= Solution
A <minimizing sequence> has an energy upper bound $C$. Set $H(t)=\Phi(t)-M\geq0$. Then $\int H(u_j')\leq C-M(b-a)=C_1$. Since $H(t)/|t|\to\infty$, for every $A>0$ there is $R_A$ such that $H(t)\geq A|t|$ for $|t|>R_A$. Consequently, for every measurable $E$,
$$
\int_E|u_j'|\leq R_A|E|+C_1/A.
$$
Taking first $A$ large and then $|E|$ small proves <uniform integrability>; taking $E=(a,b)$ also gives a uniform $L^1$ bound. The <Dunford-Pettis theorem> on a finite-measure space identifies bounded <uniformly integrable> subsets of $L^1$ with relatively weakly compact ones. Thus a subsequence satisfies
$$
\boxed{u_{j_k}'\rightharpoonup v\text{ in }L^1(a,b).}
$$
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