In his paper ``[Borne polynomiale pour le nombre de points rationnels des courbes][1]'', *Journal de théorie des nombre de Bordeaux*  23 no. 1 (2011), p. 251-255, Gaël Rémond has given a general explicit bound for the number of solutions of polynomial equations in two variables with coefficients in a number field,
assuming the equation has finitely many solutions, of course.
By Faltings's theorem, this is the case of your curve, so his bound applies and 
says that there are at most $n^{2^{3^{16}}}$ solutions.

NB. The exponent of $n$ is equal to $1.721783764\, 10^{12958354}$ and this bound is both unpractical and certainly non-optimal.

  [1]: http://dx.doi.org/10.5802/jtnb.759