Louisiana’s gubernatorial contest kicked off in earnest Tuesday with the first day for candidates to qualify for the Oct. 12 election and with Republicans looking for the right person to oust incumbent Democratic Gov. John Bel Edwards.

Republicans’ two best hopes, Baton Rouge businessman Eddie Rispone and Rep. Ralph Abraham, both qualified to run, setting up what analysts expect to be a three-way race in the jungle primary.

Under Louisiana’s system, all candidates compete in the same election. If nobody wins a majority, the top two, regardless of party, move to a runoff.

“I qualified for governor today and I intend to win,” Mr. Abraham said.

His campaign said he soon will launch a $2 million statewide television campaign to drive his message on lower taxes and more jobs.

Mr. Rispone, whose campaign is flush with $10 million of his own money, released his third statewide ad this week.

In the 30-second ad, Mr. Rispone reiterates his theme he is a “conservative outsider” who will stand with President Trump to ban sanctuary cities, while establishing work requirements for welfare recipients and pushing a tax reform proposal that would be put to a popular vote.

It’s a plan, the ad’s narrator says, “so easy a career politician could do it.”

“But they never will,” Mr. Rispone then says, chuckling.

Mr. Edwards, seeking reelection in a largely conservative state where Mr. Trump enjoys strong approval ratings, has taken a very different approach.

Flush with considerable money of its own, his campaign’s sustained advertising has been on Louisiana’s televisions for weeks, striking a largely nonpartisan tone and depicting the governor as a moderate who has successfully steered state finances from a deficit to a surplus.

Judging from his informal comments and his occasional bickering with the Republican legislative majority, Mr. Edwards might prefer a more aggressively liberal approach to governing, but his campaign pitches him as middle-of-the-road — a West Point graduate who can oversee incremental changes that keep Louisiana moving in a positive direction.

Mr. Edwards hit the same notes when he appeared to qualify to run. He credited a surging economy, rather than tax increases, that “allowed us to stabilize our budget,” and reiterated his big theme that a former government deficit is now “firmly in the rearview mirror.”

“And so the question that you need to ask — and it’s a rhetorical question, because I know the answer — is do you want to go back to the failed policies of the past? Because that’s what my opponents offer.”

Republicans say the governor is vulnerable on issues such as taxes, but his approval rating remains above water with an array of voting groups.

That has perplexed Louisiana conservatives, some of whom had urged Mr. Rispone to begin spending some of his $10-plus million earlier in the race, even though Louisiana elections traditionally are fought between Labor Day and Election Day.

Mr. Abraham has been hampered by a much smaller war chest, though he enjoys what appears to be solid support in his congressional district, which runs along the state’s eastern border and down into the top laces of Louisiana’s boot shape.

He also has the backing of some of the state’s top Republican fundraisers — both factors that should help him stay competitive as the homestretch approaches.

Other candidates who qualified on Tuesday are long shots. The qualifying period will end Thursday. The fee for filing for the governor’s race is $750 and Louisiana does not allow write-in ballots.

Gary Landrieu, a cousin of the more famous Landrieus who have been fixtures among Louisiana Democrats for decades, is running as an independent, touting a strong stance against illegal immigration and the repeal of a portion of the state’s sales tax.

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