eta_{\p}|^{2}\, dx dy+\beta^{-N}(\bar{V}+m^{2s})|\eta_{\p}(\cdot, 0)|^{2}_{2}  \right)^{\frac{N}{2s}}\beta^{\frac{N(N-2s)}{4s}}-\lambda C|\eta_{\p}(\cdot, 0)|_{p}^{p} \\
&\leq \frac{s}{N}(\zeta S_{*})^{\frac{N}{2s}}+O(\p^{N-2s})+C|\eta_{\p}(\cdot, 0)|^{2}_{2}-\lambda C|\eta_{\p}(\cdot, 0)|_{p}^{p},
\end{align*}
where we used 
\begin{align*}
\beta^{(2s-N)\frac{N}{2s}+\frac{N(N-2s)}{4s}}=\beta^{-\frac{N(N-2s)}{4s}}=\left(\zeta^{-\frac{2}{N-2s}}\right)^{-\frac{N(N-2s)}{4s}}=\zeta^{\frac{N}{2s}}.
\end{align*}
In what follows, we show that, for $\p>0$ small enough,
\begin{align}\label{ALEX}
\alpha(t_{\p})<\frac{s}{N}(\zeta S_{*})^{\frac{N}{2s}}.
\end{align} 
Assume that $N>4s$. Then $p>2>\frac{N}{N-2s}$ and using \eqref{HZ3}, we have
\begin{align*}
\alpha(t_{\p})\leq \frac{s}{N}(\zeta  S_{*})^{\frac{N}{2s}}+O(\p^{N-2s})+O(\p^{2s})-O(\p^{N-\frac{(N-2s)p}{2}}).
\end{align*}
Since
\begin{align*}
N-\frac{(N-2s)p}{2}<2s<N-2s,
\end{align*}
we infer that \eqref{ALEX} holds 
as long as $\p>0$ is sufficiently small.

If $N=4s$, then $p>2=\frac{N}{N-2s}$, and in view of \eqref{HZ3}, we obtain that 
\begin{align*}
\alpha(t_{\p})\leq \frac{s}{N}(\zeta  S_{*})^{\frac{N}{2s}}+O(\p^{2s}(1+|\log \p|))-O(\p^{4s-sp}).
\end{align*}
Observing that $p>2$ yields 
\begin{align*}
\lim_{\p\rightarrow 0} \frac{\p^{4s-sp}}{\p^{2s}(1+|\log \p|)}=\infty,
\end{align*}
we get the conclusion for $\p>0$ small enough.

Now let us consider the case $2s<N<4s$. First, we assume that $2^{*}_{s}-2<p<2^{*}_{s}$. Then,
\begin{align*}
p>2^{*}_{s}-2>\frac{N}{N-2s}
\end{align*}
which combined with \eqref{HZ3} gives 
\begin{align*}
\alpha(t_{\p})\leq \frac{s}{N}(\zeta  S_{*})^{\frac{N}{2s}}+O(\p^{N-2s})-O(\p^{N-\frac{(N-2s)p}{2}}).
\end{align*}
Since
\begin{align*}
N-\frac{(N-2s)p}{2}<N-2s<2s,
\end{align*}
we have that \eqref{ALEX} is satisfied for $\p>0$ small enough.
Second, we deal with the case $2<p\leq 2^{*}_{s}-2$. We distinguish the following subcases:
\begin{align*}
2<p<\frac{N}{N-2s}, \quad p=\frac{N}{N-2s} \quad \mbox{ and } \quad \frac{N}{N-2s}<p\leq 2^{*}_{s}-2.
\end{align*}
If $2<p<\frac{N}{N-2s}$ then, by \eqref{HZ3}, it holds 
\begin{align*}
\alpha(t_{\p})\leq \frac{s}{N}(\zeta  S_{*})^{\frac{N}{2s}}+O(\p^{N-2s})-\lambda O(\p^{\frac{(N-2s)p}{2}}),
\end{align*}
and noting that 
\begin{align*}
N-2s<\frac{(N-2s)p}{2},
\end{align*}
we can take $\lambda=\p^{-\nu}$, with $\nu>\frac{(N-2s)(p-2)}{2}$, to get the required conclusion.

If $p=\frac{N}{N-2s}$, then by \eqref{HZ3} we have
\begin{align*}
\alpha(t_{\p})\leq \frac{s}{N}(\zeta S_{*})^{\frac{N}{2s}}+O(\p^{N-2s})-\lambda O(\p^{\frac{N}{2}}|\log \p|),
\end{align*}
and taking $\lambda=\p^{-\nu}$, with $\nu>2s-\frac{N}{2}$, we deduce the assertion for $\p>0$ small enough.

Finally, when $\frac{N}{N-2s}<p\leq 2^{*}_{s}-2$, it follows from \eqref{HZ3} that
\begin{align*}
\alpha(t_{\p})\leq \frac{s}{N}(\zeta S_{*})^{\frac{N}{2s}}+O(\p^{N-2s})-\lambda O(\p^{N-\frac{(N-2s)p}{2}}) ,
\end{align*}
and choosing $\lambda=\p^{-\nu}$, with $\nu>2s-\frac{(N-2s)p}{2}$, we have the thesis for $\p>0$ sufficiently small.\\
Consequently, \eqref{DEDDY} and \eqref{ALEX} yield
\begin{align*}
\max_{t\geq 0} J_{1}(t v_{\p})<\frac{s}{N} (\zeta S_{*})^{\frac{N}{2s}},
\end{align*}
and using the fact that $c_{1}\leq \max_{t\geq 0} J_{1}(t v_{\p})$, we obtain the desired estimate.
\end{proof}



\noindent
In what follows, we show that $J_{\e}$ satisfies the local compactness condition. First, we prove the boundedness of Palais-Smale sequences of $J_{\e}$. 
\begin{lem}\label{boundPS}
Let $0<c<\frac{s}{N}(\zeta S_{*})^{\frac{N}{2s}}$ and $(v_{n})\subset X_{\e}$ be a Palais-Smale sequence of $J_{\e}$ at the level $c$. Then $(v_{n})$ is bounded in $X_{\e}$.
\end{lem}
\begin{proof}
By assumptions, we know that
\begin{align}\label{PSSQ}
J_{\e}(v_{n})\rightarrow c \quad \mbox{ and } \quad J'_{\e}(v_{n})\rightarrow 0 \mbox{ in } X^{*}_{\e},
\end{align}
as  $n\rightarrow \infty$. 
By this fact, \eqref{PSSQ}, $(g_3)$, \eqref{m-ineq} and \eqref{equivalent}, for $n$ big enough, we have
\begin{align*}
c+1+\|v_{n}\|_{\e}&\geq J_{\e}(v_{n})-\frac{1}{\theta}\langle J'_{\e}(v_{n}), v_{n}\rangle\\
&=\left(\frac{1}{2}-\frac{1}{\theta}\right) \|v_{n}\|^{2}_{\e}+\frac{1}{\theta} \int_{\mathbb{R}^{N}} g_{\e}(x, v_{n}(x, 0))v_{n}(x, 0)-\theta G_{\e}(x, v_{n}(x, 0))\, dx\\
&\geq \left(\frac{1}{2}-\frac{1}{\theta}\right) \|v_{n}\|^{2}_{\e}-\left(\frac{1}{2}-\frac{1}{\theta}\right) \frac{V_{1}}{\kappa} \int_{\mathbb{R}^{N}}  v_{n}^{2}(x, 0)\, dx \\
&= \left(\frac{1}{2}-\frac{1}{\theta}\right) \|v_{n}\|^{2}_{\e}-\left(\frac{1}{2}-\frac{1}{\theta}\right) \frac{V_{1}}{\kappa m^{2s}} m^{2s}\int_{\mathbb{R}^{N}}  