Let

- $E$ be the usual sobolev space $H^{1}_{0}(\Omega)$ on a smoothly bounded domain $\Omega$,
- $E_{k}$ be its subspace spanned by the first $k$ eigenfunctions of the Laplace operator, i.e. $$E_{k}:=\text{span}\{\varphi_{j}\in E: -\Delta\varphi_{j}=\lambda_{j}\varphi_{j},~j=1,2\dots,k \},$$
- $P$ be the positive cone in $E$, i.e, $$P:=\{u \in E~; u \geq 0 ~a.e.\},$$

Now set $P_{k}=P\cap E_{k}~~u^{-}=min\{u,0\}$.

**My question**: does it exist $C_{k}>0$, such that
$$\text{dist}_{L^{2}}(u,P_{k}) \leq C_{k} \text{dist}_{L^{2}}(u,P)~\text{ holds }~\forall u\in E_{k}\; ?$$
or
$$\text{dist}_{E}(u,P_{k}) \leq C_{k} \text{dist}_{E}(u,P)~\text{ holds }~\forall u\in E_{k}\; ?$$
or $$\text{dist}_{E}(u,P_{k}) \leq C_{k} \|u^{-}\|_{E}~\text{ holds }~\forall u\in E_{k}\; ?$$
If not, could you please show me a counterexample? Thanks.

Moreover, this problem is raised from the proof of lemma 3.6 in "Infinitely many solutions to perturbed elliptic equations",doi:10.1016/j.jfa.2005.06.014 enter image description here

also see lemma 5.4 in "On finding sign-changing solutions",doi:10.1016/j.jfa.2005.09.004