The stock market, of course, is a fickle environment. What’s more, thanks to financial turmoil in Europe and economic uncertainty in China, it has been especially volatile this spring. But volatility is a two-edged blade: It could take only one blowout quarter from Facebook (FB) to send its stock rocketing back toward its $38 a share offering price.

The big question now is not what went wrong with Facebook’s offering, it is whether Facebook can deliver growth strong enough to win back the confidence of investors. And that question has created a robust debate between bulls and bears on Facebook’s future, two camps in such sharp disagreement that it’s almost as if Facebook were split into two companies. The first is Facebook the social network that is losing popularity and will face the long, slow decline that MySpace suffered. The second is Facebook the innovative giant that employs much of Silicon Valley’s top talent and will be defining how we interact online for years to come.

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The disagreement can be seen in the price targets that Wall Street analysts have put on the stock in the past few weeks. On Monday, Bernstein Research saw Facebook sinking further, to $25 a share or below. A week earlier, Wedbush Securities gave the same stock a $44 a share target. Basically, the truth lies somewhere in that wide range.

In the technology press, the debate has been even more spirited, centered primarily around whether Facebook can maintain its torrid growth rate, whether it can draw more revenue out of its existing members and how it can make money from the mobile web without alienating users.

No one will deny that Facebook’s growth to date makes it one of the most successful tech companies ever. The case for Facebook says that Facebook’s active users — 901 million per month and counting — makes it a natural monopoly, and that growth is still strong in developing markets like Brazil and India. Plus, new areas revenue like payments nearly doubled in the first quarter.

The case against Facebook counters that growth is negligible in large markets like the U.S. and the U.K., that its revenue per user is significantly lower outside of the North America and Europe and that the revenue it makes from all monthly users has seen its growth slow dramatically, even among its most active users.

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Whether Facebook can extract more money from its users will depend in good part on its strategy for the mobile web. Which brings us to another sharp area of debate.

The bullish case argues that Facebook has a long-term plan to integrate its social graph onto smartphones and tablets, that the company isn’t rushing so that it can get this strategy right and that it’s building a stable of promising web properties like Instagram, Face.com and possibly the maker of the Opera browser. Facebook’s social network is likely to be integrated closelyin future versions of iOS, Apple’s (AAPL) mobile platform.

The bearish case against Facebook responds that the devil is in the details. How will the company fit its cluttery sponsored ads onto a smartphone screen without angering users? How can it woo big brands over to mobile ads when it’s having trouble keeping big advertisers on its site? And above all, how can Facebook get a strong foothold in the competitive mobile OS space? It has several options, but the best ones will require several years of work.

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Which Facebook will prevail? The one that will dominate social networking for years or the one that proves to be a glorious flash in the pan? Both the optimistic and the pessimistic takes on Facebook’s future have merit. Which one proves right will depend not so much on how the company responds than on how quickly it can address the challenges before it.

Given enough time, Facebook has the talent, the cash and the huge user base to solve all of these problems. But the longer it takes to solve them, the harder it will be to retain its dominance. And the longer Facebook’s stock price stays down, the more tempted its talented staff will be to exit to work at another growing startup, or to form one.

That’s a concern because the most likely outcome is that Facebook will continue to face challenges for the next several quarters. The company will need that much time to lay the groundwork for its next phase of growth. The good news is that, by next year, Facebook will have a lot of new ideas. The bad news is that a year on the web is a very long time.

Kevin Kelleher is a writer in the San Francisco Bay Area. You can find his bullish and bearish thoughts on Twitter.