How would you show that the functional $F$ defined as
$$F : u \mapsto \int_{(-1,1)} \big(u^{\prime \prime}(x)+u^{3}(x)\mathbb{1}_{u^{\prime}\neq0}(x) \big)^2dx$$
with $\mathbb{1}_{X}$ the indicator function of $X$, over the set
$$Y=\left\{u \in H^{2}((-1,1)),\quad \int_{\Omega}u(x)dx=0 \right\}$$
has a minimum?
The first thing I can say is that, since $F \geq 0$ then $\mu:=\inf_{u\in Y}F(u)$ exists and we have a sequence $(u_{n})_{n}\in Y^{\mathbb{N}}$ such that $\lim_{n \to \infty}F(u_{n})=\mu$. To conclude I would like to have some compactness properties and lsc, which I think I don't have.
So, my question is, does anybody know of any other method with which to approach this problem? Thank you.