Proof. (a) For each $u \in {X_\varepsilon^+}$ and t > 0, $\ell_u(0) = 0$. From theorem 4.3 of Lieb and Loss [Reference Lieb and Loss38], $(g_2)$, $(f_1)$–$(f_2)$, (2.1), and hypothesis $(V_1)$ we can deduce that there exists some constant C > 0 such that

\begin{align*}\ell_u(t) \geqslant \frac{1}{2^{q - 1}q} \|u\|_{X_\varepsilon}^q t^q - C \|u\|_{X_\varepsilon}^{2 p }t^{2 p}\ \text{for}\ 0 \lt t \lt \frac{1}{\|u\|_{X_\varepsilon}}.\end{align*}

Due to $2p \gt q$, we see that $\ell_u(t) \gt 0$ for t > 0 sufficiently small. Using $(g_3)_i$ and $(g_3)_{ii}$, we can find a constant

\begin{align*}C_u = \frac{1}{2} \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * G(\varepsilon x, u)\right)G(\varepsilon x, u) dx \gt 0\end{align*}

such that

\begin{align*}\ell_u(t) \leqslant \frac{1}{p} \left(\|u\|_{X_\varepsilon}^p + \|u\|_{X_\varepsilon}^q\right)t^q - C_u t^{2p}\end{align*}

for all t > 1. Applying $2p \gt q$ again, we know that $\ell_u(t) \lt 0$ for t > 1 large enough. Hence, $\max_{t \geqslant 0} \ell_u(t)$ is attained at some $t_u \gt 0$ verifying $\ell_u'(t_u)= 0$ and $t_u u \in \mathcal{N}_\varepsilon$. We point out that

(2.3)\begin{align} t u \in \mathcal{N}_\varepsilon \Longleftrightarrow \|u\|^q_{V_\varepsilon,q} = \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{G(\varepsilon y, t u(y))}{t^{\frac{q}{2}} |x - y|^\mu} \frac{g(\varepsilon x, t u (x))}{t^{\frac{q}{2} - 1}} u(x) dxdy - t^{p - q} \|u\|^p_{V_\varepsilon,p}. \end{align}

According to q > p and $(g_4)$, we conclude that the right-hand side in (2.3) is an increasing function with respect to t > 0. Therefore, the uniqueness of t_u is now obvious.

(b) For any $u \in S_\varepsilon^+$, in view of lemma 2.2-(a), there exists $t_u \gt 0$ such that

\begin{align*}t_u^p \|u\|_{V_\varepsilon,p}^p + t_u^q \|u\|_{V_\varepsilon,q}^q = \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * G(\varepsilon x, t_u u) \right) g(\varepsilon x, t_u u) t_u u dx.\end{align*}

By theorem 4.3 of Lieb and Loss [Reference Lieb and Loss38], together with $(g_2)$, $(f_1)$, $(f_2)$, (2.1), and hypothesis $(V_1)$, we can infer that for any σ > 0, there exists some constant $C_\sigma \gt 0$ such that for all $u \in S_\varepsilon^+$,

\begin{align*}t_u^p \|u\|_{V_\varepsilon,p}^p + t_u^q \|u\|_{V_\varepsilon,q}^q \leqslant \sigma t_u^{2 p} \|u\|_{V_\varepsilon,p}^{2 p} + C_\sigma t_u^{2 \nu} \|u\|_{V_\varepsilon,q}^{2 \nu}.\end{align*}

Assume that $t_u \leqslant 1$. Choosing $\sigma = \frac{1}{2}$, we have

\begin{align*} Ct_u^q & \leqslant \frac{1}{2} t_u^q \left(\|u\|_{V_\varepsilon,p}^q + \|u\|_{V_\varepsilon,q}^q\right) \ \text{for some constant } C \gt 0\\ & \leqslant \frac{1}{2} t_u^p \|u\|_{V_\varepsilon,p}^p + t_u^q \|u\|_{V_\varepsilon,q}^q\ (\text{since } q \gt p,\, t_u \leqslant 1 \ \text{and } 1 = \|u\|_{X_\varepsilon} \geqslant \|u\|_{V_\varepsilon,p})\\ & \leqslant C_{1 / 2} t_u^{2 \nu}\ (\text{since } 1 = \|u\|_{X_\varepsilon} \geqslant \|u\|_{V_\varepsilon,q}),\\ \Rightarrow\ \ & t_u \geqslant \tau \ \text{for some constant } \tau \gt 0\ (\text{since } \nu \gt q), \ \text{where } \tau \ \text{is independent of } u. \end{align*}

Assume that $t_u \gt 1$. Taking σ = 1 and applying $1 = \|u\|_{X_\varepsilon} \geqslant \|u\|_{V_\varepsilon,p}$, we get

\begin{align*} C t_u^p & \leqslant t_u^p \left(\|u\|_{V_\varepsilon, p}^q + \|u\|_{V_\varepsilon, q}^q\right) \ \text{for some constant } C \gt 0\\ & \leqslant t_u^p \|u\|_{V_\varepsilon,p}^p + t_u^q \|u\|_{V_\varepsilon,q}^q \ (\text{since } q \gt p \ \text{and } 1 = \|u\|_{X_\varepsilon} \geqslant \|u\|_{V_\varepsilon,p})\\ & \leqslant (1 + C_1) t_u^{2 \nu}\ (\text{since } q \gt p,\, t_u \gt 1 \ \text{and } 1 = \|u\|_{X_\varepsilon} \geqslant \|u\|_{V_\varepsilon,p},\ \|u\|_{V_\varepsilon,q}),\\ \Rightarrow\ \ & t_u \geqslant \tau \ \text{for some constant } \tau \gt 0 \ (\text{since } \nu \gt q), \ \text{where } \tau \ \text{is independent of } u. \end{align*}

So, there exists τ > 0 independent of u such that $t_u \geqslant \tau$ for all $u \in S_\varepsilon^+.$

Let $\mathcal{W} \subset S_\varepsilon^+$ be a compact set. Arguing by contradiction, we may assume that there exists a sequence $\{u_n\}_{n \in \mathbb{N}} \subset \mathcal{W}$ such that $1 \leqslant t_n :=t_{u_n} \rightarrow + \infty$ as $n \rightarrow \infty$. Since $\mathcal{W}$ is a compact set, there is $u \in \mathcal{W}$ such that $u_n \rightarrow u$ in X_ɛ as $n \rightarrow \infty$. As in the proof of lemma 2.2-(a), together with Fatou’s Lemma, we can see that

(2.4)\begin{align} J_\varepsilon(t_n u_n) \rightarrow - \infty \ \text{as } n \rightarrow \infty. \end{align}

In addition, for any fixed $\varphi \in \mathcal{N}_\varepsilon$, we have $\langle J_\varepsilon'(\varphi), \varphi \rangle = 0$. Combining $(g_3)_i$ with $(g_3)_{ii}$, we have

\begin{align*} J_\varepsilon(\varphi) & = J_\varepsilon(\varphi) - \frac{1}{2 p} \langle J_\varepsilon'(\varphi), \varphi \rangle \\ & \geqslant \left(\frac{1}{q} - \frac{1}{2p}\right) \left(\|\varphi\|_{V_\varepsilon,p}^p + \|\varphi\|_{V_\varepsilon,q}^q\right). \end{align*}

Taking $\varphi_n = t_n u_n \in \mathcal{N}_\varepsilon$ in the above inequality, we get

\begin{align*} J_\varepsilon(t_n u_n) & \geqslant \left(\frac{1}{q} - \frac{1}{2p}\right) \left(t_n^p \|u_n\|_{V_\varepsilon,p}^p + t_n^q \|u_n\|_{V_\varepsilon,q}^q\right)\\ & \geqslant \left(\frac{1}{q} - \frac{1}{2p}\right) \left(t_n^p \|u_n\|_{V_\varepsilon,p}^q + t_n^q \|u_n\|_{V_\varepsilon,q}^q\right)\\ & (\text{since } q \gt p,\ t_n \geqslant 1 \ \text{and } 1 = \|u_n\|_{X_\varepsilon} \geqslant \|u_n\|_{V_\varepsilon,p})\\ & \geqslant C t_n^p \ \text{for some constant } C \gt 0,\\ \Rightarrow\ \ & - \infty \geqslant + \infty \ (\text{see (2.4) and use the assumption } t_n \rightarrow + \infty \ \text{as } n \rightarrow \infty). \end{align*}

This is a contradiction.

(c) As in lemma 2.2-(a), we can define the maps $\hat{m}_\varepsilon : {X_\varepsilon^+} \mapsto \mathcal{N}_\varepsilon$ and $m_\varepsilon : S_\varepsilon^+ \mapsto \mathcal{N}_\varepsilon $ by

(2.5)\begin{align} \hat{m}_\varepsilon(u) = t_u u \quad \ \text{and } \quad m_\varepsilon = \hat{m}_\varepsilon |_{S_\varepsilon^+}. \end{align}

Firstly, we note that $\hat{m}_\varepsilon$, m_ɛ, and $m_\varepsilon^{- 1}$ are well-defined. Indeed, using lemma 2.2 (a), for any fixed $u \in {X_\varepsilon^+}$ it follows that there exists a unique $\hat{m}_\varepsilon(u) \in \mathcal{N}_\varepsilon$. In addition, if $u \in \mathcal{N}_\varepsilon$, and so $u \in {X_\varepsilon^+}$. Otherwise, we obtain $|{\mathrm{supp}\,}(u^+)| = 0$. The above equality, hypothesis $(V_1)$, the definition of g yield that u = 0. This is impossible since u ≠ 0. Thus, the inverse of m_ɛ is given by $m_\varepsilon^{- 1}(u) = \frac{u}{\|u\|_{X_\varepsilon}} \in S_\varepsilon^+$ for all $u \in \mathcal{N}_\varepsilon$. Consequently, $m_\varepsilon^{- 1}$ is well-defined and continuous. On the other hand, for all $u \in S_\varepsilon^+$, we can deduce that

\begin{align*}m_\varepsilon^{- 1}(m_\varepsilon(u)) = m_\varepsilon^{- 1}(t_u u) = \frac{t_u u}{\|t_u u\|_{X_\varepsilon}} = \frac{u}{\|u\|_{X_\varepsilon}} = u.\end{align*}

This yields that m_ɛ is bijection.

Next, we show that $\hat{m}_\varepsilon$ is a continuous map. To this end, let $\{u_n,u\}_{n \in \mathbb{N}} \subset {X_\varepsilon^+}$ such that $u_n \rightarrow u$ in X_ɛ as $n \rightarrow \infty$. On account of the fact that $\hat{m}_\varepsilon (t u) = \hat{m}_\varepsilon(u)$ for all t > 0, we can assume that $\|u_n\|_{X_\varepsilon} = \|u\|_{X_\varepsilon} = 1$ for all $n \in \mathbb{N}$. According to lemma 2.2 (b), we know that there exists $t_n :=t_{u_n} \rightarrow t_0 \gt 0$ as $n \rightarrow \infty$ such that $t_n u_n \in \mathcal{N}_\varepsilon$, then we have

\begin{align*}t_n^p \|u_n\|_{V_\varepsilon,p}^p + t_n^q \|u_n\|_{V_\varepsilon,q}^q = \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * G(\varepsilon x, t_n u_n)\right) g(\varepsilon x, t_nu_n) t_n u_n dx.\end{align*}

In the above relation, we pass to the limit as $n \rightarrow \infty$. Then,

\begin{align*}t_0^p \|u\|_{V_\varepsilon,p}^p + t_0^q \|u\|_{V_\varepsilon,q}^q = \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * G(\varepsilon x, t_0 u)\right) g(\varepsilon x, t_0u) t_0 u dx.\end{align*}

This implies that $t_0 u \in \mathcal{N}_\varepsilon$. From lemma 2.2 (a), we know that $t_u = t_0$. Hence, it follows that $\hat{m}_\varepsilon(u_n) \rightarrow \hat{m}_\varepsilon(u)$ in ${X_\varepsilon^+}$ as $n \rightarrow \infty$. Thus, $\hat{m}_\varepsilon$ and m_ɛ are continuous mappings.

(d) For ɛ > 0, $0 \lt t \lt 1$, and $u \in S_\varepsilon^+$, using theorem 4.3 of Lieb and Loss [Reference Lieb and Loss38], $(g_2)$, $(f_1)$–$(f_2)$, (2.1), and hypothesis $(V_1)$ again, we can conclude that there exists C > 0 such that

\begin{align*}J_\varepsilon(t u) \geqslant \frac{1}{2^{q - 1}q} t^{q} - C t^{2 p}.\end{align*}

Thus, we can find $\varrho_0 \gt 0$ such that $J_\varepsilon(t u) \geqslant \varrho_0$ for $0 \lt t \lt 1$ sufficiently small (since $2 p \gt q \gt 0$), where ϱ ₀ is independent of $\varepsilon,$ K, and a. In addition, by lemma 2.2 (a), (b), and (c), we observe (see the study by Szulkin and Weth [Reference Szulkin, Weth, Gao and Motreanu55]) that

\begin{align*}c_\varepsilon = \inf_{u \in \mathcal{N}_\varepsilon} J_\varepsilon(u) = \inf_{u \in {X_\varepsilon^+}} \max_{t \gt 0} J_\varepsilon(t u) = \inf_{u \in S_\varepsilon^+} \max_{t \gt 0} J_\varepsilon(t u).\end{align*}

So, $J_\varepsilon(u) |_{\mathcal{N}_\varepsilon} \geqslant\varrho_0$.

(e) Let $\left\{u_n\right\}_{n \in \mathbb{N}} \subset S_\varepsilon^+$ be a sequence such that ${\mathrm{dist}\,}(u_n,\partial S_\varepsilon^+) \rightarrow 0$ as $n \rightarrow \infty$. For any $\varphi \in \partial S_\varepsilon^+$ and $n \in \mathbb{N}$, then we obtain $|u_n^+| \leqslant |u_n - \varphi|$ a.e. in $\Lambda_\varepsilon$. Hence, from $(V_1)$ and Sobolev embedding, for any $r \in [p,q_s^*]$ and $n \in \mathbb{N}$, it follows that

\begin{align*} |u_n^+|_{L^r(\Lambda_\varepsilon)} & \leqslant \inf_{\varphi \in \partial S_\varepsilon^+} |u_n - \varphi|_{L^r(\Lambda_\varepsilon)} \leqslant C_r \inf_{\varphi \in \partial S_\varepsilon^+} \|u_n - \varphi\|_{X_\varepsilon}. \end{align*}

Taking into account $\|u_n\|_{X_\varepsilon} = 1$ and using hypothesis $(V_1)$, we have

\begin{align*} \|u_n\|_{W^{s,p}(\mathbb{R}^N)}^p + \|u_n\|_{W^{s,q}(\mathbb{R}^N)}^q & \leqslant \frac{1}{\min\left\{V_0,1\right\}}\left(\|u_n\|_{V_\varepsilon,p}^p + \|u_n\|_{V_\varepsilon,q}^q\right) \\ &\leqslant \frac{1}{\min\left\{V_0,1\right\}}\left(\|u_n\|_{X_\varepsilon}^p + \|u_n\|_{X_\varepsilon}^q\right) \\ &\leqslant \frac{2}{V_0} + 2. \end{align*}

Note that $0 \lt \mu \lt sp$ and $\nu \lt (N - \mu) q / (N - sq).$ Then, for all t > 0, we can deduce from lemma 2.6, $(V_1)$, $(g_3)_{ii}$, $(g_2)$, and $(f_1)$–$(f_2)$ that

\begin{align*} & \quad \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * G(\varepsilon x, tu_n) \right) G(\varepsilon x, tu_n) dx \\ & = \frac{K}{2}\int_{\Lambda_\varepsilon^c} G(\varepsilon x, tu_n) dx + \frac{K}{2} \int_{\Lambda_\varepsilon} G(\varepsilon x, tu_n) dx\\ & \leqslant \frac{V_0}{2p} \int_{\Lambda_\varepsilon^c} \left(t^p|u_n|^p + t^q|u_n|^q \right) dx+ \frac{K}{2} \int_{\Lambda_\varepsilon} F(t u_n^+) dx\\ & \leqslant \frac{t^p}{2p} \int_{\mathbb{R}^N} V(\varepsilon x)|u_n|^p dx+ \frac{t^q}{2p} \int_{\mathbb{R}^N} V(\varepsilon x)|u_n|^q dx + C_1 t^p \int_{\Lambda_\varepsilon} (u_n^+)^p dx\nonumber\\ & + C_2 t^\nu \int_{\Lambda_\varepsilon} (u_n^+)^\nu dx\\ & \leqslant \frac{t^p}{2p} \int_{\mathbb{R}^N} V(\varepsilon x)|u_n|^p dx+ \frac{t^q}{2p} \int_{\mathbb{R}^N} V(\varepsilon x)|u_n|^q dx + \hat{C}_p t^{p} {\mathrm{dist}\,}(u_n, \partial S_\varepsilon^+)^{p} \nonumber\\ & + \hat{C}_\nu t^\nu {\mathrm{dist}\,}(u_n, \partial S_\varepsilon^+)^\nu, \end{align*}

where C ₁, C ₂, $\hat{C}_p$, and $\hat{C}_\nu$ are some positive constants. So, we have

(2.6)\begin{align} &\int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * G(\varepsilon x, tu_n) \right) G(\varepsilon x, tu_n) dx\nonumber\\ \leqslant \ & \frac{t^p}{2p} \int_{\mathbb{R}^N} V(\varepsilon x)|u_n|^p dx+ \frac{t^q}{2p} \int_{\mathbb{R}^N} V(\varepsilon x)|u_n|^q dx + o_n(1), \end{align}

as $n \rightarrow \infty$. Moreover, for any t > 1, we infer that

(2.7)\begin{align} & \quad \frac{t^p}{p} \|u_n\|_{V_\varepsilon,p}^p + \frac{t^q}{q} \|u_n\|_{V_\varepsilon,q}^q - \frac{t^p}{2p} \int_{\mathbb{R}^N} V(\varepsilon x)|u_n|^p dx- \frac{t^q}{2p} \int_{\mathbb{R}^N} V(\varepsilon x)|u_n|^q dx\nonumber\\ & = \frac{t^p}{p} [u_n]_{s,p}^p + \frac{1}{2p} t^p \int_{\mathbb{R}^N} V(\varepsilon x) |u_n|^p dx + \frac{t^q}{q} [u_n]_{s,q}^q + \left(\frac{1}{q} - \frac{1}{2p}\right) t^q \int_{\mathbb{R}^N} V(\varepsilon x)|u_n|^q dx\nonumber\\ & \geqslant \frac{1}{2p} t^p \|u_n\|_{V_\varepsilon,p}^p + \left(\frac{1}{q} - \frac{1}{2p}\right) t^q \|u_n\|_{V_\varepsilon,q}^q\ \left(\text{since } 2p \gt q \right)\nonumber\\ & \geqslant \frac{1}{2p} t^p \|u_n\|_{V_\varepsilon,p}^q + \left(\frac{1}{q} - \frac{1}{2p}\right) t^q \|u_n\|_{V_\varepsilon,q}^q\ (\text{due to } q \gt p \ \text{and } 1 = \|u_n\|_{X_\varepsilon} \geqslant \|u_n\|_{V_\varepsilon,p}) \nonumber\\ & \geqslant \frac{1}{2^{q - 1}} \left(\frac{1}{q} - \frac{1}{2p}\right) t^p\ (\text{since } 2p \gt q \gt p \gt 1,\ t \gt 1). \end{align}

Recalling that the definition of m_ɛ and invoking relations (2.6) and (2.7), for all t > 1, we can deduce that

\begin{align*}\liminf_{n \rightarrow \infty} J_\varepsilon(m_\varepsilon(u_n)) \geqslant \liminf_{n \rightarrow \infty} J_\varepsilon(t u_n) \geqslant \frac{1}{2^{q - 1}} \left(\frac{1}{q} - \frac{1}{2p}\right) t^p.\end{align*}

The above inequality combined with the definition of J_ɛ and the arbitrariness of t > 1 means that

\begin{align*} \liminf_{n \rightarrow \infty} \left(\frac{1}{p}\|m_\varepsilon(u_n)\|_{V_\varepsilon,p}^p + \frac{1}{q} \|m_\varepsilon(u_n)\|_{V_\varepsilon,q}^q\right) \geqslant \liminf_{n \rightarrow \infty} J_\varepsilon(m_\varepsilon(u_n)) = + \infty, \end{align*}

and so $\|m_\varepsilon(u_n)\|_{X_\varepsilon} \rightarrow + \infty$ as $n \rightarrow \infty$. This proof is now complete.

Proof. In view of lemma 2.2, we only need to verify that J_ɛ possesses a mountain pass geometry, that is, the functional J_ɛ satisfies the following properties:

1. (i) there exist some constants $\rho_1 \gt 0$ and $\delta_1 \gt 0$ such that $J_\varepsilon(u) \geqslant \delta_1$ for $\|u\|_{X_\varepsilon} = \rho_1$;

2. (ii) there exists an element $ e \in {X_\varepsilon}$ with $\|e\|_{X_\varepsilon} \gt \rho_1$ such that $J_\varepsilon(e) \lt 0$.

(i) Arguing as in the proof of lemma 2.2 (a), we find some constants C > 0 and $C' \gt 0$ such that

\begin{align*}J_\varepsilon(u) \geqslant C\|u\|_{X_\varepsilon}^q - C' \|u\|_{X_\varepsilon}^{2 p}\end{align*}

for $\|u\|_{X_\varepsilon} = \rho_1 \in (0,1)$. So, using $2 p \gt q$, we obtain (i) if we take ρ ₁ small enough.

(ii) We choose a suitable function $\varphi \in C_0^\infty(\mathbb{R}^N)$ such that $\varphi \geqslant 0$, $\varphi \not\equiv 0$, and ${\mathrm{supp}}\, (\varphi) \subset \Lambda$. Then, for all ɛ sufficiently small, it is obvious that $G(\varepsilon x, \varphi) = F(\varphi)$ for all $x \in \mathbb{R}^N$. Hence, with arguments as in the proof of lemma 2.2 (a), there exist two constants

\begin{align*}C_\varphi = \frac{1}{2} \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * F(\varphi)\right)F(\varphi) dx \gt 0\end{align*}

and C > 0 such that

\begin{align*}J_\varepsilon(t \varphi) \leqslant Ct^q - C_\varphi t^{2p}\end{align*}

for all t > 1. Hence, (ii) holds true for $e = t \varphi$ and for some t > 1 sufficiently large.

According to the mountain pass theorem without the Palais–Smale condition (see the study by Brezis and Nirenberg [Reference Brezis and Nirenberg18]), we establish the existence of a Palais–Smale sequence $\{u_n\}_{n \in \mathbb{N}} \subset {X_\varepsilon}$ at the level c_ɛ. We recall that ${\mathrm{supp}}\, (\varphi) \subset \Lambda$, and then we can infer that there exists a constant ϑ > 0 such that $c_\varepsilon \lt \vartheta$ for all ɛ small enough, where ϑ is independent of ɛ, K, and a. This completes the proof of the lemma.

Proof. According to $(g_3)_{i,ii}$ and using hypothesis $(V_1)$, for any ɛ > 0 small enough, we deduce that

(2.8)\begin{align} \vartheta + o_n(1) \|u_n\|_{X_\varepsilon} & \geqslant c + o_n(1) \|u_n\|_{X_\varepsilon}\nonumber \\ & \geqslant J_\varepsilon(u_n) - \frac{1}{2 p} \langle J_\varepsilon'(u_n),u_n\rangle\nonumber\\ & \geqslant \left(\frac{1}{q} - \frac{1}{2 p}\right) \left( \|u_n\|_{V_\varepsilon,p}^p + \|u_n\|_{V_\varepsilon,q}^q\right)\nonumber\\ & + \frac{1}{2 p}\int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * G(\varepsilon x, u_n)\right) \left( g(\varepsilon x, u_n) u_n - p G(\varepsilon x, u_n) \right) dx\nonumber\\ & \geqslant \min\left\{V_0, 1\right\} \left(\frac{1}{q} - \frac{1}{2 p}\right) \left( \|u_n\|_{W^{s,p}(\mathbb{R}^N)}^p + \|u_n\|_{W^{s,q}(\mathbb{R}^N)}^q\right). \end{align}

So, for ɛ > 0 small enough, we deduce that the sequence $\{u_n\}_{n \in \mathbb{N}} \subset {X_\varepsilon}$ is bounded and

\begin{equation*}\Arrowvert u_n\Arrowvert_{W^{s,p}(\mathbb{R}^N)}^p+\Arrowvert u_n\Arrowvert_{W^{s,q}(\mathbb{R}^N)}^q\leqslant2\left(\frac1{V_0}+1\right)\frac{pq(\vartheta+1)}{2p-q}\end{equation*}

for all $n \in \mathbb{N}$ large enough.

Proof. Firstly, $(g_2)$ and $(f_1)$–$(f_2)$ imply that there exists a constant C > 0 (which is independent of ɛ) such that

(2.9)\begin{align} |G(\varepsilon x,t)| \leqslant |F(t)| \leqslant C(|t|^p + |t|^\nu) \ \text{for all}\ t \in \mathbb{R} \ \text{and for all}\ \varepsilon \gt 0. \end{align}

Therefore,

\begin{align*} \left|\frac{1}{|x|^\mu} * G(\varepsilon x, u)\right| & \leqslant \left|\frac{1}{|x|^\mu} * F(u)\right|\\ & \leqslant \int_{|x - y| \leqslant 1} \frac{|F(u(y))|}{|x - y|^\mu} dy + \int_{|x - y| \gt 1} \frac{|F(u(y))|}{|x - y|^\mu} dy\\ & \leqslant C \int_{|x -y| \leqslant 1} \frac{|u(y)|^p + |u(y)|^\nu}{|x - y|^\mu} dy \\ & \quad + C \int_{\mathbb{R}^N} \left(|u(y)|^p + |u(y)|^\nu\right) dy\ (\text{see (2.9)})\\ & \leqslant C \int_{|x -y| \leqslant 1} \frac{|u(y)|^p + |u(y)|^\nu}{|x - y|^\mu} dy \\ & \quad + C\ (\text{see (2.1) and use the definition of}\ \mathcal{B}) \end{align*}

for some constant C > 0 (which is independent of $x \in \mathbb{R}^N$ and ɛ > 0).

Let us choose

\begin{align*}t_1 \in \left(\frac{N}{N - \mu}, \frac{N}{N - sp}\right]\quad \ \text{and} \quad t_2 \in \left(\frac{N}{N - \mu},\frac{N q}{(N - sq) \nu}\right],\end{align*}

since $0 \lt \mu \lt sp$ and $\nu \lt (N - \mu) q / (N - sq).$ Then, combining the Hölder inequality, (2.1), and the definition of $\mathcal{B}$, we can easily see that

\begin{align*}\int_{|x -y| \leqslant 1} \frac{|u(y)|^p + |u(y)|^\nu}{|x - y|^\mu} dy \leqslant C \ \text{for all}\ x \in \mathbb{R}^N.\end{align*}

for some constant C > 0 (which is independent of $x \in \mathbb{R}^N$ and ɛ > 0). So, from the above information, we complete the proof of this lemma.

Proof. For all ɛ > 0 sufficiently small, let $\{u_n\}_{n \in \mathbb{N}} \subset {X_\varepsilon}$ be a Palais–Smale sequence of the functional J_ɛ at the level c. From lemma 2.5, we know that $\{u_n\}_{n \in \mathbb{N}} \subset {X_\varepsilon}$ is bounded for all ɛ > 0 small enough. So, passing to a subsequence, we may assume that there exists some $u \in {X_\varepsilon}$ such that $u_n \xrightarrow w u$ in X_ɛ, $u_n(x) \rightarrow u(x)$ a.e. in $\mathbb{R}^N$ and $u_n \rightarrow u$ in $L^r_{\mathrm{loc}}(\mathbb{R}^N)$ for all $r \in [1,q_s^*)$ as $n \rightarrow \infty$. We first assert that the following property is fulfilled.

Claim 1: The following properties hold up to a subsequence:

1. (a) $\lim_{n \rightarrow \infty}\int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * \left(G(\varepsilon x,u_n) -G(\varepsilon x,u)\right)\right) g(\varepsilon x,u) \varphi dx = 0$ for all $\varphi \in C^\infty_0(\mathbb{R}^N)$;

2. (b) $\lim_{n \rightarrow \infty}\int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * G(\varepsilon x,u_n)\right) \left(g(\varepsilon x,u_n) - g(\varepsilon x,u) \right) \varphi dx = 0$ for all $\varphi \in C^\infty_0(\mathbb{R}^N)$;

3. (c) we have

   \begin{align*} &\lim_{n \rightarrow \infty} \int_{\mathbb{R}^N}\left(\frac{1}{|x|^\mu} * G(\varepsilon x, u_n)\right) g(\varepsilon x,u_n) \varphi dx\\ =\ & \int_{\mathbb{R}^N}\left(\frac{1}{|x|^\mu} * G(\varepsilon x, u)\right) g(\varepsilon x,u) \varphi dx \ \text{for all}\ \varphi \in C^\infty_0(\mathbb{R}^N). \end{align*}

(a) Note that

\begin{align*} \int_{\mathbb{R}^N} |G(\varepsilon x,u_n)|^{\frac{2 N}{2 N - \mu}} dx & \leqslant \int_{\mathbb{R}^N} \left|C\left(|u_n|^p + |u_n|^{\nu} \right)\right|^{\frac{2 N}{2 N - \mu}} dx \ (\text{see (2.9)})\\ & \leqslant C \int_{\mathbb{R}^N} \left(|u_n|^{\frac{2 N p}{2 N - \mu}} + |u_n|^{\frac{2 N \nu}{2 N - \mu}}\right) dx\\ & \leqslant C \left(\|u_n\|_{X_\varepsilon}^{2 N p /(2 N - \mu)} + \|u_n\|_{X_\varepsilon}^{2 N \nu /(2 N - \mu)} \right)\ (\text{see (2.1)})\\ & \leqslant C \ (\text{due to the boundedness of the sequence}\ \{u_n\}_{n \in \mathbb{N}} \subset {X_\varepsilon}). \end{align*}

On the other hand, since $u_n(x) \rightarrow u(x)$ a.e. in $\mathbb{R}^N$ as $n \rightarrow \infty$, we obtain

\begin{align*}G(\varepsilon x, u_n) \rightarrow G(\varepsilon x, u) \ \text{a.e. in}\ \mathbb{R}^N \ \text{as}\ n \rightarrow \infty.\end{align*}

From proposition 5.4.7 in the study by Willem [Reference Willem58], it follows that

(2.10)\begin{align} G(\varepsilon \cdot,u_n) \xrightarrow w G(\varepsilon \cdot, u) \ \text{in}\ L^{\frac{2 N}{2N - \mu}}(\mathbb{R}^N) \ \text{as}\ n \rightarrow \infty. \end{align}

Let us define

\begin{align*}H(w) := \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * w(x)\right) g(\varepsilon x,u) \varphi dx \ \text{for all}\ w \in L^{\frac{2 N}{2 N - \mu}}(\mathbb{R}^N).\end{align*}

Then, it follows from theorem 4.3 of [Reference Lieb and Loss38], $(g_2)$, $(f_1)$–$(f_2)$, Hölder’s inequality, (2.1), and $u\in {X_\varepsilon}$ that H is linear bounded functional. Combining this with (2.10), we derive that (a) holds true.

(b) Using the boundedness of the sequence $\{u_n\}_{n \in \mathbb{N}} \subset {X_\varepsilon}$, together with theorem 4.3 in the study by Lieb and Loss [Reference Lieb and Loss38], $(g_2)$, $(f_1)$–$(f_2)$, and (2.1), we see that $G(\varepsilon \cdot,u_n)$ is also bounded in $L^{\frac{2 N}{2 N -\mu}}(\mathbb{R}^N)$. Then, applying the boundedness of the sequence $\{u_n\}_{n \in \mathbb{N}} \subset {X_\varepsilon}$ again and combining with the compact embeddings and Dominated Convergence Theorem, we deduce that (b) is also true.

(c) This result is a consequence of (a) and (b).

Next, we consider the following sequence:

\begin{align*}b_n(x,y) := \frac{|u_n(x) - u_n(y)|^{p - 2} (u_n(x) - u_n(y))}{|x - y|^{\frac{N + s p}{p'}}}\end{align*}

and we also introduce the following function

\begin{align*}b(x,y) := \frac{|u(x) - u(y)|^{p - 2} (u(x) - u(y))}{|x - y|^{\frac{N + s p}{p'}}},\end{align*}

where $p' = \frac{p}{p - 1}$. Then, $\{b_n\}_{n \in \mathbb{N}} \subset L^{p'}(\mathbb{R}^{2 N})$ is bounded and $b_n(x,y) \rightarrow b(x,y)$ a.e. in $\mathbb{R}^{2 N}$ as $n \rightarrow \infty$. Then, we deduce from proposition 5.4.7 in the study by Willem [Reference Willem58] that

(2.11)\begin{align} b_n \xrightarrow w b \ \text{in}\ L^{p'}(\mathbb{R}^{2 N}) \ \text{as } n \rightarrow \infty. \end{align}

On the other hand, for all $\varphi \in C^\infty_0(\mathbb{R}^N)$, we see that

\begin{align*}\frac{\varphi(x) - \varphi(y)}{|x - y|^{\frac{N + s p}{p}}} \in L^p(\mathbb{R}^{2 N}).\end{align*}

From this and (2.11), we have

\begin{align*} &\lim_{n \rightarrow \infty}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u_n(x) - u_n(y)|^{p - 2} (u_n(x) - u_n(y)) (\varphi(x) - \varphi(y))}{|x - y|^{N + s p}} dxdy\\ =\ & \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u(x) - u(y)|^{p - 2} (u(x) - u(y)) (\varphi(x) - \varphi(y))}{|x - y|^{N + s p}} dxdy. \end{align*}

In a similar fashion, we also have

\begin{align*} &\lim_{n \rightarrow \infty}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u_n(x) - u_n(y)|^{q - 2} (u_n(x) - u_n(y)) (\varphi(x) - \varphi(y))}{|x - y|^{N + s q}} dxdy\\ =\ & \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u(x) - u(y)|^{q - 2} (u(x) - u(y)) (\varphi(x) - \varphi(y))}{|x - y|^{N + s q}} dxdy. \end{align*}

Finally, it is easy to check that

\begin{align*} & \lim_{n \rightarrow \infty}\int_{\mathbb{R}^N} V(\varepsilon x,u_n) |u_n|^{p -2} u_n \varphi dx= \int_{\mathbb{R}^N} V(\varepsilon x,u) |u|^{p -2} u \varphi dx,\\ & \lim_{n \rightarrow \infty}\int_{\mathbb{R}^N} V(\varepsilon x,u_n) |u_n|^{q -2} u_n \varphi dx= \int_{\mathbb{R}^N} V(\varepsilon x,u) |u|^{q -2} u \varphi dx. \end{align*}

From the above information, together with the fact that $\langle J_\varepsilon'(u_n),\varphi \rangle \rightarrow 0$ as $n \rightarrow \infty$, it follows that $\langle J_\varepsilon'(u),\varphi \rangle = 0$ for all $\varphi \in C^\infty_0(\mathbb{R}^N)$. Using the density of $C^\infty_0(\mathbb{R}^N)$ in X_ɛ, we know that u is a critical point of the functional J_ɛ. Consequently, $\langle J_\varepsilon'(u),u\rangle = 0$.

In order to show that the Palais–Smale sequence satisfies the Palais–Smale condition, we need to establish the following property.

Claim 2: We have

\begin{align*} & \limsup_{n \rightarrow \infty} \int_{B_R^c} \left(\int_{\mathbb{R}^N} \left( \frac{|u_n(x) - u_n(y)|^p}{|x - y|^{N + s p}} + \frac{|u_n(x) - u_n(y)|^q}{|x - y|^{N + s q}} \right) dy + V(\varepsilon x) \left(|u_n|^p + |u_n|^q\right) \right) dx\\ &\rightarrow 0 \ \text{as } R \rightarrow + \infty. \end{align*}

We first observe that, for all ɛ > 0 small enough, there exists $ n_0 \in \mathbb{N}$ such that

\begin{align*}\sup_{n \geqslant n_0} \left|\frac{1}{|x|^\mu} * G(\varepsilon x, u_n)\right|_{L^\infty(\mathbb{R}^N)} \lt \frac{K}{2}.\end{align*}

For any R > 0, let $\eta_R \in C^\infty(\mathbb{R}^N)$ be such that $0 \leqslant \eta_R \leqslant 1$, $\eta_R = 0$ in $B_{\frac{R}{2}}$, $\eta_R = 1$ in $B_R^c$, and $|\nabla \eta_R| \leqslant C / R$ for some constant C > 0 (which is independent of R). Taking into account the boundedness of the sequence $\{\eta_R u_n\}_{n \in \mathbb{N}} \subset {X_\varepsilon}$, we see that $\langle J_\varepsilon'(u_n),\eta_R u_n \rangle \rightarrow 0$ as $n \rightarrow \infty$. It follows that

\begin{align*} & \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u_n(x) - u_n(y)|^p \eta_R(x)}{|x - y|^{N + s p}} dxdy + \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u_n(x) - u_n(y)|^q \eta_R(x)}{|x - y|^{N + s q}} dxdy\\ +\ & \int_{\mathbb{R}^N} V(\varepsilon x) \left(|u_n|^p + |u_n|^q\right) \eta_R dx\\ = \ & o_n(1) - \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u_n(x) - u_n(y)|^{p - 2} (u_n(x) - u_n(y))(\eta_R(x) - \eta_R(y)) u_n(y)}{|x - y| ^{N + s p}} dxdy\\ - \ & \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u_n(x) - u_n(y)|^{q - 2} (u_n(x) - u_n(y))(\eta_R(x) - \eta_R(y)) u_n(y)}{|x - y| ^{N + s q}} dxdy\\ + \ & \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * G(\varepsilon x,u_n)\right) g(\varepsilon x,u_n) u_n \eta_R dx. \end{align*}

Fix ɛ > 0 small enough. Let R > 0 sufficiently large such that $\Lambda_\varepsilon \subset B_{\frac{R}{2}}$. Using the definitions of η_R and K, together with $(g_3)_{ii}$, for $n \geqslant n_0$ large enough, we have

(2.12)\begin{align} & \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u_n(x) - u_n(y)|^p \eta_R(x)}{|x - y|^{N + s p}} dxdy + \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u_n(x) - u_n(y)|^q \eta_R(x)}{|x - y|^{N + s q}} dxdy\nonumber\\ + \ & \frac{1}{2} \int_{\mathbb{R}^N} V(\varepsilon x) \left(|u_n|^p + |u_n|^q\right) \eta_R dx\nonumber\\ \leqslant\ & o_n(1) - \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u_n(x) - u_n(y)|^{p - 2} (u_n(x) - u_n(y))(\eta_R(x) - \eta_R(y)) u_n(y)}{|x - y| ^{N + s p}} dxdy\nonumber\\ - \ & \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u_n(x) - u_n(y)|^{q - 2} (u_n(x) - u_n(y))(\eta_R(x) - \eta_R(y)) u_n(y)}{|x - y| ^{N + s q}} dxdy. \end{align}

From the Hölder inequality and the boundedness of the sequence $\{u_n\}_{n \in \mathbb{N}} \subset {X_\varepsilon}$, it follows that

(2.13)\begin{align} & \left| \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u_n(x) - u_n(y)|^{p - 2} (u_n(x) - u_n(y))(\eta_R(x) - \eta_R(y)) u_n(y)}{|x - y| ^{N + s p}} dxdy \right| \nonumber\\ \leqslant \ & C \left(\int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|\eta_R(x) - \eta_R(y)|^p |u_n(x)|^p}{|x - y|^{N + sp}} dxdy\right)^{\frac{1}{p}}\ (\text{for some constant } C \gt 0). \end{align}

In addition, by the definition of η_R, polar coordinates, and the boundedness of the sequence $\{u_n\}_{n \in \mathbb{N}} \subset {X_\varepsilon}$, we have

(2.14)\begin{align} & \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|\eta_R(x) - \eta_R(y)|^p |u_n(x)|^p}{|x - y|^{N + sp}} dxdy\nonumber\\ \leqslant \ & \int_{\mathbb{R}^N} \int_{|x - y| \gt R} \frac{|\eta_R(x) - \eta_R(y)|^p |u_n(x)|^p}{|x - y|^{N + sp}} dxdy \nonumber\\ &+ \int_{\mathbb{R}^N} \int_{|x - y| \leqslant R} \frac{|\eta_R(x) - \eta_R(y)|^p |u_n(x)|^p}{|x - y|^{N + sp}} dxdy\nonumber\\ \leqslant \ & C \int_{\mathbb{R}^N} \int_{|z| \gt R} \frac{|u_n(x)|^p}{|z|^{N + s p}} dxdz + \frac{C}{R^p} \int_{\mathbb{R}^N} \int_{|z| \leqslant R} \frac{|u_n(x)|^p}{|z|^{N + s p - p}} dxdz\nonumber\\ \leqslant \ & \frac{C}{R^{s p}} \int_{\mathbb{R}^N} |u_n|^p dx + \frac{C}{R^p} R^{- s p + p} \int_{\mathbb{R}^N} |u_n|^p dx\nonumber\\ \leqslant \ & \frac{C}{R^{s p}}. \end{align}

Using (2.13) and (2.14), we see that

(2.15)\begin{align} \left| \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u_n(x) - u_n(y)|^{p - 2} (u_n(x) - u_n(y))(\eta_R(x) - \eta_R(y)) u_n(y)}{|x - y| ^{N + s p}} dxdy \right| \leqslant \frac{C}{R^s}. \end{align}

Also, we have

(2.16)\begin{align} \left| \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{|u_n(x) - u_n(y)|^{q - 2} (u_n(x) - u_n(y))(\eta_R(x) - \eta_R(y)) u_n(y)}{|x - y| ^{N + s q}} dxdy \right| \leqslant \frac{C}{R^s}. \end{align}

So, we deduce from (2.12), (2.15), and (2.16) that Claim 2 holds true.

Using Claim 2 and applying the locally compact embedding $X_\varepsilon \hookrightarrow L^p_{\mathrm{loc}}(\mathbb{R}^N)$, we can derive that $u_n \rightarrow u$ in $L^p(\mathbb{R}^N)$ as $n \rightarrow \infty$. In addition, we deduce from the interpolation inequality that $u_n \rightarrow u$ in $L^r(\mathbb{R}^N)$ as $n \rightarrow \infty$ for all $r \in [p,q_s^*)$. Then, from theorem 4.3 in the study by Lieb–Loss [Reference Lieb and Loss38], $(g_2)$, $(f_1)$–$(f_2)$, the Dominated Convergence Theorem, and the boundedness of the sequence $\{u_n\}_{n \in \mathbb{N}} \subset {X_\varepsilon}$, it follows that

(2.17)\begin{align} \lim_{n \rightarrow \infty}\int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * G(\varepsilon x,u_n)\right) g(\varepsilon x,u_n) u_n dx = \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * G(\varepsilon x,u)\right) g(\varepsilon x,u) u dx. \end{align}

Also, we have

\begin{align*} & \langle J_\varepsilon'(u_n),u_n\rangle = o_n(1) \ \text{as } n \rightarrow \infty,\\ \Rightarrow\ \ & \|u_n\|_{V_\varepsilon,p}^p + \|u_n\|_{V_\varepsilon,q}^q = \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * G(\varepsilon x,u_n)\right) g(\varepsilon x,u_n) u_n dx + o_n(1) \ \text{as } n \rightarrow \infty \end{align*}

and

\begin{align*} & \langle J_\varepsilon'(u),u\rangle = 0,\\ \Rightarrow\ \ & \|u\|_{V_\varepsilon,p}^p + \|u\|_{V_\varepsilon,q}^q = \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * G(\varepsilon x,u)\right) g(\varepsilon x,u) u dx. \end{align*}

Hence, the above fact and (2.17) imply that

\begin{align*}\|u_n\|_{V_\varepsilon,p}^p + \|u_n\|_{V_\varepsilon,q}^q = \|u\|_{V_\varepsilon,p}^p + \|u\|_{V_\varepsilon,q}^q + o_n(1) \ \text{as } n \rightarrow \infty.\end{align*}

Furthermore, according to lemma 2.10 (a), (b), we see that

\begin{align*}\|u_n - u\|_{V_\varepsilon,p}^p = \|u_n\|_{V_\varepsilon,p}^p - \|u\|_{V_\varepsilon,p}^p + o_n(1)\end{align*}

and

\begin{align*}\|u_n - u\|_{V_\varepsilon,q}^q = \|u_n\|_{V_\varepsilon,q}^q - \|u\|_{V_\varepsilon,q}^q + o_n(1),\end{align*}

as $n \rightarrow \infty$. Thus, we conclude that

\begin{align*}\|u_n - u\|_{V_\varepsilon,p}^p + \|u_n - u\|_{V_\varepsilon,q}^q = o_n(1).\end{align*}

This fact means that $\|u_n - u\|_{X_\varepsilon} \rightarrow 0$ as $n \rightarrow \infty$. Now, we complete the proof of the lemma.

Proof. Using lemmas 2.4 and 2.7, we can employ the Mountain Pass Theorem to infer that for all ɛ > 0 sufficiently small, there exists $u_\varepsilon \in {X_\varepsilon}$ such that $J_\varepsilon'(u_\varepsilon) = 0$ and $J_\varepsilon(u_\varepsilon) = c_\varepsilon$. Furthermore, choosing $u_\varepsilon^- := \min\{u_\varepsilon,0\}$ and recalling that $g(\varepsilon \cdot,t) = 0$ for all $t \leqslant 0$ and $\langle J_\varepsilon'(u_\varepsilon),u_\varepsilon^- \rangle = 0$, we can infer that

\begin{align*} & \|u_\varepsilon^-\|_{V_\varepsilon,p}^p + \|u_\varepsilon^-\|_{V_\varepsilon,q}^q \leqslant 0,\\ \Rightarrow\ \ & u_\varepsilon^- = 0,\\ \Rightarrow\ \ & u_\varepsilon \geqslant 0 \ \text{and } u_\varepsilon \not\equiv 0. \end{align*}

This ends the proof of the lemma.

Proof. Let $\{u_n\}_{n \in \mathbb{N}} \subset S_\varepsilon^+$ be a Palais–Samle sequence for the functional ψ_ɛ at the level c, that is,

\begin{align*}\psi_\varepsilon(u_n) \rightarrow c \ \text{in } \mathbb{R} \quad \ \text{and } \quad \psi_\varepsilon'(u_n) \rightarrow 0 \ \text{in }T_{u_n} (S_\varepsilon^+)^* \ \text{as } n \rightarrow \infty.\end{align*}

Using lemma 2.3 (c), we see that $\left\{m_\varepsilon(u_n)\right\}_{n \in \mathbb{N}} \subset {X_\varepsilon}$ is also a Palais–Samle sequence for the functional J_ɛ at the level c. Therefore, we can deduce from lemma 2.7 that the functional J_ɛ verifies the Palais–Smale condition. Hence, passing to a subsequence, we can find some $u \in S_\varepsilon^+$ such that $m_\varepsilon(u_n) \rightarrow m_\varepsilon(u)$ in X_ɛ as $n \rightarrow \infty$. Combining this fact with lemma 2.2 (c), we conclude that $u_n \rightarrow u$ in $S_\varepsilon^+$ as $n \rightarrow \infty$.

Proof. We first point out that the proofs of (a) and (b) follow from the Brezis–Lieb Lemma (see [Reference Brezis and Lieb17]). Moreover, arguing as in the proof lemma 3.2 of Mercuri and Willem [Reference Mercuri and Willem43], we see that (c) holds true.

Next, we give the details of the proofs of (d), (e), and (f) for the convenience of readers.

(d) Case 1: $2 \leqslant t \in \left\{p,q\right\}$.

We deduce from the Mean Value Theorem, the Young inequality, and $t \geqslant 2$ that for any fixed σ > 0, there exists $C_\sigma \gt 0$ such that

(2.18)\begin{align} \left||a + b|^{t - 2}(a + b) - \left|a\right|^{t - 2} a\right| \leqslant \sigma |a|^{t - 1} + C_\sigma |b|^{t - 1} \quad \ \text{for all } a,\ b \in \mathbb{R}. \end{align}

In (2.18), we take

\begin{align*}a = \frac{w_n(x) - w_n(y)}{|x - y|^{\frac{N + s t}{t}}} \quad \ \text{and }\quad b = \frac{u(x) - u(y)}{|x - y|^{\frac{N + s t}{t}}},\end{align*}

and we have

\begin{align*} \Bigg| \frac{\mathcal{A}_t( u_n(x) - u_n(y))}{|x - y|^{\frac{N + s t}{t / (t - 1)}}} - \frac{\mathcal{A}_t(w_n(x) - w_n(y))}{|x - y|^{\frac{N + s t}{t / (t - 1)}}}\Bigg| & \leqslant\sigma \frac{|w_n(x) - w_n(y)|^{t - 1}}{|x - y|^{\frac{N + s t}{t / (t - 1)}}} \\ & \quad + C_\sigma \frac{|u(x) - u(y)|^{t - 1}}{|x - y|^{\frac{N + s t}{t / (t - 1)}}} . \end{align*}

Now, we introduce the mapping $H_{\sigma,n}^1 : \mathbb{R}^{2 N} \mapsto \mathbb{R}_+$ given by

\begin{align*} &H_{\sigma,n}^1(x,y) := \max\Bigg\{\left|\frac{\mathcal{A}_t(u_n(x) - u_n(y))}{|x - y|^{\frac{N + st}{t / (t - 1)}}} - \frac{\mathcal{A}_t(w_n(x) - w_n(y))}{|x - y|^{\frac{N + st}{t / (t - 1)}}} - \frac{\mathcal{A}_t(u(x) - u(y))}{|x - y|^{\frac{N + st}{t / (t - 1)}}}\right|\\ &- \sigma \frac{|w_n(x) - w_n(y)|^{t - 1}}{|x - y|^{\frac{N + s t}{t / (t - 1)}}},0 \Bigg\}. \end{align*}

Then, $H_{\sigma,n}^1(x,y) \rightarrow 0$ a.e. in $\mathbb{R}^{2 N}$ as $n \rightarrow \infty$ and

\begin{align*}0 \leqslant H_{\sigma,n}^1(x,y) \leqslant C \frac{|u(x) - u(y)|^{t - 1}}{|x - y|^{\frac{N + s t}{t / (t - 1)}}} \in L^{\frac{t}{t - 1}}(\mathbb{R}^{2 N})\end{align*}

for some constant C > 0. Hence, from the Dominated Convergence Theorem, we obtain

\begin{align*}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \left|H_{\sigma,n}^1\right|^{\frac{t}{t - 1}} dxdy \rightarrow 0 \ \text{in } \mathbb{R} \ \text{as } n \rightarrow \infty.\end{align*}

In addition, according to the definition of $H_{\sigma,n}^1$, we have

\begin{align*} & \left|\frac{\mathcal{A}_t(u_n(x) - u_n(y))}{|x - y|^{\frac{N + st}{(t / (t - 1))}}} - \frac{\mathcal{A}_t(w_n(x) - w_n(y))}{|x - y|^{\frac{N + st}{(t / (t - 1))}}} - \frac{\mathcal{A}_t(u(x) - u(y))}{|x - y|^{\frac{N + st}{(t / (t - 1))}}}\right|^\frac{t}{t - 1} \\ \leqslant\ & C \left(\sigma^\frac{t}{t - 1} \frac{|w_n(x) - w_n(y)|^t}{|x - y|^{N + s t}} + (H_{\sigma,n}^1(x,y))^\frac{t}{t - 1}\right) \end{align*}

for some constant C > 0. This implies that

\begin{align*} & \quad \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \left|\frac{\mathcal{A}_t(u_n(x) - u_n(y))}{|x - y|^{\frac{N + st}{t / (t - 1)}}} - \frac{\mathcal{A}_t(w_n(x) - w_n(y))}{|x - y|^{\frac{N + st}{t / (t - 1)}}} - \frac{\mathcal{A}_t(u(x) - u(y))}{|x - y|^{\frac{N + st}{t / (t - 1)}}} \right|^{\frac{t}{t - 1}}dxdy\\ & = o_n(1)\end{align*}

holds true for $n \in \mathbb{N}$ large enough.

Case 2: $1 \lt t \lt 2$.

Invoking lemma 3.1 in the study by Mercuri and Willem [Reference Mercuri and Willem43] and applying the Dominated Convergence Theorem, for $n \in \mathbb{N}$ large enough, we obtain that

\begin{align*} & \quad \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \left|\frac{\mathcal{A}_t(u_n(x) - u_n(y))}{|x - y|^{\frac{N + st}{t / (t - 1)}}} - \frac{\mathcal{A}_t(w_n(x) - w_n(y))}{|x - y|^{\frac{N + st}{t / (t - 1)}}} - \frac{\mathcal{A}_t(u(x) - u(y))}{|x - y|^{\frac{N + st}{t / (t - 1)}}} \right|^{\frac{t}{t - 1}}dxdy \\ & = o_n(1)\end{align*}

holds true.

Now, combining Case 1 and Case 2, we complete the proof of (d).

(e) Using the Mean Value Theorem, hypotheses $(f_1)$–$(f_2)$, and Young’s inequality, we can infer that for any σ > 0, there exists $C_\sigma \gt 0$ such that

\begin{align*}\left|F(u_n) - F(w_n) -F(u)\right|^{\frac{2 N }{2 N - \mu}} & \leqslant \sigma \left(|u_n|^{\frac{2 N p }{2 N - \mu}} + |u_n|^{\frac{2 N \nu}{2 N - \mu}} \right) \\ & \quad + C_\sigma\left(|u|^{\frac{2 N p }{2 N - \mu}} + |u|^{\frac{2 N \nu}{2 N - \mu}}\right).\end{align*}

Define the following mapping:

\begin{align*}H_{\sigma,n}^2 := \max\left\{\left|F(u_n) - F(w_n) -F(u)\right|^{\frac{2 N }{2 N - \mu}} - \sigma \left(|u_n|^{\frac{2 N p }{2 N - \mu}} + |u_n|^{\frac{2 N \nu}{2 N - \mu}} \right),0\right\}.\end{align*}

Then, arguing as in the proof of lemma 2.10 (d) and using the boundedness of the sequence $\left\{u_n\right\}_{n \in \mathbb{N}} \subset {X_\varepsilon}$, for $n \in \mathbb{N}$ sufficiently large, we can derive that

(2.19)\begin{align} \int_{\mathbb{R}^N}\left|F(u_n) - F(w_n) -F(u)\right|^{\frac{2 N }{2 N - \mu}} dx = o_n(1). \end{align}

On the other hand, we can deduce from theorem 4.3 in the study by Lieb and Loss [Reference Lieb and Loss38] that

(2.20)\begin{align} \int_{\mathbb{R}^N}\left|\frac{1}{|x|^\mu} *\left( F(u_n) - F(w_n) - F(u)\right)\right|^{\frac{2 N }{\mu}} dx = o_n(1) \end{align}

for $n \in \mathbb{N}$ large enough.

We note that

\begin{align*} &\int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * F(u_n)\right) F(u_n) dx - \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * F(w_n)\right) F(w_n) dx\\ =\ & \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * \left(F(u_n) - F(w_n)\right)\right) \left(F(u_n) - F(w_n)\right) dx \\ + \ & 2 \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * \left(F(u_n) - F(w_n)\right)\right) F(w_n) dx. \end{align*}

Using the above equality and taking into account that $F(w_n) \xrightarrow w 0$ in $L^{\frac{2 N }{2 N - \mu}}(\mathbb{R}^N)$ as $n \rightarrow \infty$, together with (2.19)–(2.20), we can show that (e) holds true.

(f) We first prove that

(2.21)\begin{align} \sup_{\|\varphi\|_{X_\varepsilon} \leqslant 1} \int_{\mathbb{R}^N} \left|\left(f(u_n) - f(w_n) - f(u)\right) \varphi\right|^{\frac{2 N }{2 N - \mu}} dx = o_n(1) \end{align}

for $n \in \mathbb{N}$ large enough.

For any fixed $0 \lt \sigma \lt 1$, we deduce from hypothesis $(f_1)$ that there exists $0 \lt \lambda_0 := \lambda_0(\sigma) \lt 1$ such that

(2.22)\begin{align} f(t) \leqslant \sigma |t|^{p - 1} \quad \ \text{for all } |t| \leqslant 2 \lambda_0. \end{align}

In addition, by hypothesis $(f_2)$, we can find $\lambda_1 := \lambda_1(\sigma) \gt 2$ such that

(2.23)\begin{align} |f(t)| \leqslant \sigma |t|^{\nu - 1} \quad \ \text{for all } |t| \geqslant \lambda_1 - 1. \end{align}

Next, by the continuity of f, there is a positive constant $\gamma :=\gamma(\sigma) \lt \lambda_0$ such that

(2.24)\begin{align} |f(t_1) - f(t_2)| \leqslant \lambda_0^{p - 1} \sigma \quad \ \text{for all } |t_1 - t_2| \leqslant \gamma,\ |t_1|,\, |t_2| \leqslant \lambda_1 + 1. \end{align}

Additionally, combining $(f_1)$ and $(f_2)$, we can conclude that there exists $C(\sigma) \gt 0$ such that

(2.25)\begin{align} f(t) \leqslant C(\sigma) |t|^{p - 1} + \sigma |t|^{\nu - 1} \quad\ \text{for all } t \in \mathbb{R}. \end{align}

We now estimate the following term:

\begin{align*}\int_{\mathbb{R}^N \setminus B_R}\left|(f(u_n) - f(w_n) - f(u))\varphi\right|^{\frac{2 N}{2 N - \mu}} dx.\end{align*}

Since $u \in {X_\varepsilon}$, together with (2.25), the Hölder inequality (2.1), and $(V_1)$, we know that there exists $R = R(\sigma) \gt 0$ such that

(2.26)\begin{align} & \quad \int_{\mathbb{R}^N \setminus B_R} \left|f(u) \varphi\right|^{\frac{2 N}{2 N - \mu}} dx \nonumber\\ & \leqslant C \left(\int_{\mathbb{R}^N \setminus B_R} |u|^{\frac{2 N (p - 1)}{2 N - \mu}} |\varphi|^{\frac{2 N}{2 N - \mu}} dx + \int_{\mathbb{R}^N \setminus B_R} |u|^{\frac{2 N (\nu - 1)}{2 N - \mu}} |\varphi|^{\frac{2 N}{2 N - \mu}} dx \right)\nonumber\\ &\leqslant C \left(\left(\int_{\mathbb{R}^N \setminus B_R} |u|^{\frac{2 Np}{2 N - \mu}} dx\right)^{\frac{p - 1}{p}} + \left(\int_{\mathbb{R}^N \setminus B_R} |u|^{\frac{2 N \nu}{2 N - \mu}} \right)^{\frac{\nu - 1}{\nu}} \right)\|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}}\nonumber\\ & \leqslant C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}} \end{align}

for some constant C > 0.

Let us define

\begin{align*}D_{u_n}^1 := \left\{x \in \mathbb{R}^N \setminus B_R : |u_n(x)| \leqslant \lambda_0\right\}.\end{align*}

According to (2.22), the Hölder inequality, (2.1), $(V_1)$, and the boundedness of the sequence $\left\{u_n\right\}_{n \in \mathbb{N}} \subset {X_\varepsilon}$, we have

(2.27)\begin{align} &\int_{D_{u_n}^1 \cap |u| \leqslant \gamma} |(f(u_n) - f(w_n))\varphi|^{\frac{2 N}{2 N- \mu}} dx \nonumber\\ \leqslant\ & \int_{D_{u_n}^1 \cap |u| \leqslant \gamma} \left( \sigma |u_n|^{p - 1} + \sigma |w_n|^{p - 1}\right)^\frac{2 N}{2 N - \mu}|\varphi|^{\frac{2 N}{2 N - \mu}} dx\nonumber\\ \leqslant \ & C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}} \end{align}

for some constant C > 0.

Set

\begin{align*}D_{u_n}^2 := \left\{x \in \mathbb{R}^N \setminus B_R : |u_n(x)| \geqslant \lambda_1\right\}.\end{align*}

Arguing as in (2.27), together with (2.23), we can obtain

(2.28)\begin{align} &\int_{D_{u_n}^2 \cap |u| \leqslant \gamma} |(f(u_n) - f(w_n))\varphi|^{\frac{2 N}{2 N- \mu}} dx \leqslant C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}}. \end{align}

for some positive constant C > 0.

Now, we introduce the following set:

\begin{align*}D_{u_n}^3 := \left\{x \in \mathbb{R}^N \setminus B_R : \lambda_0 \leqslant |u_n(x)| \leqslant \lambda_1\right\}.\end{align*}

It is easy to check that $|D_{u_n}^3| \lt + \infty$, since $u_n \in {X_\varepsilon}$. So, from (2.24), it follows that

(2.29)\begin{align} &\int_{D_{u_n}^3 \cap |u| \leqslant \gamma} |(f(u_n) - f(w_n))\varphi|^{\frac{2 N}{2 N- \mu}} dx \leqslant C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}} \end{align}

for some constant C > 0.

Thus, we deduce from (2.27)–(2.29) that

(2.30)\begin{align} \int_{(\mathbb{R}^N \setminus B_R) \cap |u| \leqslant \gamma}\left|(f(u_n) - f(w_n))\varphi\right|^{\frac{2 N}{2 N - \mu}} dx \leqslant C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}} \quad \ \text{for all } n \in \mathbb{N}. \end{align}

Thanks to $u \in {X_\varepsilon}$, we have $|(\mathbb{R}^N \setminus B_R) \cap \{|u| \geqslant \gamma\}| \rightarrow 0$ as $R \rightarrow \infty$. Using (2.25) again, we deduce that there exists $R := R(\sigma) \gt 0$ such that

\begin{align*} &\int_{(\mathbb{R}^N \setminus B_R) \cap |u| \geqslant \gamma} |(f(u_n) - f(w_n)) \varphi|^{\frac{2 N}{2 N - \mu}} dx\\ \leqslant \ & \int_{(\mathbb{R}^N \setminus B_R) \cap |u| \geqslant \gamma} \left(\sigma \left(|u_n|^{\nu - 1} + |w_n|^{\nu - 1}\right)|\varphi| + C(\sigma)\left(|u_n|^{p - 1} + |w_n|^{p - 1}\right)|\varphi|\right)^{\frac{2 N}{2 N - \mu}} dx\\ \leqslant \ & C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}} + C \int_{(\mathbb{R}^N \setminus B_R) \cap |u| \geqslant \gamma} \left(C(\sigma)\left(|u_n|^{p - 1} + |w_n|^{p - 1}\right)|\varphi|\right)^{\frac{2 N}{2 N - \mu}} dx\\ \leqslant \ & C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}} + (2C(\sigma))^{\frac{2 N}{2 N - \mu}} \int_{(\mathbb{R}^N \setminus B_R) \cap |u| \geqslant \gamma} \left(|u_n|^{\frac{2 N (p - 1)}{2 N - \mu}} + |w_n|^{\frac{2 N(p - 1)}{2 N - \mu}}\right)|\varphi|^{\frac{2 N}{2 N - \mu}} dx\\ \leqslant \ & C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}} + C(C (\sigma))^{\frac{2 N}{2 N - \mu}} \left(\int_{(\mathbb{R}^N \setminus B_R) \cap |u| \geqslant \gamma} |\varphi|^{\frac{2 N p}{2 N - \mu}} dx\right)^{\frac{1}{p}}\\ \leqslant \ & C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}} + C(C (\sigma))^{\frac{2 N}{2 N - \mu}} \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}} |(\mathbb{R}^N \setminus B_R) \cap \{|u| \geqslant \gamma\}|^{\frac{2 s p - \mu}{p (2 N - \mu)}}\\ \leqslant \ & C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}}, \end{align*}

where C is a positive constant. Combining this inequality with (2.26) and (2.30), we deduce that there is a constant C > 0 such that for all $n \in \mathbb{N}$,

(2.31)\begin{align} \int_{\mathbb{R}^N \setminus B_R} \left|(f(u_n) - f(w_n) - f(u)) \varphi\right|^{\frac{2 N}{2 N - \mu}} dx \leqslant C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}}. \end{align}

Recalling that $u_n \xrightarrow w u$ in X_ɛ as $n \rightarrow \infty$ and passing to a subsequence (still denoted by $\left\{u_n\right\}_{n \in \mathbb{N}}$), we can assume that $u_n \rightarrow u$ strongly in $L^{\frac{2 N p}{2 N - \mu}}(B_R)$ (since $0 \lt \mu \lt 2 sp$) and there is a function $d \in L^{\frac{2 N p}{2 N - \mu}}(B_R)$ such that $|u_n(x)|$, $|u(x)| \leqslant d(x)$ a.e. $x \in B_R$.

For $n \in \mathbb{N}$ large enough, we have

(2.32)\begin{align} \int_{B_R} |f(w_n) \varphi|^{\frac{2 N}{2 N - \mu}} & dx \leqslant \int_{B_R} \left( C(\sigma) |w_n)|^{p - 1} + \sigma |w_n|^{\nu - 1}\right)^{\frac{2 N}{2 N - \mu}} |\varphi|^{\frac{2 N}{2 N - \mu}}dx\nonumber\\ &\leqslant C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}} + (2C(\sigma))^{\frac{2 N}{2 N - \mu}} \left(\int_{B_R}|w_n|^{\frac{2 N p}{2 N - \mu}}\right)^{\frac{p - 1}{p}} |\varphi|_{\frac{2 N p}{2 N - p}}^{\frac{2 N}{2 N - \mu}}\nonumber\\ & \leqslant C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}}, \end{align}

where C > 0 is some constant.

Set $D_{u_n}^4 :=\left\{x \in B_R : |u_n(x) - u(x)| \geqslant 1\right\}$. So, we get

\begin{align*} &\int_{D_{u_n}^4} |(f(u_n) - f(u))\varphi|^{\frac{2 N}{2 N - \mu}} dx \\ \leqslant \ & \int_{D_{u_n}^4} \left(C(\sigma) (|u_n|^{p - 1} + |u|^{p - 1}) +\sigma (|u_n|^{\nu - 1} + |u|^{\nu - 1})\right)^{\frac{2 N}{2 N - \mu}} |\varphi|^{\frac{2 N}{2 N - \mu}} dx\\ \leqslant \ & C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}} + C(2 C(\sigma))^{\frac{2 N}{2 N - \mu}} \left(\int_{D_{u_n}^4} d^{\frac{2 N p}{2 N - \mu}} dx\right)^\frac{p - 1}{p} \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}}, \end{align*}

where C > 0 is some constant. We observe that $|D_{u_n}^4| \rightarrow 0$ in $\mathbb{R}$ as $n \rightarrow \infty$, and then we can deduce that there exists some constant C > 0 such that

(2.33)\begin{align} \int_{D_{u_n}^4} |(f(u_n) - f(u))\varphi|^{\frac{2 N}{2 N - \mu}} dx \leqslant C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}}, \end{align}

provided that $n \in \mathbb{N}$ is sufficiently large.

In addition, from $u \in {X_\varepsilon}$, it follows that

\begin{align*}\left|\left\{x \in \mathbb{R}^N : |u(x)| \geqslant L\right\}\right| \rightarrow 0 \ \text{in } \mathbb{R}^N \ \text{as } L \rightarrow +\infty.\end{align*}

Using the above fact and invoking (2.25) again, we can infer that there exists $L :=L(\sigma) \gt 0$ such that

(2.34)\begin{align} &\int_{(B_R \setminus D_{u_n}^4) \cap \left\{x \in \mathbb{R}^N : |u(x)| \geqslant L\right\}} |(f(u_n) - f(u))\varphi|^{\frac{2 N}{2 N - \mu}} dx\nonumber\\ \leqslant \ & \int_{(B_R \setminus D_{u_n}^4) \cap \left\{x \in \mathbb{R}^N : |u(x)| \geqslant L\right\}} \left|\left(C(\sigma) (|u_n|^{p - 1} + |u|^{p - 1}) + \sigma(|u_n|^{\nu - 1} + |u|^{\nu - 1})\right)\varphi\right|^{\frac{2 N}{2 N - \mu}} dx\nonumber\\ \leqslant \ & C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}} + C(C (\sigma))^{\frac{2 N}{2 N - \mu}} \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}} \left|(B_R \setminus D_{u_n}^4) \cap \left\{x \in \mathbb{R}^N : |u(x)| \geqslant L\right\}\right|^{\frac{2 s p - \mu}{p (2 N - \mu)}}\nonumber\\ \leqslant \ & C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}}, \end{align}

where C > 0 is a constant. Additionally, we can deduce from the Dominated Convergence Theorem that

\begin{align*} \int_{(B_R \setminus D_{u_n}^4) \cap \left\{x \in \mathbb{R}^N : |u(x)| \leqslant L\right\}} |f(u_n) - f(u)|^{\frac{2 N p}{(2 N - \mu)(p - 1)}} dx = o_n(1) \end{align*}

for $n \in \mathbb{N}$ large enough.

It follows that

(2.35)\begin{align} \int_{(B_R \setminus D_{u_n}^4) \cap \left\{x \in \mathbb{R}^N : |u(x)| \leqslant L\right\}} |(f(u_n) - f(u))\varphi|^{\frac{2 N }{2 N - \mu}} dx \leqslant C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}} \end{align}

for some constant C > 0.

Hence, we deduce from (2.32), (2.33), (2.34), and (2.35) that there exists some constant C > 0 such that

(2.36)\begin{align} \int_{B_R} \left|(f(u_n) - f(w_n) - f(u))\varphi\right|^{\frac{2 N}{2 N - \mu}} dx \leqslant C \sigma \|\varphi\|_{X_\varepsilon}^{\frac{2 N}{2 N - \mu}} \end{align}

for large enough $n \in \mathbb{N}$.

Putting together (2.31) and (2.36), we conclude that (2.21) holds true.

On account of the fact that $\frac{1}{|x|^\mu}*F(u) \in L^\frac{2 N}{\mu}(\mathbb{R}^N)$, for any $0 \lt \sigma \lt 1$, there exists $R_1 :=R_1(\sigma) \gt 0$ such that

\begin{align*} \left(\int_{\mathbb{R}^N \setminus B_{R_1}} \left|\frac{1}{|x|^\mu}*F(u)\right|^\frac{2 N}{\mu} dx\right)^\frac{\mu}{2 N} \lt \sigma. \end{align*}

Using the above all information and by a straightforward computation, for $n \in \mathbb{N}$ large enough, we can conclude that

(2.37)\begin{align} & \left|\int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu}*F(u)\right) f(w_n) \varphi dx\right| \nonumber\\ \leqslant \ & \int_{B_{R_1}} \left|\left(\frac{1}{|x|^\mu}*F(u)\right) f(w_n) \varphi \right| dx + \int_{\mathbb{R}^N \setminus B_{R_1}} \left|\left(\frac{1}{|x|^\mu}*F(u)\right) f(w_n) \varphi \right| dx \nonumber\\ \leqslant \ & C_u \left(\int_{B_{R_1}} |f(w_n) \varphi|^\frac{2 N}{2 N - \mu} dx\right)^{\frac{2 N -\mu}{2 N}}\nonumber\\ +\ & \left(\int_{\mathbb{R}^N \setminus B_{R_1}} \left|\frac{1}{|x|^\mu}*F(u)\right|^\frac{2 N}{\mu} dx\right)^\frac{\mu}{2 N} \left(\int_{\mathbb{R}^N} |f(w_n) \varphi|^{\frac{2 N}{2N - \mu}} dx \right)^\frac{2 N - \mu}{2 N}\nonumber\\ \leqslant \ & C \sigma \|\varphi\|_{X_\varepsilon}, \end{align}

where C_u and C are positive constants.

Next, we prove the following relation:

(2.38)\begin{align} \left|\int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu}*F(w_n)\right) f(u) \varphi dx \right| \leqslant C \sigma\|\varphi\|_{X_\varepsilon} \end{align}

for $n \in \mathbb{N}$ sufficiently large.

Indeed, it is easy to check that, for any $\sigma \in (0,1)$ and some $R_2 :=R(\sigma) \gt 0$, there exists some constant C > 0 such that

(2.39)\begin{align} \int_{B_{R_2}} |F(w_n)| \left(\int_{B_{R_2}} \frac{|f(u(x)) \varphi(x)|}{|x - y|^\mu} dx\right) dy \leqslant C \sigma \|\varphi\|_{X_\varepsilon}, \end{align}

(2.41)\begin{align} \int_{\mathbb{R}^N} |F(w_n)|\left(\int_{\mathbb{R}^N \setminus B_{R_2}}\frac{|f(u(x)) \varphi(x)|}{|x - y|^\mu} dx \right) dy \leqslant C \sigma \|\varphi\|_{X_\varepsilon} \end{align}

for $n \in \mathbb{N}$ large enough.

To prove (2.38), now we need to estimate the following part:

\begin{align*}\Phi_n :=\int_{\mathbb{R}^N \setminus B_{R_2}} |F(w_n)| \left(\int_{B_{R_2}} \frac{|f(u(x)) \varphi(x)|}{|x - y|^\mu} dx\right) dy.\end{align*}

To this end, we divide this into two parts for discussion.

($*$) If $f(u(x)) \varphi(x) = 0$ a.e. on $B_{R_2}$, so, we have

\begin{align*}\Phi_n \leqslant C \sigma \|\varphi\|_{X_\varepsilon} \ \text{for all } \sigma \gt 0 \ \text{and for some constant } C \gt 0.\end{align*}

($**$) If $|\left\{x \in B_{R_2} : f(u(x)) \varphi(x) \neq 0\right\}| \gt 0$, that is,

\begin{align*}\int_{B_{R_2}} |f(u) \varphi |^{\frac{6N}{6N - \mu}} dx \gt 0.\end{align*}

In addition, we can easily derive that

\begin{align*} \int_{B_{R_2}} |f(u)\varphi|^{\frac{6N}{6N - \mu}} dx \leqslant C \|\varphi\|_{X_\varepsilon}^{\frac{6N}{6N - \mu}}|B_{R_2}|^{\frac{2\mu}{6N - \mu}} \ \text{for some constant } C \gt 0.\end{align*}

Let us define

\begin{align*} d_\sigma := \left(\frac{|f(u)\varphi|_{L^{\frac{6N}{6N - \mu}}(B_{R_2})}}{\sigma \|\varphi\|_{X_\varepsilon}}\right)^{\frac{3}{\mu}} \ \text{and }\widehat{R}_2 := R_2 + \sigma^{-\frac{3}{\mu}} |B_{R_2}|^{\frac{6}{6N - \mu}} C^{\frac{6 N - \mu}{2 N\mu}}. \end{align*}

In the case ($**$), we can apply the above relations, theorem 4.3 in the study by Lieb and Loss [Reference Lieb and Loss38], the Sobolev continuous embedding and the local compactness Sobolev embedding to infer that, for $n \in \mathbb{N}$ sufficiently large,

(2.42)\begin{align} \Phi_n & = \int_{B_{R_2}} |f(u) \varphi| \left(\int_{\mathbb{R}^N \setminus B_{R_2}} \frac{|F(w_n(y))|}{|x - y|^\mu} dy\right) dx\nonumber\\ & =\int_{B_{R_2}} |f(u) \varphi| \left(\int_{\mathbb{R}^N \setminus B_{R_2 + d_\sigma}} \frac{|F(w_n(y))|}{|x - y|^\mu} dy\right) dx\nonumber\\ & + \int_{B_{R_2}} |f(u) \varphi| \left(\int_{ {B_{R_2 + d_\sigma}} \setminus B_{R_2}} \frac{|F(w_n(y))|}{|x - y|^\mu} dy\right) dx\nonumber\\ & \leqslant C \frac{\sigma\|\varphi\|_{X_\varepsilon}}{|f(u)\varphi|_{L^{\frac{6N}{6N - \mu}}(B_{R_2})}}\int_{B_{R_2}} |f(u)\varphi| \left(\int_{\mathbb{R}^N \setminus B_{R_2 + d_\sigma}} \frac{|F(w_n(y))|}{|x - y|^{2\mu / 3}} dy\right) dx\nonumber\\ & + \int_{B_{R_2}} |f(u) \varphi| \left(\int_{ {B_{R_2 + d_\sigma}} \setminus B_{R_2}} \frac{|F(w_n(y))|}{|x - y|^\mu} dy\right) dx\nonumber\\ & \leqslant C \frac{\sigma\|\varphi\|_{X_\varepsilon}}{|f(u)\varphi|_{L^{\frac{6N}{6N - \mu}}(B_{R_2})}}\int_{B_{R_2}} |f(u)\varphi| \left(\int_{\mathbb{R}^N} \frac{|F(w_n(y))|}{|x - y|^{2\mu / 3}} dy\right) dx\nonumber\\ & + \int_{B_{R_2}} |f(u) \varphi| \left(\int_{ {B_{R_2 + d_\sigma}} \setminus B_{R_2}} \frac{|F(w_n(y))|}{|x - y|^\mu} dy\right) dx\nonumber\\ & \leqslant C \frac{\sigma\|\varphi\|_{X_\varepsilon}}{|f(u)\varphi|_{L^{\frac{6N}{6N - \mu}}(B_{R_2})}} |f(u)\varphi|_{L^{\frac{6N}{6N - \mu}}(B_{R_2})} \nonumber\\ & + \int_{B_{R_2}} |f(u) \varphi| \left(\int_{ {B_{\widehat{R}_2}}\setminus B_{R_2}} \frac{|F(w_n(y))|}{|x - y|^\mu} dy\right) dx\nonumber\\ & \leqslant C\sigma \|\varphi\|_{X_\varepsilon} + \int_{B_{R_2}} |f(u) \varphi| \left(\int_{ {B_{\widehat{R}_2}}\setminus B_{R_2}} \frac{|F(w_n(y))|}{|x - y|^\mu} dy\right) dx\nonumber\\ & \leqslant C \sigma\|\varphi\|_{X_\varepsilon}. \end{align}

Thus, relations (2.39), (2.41), and (2.42) and ($*$) imply that (2.38) holds true.

Finally, we point out that

\begin{align*} & \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * F(u_n)\right) f(u_n) \varphi dx - \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * F(w_n)\right) f(w_n) \varphi dx\\ =\ & \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * \left(F(u_n) - F(w_n)\right)\right) (f(u_n) - f(w_n))\varphi dx\\ + \ & \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * (F(u_n) - F(w_n))\right) f(w_n) \varphi dx \\ + \ & \int_{\mathbb{R}^N} \left(\frac{1}{|x|^\mu} * F(w_n)\right) (f(u_n) - f(w_n)) \varphi dx. \end{align*}

Then, the above equality combined with (2.19), (2.20), (2.21), (2.37), and (2.38); theorem 4.3 in the study by Lieb and Loss [Reference Lieb and Loss38]; the Hölder inequality; the boundedness of the sequence $\left\{u_n\right\}_{n \in\mathbb{N}} \subset {X_\varepsilon}$; and the fact that $\|\varphi\|_{X_\varepsilon} \leqslant 1$ yields that (f) is true. This proof is now complete.