We consider the following non-autonomous variational problem: \[\textrm{minimize\,} \left\{F(v)=\int_a^b f(x,v(x),v'(x))\ \mathrm dx\,:\,v\in \Omega \right\} \] where $\Omega:=\{v\in W^{1,1}(a,b),\ v(a)=A, \ v(b)=B,\ v(x)\in I \}$. The Lagrangian \(F\) is assumed to have just a "non-everywhere" superlinear growth, being allowed to vanish at some $x_0\in [a,b]$, or $s_0\in I$. We prove some sufficient conditions ensuring the coercivity of the functional $F$. As a consequence, when $f$ is convex with respect to the last variable, the existence of the minimum can be immediately derived.