I have a functional $\mathcal{V}: H_0^1((0,1)) \to \mathbb{R}$ defined by
$$
\mathcal{V}(u) := \int_0^1 \! \left( -u_x^2 + u^2 - u^3 \right) \, \mathrm{d}x.
$$
I would like to show that there exists $\delta > 0$ such that for all $u$ satisfying $\|u\|_2 < \delta$, $\|\cdot\|_2$ denoting the $L^2$ norm, we obtain
$$
\mathcal{V}(u) \le 0.
$$
I know that I can bound the first term of the integrand via the Poincaré inequality so that I obtain $$ \mathcal{V}(u) \le \left( 1 - \pi^2 \right) \| u \|_2^2 - \int_0^1 \! u^3 \, \mathrm{d}x. $$ I am not sure what can be done about the remaining cubic term (if anything?).