In this section, we give the proof of Theorem 1.1. Next, let us recall some properties of the variables \(f : {\mathbb{R}}\rightarrow {\mathbb{R}}\). These properties have been proved in [7, 12].
Lemma 2.1
([7, 12])
The function\(f(t)\)and its derivative satisfy the following properties:
- (1):
-
\(f(t)/t\rightarrow 1\)as\(t\rightarrow 0\);
- (2):
-
\(f(t)\leq |t|\)for any\(t\in {\mathbb{R}}\);
- (3):
-
\(f(t)\leq 2^{\frac{1}{4}}\sqrt{|t|}\)for all\(t\in {\mathbb{R}}\);
- (4):
-
\(f^{2}(t)/2\leq tf(t)f'(t)\leq f^{2}(t)\)for all\(t\in {\mathbb{R}}\);
- (5):
-
there exists a constant\(C>0\)such that
$$ \bigl\vert f(t) \bigr\vert \geq \textstyle\begin{cases} C \vert t \vert , & \textit{if } \vert t \vert \leq 1, \\ C \vert t \vert ^{\frac{1}{2}},& \textit{if } \vert t \vert \geq 1; \end{cases} $$
- (6):
-
\(|f(t)f'(t)|\leq \frac{1}{\sqrt{2}}\)for all\(t\in {\mathbb{R}}\).
By the standard argument in [16, 19], we have the following Pohozaev type identity.
Lemma 2.2
If\(v\in H^{1}({\mathbb{R}}^{N})\)is a critical point of (1.3), thenvsatisfies
$$ \mathcal{P}(v):=\frac{N-2}{2} \int _{{\mathbb{R}}^{N}} \vert \nabla v \vert ^{2}+ \frac{N}{2} \int _{{\mathbb{R}}^{N}}f^{2}(v)-\frac{N+\alpha }{2} \int _{{ \mathbb{R}}^{N}}\bigl(I_{\alpha }*G\bigl(f(v)\bigr)\bigr)G \bigl(f(v)\bigr)=0. $$
(2.1)
Motivated by [18], by a simple calculation, for any \(t\in [0,1)\cup (1,+\infty )\), one has
$$ \begin{gathered} \beta (t):=\alpha +2-(N+\alpha )t^{N-2}+(N-2)t^{N+\alpha }>0 \quad \text{and}\\ \alpha -(N+\alpha )t^{N}-N\bigl(1-t^{N+\alpha } \bigr)>0. \end{gathered} $$
(2.2)
Lemma 2.3
Assume that\((g_{1})\)–\((g_{4})\)hold. Then, for all\(v\in H^{1}({\mathbb{R}}^{N})\)and\(t>0\),
$$ \varPhi (v)\geq \varPhi (v_{t})+\frac{1-t^{N+\alpha }}{N+\alpha } \mathcal{Q}(v) + \frac{\beta (t)}{2(N+\alpha )} \Vert \nabla v \Vert ^{2}_{2}. $$
Proof
From (1.4), we have
$$ \varPhi (v_{t})=\frac{t^{N-2}}{2} \int _{{\mathbb{R}}^{N}} \vert \nabla v \vert ^{2}+ \frac{t^{N}}{2} \int _{{\mathbb{R}}^{N}}f^{2}(v)- \frac{t^{N+\alpha }}{2} \int _{{\mathbb{R}}^{N}}\bigl(I_{\alpha }*G\bigl(f(v)\bigr)\bigr)G \bigl(f(v)\bigr). $$
Thus, by (2.2), we have
$$ \begin{aligned} &\varPhi (v)-\varPhi (v_{t})\\ &\quad =\frac{1-t^{N-2}}{2} \int _{{\mathbb{R}}^{N}} \vert \nabla v \vert ^{2} + \frac{1-t^{N}}{2} \int _{{\mathbb{R}}^{N}}f^{2}(v)- \frac{1-t^{N+\alpha }}{2} \int _{{\mathbb{R}}^{N}}\bigl(I_{\alpha }*G\bigl(f(v)\bigr)\bigr)G \bigl(f(v)\bigr) \\ &\quad =\frac{1-t^{N+\alpha }}{N+\alpha } \mathcal{Q}(v)+ \frac{\alpha +2-(N+\alpha )t^{N-2}+(N-2)t^{N+\alpha } }{2(N+\alpha )} \Vert \nabla v \Vert ^{2}_{2} \\ &\qquad {}+ \frac{\alpha -(N+\alpha )t^{N}-N(1-t^{N+\alpha })}{N+\alpha } \bigl\Vert f(v) \bigr\Vert ^{2}_{2} \\ &\quad \geq \frac{1-t^{N+\alpha }}{N+\alpha } \mathcal{Q}(v) + \frac{\beta (t)}{2(N+\alpha )} \Vert \nabla v \Vert ^{2}_{2}. \end{aligned} $$
The proof is completed. □
Corollary 2.4
Assume that\((g_{1})\)–\((g_{4})\)hold. Then, for any\(v\in \mathcal{Q}\), \(\varPhi (v)=\max_{t>0}\varPhi (v_{t})\).
Lemma 2.5
Assume that\((g_{1})\)–\((g_{4})\)hold. Then, for any\(\varTheta \neq \emptyset \)and the set
$$ \bigl\{ v\in H^{1}\bigl({\mathbb{R}}^{N}\bigr)\backslash \{0 \} : \mathcal{P}(v) \leq 0 \bigr\} \subset \varTheta . $$
Proof
By using \((g_{4})\) and the method in [17] and [18], it follows that \(\varTheta \neq \emptyset \). Next, for any \(v\in H^{1}({\mathbb{R}}^{N})\backslash \{0\}\), it follows from \(\mathcal{P}(v)\leq 0\) that
$$ \frac{N}{2} \int _{{\mathbb{R}}^{N}}f^{2}(v)-\frac{N+\alpha }{2} \int _{{ \mathbb{R}}^{N}}\bigl(I_{\alpha }*G\bigl(f(v)\bigr)\bigr)G \bigl(f(v)\bigr) \leq -\frac{N-2}{2} \int _{{ \mathbb{R}}^{N}} \vert \nabla v \vert ^{2}< 0, $$
which shows that \(v\in \varTheta \). The proof is completed. □
Lemma 2.6
Assume that\((g_{1})\)–\((g_{4})\)hold. Then, for any\(v\in \varTheta \), there exists unique\(t_{v}>0\)such that\(v_{t_{v}}\in \mathcal{Q}\).
Proof
Let \(v\in \varTheta \) be fixed. Set \(\varGamma (t):=\varPhi (v_{t}) \). Then it follows from \(\varGamma '(t)=0\) that
$$ \begin{aligned}\frac{N-2}{2}t^{N-3} \int _{{\mathbb{R}}^{N}} \vert \nabla v \vert ^{2} + \frac{N t^{N-1}}{2} \int _{{\mathbb{R}}^{N}}f^{2}(v) - \frac{(N+\alpha )t^{N+\alpha -1}}{2} \int _{{\mathbb{R}}^{N}}(I_{\alpha }*G\bigl(f(v)\bigr)G\bigl(f(v) \bigr)=0. \end{aligned} $$
Then
$$ \frac{N-2}{2}t^{N-2} \int _{{\mathbb{R}}^{N}} \vert \nabla v \vert ^{2} + \frac{N t^{N}}{2} \int _{{\mathbb{R}}^{N}}f^{2}(v) - \frac{(N+\alpha )t^{N+\alpha }}{2} \int _{{\mathbb{R}}^{N}}(I_{\alpha }*G\bigl(f(v)\bigr)G\bigl(f(v) \bigr)=0, $$
which implies that \(\mathcal{P}(v_{t})=0\Leftrightarrow v_{t}\in \mathcal{Q} \). It is easy to check that \(\lim_{t\rightarrow 0}\varGamma (t)=0\), \(\varGamma (t)>0\) for \(t>0\) enough small and \(\varGamma (t)<0\) for t large. Thus \(\max_{t>0}\varGamma (t)\) is achieved at some \(t_{v}>0\) such that \(\varGamma '(t_{v})=0\) and \(v_{t_{v}}\in \mathcal{Q}\).
Next, we will prove the uniqueness. For any given \(v\in \varTheta \), if there exist \(t_{1},t_{2}>0\) such that \(v_{t_{1}},v_{t_{2}}\in \mathcal{Q}\). Thus \(\mathcal{P}(v_{t_{1}})=\mathcal{P} (v_{t_{2}})=0\). Therefore, we have
$$ \begin{aligned}\varPhi (v_{t_{1}})&\geq \varPhi (v_{t_{2}})+ \frac{t^{N}_{1}-t^{N}_{2}}{(N+\alpha )t^{N}_{1}}\mathcal{P} (v_{t_{1}}) +\frac{\beta (t_{2}/t_{1})}{2(N+\alpha )} \Vert \nabla v_{t_{1}} \Vert ^{2}_{2}= \varPhi (v_{t_{2}}) +\frac{\beta (t_{2}/t_{1})}{2(N+\alpha )} \Vert \nabla v_{t_{1}} \Vert ^{2}_{2} \end{aligned} $$
and
$$ \begin{aligned}\varPhi (v_{t_{2}})&\geq \varPhi (v_{t_{1}})+ \frac{t^{N}_{2}-t^{N}_{1}}{(N+\alpha )t^{N}_{2}}\mathcal{P} (v_{t_{2}}) +\frac{\beta (t_{1}/t_{2})}{2(N+\alpha )} \Vert \nabla v_{t_{2}} \Vert ^{2}_{2}= \varPhi (v_{t_{1}}) +\frac{\beta (t_{1}/t_{2})}{2(N+\alpha )} \Vert \nabla v_{t_{2}} \Vert ^{2}_{2}, \end{aligned} $$
which shows that \(t_{1}=t_{2}\). Thus \(t_{v}>0\) is unique for \(v\in \mathcal{Q}\). This completes the proof. □
Lemma 2.7
Assume that\((g_{1})\)–\((g_{3})\)hold, then\(\mathcal{Q}\neq \emptyset \)and
$$ \inf_{\mathcal{M}}\varPhi :=c=\inf_{v\in \varTheta }\max _{t>0}\varPhi (v_{t}). $$
Proof
This result is a consequence of Corollary 2.4, Lemma 2.5, and Lemma 2.6. The proof is completed. □
By a standard argument in [19], we can get the following Brezis–Lieb lemma.
Lemma 2.8
Assume that\((g_{1})\)–\((g_{4})\)hold. If\(v_{n}\rightharpoonup v_{0}\)in\(H^{1}({\mathbb{R}}^{N})\), then
$$ \varPhi (v_{n})=\varPhi (v_{0})+\varPhi (v_{n}-v_{0})+o_{n}(1) $$
and
$$ \mathcal{P}(v_{n})=\mathcal{P}(v_{0})+ \mathcal{P}(v_{n}-v_{0})+o_{n}(1). $$
Lemma 2.9
Assume that\((g_{1})\)–\((g_{4})\)hold. Then
-
(i)
there exists\(\rho >0\)such that\(\|\nabla v\|_{2}\geq \rho \)for any\(v\in \mathcal{Q}\);
-
(ii)
\(c=\inf_{\mathcal{Q}}\varPhi >0\).
Proof
(i) By \((g_{3})\), for any \(\varepsilon >0\), there exists \(C^{1}_{\varepsilon }>0\) such that
$$ \bigl\vert G\bigl(f(v)\bigr) \bigr\vert ^{\frac{2N}{N+\alpha }}\leq \varepsilon \vert v \vert ^{2} +C^{1}_{\varepsilon } \vert s \vert ^{2^{*}}\quad \text{and}\quad \bigl\vert G\bigl(f(v) \bigr) \bigr\vert ^{ \frac{2N}{N+\alpha }} \leq \varepsilon \vert v \vert ^{2}+\varepsilon \vert s \vert ^{2^{*}}+C^{1}_{\varepsilon } \vert s \vert ^{p}, $$
(2.3)
where \(p\in (2,2^{*})\). For any \(v\in \mathcal{Q}\), we have that \(\mathcal{P}(v)=0\). By the Sobolev embedding theorem, the Hardy–Littlewood–Sobolev inequality in [15], (2.3), and \((g_{1})\), we get
$$\begin{aligned} &\frac{(N-2)}{2} \int _{{\mathbb{R}}^{N}} \vert \nabla v \vert ^{2}+ \frac{N}{2} \int _{{\mathbb{R}}^{N}}f^{2}(v) \\ &\quad =\frac{N+\alpha }{2} \int _{{ \mathbb{R}}^{N}}\bigl(I_{\alpha }*G\bigl(f(v)\bigr)\bigr)G \bigl(f(v)\bigr) \\ &\quad \leq C \biggl(\varepsilon \int _{{\mathbb{R}}^{N}} \bigl\vert f(v) \bigr\vert ^{2}+C^{1}_{\varepsilon } \int _{{\mathbb{R}}^{N}} \vert v \vert ^{2^{*}} \biggr) \leq C \biggl( \varepsilon \int _{{\mathbb{R}}^{N}} \bigl\vert f(v) \bigr\vert ^{2}+C^{1}_{\varepsilon } \biggl( \int _{{\mathbb{R}}^{N}} \vert \nabla v \vert ^{2} \biggr)^{2^{*}/2} \biggr), \end{aligned}$$
which shows that there exists \(\rho >0\) such that \(\|\nabla v\|^{2}\geq \rho \) for any \(v\in \mathcal{Q}\).
(ii) For any \(v\in \mathcal{Q}\), from Lemma 2.2, we have
$$ \begin{aligned}\varPhi (v)&=\varPhi (v)-\frac{1}{N+\alpha } \mathcal{P}(v)\geq \frac{\alpha +2}{2(N+\alpha )} \Vert \nabla v \Vert ^{2}_{2}. \end{aligned} $$
(2.4)
This completes the proof. □
Lemma 2.10
Assume that\((g_{1})\)–\((g_{3})\)hold. Thencis achieved.
Proof
Let \(\{v_{n}\}\subset \mathcal{Q}\) be a minimizer for c, that is, \(\mathcal{P}(v_{n})=0\) and \(\varPhi (v_{n})\rightarrow c\) as \(n\rightarrow \infty \). By (2.4), one has
$$ c+o_{n}(1)=\varPhi (v_{n})-\frac{1}{N+\alpha } \mathcal{P}(v_{n}) = \frac{\alpha +2}{2(N+\alpha )} \Vert \nabla v_{n} \Vert ^{2}_{2}+ \frac{N}{2(N+\alpha )} \int _{{\mathbb{R}}^{N}}f^{2}(v_{n}), $$
which shows that \(\int _{{\mathbb{R}}^{N}}|\nabla v_{n}|^{2}+\int _{{\mathbb{R}}^{N}}f^{2}(v_{n})\) is bounded and thus \(\{v_{n}\}\) is bounded in \(D^{1,2}({\mathbb{R}}^{N})\). By the Sobolev inequality, Lemma 2.1-(5), it follows that
$$ \int _{ \vert v_{n} \vert \leq 1}v^{2}_{n}\leq \int _{{\mathbb{R}}^{N}}f^{2}(v_{n}) \quad \text{and} \quad \int _{ \vert v_{n} \vert >1}v^{2}_{n}\leq \int _{ \vert v_{n} \vert >1}v^{2^{*}}_{n} \leq C \biggl( \int _{{\mathbb{R}}^{N}} \vert \nabla v_{n} \vert ^{2} \biggr)^{2^{*}/2}. $$
Therefore
$$ \begin{aligned} \int _{{\mathbb{R}}^{N}}v^{2}_{n}= \int _{ \vert v_{n} \vert \leq 1}v^{2}_{n}+ \int _{ \vert v_{n} \vert >1}v^{2}_{n} \leq \int _{{\mathbb{R}}^{N}}f^{2}(v_{n}) +C \biggl( \int _{{\mathbb{R}}^{N}} \vert \nabla v_{n} \vert ^{2} \biggr)^{2^{*}/2}. \end{aligned} $$
(2.5)
From (2.5), we infer that there exists \(C>0\) such that \(\int _{{\mathbb{R}}^{N}}v_{n}^{2}\leq C\). Up to a subsequence, there exists \(v_{0}\in H^{1}({\mathbb{R}}^{N})\) such that \(v_{n}\rightharpoonup v_{0}\) in \(H^{1}({\mathbb{R}}^{N})\), \(v_{n}\rightarrow v_{0}\) in \(L^{r}_{\mathrm{loc}}({\mathbb{R}}^{N})\) for \(r\in [2,2^{*})\) and \(v_{n}\rightarrow v_{0}\) a.e. on \({\mathbb{R}}^{N}\).
Now, we claim that there exist \(\delta >0\) and \(\{y_{n}\}\subset {\mathbb{R}}^{N}\) such that \(\int _{B_{1}(y_{n})}|v_{n}|^{2}>\delta \). Assume by contradiction, by Lion’s concentration compactness lemma in [19], that \(v_{n}\rightarrow 0\) in \(L^{r}({\mathbb{R}}^{N})\) for \(2< r<2^{*}\). Moreover, by \(\mathcal{P}(v_{n})=0\), we know that
$$ \begin{aligned} 0&\leftarrow \int _{{\mathbb{R}}^{N}}(I_{\alpha }*G(f(v_{n})))G(f(v_{n}))\\ &= \frac{N-2}{N+\alpha } \int _{{\mathbb{R}}^{N}} \vert \nabla v_{n} \vert ^{2}+ \frac{N}{N+\alpha } \int _{{\mathbb{R}}^{N}}f^{2}(v_{n}) \geq \frac{N-2}{N+\alpha }\rho ^{2}>0, \end{aligned} $$
as \(n\rightarrow +\infty \). This is a contradiction. Thus there exist \(\delta >0\) and \(\{y_{n}\}\subset {\mathbb{R}}^{N}\) such that \(\int _{B_{1}(y_{n})}|v_{n}|^{2}>\delta \). Set \(\bar{v}_{n}(x)=v_{n}(x+y_{n})\). Then \(\|\bar{v}_{n}\|=\|v_{n}\|\). Thus, up to a subsequence, there exists \(\bar{v}_{0}\in H^{1}({\mathbb{R}}^{N})\backslash \{0\}\) such that \(\bar{v}_{n}\rightharpoonup \bar{v}_{0}\) in \(H^{1}({\mathbb{R}}^{N})\), \(\bar{v}_{n}\rightarrow \bar{v}_{0}\) in \(L^{r}_{\mathrm{loc}}({\mathbb{R}}^{N})\) for \(r\in [2,2^{*})\), and \(\bar{v}_{n}\rightarrow \bar{v}_{0}\) a.e. on \({\mathbb{R}}^{N}\). By translation invariance, one has
$$ \varPhi (\bar{v}_{n})\rightarrow c, \qquad \mathcal{P}( \bar{v}_{n}) \rightarrow 0, \quad \text{as } n\rightarrow +\infty $$
(2.6)
and \(\int _{B_{1}(0)}|\bar{v}_{n}|^{2}>\delta \). Set \(\bar{w}_{n}:=\bar{v}_{n}-\bar{v}_{0}\). Thus Lemma 2.7 yields that
$$ c=\varPhi (\bar{v}_{0})+\varPhi (\bar{w}_{n})+o_{n}(1) \quad \text{and} \quad 0= \mathcal{P}(\bar{v}_{0})+\mathcal{P}( \bar{w}_{n})+o_{n}(1). $$
(2.7)
If there exists a subsequence \(\{\bar{w}_{n_{i}}\}\) of \(\{\bar{w}_{n}\}\) such that \(\bar{w}_{n_{i}}=0\), then up to a subsequence, we have
$$ \varPhi (\bar{v}_{0})=c,\qquad \mathcal{P}( \bar{v}_{0})=0. $$
(2.8)
Next, we assume that \(\bar{w}_{n}\neq 0\). We claim that \(\mathcal{P}(\bar{v}_{0})\leq 0\). Otherwise, if \(\mathcal{P}(\bar{v}_{0})>0\), it follows from (2.7) that \(\mathcal{P}(\bar{w}_{n})<0\) for n large. By virtue of Lemma 2.6, there exists \(t_{n}>0\) such that \((\bar{w}_{n})_{t_{n}}\in \mathcal{Q}\). By (2.7) and Lemma 2.2, we get
$$ \begin{aligned} &c-\frac{N-2}{N+\alpha } \int _{{\mathbb{R}}^{N}} \vert \nabla \bar{v}_{0} \vert ^{2}- \frac{N}{N+\alpha } \int _{{\mathbb{R}}^{N}}f^{2}(\bar{v}_{0})+o_{n}(1) \\ &\quad = \varPhi (\bar{w}_{n})-\frac{1}{N+\alpha }\mathcal{P}( \bar{w}_{n})\geq \varPhi \bigl((\bar{w}_{n})_{t_{n}}\bigr)- \frac{t^{N+\alpha }_{n}}{N+\alpha } \mathcal{P}(\bar{w}_{n}) \geq c-\frac{t^{N+\alpha }_{n}}{N+\alpha } \mathcal{P}(\bar{w}_{n})\geq c, \end{aligned} $$
which is a contradiction due to \(\int _{{\mathbb{R}}^{N}}|\nabla \bar{v}_{0}|^{2}>0\). Thus \(\mathcal{P}(\bar{v}_{0})\leq 0\). Since \(\bar{v}_{0}\neq 0\), in view of Lemma 2.6, there exists \(t_{0}>0\) such that \((\bar{v}_{0})_{t_{0}}\in \mathcal{Q}\). By Lemma 2.3 and the weak semi-continuity of norm, we have
$$ \begin{aligned}c&=\lim_{n\rightarrow \infty } \biggl[\varPhi ( \bar{v}_{n}) - \frac{1}{N+\alpha }\mathcal{P}(\bar{v}_{n}) \biggr] \\ &=\frac{N-2}{N+\alpha }\lim_{n\rightarrow \infty } \int _{{\mathbb{R}}^{N}} \vert \nabla \bar{v}_{n} \vert ^{2} +\frac{N}{N+\alpha }\lim_{n\rightarrow \infty } \int _{{\mathbb{R}}^{N}}f^{2}(\bar{v}_{n}) \\ &\geq \frac{N-2}{N+\alpha } \int _{{\mathbb{R}}^{N}} \vert \nabla \bar{v}_{0} \vert ^{2}+ \frac{N}{N+\alpha } \int _{{\mathbb{R}}^{N}}f^{2}(\bar{v}_{0}) \\ &\geq \varPhi (\bar{v}_{0}) -\frac{t^{N+\alpha }_{0}}{N+\alpha } \mathcal{P}( \bar{v}_{0}) \\ &\geq \varPhi \bigl((\bar{v}_{0})_{t_{0}}\bigr) - \frac{t^{N+\alpha }_{0}}{N+\alpha } \mathcal{P}(\bar{v}_{0}) \\ &\geq c-\frac{t^{N+\alpha }_{0}}{N+\alpha } \mathcal{P}(\bar{v}_{0}) \geq c, \end{aligned} $$
which implies that (2.8) holds. The proof is completed. □
By a standard argument in [14, 18, 19], we can get the following lemma.
Lemma 2.11
Assume that\((g_{1})\)–\((g_{4})\)hold. If\(\tilde{v}\in \mathcal{Q}\)and\(\varPhi (\tilde{v})=c\), thenṽis a critical point ofΦ.
Proof of Theorem 1.1
By Lemma 2.7, Lemma 2.10, and Lemma 2.11, there exists \(v_{0}\in \mathcal{Q}\) such that
$$ \varPhi (v_{0})=c=\inf_{v\in \varTheta }\max_{t>0} \varPhi (v_{t}),\quad \varPhi '(v_{0})=0. $$
This completes the proof. □