  Let $X$, $D$ be a smooth projective surface and a simple normal crossing
divisor on $X$, respectively. Suppose $\kappa (X, K_X + D)\ge 0$, let $C$ be an
irreducible curve on $X$ whose support is not contained in $D$ and $\alpha$ a
rational number in $ [ 0, 1 ]$. Following Miyaoka, we define an orbibundle
$\mathcal{E}_\alpha$ as a suitable free subsheaf of log differentials on a
Galois cover of $X$. Making use of $\mathcal{E}_\alpha$ we prove a
Bogomolov-Miyaoka-Yau inequality for the couple $(X, D+\alpha C)$. Suppose
moreover that $K_X+D$ is big and nef and $(K_X+D)^2 $ is greater than
$e_{X\setminus D}$, namely the topological Euler number of the open surface
$X\setminus D$. As a consequence of the inequality, by varying $\alpha$, we
deduce a bound for $(K_X+D)\cdot C)$ by an explicit function of the invariants:
$(K_X+D)^2$, $e_{X\setminus D}$ and $e_{C \setminus D} $, namely the
topological Euler number of the normalization of $C$ minus the points in the
set theoretic counterimage of $D$. We finally deduce that on such surfaces
curves with $- e_{C\setminus D}$ bounded form a bounded family, in particular
there are only a finite number of curves $C$ on $X$ such that $- e_{C\setminus
D}\le 0$.
