber \\\le & {} [N-2+2+NV_{\infty }]\Vert u\Vert ^2:=\gamma _2\Vert u\Vert ^2, \quad \ \forall \ u\in H^1(\mathbb {R}^N). \end{aligned}$$
(2.15)

Next, we prove the first inequality in (2.9). Arguing by contradiction, suppose that there exists a sequence \(\{u_n\}\subset H^1(\mathbb {R}^N)\) such that

$$\begin{aligned} \Vert u_n\Vert =1, \ \ \ \ (N-2)\Vert \nabla u_n\Vert _2^2+\int _{\mathbb {R}^N}\left[ NV(x)+\nabla V(x)\cdot x\right] u_n^2\mathrm {d}x=o(1). \end{aligned}$$
(2.16)

Thus there exists \({\bar{u}}\in H^1(\mathbb {R}^N)\) such that \(u_n\rightharpoonup {\bar{u}}\) in \(H^1(\mathbb {R}^N)\). Then \(u_n\rightarrow {\bar{u}}\) in \(L_{\mathrm {loc}}^s(\mathbb {R}^N)\) for \(2\le s<2^*\) and \(u_n\rightarrow {\bar{u}}\) a.e. in \(\mathbb {R}^N\). By (V1) and (2.14), one has

$$\begin{aligned} V(x)\rightarrow V_{\infty }, \ \ |\nabla V(x)\cdot x|\rightarrow 0\ \mathrm{as}\ |x|\rightarrow \infty . \end{aligned}$$
(2.17)

This implies that there exists a constant \(R_0>0\) such that

$$\begin{aligned} NV(x)+\nabla V(x)\cdot x\ge \frac{N}{2}V_{\infty }, \ \ \ \ \forall \ |x|\ge R_0. \end{aligned}$$
(2.18)

Since \(u_n\rightarrow {\bar{u}}\) in \(L^2(B_{R_0}(0))\), it follows from (2.5), (2.14), (2.15), (2.16), (2.18), the weak semicontinuity of norm and Fatou’s Lemma that

$$\begin{aligned} 0= & {} \lim _{n\rightarrow \infty }\left\{ (N-2)\Vert \nabla u_n\Vert _2^2+\int _{|x|< R_0} \left[ NV(x)+\nabla V(x)\cdot x\right] u_n^2\mathrm {d}x\right. \\&\ \ \left. +\int _{|x|\ge R_0}\left[ NV(x)+\nabla V(x)\cdot x\right] u_n^2\mathrm {d}x\right\} \\\ge & {} (N-2)\Vert \nabla {\bar{u}}\Vert _2^2+\int _{|x|< R_0}\left[ NV(x)+\nabla V(x)\cdot x\right] {\bar{u}}^2\mathrm {d}x +\frac{NV_{\infty }}{2}\liminf _{n\rightarrow \infty }\int _{|x|\ge R_0}u_n^2\mathrm {d}x\\\ge & {} \int _{|x|< R_0}\left[ \frac{(N-2)^3}{4|x|^2} +NV(x)+\nabla V(x)\cdot x\right] {\bar{u}}^2\mathrm {d}x +\frac{NV_{\infty }}{2}\int _{|x|\ge R_0}{\bar{u}}^2\mathrm {d}x\\\ge & {} \frac{NV_{\infty }}{2}\Vert {\bar{u}}\Vert _2^2, \end{aligned}$$

which implies \({\bar{u}}=0\). Thus, from (V1) and (2.14), one has

$$\begin{aligned} \int _{\mathbb {R}^N}\left[ N(V(x)-V_{\infty })+\nabla V(x)\cdot x\right] u_n^2\mathrm {d}x=o(1). \end{aligned}$$
(2.19)

Both (2.16) and (2.19) imply

$$\begin{aligned} o(1)= & {} (N-2)\Vert \nabla u_n\Vert _2^2+\int _{\mathbb {R}^N}\left[ NV(x)+\nabla V(x)\cdot x\right] u_n^2\mathrm {d}x\nonumber \\= & {} (N-2)\Vert \nabla u_n\Vert _2^2+NV_{\infty }\Vert u_n\Vert _2^2+o(1)\nonumber \\\ge & {} \min \{N-2, NV_{\infty }\}\Vert u_n\Vert ^2+o(1)\nonumber \\= & {} \min \{N-2, NV_{\infty }\}+o(1). \end{aligned}$$

This contradiction shows that there exists \(\gamma _1\) such that the first inequality in (2.9) holds. \(\square \)

Based on the above lemmas, we establish the following important property for \({\mathcal {M}}\).

Lemma 2.6

Assume that (V1), (V3), (F1), (F2) and (G1) hold. Then for any \(u\in H^1(\mathbb {R}^N){\setminus }\{0\}\), there exists a unique \(t_u>0\) such that \(u_{t_u}\in {\mathcal {M}}\).

Proof

Let \(u\in H^1(\mathbb {R}^N){\setminus }\{0\}\) be fixed and define a function \(\zeta (t):={\mathcal {I}}(u_t)\) on \((0, \infty )\). Clearly, by (1.7) and (2.6), we have

$$\begin{aligned} \zeta '(t)=0&\Leftrightarrow \ \ \frac{N-2}{2}t^{N-2}\Vert \nabla u\Vert _2^2+\frac{t^N}{2}\int _{{\mathbb {R}}^N}[NV(tx) +\nabla V(tx)\cdot (tx)]u^2\mathrm {d}x\nonumber \\&\ \ \ \ \ \ -\frac{(N+\alpha )t^{N+\alpha }}{2}\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x -Nt^N\int _{\mathbb {R}^N}G(u)\mathrm {d}x=0\nonumber \\&\ \Leftrightarrow \ \ {\mathcal {P}}(u_t)=0 \ \ \Leftrightarrow \ \ u_t\in {\mathcal {M}}. \end{aligned}$$
(2.20)

By (V1), (F1) and (G1), one has \(\zeta (t)=0\) and \(\zeta (t)>0\) for \(t>0\) small. Noting that (F2) implies \(\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x>0\), we get easily \(\zeta (t)<0\) for t large. Therefore \(\max _{t\in (0, \infty )}\zeta (t)\) is achieved at \(t_u>0\) so that \(\zeta '(t_u)=0\) and \(u_{t_u}\in {\mathcal {M}}\).

Next we claim that \(t_u\) is unique for any \(u\in H^1(\mathbb {R}^N){\setminus }\{0\}\). Otherwise, for some \(u\in H^1(\mathbb {R}^N){\setminus }\{0\}\), there exists two positive constants \(t_1\ne t_2\) such that \(u_{t_1}, u_{t_2} \in {\mathcal {M}}\), and so \({\mathcal {P}}\left( u_{t_1}\right) ={\mathcal {P}}\left( u_{t_2}\right) =0\). From (2.2) and (2.4), we have

$$\begin{aligned} {\mathcal {I}}\left( u_{t_1}\right) > {\mathcal {I}}\left( u_{t_2}\right) +\frac{t_1^N-t_2^N}{Nt_1^N}{\mathcal {P}}\left( u_{t_1}\right) = {\mathcal {I}}\left( u_{t_2}\right) \end{aligned}$$

and

$$\begin{aligned} {\mathcal {I}}\left( u_{t_2}\right) > {\mathcal {I}}\left( u_{t_1}\right) +\frac{t_2^N-t_1^N}{Nt_2^N}{\mathcal {P}}\left( u_{t_2}\right) ={\mathcal {I}}\left( u_{t_1}\right) . \end{aligned}$$

This contradiction shows that \(t_u> 0\) is unique for any \(u\in H^1(\mathbb {R}^N){\setminus }\{0\}\). \(\square \)

Corollary 2.7

Assume that (F1), (F2) and (G1) hold. Then for any \(u\in H^1(\mathbb {R}^N){\setminus }\{0\}\), there exists a unique \(t_u>0\) such that \(u_{t_u}\in {\mathcal {M}}^{\infty }\).

From Corollary 2.4, Lemma 2.6 and Corollary 2.7, we have \({\mathcal {M}}\ne \emptyset \), \({\mathcal {M}}^{\infty }\ne \emptyset \) and the following minimax characterization.

Lemma 2.8

Assume that (V1), (V3), (F1), (F2) and (G1) hold. Then

$$\begin{aligned} \inf _{u\in {\mathcal {M}}}{\mathcal {I}}(u) =m=\inf _{u\in H^1(\mathbb {R}^N){\setminus }\{0\}}\max _{t > 0}{\mathcal {I}}(u_t). \end{aligned}$$

Lemma 2.9

Assume that (V1)-(V3), (F1), (F2) and (G1) hold. Then

  1. (i)

    there exists \(\rho _0>0\) such that \(\Vert u\Vert \ge \rho _0, \ \forall \ u\in {\mathcal {M}}\);

  2. (ii)

    \(m=\inf _{u\in {\mathcal {M}}}{\mathcal {I}}(u)>0\).

Proof

(i). Since \({\mathcal {P}}(u)=0\) for all \( u\in {\mathcal {M}}\), by (1.7), (1.15), (2.9) and Sobolev embedding theorem, one has

$$\begin{aligned} \frac{\gamma _1}{2}\Vert u\Vert ^2\le & {} \frac{N-2}{2}\Vert \nabla u\Vert _2^2+\frac{1}{2}\int _{{\mathbb {R}}^N}[NV(x)+\nabla V(x)\cdot x]u^2\mathrm {d}x \nonumber \\= & {} \frac{N+\alpha }{2}\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x +N\int _{\mathbb {R}^N}G(u)\mathrm {d}x\nonumber \\\le & {} \Vert u\Vert ^{2(N+\alpha )/N}+C_1\Vert u\Vert ^{2(N+\alpha )/(N-2)}+\frac{\gamma _1}{4}\Vert u\Vert ^2 +C_2\Vert u\Vert ^{2N/(N-2)},\nonumber \\ \end{aligned}$$
(2.21)

which implies

$$\begin{aligned} \Vert u\Vert \ge \rho _0:=\min \left\{ 1, \left[ \frac{\gamma _1}{4(1+C_1+C_2)}\right] ^{\max \{N/2\alpha ,(N-2)/4\}}\right\} , \ \ \ \ \forall \ u\in {\mathcal {M}}. \end{aligned}$$
(2.22)

(ii). Let \(\{u_n\}\subset {\mathcal {M}}\) be such that \({\mathcal {I}}(u_n)\rightarrow m\). There are two possible cases:

(1) \(\inf _{n\in \mathbb {N}}\Vert \nabla u_n\Vert _2>0\) and (2) \(\inf _{n\in \mathbb {N}}\Vert \nabla u_n\Vert _2=0\).

Case (1) \(\inf _{n\in \mathbb {N}}\Vert \nabla u_n\Vert _2:=\varrho _0>0\). Note that by (1.2) and (1.7), one has

$$\begin{aligned} \begin{aligned} {\mathcal {I}}(u)-\frac{1}{N}{\mathcal {P}}(u) \ =&\ \frac{1}{N}\Vert \nabla u\Vert _2^2-\frac{1}{2N}\int _{\mathbb {R}^N}\nabla V(x)\cdot xu^2\mathrm {d}x\\&\ \ +\frac{\alpha }{2N}\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x, \ \ \ \ \forall \ u\in H^1(\mathbb {R}^N). \end{aligned} \end{aligned}$$
(2.23)

If (i) of (V2) holds, then it follows from Hardy inequality (2.5) that

$$\begin{aligned} \int _{{\mathbb {R}}^N}\nabla V(x)\cdot x u^2\mathrm {d}x \le \frac{\theta (N-2)^2}{2}\int _{{\mathbb {R}}^N}\frac{u^2}{|x|^2}\mathrm {d}x \le 2\theta \Vert \nabla u\Vert _2^2, \ \ \ \ \forall \ u\in H^1(\mathbb {R}^N). \end{aligned}$$
(2.24)

If (ii) of (V2) holds, then it follows from the Sobolev embedding inequality that

$$\begin{aligned}&\int _{{\mathbb {R}}^N}\nabla V(x)\cdot xu^2\mathrm {d}x\nonumber \\&\quad \le \left( \int _{{\mathbb {R}}^N}\left| \max \{\nabla V(x)\cdot x,0\}\right| ^{N/2}\mathrm {d}x\right) ^{2/N} \left( \int _{{\mathbb {R}}^N}|u|^{2N/(N-2)}\mathrm {d}x\right) ^{(N-2)/N}\nonumber \\&\quad \le \frac{\left\| \max \{\nabla V(x)\cdot x,0\}\right\| _{N/2}}{S}\Vert \nabla u\Vert _2^2 \le 2\theta \Vert \nabla u\Vert _2^2, \ \ \ \ \forall \ u\in H^1(\mathbb {R}^N). \end{aligned}$$
(2.25)

By (2.23) and (2.24) or (2.25), we have

$$\begin{aligned} m+o(1)={\mathcal {I}}(u_n)={\mathcal {I}}(u_n)-\frac{1}{N}{\mathcal {P}}(u_n) \ge \frac{1-\theta }{N}\Vert \nabla u_n\Vert _2^2\ge \frac{1-\theta }{N}\varrho _0^2. \end{aligned}$$

Case (2) \(\inf _{n\in \mathbb {N}}\Vert \nabla u_n\Vert _2=0\). In this case, by (2.22), passing to a subsequence, one has

$$\begin{aligned} \Vert \nabla u_n\Vert _2\rightarrow 0, \ \ \ \ \Vert u_n\Vert _2\ge \frac{1}{2}\rho _0. \end{aligned}$$
(2.26)

By (1.15) and the Sobolev inequality, one has for all \(u\in H^1(\mathbb {R}^N)\),

$$\begin{aligned}&\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x\nonumber \\&\quad \le C_3\left( \Vert u\Vert _2^{2(N+\alpha )/N}+\Vert u\Vert _{2^*}^{2(N+\alpha )/(N-2)}\right) \nonumber \\&\quad \le C_3\left( \Vert u\Vert _2^{2(N+\alpha )/N}+S^{-(N+\alpha )/(N-2)}\Vert \nabla u\Vert _{2}^{2(N+\alpha )/(N-2)}\right) . \end{aligned}$$
(2.27)

By (V1), there exists \(R>0\) such that \(V(x)\ge \frac{V_{\infty }}{2}\) for \(|x|\ge R\). This implies

$$\begin{aligned} \int _{|tx|\ge R}V(tx)u^2\mathrm {d}x\ge \frac{V_{\infty }}{2}\int _{|tx|\ge R}u^2\mathrm {d}x, \ \ \ \ \forall \ t>0,\ u\in H^1(\mathbb {R}^N). \end{aligned}$$
(2.28)

Making use of the Hölder inequality and the Sobolev inequality, we get

$$\begin{aligned} \int _{|tx|< R}u^2\mathrm {d}x\le & {} \left( \frac{\omega _N R^N}{t^N}\right) ^{(2^*-2)/2^*}\left( \int _{|tx|< R}u^{2^*}\mathrm {d}x\right) ^{2/2^*}\nonumber \\\le & {} \omega _N^{2/N} R^2t^{-2}S^{-1}\Vert \nabla u\Vert _2^2, \ \ \ \ \forall \ t>0,\ u\in H^1(\mathbb {R}^N), \end{aligned}$$
(2.29)

where \(\omega _N\) denotes the volume of the unit ball of \(\mathbb {R}^N\). Let

$$\begin{aligned} \delta _0=\min \left\{ V_{\infty }, SR^{-2}\omega _N^{-2/N}\right\} \end{aligned}$$
(2.30)

and

$$\begin{aligned} t_n=\left( \frac{\delta _0}{8C_3}\right) ^{1/\alpha }\Vert u_n\Vert _2^{-2/N}. \end{aligned}$$
(2.31)

By (G1) and the Sobolev inequality, one has

$$\begin{aligned} \int _{\mathbb {R}^N}G(u)\mathrm {d}x\le \frac{\delta _0}{8}\Vert u\Vert _2^2+C_4\Vert u\Vert _{2^*}^{2^*} \le \frac{\delta _0}{8}\Vert u\Vert _2^2+C_4S^{-N/(N-2)}\Vert \nabla u\Vert _{2}^{2N/(N-2)}. \end{aligned}$$
(2.32)

Since (2.26) implies \(\{t_n\}\) is bounded, then it follows from (2.6), (2.8), (2.26), (2.27), (2.28), (2.29), (2.30), (2.31) and (2.32) that

$$\begin{aligned} m+o(1)= & {} {\mathcal {I}}(u_n)\ge {\mathcal {I}}\left( (u_n)_{t_n}\right) \nonumber \\= & {} \frac{t_n^{N-2}}{2}\Vert \nabla u_n\Vert _2^2+\frac{t_n^N}{2}\int _{{\mathbb {R}}^N}V(t_nx)u_n^2\mathrm {d}x -\frac{t_n^{N+\alpha }}{2}\int _{\mathbb {R}^N}(I_{\alpha }*F(u_n))F(u_n)\mathrm {d}x\nonumber \\&\ \ -t_n^N\int _{\mathbb {R}^N}G(u_n)\mathrm {d}x\nonumber \\\ge & {} \frac{S}{2R^2\omega _N^{2/N}}t_n^N\int _{|t_nx|< R}u_n^2\mathrm {d}x +\frac{1}{4}V_{\infty }t_n^N\int _{|t_nx|\ge R}u_n^2\mathrm {d}x\nonumber \\&\ \ -\frac{1}{2}C_3t_n^{N+\alpha }\Vert u_n\Vert _2^{2(N+\alpha )/N} -\frac{C_3}{2S^{(N+\alpha )/(N-2)}}t_n^{N+\alpha }\Vert \nabla u_n\Vert _{2}^{2(N+\alpha )/(N-2)}\nonumber \\&\ \ -\frac{\delta _0t_n^N}{8}\Vert u_n\Vert _2^2-C_4S^{-N/(N-2)}t_n^N\Vert \nabla u_n\Vert _{2}^{2N/(N-2)}\nonumber \\\ge & {} \frac{1}{8}\delta _0t_n^N\Vert u_n\Vert _2^2-\frac{1}{2}C_3t_n^{N+\alpha }\Vert u_n\Vert _2^{2(N+\alpha )/N}+o(1)\nonumber \\= & {} \frac{1}{8}t_n^N\Vert u_n\Vert _2^2\left( \delta _0-4C_3t_n^{\alpha }\Vert u_n\Vert _2^{2\alpha /N}\right) +o(1)\nonumber \\= & {} \frac{\delta _0}{16}\left( \frac{\delta _0}{8C_3}\right) ^{N/\alpha }+o(1). \end{aligned}$$
(2.33)

Cases (1) and (2) show that \(m=\inf _{u\in {\mathcal {M}}}{\mathcal {I}}(u)>0\). \(\square \)

Lemma 2.10

Assume that (V1)–(V3), (F1), (F2) and (G1) hold. Then \(m\le m^{\infty }\).

Proof

Arguing by contradiction, we assume that \(m> m^{\infty }\). Let \(\varepsilon :=m-m^{\infty }\). Then there exists \(u_{\varepsilon }^{\infty }\) such that

$$\begin{aligned} u_{\varepsilon }^{\infty }\in {\mathcal {M}}^{\infty } \ \ \ \ \text{ and } \ \ \ \ m^{\infty }+\frac{\varepsilon }{2}>{\mathcal {I}}^{\infty }(u_{\varepsilon }^{\infty }). \end{aligned}$$
(2.34)

In view of Lemma 2.6, there exists \(t_{\varepsilon }>0\) such that \((u_{\varepsilon }^{\infty })_{t_{\varepsilon }}\in {\mathcal {M}}\). Thus, it follows from (V1), (1.2), (1.10), (2.7) and (2.34) that

$$\begin{aligned} m^{\infty }+\frac{\varepsilon }{2}>{\mathcal {I}}^{\infty }(u_{\varepsilon }^{\infty }) \ge {\mathcal {I}}^{\infty }\left( (u_{\varepsilon }^{\infty })_{t_{\varepsilon }}\right) \ge {\mathcal {I}}\left( (u_{\varepsilon }^{\infty })_{t_{\varepsilon }}\right) \ge m. \end{aligned}$$

This contradiction shows the conclusion of Lemma 2.10 is true. \(\square \)

Lemma 2.11

Assume that (V1)–(V3), (F1), (F2) and (G1) hold. Then m is achieved.

Proof

In view of Lemma 2.9, we have \(m>0\). Let \(\{u_n\}\subset {\mathcal {M}}\) be such that \({\mathcal {I}}(u_n)\rightarrow m\). Since \({\mathcal {P}}(u_n)=0\), it follows from (2.23) and (2.24) or (2.25) that

$$\begin{aligned} m+o(1)= {\mathcal {I}}(u_n)\ge \frac{1-\theta }{N}\Vert \nabla u_n\Vert _2^2. \end{aligned}$$
(2.35)

Moreover, from (V2), (G1), (1.2), (1.7), (1.14), (2.5), the Sobolev embedding inequality and (2.24) or (2.25), we derive

$$\begin{aligned} {\mathcal {I}}(u_n)= & {} {\mathcal {I}}(u_n)-\frac{1}{N+\alpha }{\mathcal {P}}(u_n)\nonumber \\= & {} \frac{2+\alpha }{2(N+\alpha )}\Vert \nabla u_n\Vert _2^2 +\frac{1}{2(N+\alpha )}\int _{\mathbb {R}^N}\left[ \alpha V(x) -\nabla V(x)\cdot x\right] u_n^2\mathrm {d}x\nonumber \\&\ \ -\frac{\alpha }{N+\alpha }\int _{\mathbb {R}^N}G(u_n)\mathrm {d}x\nonumber \\\ge & {} \frac{\alpha }{2(N+\alpha )}\int _{\mathbb {R}^N}\left[ |\nabla u_n|^2 +V(x)u_n^2\right] \mathrm {d}x -\frac{\alpha \gamma _0}{4(N+\alpha )}\Vert u_n\Vert _2^2 -C_5\Vert u_n\Vert _{2^*}^{2^*}\nonumber \\\ge & {} \frac{\alpha \gamma _0}{4(N+\alpha )}\Vert u_n\Vert ^2 -C_5S^{-2^*/2}\Vert \nabla u_n\Vert _{2}^{2^*}, \end{aligned}$$
(2.36)

which, together with (2.35), implies that \(\{u_n\}\) is bounded in \(H^1(\mathbb {R}^N)\). Passing to a subsequence, we have \(u_n\rightharpoonup {\bar{u}}\) in \(H^1(\mathbb {R}^N)\). Then \(u_n\rightarrow {\bar{u}}\) in \(L_{\mathrm {loc}}^s(\mathbb {R}^N)\) for \(2\le s<2^*\) and \(u_n\rightarrow {\bar{u}}\) a.e. in \(\mathbb {R}^N\). Inspired by [28, Lemma 3.2], we now distinguish the following two cases: (i) \({\bar{u}}=0\) and (ii). \({\bar{u}}\ne 0\).

Case (i). \({\bar{u}}=0\), i.e. \(u_n\rightharpoonup 0\) in \(H^1(\mathbb {R}^N)\). Then \(u_n\rightarrow 0\) in \(L_{\mathrm {loc}}^s(\mathbb {R}^N)\) for \(2\le s<2^*\) and \(u_n\rightarrow 0\) a.e. in \(\mathbb {R}^N\). By (2.17), it is easy to show that

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{\mathbb {R}^N}[V_{\infty }-V(x)]u_n^2\mathrm {d}x= \lim _{n\rightarrow \infty }\int _{\mathbb {R}^N}\nabla V(x)\cdot xu_n^2\mathrm {d}x=0. \end{aligned}$$
(2.37)

From (1.2), (1.7), (1.10), (1.11) and (2.37), one can get

$$\begin{aligned} {\mathcal {I}}^{\infty }(u_n)\rightarrow m, \ \ \ \ {\mathcal {P}}^{\infty }(u_n)\rightarrow 0. \end{aligned}$$
(2.38)

From Lemma 2.11 (i), (1.11) and (2.38), one has

$$\begin{aligned}&\min \{N-2, NV_{\infty }\}\rho _0^2 \le \min \{N-2, NV_{\infty }\}\Vert u_n\Vert ^2\nonumber \\&\quad \le (N-2)\Vert \nabla u_n\Vert _2^2+NV_{\infty }\Vert u_n\Vert _2^2\nonumber \\&\quad = (N+\alpha )\int _{\mathbb {R}^N}(I_{\alpha }*F(u_n))F(u_n)\mathrm {d}x +2N\int _{\mathbb {R}^N}G(u_n)\mathrm {d}x+o(1). \end{aligned}$$
(2.39)

Using (1.15), (2.39) and Lions’ concentration compactness principle [32, Lemma 1.21], we can prove that there exist \(\delta >0\) and a sequence \(\{y_n\}\subset \mathbb {R}^N\) such that \(\int _{B_1(y_n)}|u_n|^2\mathrm {d}x> \delta \). Let \({\hat{u}}_n(x)=u_n(x+y_n)\). Then we have \(\Vert {\hat{u}}_n\Vert =\Vert u_n\Vert \) and

$$\begin{aligned} {\mathcal {I}}^{\infty }({\hat{u}}_n)\rightarrow m, \ \ \ \ {\mathcal {P}}^{\infty }({\hat{u}}_n)= o(1), \ \ \ \ \int _{B_1(0)}|{\hat{u}}_n|^2\mathrm {d}x> \delta . \end{aligned}$$
(2.40)

Therefore, there exists \({\hat{u}}\in H^1(\mathbb {R}^N){\setminus } \{0\}\) such that, passing to a subsequence,

$$\begin{aligned} \left\{ \begin{array}{ll} {\hat{u}}_n\rightharpoonup {\hat{u}}, &{} \text{ in } \ H^1(\mathbb {R}^N); \\ {\hat{u}}_n\rightarrow {\hat{u}}, &{} \text{ in } \ L_{\mathrm {loc}}^s(\mathbb {R}^N), \ \forall \ s\in [1, 2^*);\\ {\hat{u}}_n\rightarrow {\hat{u}}, &{} \text{ a.e. } \text{ on } \ \mathbb {R}^N. \end{array} \right. \end{aligned}$$
(2.41)

Let \(w_n={\hat{u}}_n-{\hat{u}}\). Then (2.41) and the Brezis–Lieb type Lemma (see [18, Lemmas 5.1–5.3], [27, Lemmas 2.7 and 2.8] and [32]) lead to

$$\begin{aligned} {\mathcal {I}}^{\infty }({\hat{u}}_n)={\mathcal {I}}^{\infty }({\hat{u}})+{\mathcal {I}}^{\infty }(w_n)+o(1) \end{aligned}$$
(2.42)

and

$$\begin{aligned} {\mathcal {P}}^{\infty }({\hat{u}}_n) = {\mathcal {P}}^{\infty }({\hat{u}})+{\mathcal {P}}^{\infty }(w_n)+o(1). \end{aligned}$$
(2.43)

Set

$$\begin{aligned} \begin{aligned} \Psi _0(u)&= {\mathcal {I}}^{\infty }(u)-\frac{1}{N}{\mathcal {P}}^{\infty }(u)\\&= \frac{1}{N}\Vert \nabla u\Vert _2^2+\frac{\alpha }{2N}\int _{\mathbb {R}^N} (I_{\alpha }*F(u))F(u)\mathrm {d}x , \ \ \ \ \forall \ u\in H^1(\mathbb {R}^N). \end{aligned} \end{aligned}$$
(2.44)

From (2.40), (2.42), (2.43) and (2.44), one has

$$\begin{aligned} \Psi _0(w_n)=m-\Psi _0({\hat{u}})+o(1), \ \ \ \ {\mathcal {P}}^{\infty }(w_n) = -{\mathcal {P}}^{\infty }({\hat{u}})+o(1). \end{aligned}$$
(2.45)

If there exists a subsequence \(\{w_{n_i}\}\) of \(\{w_n\}\) such that \(w_{n_i}=0\), then going to this subsequence, we have

$$\begin{aligned} {\mathcal {I}}^{\infty }({\hat{u}})=m, \ \ \ \ {\mathcal {P}}^{\infty }({\hat{u}})=0. \end{aligned}$$
(2.46)

Next, we assume that \(w_n\ne 0\). We claim that \({\mathcal {P}}^{\infty }({\hat{u}})\le 0\). Otherwise, if \({\mathcal {P}}^{\infty }({\hat{u}})>0\), then (2.45) implies \({\mathcal {P}}^{\infty }(w_n) < 0\) for large n. In view of Corollary 2.7, there exists \(t_n>0\) such that \((w_n)_{t_n}\in {\mathcal {M}}^{\infty }\) for large n. From (2.7), (2.44) and (2.45), we obtain

$$\begin{aligned} m-\Psi _0({\hat{u}})+o(1)= & {} \Psi _0(w_n) = {\mathcal {I}}^{\infty }(w_n)-\frac{1}{N}{\mathcal {P}}^{\infty }(w_n)\nonumber \\\ge & {} {\mathcal {I}}^{\infty }\left( {(w_n)}_{t_n}\right) -\frac{t_n^N}{N}{\mathcal {P}}^{\infty }(w_n)\nonumber \\\ge & {} m^{\infty }-\frac{t_n^N}{N}{\mathcal {P}}^{\infty }(w_n)\ge m^{\infty }, \end{aligned}$$

which implies \({\mathcal {P}}^{\infty }({\hat{u}})\le 0\) due to \(m\le m^{\infty }\) and \(\Psi _0({\hat{u}})>0\). Since \({\hat{u}}\ne 0\) and \({\mathcal {P}}^{\infty }({\hat{u}})\le 0\), in view of Corollary 2.7, there exists \(t_{\infty }>0\) such that \({\hat{u}}_{t_{\infty }}\in {\mathcal {M}}^{\infty }\). From (2.1), (2.7), (2.40), (2.44), the weak semicontinuity of norm and Fatou’s lemma, one has

$$\begin{aligned} m= & {} \lim _{n\rightarrow \infty } \left[ {\mathcal {I}}^{\infty }({\hat{u}}_n)-\frac{1}{N}{\mathcal {P}}^{\infty }({\hat{u}}_n)\right] \nonumber \\= & {} \lim _{n\rightarrow \infty }\Psi _0({\hat{u}}_n)\ge \Psi _0({\hat{u}})\nonumber \\= & {} {\mathcal {I}}^{\infty }({\hat{u}})-\frac{1}{N}{\mathcal {P}}^{\infty }({\hat{u}})\ge {\mathcal {I}}^{\infty }\left( {{\hat{u}}}_{t_{\infty }}\right) -\frac{(t_{\infty })^N}{N}{\mathcal {P}}^{\infty }({\hat{u}})\nonumber \\\ge & {} m^{\infty }-\frac{(t_{\infty })^N}{N}{\mathcal {P}}^{\infty }({\hat{u}}) \ge m-\frac{(t_{\infty })^N}{N}{\mathcal {P}}^{\infty }({\hat{u}})\ge m, \end{aligned}$$

which implies (2.46) holds also. In view of Lemma 2.6, there exists \({\hat{t}}>0\) such that \({\hat{u}}_{{\hat{t}}}\in {\mathcal {M}}\), moreover, it follows from (V1), (1.2), (1.10), (2.46) and Corollary 2.3 that

$$\begin{aligned} m\le {\mathcal {I}}({\hat{u}}_{{\hat{t}}})\le {\mathcal {I}}^{\infty }({\hat{u}}_{{\hat{t}}})\le {\mathcal {I}}^{\infty }({\hat{u}})=m. \end{aligned}$$

This shows that m is achieved at \({\hat{u}}_{{\hat{t}}}\in {\mathcal {M}}\).

Case (ii). \({\bar{u}}\ne 0\). Let \(v_n=u_n-{\bar{u}}\). Then the Brezis–Lieb type Lemma (see [18, Lemmas 5.1-5.3], [27, Lemmas 2.7 and 2.8] and [32]) leads to

$$\begin{aligned} {\mathcal {I}}(u_n)={\mathcal {I}}({\bar{u}})+{\mathcal {I}}(v_n)+o(1) \end{aligned}$$
(2.47)

and

$$\begin{aligned} {\mathcal {P}}(u_n)={\mathcal {P}}({\bar{u}})+{\mathcal {P}}(v_n)+o(1). \end{aligned}$$
(2.48)

Set

$$\begin{aligned} \Psi (u)={\mathcal {I}}(u)-\frac{1}{N}{\mathcal {P}}(u), \ \ \ \ \forall \ u\in H^1(\mathbb {R}^N). \end{aligned}$$
(2.49)

Since \({\mathcal {I}}(u_n)\rightarrow m\) and \({\mathcal {P}}(u_n)=0\), then it follows from (2.47), (2.48) and (2.49) that

$$\begin{aligned} \Psi (v_n)=m-\Psi ({\bar{u}})+o(1), \ \ \ \ {\mathcal {P}}(v_n) = -{\mathcal {P}}({\bar{u}})+o(1). \end{aligned}$$
(2.50)

If there exists a subsequence \(\{v_{n_i}\}\) of \(\{v_n\}\) such that \(v_{n_i}=0\), then going to this subsequence, we have

$$\begin{aligned} {\mathcal {I}}({\bar{u}})=m, \ \ \ \ {\mathcal {P}}({\bar{u}})=0, \end{aligned}$$
(2.51)

which implies the conclusion of Lemma 2.11 holds. Next, we assume that \(v_n\ne 0\). We claim that \({\mathcal {P}}({\bar{u}})\le 0\). Otherwise \({\mathcal {P}}({\bar{u}})>0\), then (2.50) implies \({\mathcal {P}}(v_n) < 0\) for large n. In view of Lemma 2.6, there exists \(t_n>0\) such that \((v_n)_{t_n}\in {\mathcal {M}}\) for large n. From (2.4), (2.49) and (2.50), we obtain

$$\begin{aligned} m-\Psi ({\bar{u}})+o(1)= & {} \Psi (v_n) = {\mathcal {I}}(v_n)-\frac{1}{N}{\mathcal {P}}(v_n)\nonumber \\\ge & {} {\mathcal {I}}\left( {(v_n)}_{t_n}\right) -\frac{t_n^N}{N}{\mathcal {P}}(v_n)\nonumber \\\ge & {} m-\frac{t_n^N}{N}{\mathcal {P}}(v_n)\ge m, \end{aligned}$$

which implies \({\mathcal {P}}({\bar{u}})\le 0\) due to \(\Psi ({\bar{u}})>0\). Since \({\bar{u}}\ne 0\) and \({\mathcal {P}}({\bar{u}})\le 0\), in view of Lemma 2.6, there exists \({\bar{t}}>0\) such that \({\bar{u}}_{{\bar{t}}}\in {\mathcal {M}}\). From (2.4), (2.23), (2.49), the weak semicontinuity of norm, Fatou’s lemma and (2.24) or (2.25), one has

$$\begin{aligned} m= & {} \lim _{n\rightarrow \infty } \left[ {\mathcal {I}}(u_n)-\frac{1}{N}{\mathcal {P}}(u_n)\right] = \lim _{n\rightarrow \infty } \Psi (u_n)\ge \Psi ({\bar{u}})\nonumber \\= & {} {\mathcal {I}}({\bar{u}})-\frac{1}{N}{\mathcal {P}}({\bar{u}})\ge {\mathcal {I}}\left( {{\bar{u}}}_{{\bar{t}}}\right) -\frac{{\bar{t}}^N}{N}{\mathcal {P}}({\bar{u}})\nonumber \\\ge & {} m-\frac{{\bar{t}}^N}{N}{\mathcal {P}}({\bar{u}})\ge m, \end{aligned}$$

which implies (2.51) also holds. \(\square \)

Lemma 2.12

Assume that (V1)–(V3), (F1), (F2) and (G1) hold. If \({\bar{u}}\in {\mathcal {M}}\) and \({\mathcal {I}}({\bar{u}})=m\), then \({\bar{u}}\) is a critical point of \({\mathcal {I}}\).

Proof

Following the idea of [5, Lemma 2.14], we can prove the above conclusion. For the sake of completeness, we give some details. Assume that \({\mathcal {I}}'({\bar{u}})\ne 0\). Then there exist \(\delta >0\) and \(\varrho >0\) such that

$$\begin{aligned} \Vert u-{\bar{u}}\Vert \le 3\delta \Rightarrow \Vert {\mathcal {I}}'(u)\Vert \ge \varrho . \end{aligned}$$
(2.52)

As in the proof of [5, (2.40)], one has

$$\begin{aligned} \lim _{t\rightarrow 1}\left\| {\bar{u}}_t-{\bar{u}}\right\| =0. \end{aligned}$$
(2.53)

Thus, there exists \(\delta _1\in (0, 1/4)\) such that

$$\begin{aligned} |t-1|<\delta _1\Rightarrow \left\| {\bar{u}}_t-{\bar{u}}\right\| < \delta . \end{aligned}$$
(2.54)

In view of (2.2) and (2.4), one has

$$\begin{aligned} {\mathcal {I}}\left( {\bar{u}}_t\right) \le {\mathcal {I}}({\bar{u}})-\frac{\beta (t)}{2N} \int _{\mathbb {R}^N}(I_{\alpha }*F({\bar{u}}))F({\bar{u}})\mathrm {d}x < m, \ \ \ \ \forall \ t\in (0,1)\cup (1,\infty ). \end{aligned}$$
(2.55)

From (F1), (F2), (G1) and (1.7), there exist \(T_1\in (0,1)\) and \(T_2\in (1, \infty )\) such that

$$\begin{aligned} {\mathcal {P}}\left( {\bar{u}}_{T_1}\right) >0, \ \ \ \ {\mathcal {P}}\left( {\bar{u}}_{T_2}\right) <0. \end{aligned}$$
(2.56)

Moreover, (2.55) implies

$$\begin{aligned} \chi :=\max \left\{ {\mathcal {I}}\left( {\bar{u}}_{T_1}\right) , {\mathcal {I}}\left( {\bar{u}}_{T_2}\right) \right\} <m. \end{aligned}$$
(2.57)

Let \(\varepsilon :=\min \{(m-\chi )/3, 1, \varrho \delta /8\}\) and \(S:=B({\bar{u}}, \delta )\). Then [32, Lemma 2.3] yields a deformation \(\eta \in {\mathcal {C}}([0, 1]\times H^1(\mathbb {R}^N), H^1(\mathbb {R}^N))\) such that

  1. (i)

    \(\eta (1, u)=u\) if \({\mathcal {I}}(u)<m-2\varepsilon \) or \({\mathcal {I}}(u)>m+2\varepsilon \);

  2. (ii)

    \(\eta \left( 1, {\mathcal {I}}^{m+\varepsilon }\cap B({\bar{u}}, \delta )\right) \subset {\mathcal {I}}^{m-\varepsilon }\);

  3. (iii)

    \({\mathcal {I}}(\eta (1, u))\le {\mathcal {I}}(u), \ \forall \ u\in H^1(\mathbb {R}^N)\);

  4. (iv)

    \(\eta (1, u)\) is a homeomorphism of \(H^1(\mathbb {R}^N)\).

By Corollary 2.4, \({\mathcal {I}}\left( {\bar{u}}_t\right) \le {\mathcal {I}}({\bar{u}})=m\) for \(t> 0\), then it follows from (2.54) and (ii) that

$$\begin{aligned} {\mathcal {I}}\left( \eta \left( 1, {\bar{u}}_t\right) \right) \le m-\varepsilon , \ \ \ \ \forall \ t> 0, \ \ |t-1|< \delta _1. \end{aligned}$$
(2.58)

On the other hand, by (iii) and (2.55), one has

$$\begin{aligned} {\mathcal {I}}\left( \eta \left( 1, {\bar{u}}_t\right) \right) \le {\mathcal {I}}\left( {\bar{u}}_t\right) < m, \ \ \ \ \forall \ t> 0, \ \ |t-1|\ge \delta _1. \end{aligned}$$
(2.59)

Combining (2.58) with (2.59), we have

$$\begin{aligned} \max _{t\in [T_1, T_2]}{\mathcal {I}}\left( \eta \left( 1, {\bar{u}}_t\right) \right) <m. \end{aligned}$$
(2.60)

Define \(\Phi _0(t):={\mathcal {P}}\left( \eta \left( 1, {\bar{u}}_t\right) \right) \) for \(t> 0\). It follows from (2.55) and i) that \(\eta (1, {\bar{u}}_t)={\bar{u}}_t\) for \(t=T_1\) and \(t=T_2\), which, together with (2.56), implies

$$\begin{aligned} \Phi _0(T_1)={\mathcal {P}}\left( {\bar{u}}_{T_1}\right) >0, \ \ \ \ \Phi _0(T_2)={\mathcal {P}}\left( {\bar{u}}_{T_2}\right) <0. \end{aligned}$$

Since \(\Phi _0(t)\) is continuous on \((0, \infty )\), then we have that \(\eta \left( 1, {\bar{u}}_t\right) \cap {\mathcal {M}}\ne \emptyset \) for some \(t_0\in [T_1, T_2]\), which contradicts to the definition of m. \(\square \)

Proof of Theorem 1.3

In view of Lemmas 2.9, 2.11 and 2.12 , there exists \({\bar{u}}\in {\mathcal {M}}\) such that

$$\begin{aligned} {\mathcal {I}}({\bar{u}})=m=\inf _{u\in H^1(\mathbb {R}^N){\setminus }\{0\}}\max _{t> 0}{\mathcal {I}}(u_t)>0, \ \ \ \ {\mathcal {I}}'({\bar{u}})=0. \end{aligned}$$

This shows that \({\bar{u}}\) is a ground state solution of (1.1).

3 Proof of Theorem 1.1

In this section, we give the proof of Theorem 1.1. To find critical points of \({\mathcal {I}}\), we apply the following proposition due to Jeanjean and Toland [13].

Proposition 3.1

Let X be a Banach space and let \(J\subset \mathbb {R}^+\) be an interval, and

$$\begin{aligned} \Phi _{\lambda }(u)=A(u)-\lambda B(u), \ \ \ \ \forall \ \lambda \in J, \end{aligned}$$

be a family of \({\mathcal {C}}^1\)-functional on X such that

  1. (i)

    either \(A(u)\rightarrow +\infty \) or \(B(u)\rightarrow +\infty \), as \(\Vert u\Vert \rightarrow \infty \);

  2. (ii)

    B maps every bounded set of X into a set of \(\mathbb {R}\) bounded below;

  3. (iii)

    there are two points \(v_1, v_2\) in X such that

    $$\begin{aligned} {\tilde{c}}_{\lambda }:=\inf _{\gamma \in {\tilde{\Gamma }}}\max _{t\in [0, 1]}\Phi _{\lambda }(\gamma (t))>\max \{\Phi _{\lambda }(v_1), \Phi _{\lambda }(v_2)\}, \end{aligned}$$
    (3.1)

where

$$\begin{aligned} {\tilde{\Gamma }}=\left\{ \gamma \in {\mathcal {C}}([0, 1], X): \gamma (0)=v_1, \gamma (1)=v_2\right\} . \end{aligned}$$

Then, for almost every \(\lambda \in J\), there exists a sequence \(\{u_n(\lambda )\}\) such that

  1. (i)

    \(\{u_n(\lambda )\}\) is bounded in X;

  2. (ii)

    \(\Phi _{\lambda }(u_n(\lambda ))\rightarrow c_{\lambda }\);

  3. (iii)

    \(\Phi _{\lambda }'(u_n(\lambda ))\rightarrow 0\) in \(X^*\), where \(X^*\) is the dual of X.

To apply Proposition 3.1, for \(\lambda \in [1/2,1]\), we consider two families of functionals \({\mathcal {I}}_{\lambda } : H^1(\mathbb {R}^N) \rightarrow \mathbb {R}\) defined by

$$\begin{aligned} {\mathcal {I}}_{\lambda }(u)=\frac{1}{2}\int _{\mathbb {R}^N}\left( |\nabla u|^2+V(x)u^2\right) \mathrm {d}x -\lambda \int _{\mathbb {R}^N}\left[ \frac{1}{2}(I_{\alpha }*F(u))F(u)+G(u)\right] \mathrm {d}x \end{aligned}$$
(3.2)

and

$$\begin{aligned} {\mathcal {I}}_{\lambda }^{\infty }(u)=\frac{1}{2}\int _{\mathbb {R}^N}\left( |\nabla u|^2+V_{\infty }u^2\right) \mathrm {d}x -\lambda \int _{\mathbb {R}^N}\left[ \frac{1}{2}(I_{\alpha }*F(u))F(u)+G(u)\right] \mathrm {d}x. \end{aligned}$$
(3.3)

Similar to the proof of [16, Proposition 3.1], we can obtain the following lemma.

Lemma 3.2

Assume that (V1), (V2), (F1), (F2) and (G1) hold. Let u be a critical point of \({\mathcal {I}}_{\lambda }\) in \(H^1(\mathbb {R}^N)\), then we have the following Pohoz̆aev type identity

$$\begin{aligned} {\mathcal {P}}_{\lambda }(u):= & {} \frac{N-2}{2}\Vert \nabla u\Vert _2^2+\frac{1}{2}\int _{\mathbb {R}^N} \left[ NV(x)+\nabla V(x)\cdot x\right] u^2\mathrm {d}x\nonumber \\&\ \ -\frac{(N+\alpha )\lambda }{2}\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x -N\lambda \int _{\mathbb {R}^N}G(u)\mathrm {d}x=0. \end{aligned}$$
(3.4)

For \(\lambda \in [1/2,1]\), we define the following functional on \(H^1(\mathbb {R}^N)\):

$$\begin{aligned} {\mathcal {P}}_{\lambda }^{\infty }(u)= & {} \frac{N-2}{2}\Vert \nabla u\Vert _2^2+NV_{\infty }\Vert u\Vert _2^2-\frac{(N+\alpha )\lambda }{2}\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x. \end{aligned}$$
(3.5)

By Corollary 2.3, we have the following lemma.

Lemma 3.3

Assume that (F1), (F2) and (G1) hold. Then

$$\begin{aligned} {\mathcal {I}}_{\lambda }^{\infty }(u)\ge & {} {\mathcal {I}}_{\lambda }^{\infty }\left( u_t\right) +\frac{1-t^N}{N}{\mathcal {P}}_{\lambda }^{\infty }(u) +\frac{2-Nt^{N-2}+(N-2)t^N}{2N}\Vert \nabla u\Vert _2^2,\nonumber \\&\ \ \ \ \ \ \ \ \forall \ u\in H^1(\mathbb {R}^N), \ \ t > 0. \end{aligned}$$
(3.6)

In view of Corollary 1.4, \({\mathcal {I}}_1^{\infty }={\mathcal {I}}^{\infty }\) has a minimizer \(u_1^{\infty }\ne 0\) on \({\mathcal {M}}_1^{\infty } ={\mathcal {M}}^{\infty }\), i.e.

$$\begin{aligned} u_1^{\infty }\in {\mathcal {M}}_1^{\infty }, \ \ \ \ ({\mathcal {I}}_1^{\infty })'(u_1^{\infty })=0 \ \ \ \ \text{ and } \ \ \ \ m_1^{\infty }={\mathcal {I}}_1^{\infty }(u_1^{\infty }), \end{aligned}$$
(3.7)

where

$$\begin{aligned} {\mathcal {M}}_{\lambda }^{\infty }=\{u\in H^1(\mathbb {R}^N){\setminus } \{0\}: {\mathcal {P}}_{\lambda }^{\infty }(u)=0\} \end{aligned}$$
(3.8)

and

$$\begin{aligned} m_{\lambda }^{\infty } =\inf _{u\in {\mathcal {M}}_{\lambda }^{\infty }}{\mathcal {I}}_{\lambda }^{\infty }(u) =\inf _{u\in H^1(\mathbb {R}^N){\setminus }\{0\}}\max _{t > 0}{\mathcal {I}}_{\lambda }^{\infty }(u_t). \end{aligned}$$
(3.9)

Since (1.9) is autonomous, \(V\in {\mathcal {C}}(\mathbb {R}^N, \mathbb {R})\) and \(V(x)\le V_{\infty }\) but \(V(x)\not \equiv V_{\infty }\), then there exist \({\bar{x}}\in \mathbb {R}^N\) and \({\bar{r}}>0\) such that

$$\begin{aligned} V_{\infty }-V(x)>0, \ \ |u_1^{\infty }(x)|>0\ \ \ \ a.e. \ |x-{\bar{x}}|\le {\bar{r}}. \end{aligned}$$
(3.10)

Lemma 3.4

Assume that (V1), (V2), (F1), (F2) and (G1) hold. Then

  1. (i)

    there exists a constant \(T>0\) independent of \(\lambda \) such that \({\mathcal {I}}_{\lambda }((u_1^{\infty })_{T})<0\) for all \(\lambda \in [1/2, 1]\);

  2. (ii)

    there exists a positive constant \(\kappa _0 \) independent of \(\lambda \) such that for all \(\lambda \in [1/2, 1]\),

    $$\begin{aligned} c_{\lambda }:=\inf _{\gamma \in \Gamma }\max _{t\in [0, 1]}{\mathcal {I}}_{\lambda }(\gamma (t))\ge \kappa _0 >\max \left\{ {\mathcal {I}}_{\lambda }(0), {\mathcal {I}}_{\lambda }\left( (u_1^{\infty })_{T}\right) \right\} , \end{aligned}$$

    where

    $$\begin{aligned} \Gamma =\{\gamma \in {\mathcal {C}}([0, 1], H^1(\mathbb {R}^N)): \gamma (0)=0, \gamma (1)=(u_1^{\infty })_{T}\}; \end{aligned}$$
    (3.11)
  3. (iii)

    \(c_{\lambda }\) is bounded for \(\lambda \in [1/2, 1]\);

  4. (iv)

    \(m_{\lambda }^{\infty }\) is non-increasing on \(\lambda \in [1/2, 1]\);

  5. (v)

    \(\limsup _{\lambda \rightarrow \lambda _0}c_{\lambda }\le c_{\lambda _0}\) for \(\lambda _0\in (1/2, 1]\).

Proof

Note that

$$\begin{aligned} \begin{aligned} {\mathcal {I}}_{\lambda }(u_t)&= \frac{t^{N-2}}{2}\Vert \nabla u\Vert _2^2+\frac{t^N}{2}\int _{{\mathbb {R}}^N}V(tx)u^2\mathrm {d}x -\frac{\lambda t^{N+\alpha }}{2}\int _{\mathbb {R}^N}(I_{\alpha }*F(u))F(u)\mathrm {d}x\\&\ \ \ \ -\lambda t^N\int _{\mathbb {R}^N}G(u)\mathrm {d}x, \ \ \ \ \forall \ u\in H^1(\mathbb {R}^N). \end{aligned} \end{aligned}$$
(3.12)

Since \({\mathcal {P}}_{1}^{\infty }(u_1^{\infty })=0\), we have

$$\begin{aligned} \begin{aligned}&\lambda \int _{\mathbb {R}^N}\left[ NV_{\infty }|u_1^{\infty }|^2-(N+\alpha ) (I_{\alpha }*F(u_1^{\infty }))F(u_1^{\infty })-2NG(u_1^{\infty })\right] \mathrm {d}x\\&\quad \le 2{\mathcal {P}}^{\infty }(u_1^{\infty })-(N-2)\Vert \nabla u_1^{\infty }\Vert _2^2 -(1-\lambda )\int _{\mathbb {R}^N}NV_{\infty }|u_1^{\infty }|^2\mathrm {d}x\\&\quad \le -(N-2)\Vert \nabla u_1^{\infty }\Vert _2^2<0. \end{aligned} \end{aligned}$$
(3.13)

By (3.12) and (3.13), we have

$$\begin{aligned} \begin{aligned} {\mathcal {I}}_{\lambda }((u_1^{\infty })_t)&= \frac{t^{N-2}}{2}\Vert \nabla u_1^{\infty }\Vert _2^2+\frac{t^N}{2}\int _{{\mathbb {R}}^N}V(tx)|u_1^{\infty }|^2\mathrm {d}x -\frac{\lambda t^N}{2}\int _{\mathbb {R}^N}V_{\infty }|u_1^{\infty }|^2\mathrm {d}x\\&\ \ \ \ +\lambda t^N\int _{\mathbb {R}^N}\left[ \frac{1}{2}V_{\infty }|u_1^{\infty }|^2 -\frac{N+\alpha }{2N}(I_{\alpha }*F(u_1^{\infty }))F(u_1^{\infty }) -G(u)\right] \mathrm {d}x\\&\ \ \ \ +\frac{\lambda t^{N}\left( N+\alpha -Nt^{\alpha }\right) }{2N}\int _{\mathbb {R}^N} (I_{\alpha }*F(u_1^{\infty }))F(u_1^{\infty })\mathrm {d}x\\&\le \frac{t^{N-2}}{2}\Vert \nabla u_1^{\infty }\Vert _2^2 +\frac{t^N}{4}\int _{{\mathbb {R}}^N}V_{\infty }|u_1^{\infty }|^2\mathrm {d}x\\&\ \ \ \ +\frac{t^N}{2}\int _{\mathbb {R}^N}\left[ \frac{1}{2}V_{\infty }|u_1^{\infty }|^2 -\frac{N+\alpha }{2N}(I_{\alpha }*F(u_1^{\infty }))F(u_1^{\infty }) -G(u)\right] \mathrm {d}x\\&\ \ \ \ +\frac{t^{N}\left( N+\alpha -Nt^{\alpha }\right) }{2N}\int _{\mathbb {R}^N} (I_{\alpha }*F(u_1^{\infty }))F(u_1^{\infty })\mathrm {d}x,\\&\ \ \ \ \forall \ t> 2^{1/\alpha }, \ \lambda \in [1/2, 1], \end{aligned} \end{aligned}$$

which implies that (i) holds. The proof of (ii)–(iv) in Lemma 3.4 is standard, (v) can be proved in the same way as [12, Lemma 2.3], so we omit it. \(\square \)

Lemma 3.5

Assume that (V1), (V2), (F1), (F2) and (G1) hold. Then there exists \({\bar{\lambda }}\in [1/2, 1)\) such that \(c_{\lambda }<m_{\lambda }^{\infty }\) for \(\lambda \in ({\bar{\lambda }}, 1]\).

Proof

It is easy to see that \({\mathcal {I}}_{\lambda }((u_1^{\infty })_t)\) is continuous on \(t\in (0, \infty )\). Hence for any \(\lambda \in [1/2, 1]\), we can choose \(t_{\lambda }\in (0, T)\) such that \({\mathcal {I}}_{\lambda } ((u_1^{\infty })_{t_{\lambda }}) =\max _{t\in (0,T]}{\mathcal {I}}_{\lambda }((u_1^{\infty })_t)\). Set

$$\begin{aligned} \gamma _0(t)=\left\{ \begin{array}{ll} (u_1^{\infty })_{(tT)}, \ \ &{}\text{ for } \ t>0,\\ 0, \ \ &{} \text{ for } \ t=0. \end{array}\right. \end{aligned}$$
(3.14)

Then \(\gamma _0\in \Gamma \) defined by Lemma 3.4 (ii). Moreover

$$\begin{aligned} {\mathcal {I}}_{\lambda } \left( (u_1^{\infty })_{t_{\lambda }}\right) =\max _{t\in [0,1]}{\mathcal {I}}_{\lambda }\left( \gamma _0(t)\right) \ge c_{\lambda }. \end{aligned}$$
(3.15)

Let

$$\begin{aligned} \zeta _0:=\min \{3{\bar{r}}/8(1+|{\bar{x}}|), 1/4\}. \end{aligned}$$
(3.16)

Then it follows from (3.10) and (3.16) that

$$\begin{aligned} |x-{\bar{x}}|\le \frac{{\bar{r}}}{2} \ \ \text{ and } \ \ s\in [1-\zeta _0, 1+\zeta _0] \Rightarrow |sx-{\bar{x}}|\le {\bar{r}}. \end{aligned}$$
(3.17)

Let

$$\begin{aligned} {\bar{\lambda }}:= & {} \max \left\{ \frac{1}{2}, 1-\frac{(1-\zeta _0)^N\min _{s\in [1-\zeta _0, 1+\zeta _0]} \int _{\mathbb {R}^N}\left[ V_{\infty }-V(sx)\right] |u_1^{\infty }|^2\mathrm {d}x}{T^{N}\int _{\mathbb {R}^N}\left[ T^{\alpha }(I_{\alpha }*F(u_1^{\infty }))F(u_1^{\infty }) +2|G(u_1^{\infty })|\right] \mathrm {d}x},\right. \nonumber \\&\left. 1-\frac{\min _{t\in \{1-\zeta _0,1+\zeta _0\}}\left\{ 2-Nt^{N-2}+(N-2)t^N\right\} \Vert \nabla u_1^{\infty }\Vert _2^2}{NT^{N}\int _{\mathbb {R}^N} \left[ T^{\alpha }(I_{\alpha }*F(u_1^{\infty }))F(u_1^{\infty }) +2|G(u_1^{\infty })|\right] \mathrm {d}x}\right\} . \ \ \ \end{aligned}$$
(3.18)

Then it follows from (2.1), (2.2), (3.10) and (3.17) that \(1/2\le {\bar{\lambda }}<1\). We have two cases to distinguish:

Case (i) \(t_{\lambda }\in [1-\zeta _0, 1+\zeta _0]\). From (3.2), (3.3), (3.6)–(3.15), (3.17), (3.18) and Lemma 3.4 (iv), we have

$$\begin{aligned} m_{\lambda }^{\infty }\ge & {} m_1^{\infty }={\mathcal {I}}_1^{\infty }(u_1^{\infty })\ge {\mathcal {I}}_1^{\infty }\left( (u_1^{\infty })_{t_{\lambda }}\right) \nonumber \\= & {} {\mathcal {I}}_{\lambda }\left( (u_1^{\infty })_{t_{\lambda }}\right) -\frac{(1-\lambda )t_{\lambda }^{N+\alpha }}{2}\int _{\mathbb {R}^N} (I_{\alpha }*F(u_1^{\infty }))F(u_1^{\infty })\mathrm {d}x\nonumber \\&\ \ -(1-\lambda )t_{\lambda }^{N}\int _{\mathbb {R}^N}G(u_1^{\infty })\mathrm {d}x +\frac{t_{\lambda }^N}{2}\int _{\mathbb {R}^N}[V_{\infty }-V(t_{\lambda }x)]|u_1^{\infty }|^2\mathrm {d}x\nonumber \\\ge & {} c_{\lambda }-\frac{(1-\lambda )T^{N}}{2}\int _{\mathbb {R}^N}\left[ T^{\alpha } (I_{\alpha }*F(u_1^{\infty }))F(u_1^{\infty })+2|G(u_1^{\infty })|\right] \mathrm {d}x\nonumber \\&\ \ +\frac{(1-\zeta _0)^N}{2}\min _{s\in [1-\zeta _0, 1+\zeta _0]} \int _{\mathbb {R}^N}\left[ V_{\infty }-V(sx)\right] |u_1^{\infty }|^2\mathrm {d}x\nonumber \\> & {} c_{\lambda }, \ \ \ \ \forall \ \lambda \in ({\bar{\lambda }}, 1]. \end{aligned}$$

Case (ii) \(t_{\lambda }\in (0, 1-\zeta _0)\cup (1+\zeta _0, T]\). From (2.1), (2.2), (3.2), (3.3), (3.6), (3.7), (3.15), (3.18) and Lemma 3.4 (iv), we have

$$\begin{aligned} m_{\lambda }^{\infty }\ge & {} m_1^{\infty }={\mathcal {I}}_1^{\infty }(u_1^{\infty })= {\mathcal {I}}_1^{\infty }\left( (u_1^{\infty })_{t_{\lambda }}\right) +\frac{2-Nt_{\lambda }^{N-2}+(N-2)t_{\lambda }^N}{2N}\Vert \nabla u_1^{\infty }\Vert _2^2\nonumber \\= & {} {\mathcal {I}}_{\lambda }\left( (u_1^{\infty })_{t_{\lambda }}\right) -\frac{(1-\lambda )t_{\lambda }^{N+\alpha }}{2}\int _{\mathbb {R}^N} (I_{\alpha }*F(u_1^{\infty }))F(u_1^{\infty })\mathrm {d}x\nonumber \\&\ \ -(1-\lambda )t_{\lambda }^{N}\int _{\mathbb {R}^N}G(u_1^{\infty })\mathrm {d}x +\frac{t_{\lambda }^N}{2}\int _{\mathbb {R}^N}[V_{\infty }-V(t_{\lambda }x)] |u_1^{\infty }|^2\mathrm {d}x\nonumber \\&\ \ +\frac{2-Nt_{\lambda }^{N-2}+(N-2)t_{\lambda }^N}{2N}\Vert \nabla u_1^{\infty }\Vert _2^2\nonumber \\\ge & {} c_{\lambda }-\frac{(1-\lambda )T^{N}}{2}\int _{\mathbb {R}^N}\left[ T^{\alpha } (I_{\alpha }*F(u_1^{\infty }))F(u_1^{\infty })+2|G(u_1^{\infty })|\right] \mathrm {d}x\nonumber \\&\ \ +\frac{\min _{t\in \{1-\zeta _0,1+\zeta _0\}}\left\{ 2-Nt^{N-2}+(N-2)t^N\right\} \Vert \nabla u_1^{\infty }\Vert _2^2}{2N}\nonumber \\> & {} c_{\lambda }, \ \ \ \ \forall \ \lambda \in ({\bar{\lambda }}, 1]. \end{aligned}$$

In both cases, we obtain that \(c_{\lambda }<m_{\lambda }^{\infty }\) for \(\lambda \in ({\bar{\lambda }}, 1]\). \(\square \)

Similar to the proof of [18, Proposition 3.1], we can obtain the following global compactness lemma.

Lemma 3.6

Assume that (V1), (V2), (F1), (F2) and (G1) hold. Let \(\{u_n\}\) be a bounded (PS)-sequence for \({\mathcal {I}}_{\lambda }\), for \(\lambda \in [1/2, 1]\). Then there exists a subsequence of \(\{u_n\}\), still denoted by \(\{u_n\}\), an integer \(l\in \mathbb {N}\cup \{0\}\), a sequence \(\{y_n^k\}\) and \(w^k\in H^1(\mathbb {R}^N)\) for \(1\le k\le l\), such that

  1. (i)

    \(u_n\rightharpoonup u_0\) with \({\mathcal {I}}_{\lambda }'(u_0)=0\);

  2. (ii)

    \(w^k\ne 0\) and \(({\mathcal {I}}_{\lambda }^{\infty })'(w^k)=0\) for \(1\le k\le l\);

  3. (iii)

    \(\left\| u_n-u_0-\sum _{k=1}^lw^k(\cdot +y_n^k)\right\| \rightarrow 0\);

  4. (iv)

    \({\mathcal {I}}_{\lambda }(u_n)\rightarrow {\mathcal {I}}_{\lambda }(u_0)+\sum _{i=1}^{l}{\mathcal {I}}_{\lambda }^{\infty }(w^i)\);

where we agree that in the case \(l = 0\) the above holds without \(w^k\).

Lemma 3.7

Assume that (V1), (V2), (F1), (F2) and (G1) hold. Then for almost every \(\lambda \in ({\bar{\lambda }},1]\), there exists \(u_{\lambda }\in H^1(\mathbb {R}^N){\setminus } \{0\}\) such that

$$\begin{aligned} {\mathcal {I}}_{\lambda }'(u_{\lambda })=0, \ \ \ \ {\mathcal {I}}_{\lambda }(u_{\lambda }) = c_{\lambda }. \end{aligned}$$
(3.19)

Proof

Lemma 3.4 implies that \({\mathcal {I}}_{\lambda }(u)\) satisfies the assumptions of Proposition 3.1 with \(X=H^1(\mathbb {R}^N)\), \(\Phi _{\lambda }={\mathcal {I}}_{\lambda }\) and \(J=({\bar{\lambda }},1]\). So for almost every \(\lambda \in ({\bar{\lambda }},1]\), there exists a bounded sequence \(\{u_n(\lambda )\} \subset H^1(\mathbb {R}^N)\) (for simplicity, we denote it by \(\{u_n\}\)) such that

$$\begin{aligned} {\mathcal {I}}_{\lambda }(u_n)\rightarrow c_{\lambda }>0, \ \ \ \ {\mathcal {I}}_{\lambda }'(u_n) \rightarrow 0. \end{aligned}$$
(3.20)

By Lemma 3.6, there exist a subsequence of \(\{u_n\}\), still denoted by \(\{u_n\}\), \(u_{\lambda }\in H^1(\mathbb {R}^N)\), an integer \(l\in \mathbb {N}\cup \{0\}\), and \(w^1, \ldots , w^l\in H^1(\mathbb {R}^N){\setminus } \{0\}\) such that

$$\begin{aligned}&u_n\rightharpoonup u_{\lambda }\ \ \text{ in } \ H^1(\mathbb {R}^N), \ \ \ \ {\mathcal {I}}_{\lambda }'(u_{\lambda })=0, \end{aligned}$$
(3.21)
$$\begin{aligned}&({\mathcal {I}}_{\lambda }^{\infty })'(w^k)=0, \ \ \ \ {\mathcal {I}}_{\lambda }^{\infty }(w^k)\ge m_{\lambda }^{\infty },\ \ \ \ 1\le k\le l \end{aligned}$$
(3.22)

and

$$\begin{aligned} c_{\lambda }= {\mathcal {I}}_{\lambda }(u_{\lambda })+\sum _{k=1}^{l}{\mathcal {I}}_{\lambda }^{\infty }(w^k). \end{aligned}$$
(3.23)

Since \({\mathcal {I}}_{\lambda }'(u_{\lambda })=0\), then it follows from Lemma 3.2 that

$$\begin{aligned} {\mathcal {P}}_{\lambda }(u_{\lambda })= & {} \frac{N-2}{2}\Vert \nabla u_{\lambda }\Vert _2^2+\frac{1}{2}\int _{\mathbb {R}^N}\left[ NV(x)+\nabla V(x)\cdot x\right] u_{\lambda }^2\mathrm {d}x\nonumber \\&\ \ -\frac{(N+\alpha )\lambda }{2}\int _{\mathbb {R}^N}(I_{\alpha }*F(u_{\lambda }))F(u_{\lambda })\mathrm {d}x -N\lambda \int _{\mathbb {R}^N}G(u_{\lambda })\mathrm {d}x=0.\nonumber \\ \end{aligned}$$
(3.24)

Since \(\Vert u_n\Vert \nrightarrow 0\), we deduce from (3.22) and (3.23) that if \(u_{\lambda }=0\) then \(l\ge 1\) and

$$\begin{aligned} c_{\lambda } = {\mathcal {I}}_{\lambda }(u_{\lambda })+\sum _{k=1}^{l}{\mathcal {I}}_{\lambda }^{\infty }(w^k) \ge m_{\lambda }^{\infty }, \end{aligned}$$

which contradicts Lemma 3.5. Thus \(u_{\lambda }\ne 0\). It follows from (3.2), (3.24) and (2.24) or (2.25) that

$$\begin{aligned} {\mathcal {I}}_{\lambda }(u_{\lambda })= & {} {\mathcal {I}}_{\lambda }(u_{\lambda })-\frac{1}{N}{\mathcal {P}}_{\lambda }(u_{\lambda })\nonumber \\= & {} \frac{1}{N}\Vert \nabla u_{\lambda }\Vert _2^2-\frac{1}{2N}\int _{\mathbb {R}^N}\nabla V(x)\cdot xu_{\lambda }^2\mathrm {d}x +\frac{\alpha \lambda }{2N}\int _{\mathbb {R}^N}(I_{\alpha }*F(u_{\lambda }))F(u_{\lambda })\mathrm {d}x\nonumber \\\ge & {} \frac{1-\theta }{N}\Vert \nabla u_{\lambda }\Vert _2^2>0. \end{aligned}$$
(3.25)

From (3.23) and (3.25), one has

$$\begin{aligned} c_{\lambda } = {\mathcal {I}}_{\lambda }(u_{\lambda })+\sum _{k=1}^{l}{\mathcal {I}}_{\lambda }^{\infty }(w^k) > lm_{\lambda }^{\infty }. \end{aligned}$$
(3.26)

By Lemma 3.5, we have \(c_{\lambda }<m_{\lambda }^{\infty }\) for \(\lambda \in ({\bar{\lambda }}, 1]\), which, together with (3.26), implies that \(l=0\) and \({\mathcal {I}}_{\lambda }(u_{\lambda }) = c_{\lambda }\). \(\square \)

Lemma 3.8

Assume that (V1), (V2), (F1), (F2) and (G1) hold. Then there exists \({\bar{u}}\in H^1(\mathbb {R}^N){\setminus } \{0\}\) such that

$$\begin{aligned} {\mathcal {I}}'({\bar{u}})=0, \ \ \ \ 0<{\mathcal {I}}({\bar{u}}) \le c_1. \end{aligned}$$
(3.27)

Proof

In view of Lemma 3.4 (iii) and Lemma 3.7, there exist two sequences \(\{\lambda _n\}\subset ({\bar{\lambda }}, 1]\) and \(\{u_{\lambda _n}\}\subset H^1(\mathbb {R}^N){\setminus } \{0\}\), denoted by \(\{u_n\}\), such that

$$\begin{aligned} \lambda _n\rightarrow 1, \ \ \ \ c_{\lambda _n}\rightarrow c_*,\ \ \ \ {\mathcal {I}}_{\lambda _n}'(u_n)=0, \ \ \ \ {\mathcal {I}}_{\lambda _n}(u_n) = c_{\lambda _n}. \end{aligned}$$
(3.28)

Then it follows from (3.28) and Lemma 3.2 that \({\mathcal {P}}_{\lambda _n}(u_n)=0\). By (3.2), (3.4), (3.25), (3.28) and Lemma 3.4 (iii), one has

$$\begin{aligned} C_4 \ge c_{\lambda _n}={\mathcal {I}}_{\lambda _n}(u_n)-\frac{1}{N}{\mathcal {P}}_{\lambda _n}(u_n) \ge \frac{1-\theta }{N}\Vert \nabla u_n\Vert _2^2. \end{aligned}$$
(3.29)

As in the proof of (2.36), we deduce that \(\{\Vert u_n\Vert \}\) is bounded in \(H^1(\mathbb {R}^N)\). In view of Lemma 3.4 (v), we have \(\lim _{n\rightarrow \infty }c_{\lambda _n}=c_*\le c_1\). Hence, it follows from (3.2) and (3.28) that

$$\begin{aligned} {\mathcal {I}}(u_n)\rightarrow c_*, \ \ \ \ {\mathcal {I}}'(u_n)\rightarrow 0. \end{aligned}$$
(3.30)

This shows that \(\{u_n\}\) satisfies (3.20) with \(c_{\lambda }=c_*\). In view of the proof of Lemma 3.7, we can show that there exists \({\bar{u}}\in H^1(\mathbb {R}^N){\setminus } \{0\}\) such that (3.27) holds. \(\square \)

Proof of Theorem  1.1

Let \({\hat{m}}:=\inf _{u\in {\mathcal {K}}}{\mathcal {I}}(u)\). Then Lemma 3.8 shows that \({\mathcal {K}}\ne \emptyset \) and \({\hat{m}}\le c_1\). For any \(u\in {\mathcal {K}}\), Lemma 3.2 implies \({\mathcal {P}}(u)={\mathcal {P}}_1(u)=0\). Hence it follows from (3.25) that \({\mathcal {I}}(u)={\mathcal {I}}_1(u)>0\) for all \(u\in {\mathcal {K}}\), and so \({\hat{m}}\ge 0\). Let \(\{u_n\}\subset {\mathcal {K}}\) such that

$$\begin{aligned} {\mathcal {I}}'(u_n)=0, \ \ \ \ {\mathcal {I}}(u_n) \rightarrow {\hat{m}}. \end{aligned}$$
(3.31)

In view of Lemma 3.5, \({\hat{m}}\le c_1<m_1^{\infty }\). Arguing as in the proof of Lemma 3.7, we can deduce that there exists \({\bar{u}}\in H^1(\mathbb {R}^N){\setminus } \{0\}\) such that

$$\begin{aligned} {\mathcal {I}}'({\bar{u}})=0, \ \ \ \ {\mathcal {I}}({\bar{u}}) = {\hat{m}}. \end{aligned}$$
(3.32)

This shows that \({\bar{u}}\) is a ground state solution of (1.1).