Given the functional
$$J(x,x'):=\int_0^1 x^2+x'^2+(x^2-1)^2 dt$$
show that a mimimizer exists in $H^1([0,1])$.
Obviously J is bounded from below. For $\{u_n\}_{n=1}^\infty \subset H^1(\Omega)$ $$\lim_{n \to \infty} J(u_n,u_n'):=\lim_{n \to \infty}\int_0^1 u_n^2+u_n'^2+(u_n^2-1)^2 dt= \int_0^1 \lim_{n \to \infty} (u_n^2+u_n'^2+(u_n^2-1)^2 )dt \overbrace{=}^? J(u,u')$$, so J ist continuous? How can a convergent subsequence in a minimizing sequence guaranteed? Hints are very much appreciated.