Existence of minimizers for non-convex variational problems with slow growth

Consider the minimization problem $$ (*)~~~~\min\left\{\int_0^1 f(t,u'(t))dt;\ u\in W^{1,1}([0,1],\R^n),\ u(0)=u_0,\ u(1)=u_1\right\},~~~~{} $$ in which $f\colon [0,1]\times\R^n\to{\R}\cup\{+\infty\}$ is a normal integrand. Define the convex function $G\colon\R^n\to\R\cup\{+\infty\}$ by $$ G(p)\doteq\int_0^1 f^*(t,p)\,dt. $$ It is known that if the essential domain $H$ of $G$ is open, then problem (*) has a minimizer for any pair of endpoints $(u_0,u_1)$. In this paper the same result is proved under the condition that for every point $p$ in $H$, the subgradient set $\partial G(p)$ is either bounded or empty (when $H$ is open, this condition holds automatically).