I am currently reading "On a conjecture of De Giorgi and some related problems" and my question is about the proof of theorem 2.4.
Consider the following setting: Let $V: \mathbb{R}^N \mapsto \mathbb{R}$ be a smooth, bounded potential and consider the functional
$$
\mathcal{L}:H^1(\mathbb{R}^N) \setminus \{ 0\} \mapsto \mathbb{R}, \phi \mapsto \frac{\int_{\mathbb{R}^N}|\nabla \phi|^2-V|\phi|^2dx }{\int_{\mathbb{R}^N}|\phi|^2dx}
$$
How can we apply Ekelands variational principle to this functional?
My issue is only with the smoothness of $\mathcal{L}$.
The version in the paper requires $\mathcal{L} \in C^1(H^1(\mathbb{R}^N),\mathbb{R})$. However, I dont see this functional being Frechet differentiable at $0$, as it is not even continuous.
I found the following version of the Ekeland variational principle:
Let $B$ be a banach space and let $\mathcal{L}: B \mapsto \mathbb{R} \cup \{+\infty \}$ be a proper, lower semicontinuous functional on $B$, which is bounded below and Gateaux differentiable. Let $\epsilon>0$ and $\bar{\phi} \in B$ such that
$$
\mathcal{L}(\bar{\phi}) \leq \inf_B \mathcal{L} + \epsilon
$$
Then, for every $\delta>0$ there exists $\phi \in B$ such that
(i)$\mathcal{L}(\phi) \leq \mathcal{L}(\bar{\phi})$
(ii)$||\phi-\bar{\phi}||_B \leq \frac{1}{\delta}$
(iii)$||\mathcal{L}'(\phi)||_{L(B)}\leq \epsilon \delta$
In this case, we would just have to pick a suitable value for $\mathcal{L}(0)$ and hope for lower-semicontinuity as well as Gateaux differentiability at $0$. Computing the Gateaux derivative at $\phi$ in the direction $\eta$, one finds that
$$
\mathcal{L}'(\phi)(\eta)=\frac{d}{dt}\mathcal{L}(\phi+t \eta)|_{t=0}=2 \frac{\int_{\mathbb{R}^N} \nabla \phi \cdot \nabla \eta-V \phi \eta dx -\mathcal{L}(\phi)\int_{\mathbb{R}^N}\phi \eta dx }{ \int_{\mathbb{R}^N} |\phi|^2 dx }
$$
As long as $\phi \neq 0$ one can prove by applying Cauchy-Schwarz multiple times that this is indeed a continuous, linear operator.
However, if $\mathcal{L}(0) \neq \infty$ is a proper value, then our functional also needs to be Gateaux differentiable at $0$.
If $\mathcal{L}(0) =\infty$, then the functional will no longer be lower semicontinuous.
Is there any way to modify the principle such that it holds even if $\mathcal{L}$ has a singularity at $0$?
Some additional input, if it helps:
(i) $\mathcal{L}(a\phi)=\mathcal{L}(\phi)$, so the problem is scaling invariant. In particular, we can choose $\bar{\phi}$ "away from $0$
(ii) I need the statement about the Gateaux derivative, so restricting $\mathcal{L}$ to the unit sphere and using the Ekeland variational principle in its original form is not an option, unless one can recover this statement
I am thankful for every hint, comment or answer.