President-elect Donald J. Trump gave a public show of support to his embattled choice for defense secretary, Pete Hegseth, on Friday morning, saying in a social media post that he will be “fantastic” in the job and that he has a “military state of mind.”

Mr. Hegseth has spent the last two days fighting to stay as Mr. Trump’s choice to lead the Pentagon amid mounting news reports alleging troubling behaviors over time including rape, financial mismanagement and drunken behavior, which he denies.

Mr. Trump’s post on Truth Social amounted to a public dare to Republican senators, a number of whom have expressed private concerns about Mr. Hegseth, to vote against his wishes. The president-elect until now has put relatively little of his own capital on the line for Mr. Hegseth and has even floated alternative candidates, such as Gov. Ron DeSantis of Florida, for the job.

But on Friday, Mr. Trump doubled down publicly on his initial choice.

“Pete Hegseth is doing very well. His support is strong and deep, much more so than the Fake News would have you believe,” Mr. Trump wrote. “He was a great student — Princeton/Harvard educated — with a Military state of mind. He will be a fantastic, high energy, Secretary of Defense Defense, one who leads with charisma and skill. Pete is a WINNER, and there is nothing that can be done to change that!!!”

A string of damning news articles have reported problematic behavior. One woman filed a police report in 2017 accusing him of rape, which he denied and said was a consensual encounter. He entered into a settlement agreement with her that included a payment of money years later. The New York Times reported that Mr. Hegseth’s mother sent him an email in 2018 during a contentious divorce saying he had “abused” many women in different ways over the years. NBC News reported about concerns about Mr. Hegseth’s drinking, and The New Yorker reported that he had been accused of mismanagement of groups he had previously led.

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