You can't compete with that by building a better search engine. Even if you had the infrastructure, capital, and you offered Apple and Mozilla more money than they get from Google, Google could beat your offer.
The only way someone could beat Google, or even just compete with them, would be if they could make everyone understand that they can change their default search engine and give them a reason to do it and have Google screw up how they respond. I wouldn't bet on it happening.