Consider the following functional $F: H_0^1(\Omega) \times H^2 (\Omega)$ given by $$ F(u, \varphi) = \frac 12 \int_\Omega|\nabla u|^2 \ dx + \frac 12 \int_\Omega q(\varphi + \chi)u^2 \ dx - \frac 14 \int_\Omega (\Delta \varphi)^2 \ dx - \frac 14 \int_\Omega |\nabla \varphi|^2 \ dx - \frac{\alpha}{2|\Omega|}\int_\Omega\varphi \ dx $$ where $q \in C(\overline \Omega)$ and $\chi$ is of class $C^2$ in $\Omega$. Consider the restriction of the functional to the set $M \times \tilde V$, where $$ M = \left\{u \in H_0^1(\Omega) \ : \ \int_\Omega u^2 \ dx - 1 = \int_\Omega qu^2 \ dx - \alpha = 0\right\} $$ and $$ \tilde V = \left\{ \varphi \in H^2 (\Omega) \ : \ \frac{\partial \varphi}{\partial n} = 0 \text{ on } \partial \Omega \text{ and } \int_\Omega \varphi = 0 \right\}. $$
How to show that this functional is unbounded from above and from below on $M \times \tilde V$?