See Theorem 1.2 of the paper by [Bennett and Skinner] [1], which settles the problem for $p\ge 7$ (take there $C=1$ and $\alpha_0=2$).   (Earlier work of [Darmon and Granville][2] (using Faltings's theorem) showed that there are only finitely many solutions.)

[1]: https://www.math.ubc.ca/~bennett/BS.pdf

[2]: http://www.math.mcgill.ca/darmon/pub/Articles/Research/12.Granville/paper.pdf