Sarah Palin has a problem. Of course, she’s long had problems, her detractors say. But even they would concede that she has been a political celebrity virtually without peer in her party and a Republican who seemed to have perfect pitch with at least part of the conservative base.

Now, as she weighs whether to run for president in 2012, Palin has competition on both fronts. Minnesota Rep. Michele Bachmann, who appears to be moving aggressively toward a presidential candidacy, threatens to steal many of those core conservatives as Palin deliberates. Meanwhile, Donald Trump, with the flamboyance and egocentric showmanship that has marked his business career, threatens to eclipse Palin, for now, as the mega-celebrity of 2012 GOP politics.

Palin has endured some difficult months this year. Her troubles began in the aftermath of the Tucson shootings in January and her “blood libel” video, which drew criticism even from some Republicans and put her on the defensive. Her favorability ratings have fallen. In Iowa, a state that would seem tailor-made for her style of conservative politics, many GOP activists are decidedly cool to her as a possible presidential candidate — even those who say they admire her.

Her potential rivals for the GOP nomination, with a few exceptions, are all taking steps to ready their candidacies. Former Minnesota governor Tim Pawlenty filed a presidential committee with the Federal Election Commission on Monday. Former Massachusetts governor Mitt Romney is building a money network while biding his time before becoming a formal candidate. Mississippi Gov. Haley Barbour is traveling energetically to the early states as he approaches decision day. Former Pennsylvania senator Rick Santorum is virtually living in Iowa, New Hampshire and South Carolina. Former House speaker Newt Gingrich is a candidate in all but name only.

Palin has done nothing of the sort, unless she has done it by stealth. She travels to India and Israel but has made no trips to Iowa or New Hampshire. She has done no serious work in recruiting a presidential staff. Her “inner circle” remains far-flung. Only former Arkansas governor Mike Huckabee appears as ambivalent about whether to run.

Politically, Bachmann has moved to fill that vacuum on the right. In contrast with Palin, she is focused intensely on Iowa. She is making several appearances there this week, including a home-school event Wednesday and Saturday’s forum hosted by Iowa Rep. Steve King’s political action committee that will draw several other prospective candidates. She has not been shy about expressing interest in setting up an exploratory committee.

Bachmann has begun to make an impression in Iowa among both religious and fiscal conservatives. Like Palin, she excites the base with her strong, uncompromising message. She can speak tea party with the best in her party. She appears fearless, whether in dealing with House Republican leaders or potential presidential rivals. She might be at least a bit more impervious to criticism than Palin. Activists want to hear more from her.

But Bachmann is not without flaw. On a trip to New Hampshire, her recent gaffe suggesting that Lexington and Concord of Revolutionary War fame were places in the state subjected her to ridicule and will make any future missteps more costly. Given her rhetoric, she is nearly as inviting a target for the left as Palin. She has yet to demonstrate her presidential credentials, although she is not alone on that front. But none of that has slowed her down.

In the realm of celebrity politics, Palin might have met her match in The Donald. She might be happy to cede to Trump some of the baggage that comes with being a political celebrity — the flyspecking of every statement, the quickness of opponents in the blogosphere to attack, the constant mockery that she endures. But Trump has a thicker skin by far and has lived far longer in the world of gossip columnists and the wealthy and powerful. Palin remains a product of a small town in a remote state. Trump is the product of a high-flying world and plays that game with gusto.

Trump might not be a serious candidate for president. Sen. Lamar Alexander (R-Tenn.) said Trump could never be elected. Trump might never be a candidate at all. But he knows how to create interest and controversy. Palin might create controversy by accident. Trump will do it by design, as he did this week when, on ABC’s “The View,” he called on President Obama to show people his birth certificate. Until he says no to running, Trump will be a magnet for the media. For the first time, Palin might have to fight for attention.

Palin still has her megaphone. Facebook and Twitter offer platforms to insert herself into debates. She is still a Fox News contributor. On Wednesday night, she was interviewed for half an hour by Greta Van Susteren and offered her views on Libya, Israel and other topics, challenging Obama on all of them.

She can wait longer than, say, a Pawlenty to make a decision about 2012. Those loyal to her believe she can raise money quickly and prodigiously if she runs, thanks to the grass roots and the Internet. She might have little incentive to participate in some of the intramural scrimmaging that takes place in the early months of a presidential nomination battle, whether straw polls or debates with candidates spread from one side of a stage to the other. Fair enough.

But through these past months, as others have marched toward the starting gates, Palin might have lost altitude. On Fox News, she said she hasn’t made up her mind but added that she will play a role in 2012, as a candidate or non-candidate. Could that have been a serious hint that she will stay on the sidelines? Only Palin knows, but the landscape changes as she deliberates, and the competition for the space she has filled is rapidly taking shape.