I am reading the book "Struwe, Michael, Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems". I have some questions about the proof of Theorem 7.2. I hope someone can help me.
Theorem 7.2 (page 120). Let $\Omega$ be a smmothly bounded domain in $\mathbb{R}^n$, $n\geq 3$. Suppose
$g:\Omega\times\mathbb{R}\to \mathbb{R}$ is continuous and odd with primitive $G(x,u)=\int_{0}^{u}g(x,v)dv$;
$\exists p<2^{*}=\frac{2n}{n-2}， C:|g(x,u)|\leq C(1+|u|^{p-1}$ almost everywhere;
- $\exists q>2, R_{0}:0<qG(x,u)\leq g(x,u)u$ for almost every $x$, $|u|\geq R_{0}$. Moreover, suppose that $$ \frac{2p}{n(p-2)}-1>\frac{q}{q-1}.$$ Then for any $f\in L^2(\Omega)$ the problem \begin{equation*} \begin{aligned} -\triangle u&=g(\cdot,u)+f\qquad \mbox{in}~ \Omega,\\ u&=0\qquad\qquad\qquad \mbox{on}~\partial\Omega, \end{aligned} \end{equation*} has an unbounded sequence of solutions $u_{k}\in H_{0}^{1,2}(\Omega)$, $k\in \mathbb{N}$.
In the proof of above Theorem 7.2, the author wrote " Consider a pseudo-gradient vector field $v$ for the $E$ such that $$\|v(u)\|_{H^{1,2}(\Omega)}\leq 2,~\langle DE(u),v(u)\rangle\geq\min\{1,\|DE(u)\|_{H^{-1}(\Omega)}\}\|DE(u)\|_{H^{-1}(\Omega)}.$$ Since $E$ is even, we may assume $v$ to be odd. Note that \begin{align*} \langle D\tilde{E}(u),v(u)\rangle&=\langle DE(u),v(u)\rangle+\langle D\tilde{E}(u)-DE(u),v(u)\rangle\\ &\geq\min\{1,\|DE(u)\|_{H^{-1}(\Omega)}\}\|DE(u)\|_{H^{-1}(\Omega)}-2\|D\tilde{E}(u)-DE(u)\|_{H^{-1}(\Omega)}\\ &\geq\min\{1,\|D\tilde{E}(u)\|_{H^{-1}(\Omega)}\}\|D\tilde{E}(u)\|_{H^{-1}(\Omega)}-4\|D\tilde{E}(u)-DE(u)\|_{H^{-1}(\Omega)}. \end{align*} Therefore, $v$ is also a pseudo-gradient vector field for $\tilde{E}$ off $\tilde{N}_\delta$ for any $\delta\geq8$, where for $\delta>0$ the set $\tilde{N}_\delta$ is given by $$\tilde{N}_\delta=\{u\in {H_{0}^{1,2}(\Omega)}; \|D\tilde{E}(u)-DE(u)\|_{H^{-1}(\Omega)}>\delta^{-1}\min \{1,\|D\tilde{E}(u)\|_{H^{-1}(\Omega)}\}\|D\tilde{E}(u)\|_{H^{-1}(\Omega)}\}.$$ Note that $\tilde{N}_\delta$ is a neighborhood of the set of critical points of $E$ and $\tilde{E}$, for any $\delta>1$ and $f\neq0$. Denote $N_\delta=\tilde{N}_\delta\cup(-\tilde{N}_\delta)$ the symmetric sets of $\tilde{N}_\delta$. Let $\eta$ be an even, locally Lipschitz function which satisfies $0\leq\eta\leq1, \eta(u)=0$ in $N_{10}$ and $\eta(u)=1$ off $N=N_{20}$. Also let $\varphi$ be a Lipschitz function which satisfies $0\leq\varphi\leq1, \varphi(s)=0$ for $s\leq 0, \varphi(s)=1$ for $s\geq1$, and let $$ e(u)=-\varphi(\max\{\tilde{E}(u),\tilde{E}(-u)\})\eta(u)v(u)$$ denote the truncated pseudo-gradient vector field on $H_{0}^{1,2}(\Omega)$.
Then，if $\Phi$ denotes the (odd) pseudo-gradient flow for $E$ induced by $e,\Phi$ will also be a pseudo-gradient flow for $\tilde{E}$ and will strictly decrease $\tilde{E}$ off $N$ with "speed" $$ \frac{d}{dt}\tilde{E}(\Phi(u,t))|_{t=0}\leq -\frac{1}{2}\min\{1,\|D\tilde{E}(u)\|^{2}\},\qquad \mbox{if}~\tilde{E}(u)\geq 1.$$ Moreover, $\Phi(\cdot,t)\in \Gamma$ for all $t\geq 0$."
My Questions：
- Why $v$ is also a pseudo-gradient vector field for $\tilde{E}$ off $\tilde{N}_\delta$ for any $\delta\geq8$？ By the definition of $\tilde{N}_{\delta}$, we can only deduce that $$ \langle D\tilde{E}(u),v(u)\rangle\geq \frac{1}{2}\min\{1,\|D\tilde{E}(u)\|_{H^{-1}(\Omega)}\}\|D\tilde{E}(u)\|_{H^{-1}(\Omega)}, $$ but the definition of pseudo-gradient vector field (Definition 3.1, page 81) requires that $$ \langle D\tilde{E}(u),v(u)\rangle> \min\{1,\|D\tilde{E}(u)\|_{H^{-1}(\Omega)}\}\|D\tilde{E}(u)\|_{H^{-1}(\Omega)}, $$ so I don't know how can we deduce that $v$ is also a pseudo-gradient vector field for $\tilde{E}$ off $\tilde{N}_\delta$ for any $\delta\geq8$.
- How can we deduce that $$ e(u)=-\varphi(\max\{\tilde{E}(u),\tilde{E}(-u)\})\eta(u)v(u)$$ is a truncated pseudo-gradient vector field on $H_{0}^{1,2}(\Omega)$?
- How can we obtain the following conclusion? $$ \frac{d}{dt}\tilde{E}(\Phi(u,t))|_{t=0}\leq -\frac{1}{2}\min\{1,\|D\tilde{E}(u)\|^{2}\},\qquad \mbox{if}~\tilde{E}(u)\geq 1.$$