both $\xi_1$ and $\xi_2$ $$\inner{L_c f}{f} \geq C \norm{f}_{L^2}^2$$ which further implies $$\inner{L_c f}{f} \geq C' \norm{f}_\h{1}^2$$ \[cor:lowerbound\]

The first part is obvious by Proposition \[prop-quadratic-bound\]. To prove the second inequality, let us express $L_c$ using as $$L_c = - c \dx \paren{\f_c^{2n} \dx\cdot} + V(x)$$ Then $$\begin{aligned}
\inner{L_c f}{f} &=&c \inner{\f_c^n \dx f}{\f_c^n \dx f} + \inner{Vf}{f} \\
&\geq& \norm{\dx f}_{L^2}^2 - \norm{V}_{L^\infty} \norm{f}_{L^2}^2\\
&\geq& \norm{\dx f}_{L^2}^2 - C^{-1} \norm{V}_{L^\infty}\inner{L_c f}{f}\end{aligned}$$ hence $$\inner{L_c f}{f} \geq \paren{1 + C^{-1}\norm{V}_{L^\infty}}^{-1} \norm{\dx f}_{L^2}^2$$ and the inequality follows.

Given a solitary wave $\f_c$ with $c>n>1$ for , there exist positive constants $C_2$, $C_3$ such that for all $u\in \h{1}$ satisfying $$\begin{aligned}
\label{eq:poly1}
\norm{u}_\h{1}&\leq& \frac{1}{2}\\
\label{eq:poly2}
\calN[\f_c +u] &=& \calN[\f_c]\\
\label{eq:poly3}
\inner{\xi_2}{u}&=&0\end{aligned}$$ $\xi_2$ given by , we have $$\inner{L_c u}{u} \geq C_2 \norm{u}_\h{1}^2 - C_3 \norm{u}_\h{1}^{3}
\label{eq:lowerbound}$$ \[prop:lowerbound\]

Let $u= u_\bot + u_\|$ where $$u_\| = \inner{u}{\xi_1}\xi_1$$ and $$u_\bot = u-u_\| = u-\inner{u}{\xi_1}\xi_1$$ Then $$\inner{L_c u}{u}  =  \inner{L_c\paren{u_\bot + u_\| }}{u_\bot +u_\|  } = \inner{L_c u_\bot}{u _\bot} + 2\inner{L_c u_\bot}{u_\| }+\inner{u_\| }{u_\| }$$ By Corollary \[cor:lowerbound\] and Lemma \[lem:invariantconstraint\], $$\begin{aligned}
\nonumber\inner{L_c u_\bot}{u _\bot} &\geq& C\norm{u_\bot}_\h{1}^2 = C\bracket{\norm{u}_\h{1}^2-2\inner{u}{u_\| }-2\inner{\dx u}{\dx u_\| } +\norm{u_\| }_\h{1}^2}\\
\nonumber &=&C\left[\norm{u}_\h{1}^2-2\inner{u}{\xi_1}^2-2\inner{ \dx u}{\xi_1}\inner{u}{\xi_1} +\inner{u}{\xi_1}^2\norm{\xi_1}_\h{1}^2 \right]\\
 &\geq& C\norm{u}_{\h{1}}^2 - D\norm{u}_\h{1}^{3}
\label{eq:lowerbound1}\end{aligned}$$

