Firming up Neapolitan Pizza Crust

500F is not really that hot, in terms of Neapolitan pizza. Traditionally, they are baked for 2 minutes or less, at 700F - 900F (371 - 482C). So, this could be a temperature, and/or a cooking surface issue. What type of oven are you using? On what type of surface is your pizza cooking?

I've heard that Neapolitan pizza is traditionally served with a soggy base in the middle. I've never been to Naples so I can't confirm that this is true, but you might just be cooking a "classic" Neopolitan pizza :)

That said, I prefer a crispy base like you! I can think of a few possible explanations for a crispy crust and a soggy base/bottom:

1. Trapped steam. Try resting your pizza for a minute on a wire rack to let steam escape from the base before serving.
2. The pizza was taken out prematurely, which prevented the pizza from developing a hearty base/crust. If you have a thick pizza, you can even end up with pizza that's doughy in the middle.
3. Wet ingredients can cause the crust to absorb all the moisture from the sauce and ingredients (use low-moisture mozzarella!)
4. Cooking on a non-conductive surface. Unlike most indoor ovens, most of the heat in an outdoor oven doesn't come from below the pizza -- it usually comes from the wood, which is lit and placed next to the cooking area. If the cooking surface isn't conductive for some reason, you might end up with a soggy bottom. Preheating a pizza stone or cooking your pizza in a cast iron pan could help transfer the heat from your oven to the base of your pizza.

An authentic Neapolitan Pizza usually has a rim that is fluffy and soft in the inside and comes with a delicate eggshell crust (cornicione) and when cutting a slice of it, it is expected to bend at the tip. So if you are aiming for a crunchy crust you probably don´t want a Neapolitana. To reach this goal you can tweak some parameters:

• Lower the hydration to less than 50%*, official Neapolitana calls for 55-62% and hydrations of 70% and above are common. The reason for this is that the water 'explodes' to steam faster at the 900°F than the dough cooks creating the big, fluffy alveoli in the rim.
• According to this lower temperatures will also help create a crunchier crust because the water evaporates slower leading to a more compact and thus stiffer result. Your 500°F should already be a good choice and I would not recommend to go even lower.
• Add 4-8% oil to your dough, this will give the texture more crunchiness as 'fries' the dough a bit. Neapolitan doug does not contain oil at all.
• Increasing the amount of salt (e.g. by 0.5%) also might help a little as it strengthens the gluten network. (For this reason doughs used for pizza acrobatics contain much more salt than doughs for baking.)

* All percentages are bakers percents.