Solution
= Solution
Choose any nonzero admissible $u_0$ and set $u=\gamma u_0$. Then
$$
I_\lambda(\gamma u_0)=\gamma^2\int_0^\pi(u_0'^2-\lambda^2u_0^2)dx-\frac{\gamma^6}{2009}\int_0^\pi u_0^6dx.
$$
The coefficient of $-\gamma^6$ is strictly positive, so the energy tends to minus infinity as $|\gamma|\to\infty$. Hence
$$
\boxed{\inf_{\mathcal A}I_\lambda=-\infty,\qquad0\text{ is not a global minimizer}.}
$$