How does it historically work out for the presidential nominee when they are challenging the incumbent of their own party?
Are there cases where this succeeded?
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Generally speaking, it's considered a sign of a weak president and a dis-unified party if an incumbent president chooses to run and gets a legitimate, mainstream primary challenger (or, basically, anyone who would register on the radar screen for getting more than an handful percentage of the vote).
I believe recent history has presidents who were challenged eventually losing the general election, even if they win the primary, so parties really, really try to avoid going through that. We also see only a couple of instances where the challenger actually is successful in the challenge (as defined by unseating the incumbent), and none where that initial strong challenger wins the nomination -
2012 - Obama incumbent, no real challenge.
2004 - "W" Bush incumbent, unopposed
1996 - Clinton incumbent, no real challenge
1992 - "HW" Bush incumbent, Pat Buchanan challenged, was not considered a serious threat, but has enough support to get the party platform to include some far-right provisions to appease him and his supporters, and he was given a major speaking spot at the nominating convention, delivering what some refer to as his famous "Brown Shirt Speech," which many credited with sending independent voters towards the Democrats. Clinton won in the general election.
1984 - Reagan incumbent, unopposed
1980 - Carter incumbent, challenged by Ted Kennedy, won primary, lost general to Reagan.
1976 - Ford incumbent, but not really - a special case because he was never elected, was only president because of resignations in disgrace of previous VP and President, faced a stiff conservative primary challenge by Ronald Reagan, and lost in the general to Carter.
1972 - Nixon incumbent, was challenged but no seriously, winning all but one delegate during the process.
1968 - Johnson incumbent, was primaried by Eugene McCarthy, barely losing New Hampshire and doing well in the polls, and then Robert Kennedy entered the race. Johnson dropped out, causing VP Humphrey to eventually throw his hat into the ring (so, technically, was successfully primaried). Kennedy was assassinated, Humphrey was already ahead by then, and got the nomination. Nixon won the general.
1964 - Johnson incumbent, having become president the previous year when Kennedy was assassinated. Beat back a relatively feeble challenge by George Wallace of Alabama. Johnson had even claimed to not be decided until just before the nominating convention, so most of his votes gathered in the primary contests were as write-ins. With the goodwill and popularity/sympathy after the Kennedy assassination, his nomination was generally considered a foregone conclusion.
1956 - Eisenhower, the war hero general who commanded all Allied forces in defeating Nazi Germany, was not seriously challenged in either a primary or the general election.
1952 - Truman was the incumbent, was challenged by Estes Kefauver from Tennessee, lost enough early contests to drop out of the race. Adlai Stevenson eventually got the nomination, lost to Eisenhower in the general
1948 - Truman was president, taking over upon FD Roosevelt's death, was not seriously challenged, won the general election in a very close race.
That covers everything back to FDR. As you can see, modern history does not smile upon the party where an incumbent is faced with a serious primary challenge.
References: Wikipedia - "Democratic Party presidential primaries" and "Republican Party presidential primaries"
The incumbent President still have to win the primary and be nominated, though they usually do not have any serious challengers. For example, Barack Obama was nominated unanimously by all 5,556 delegates at the 2012 Democratic National Convention. Should they be challenged, it is called a "primary challenge" and it is very rare as incumbent Presidents are usually seen as the leader of their party.
There have only been 5 incumbent Presidents who were denied the nomination by their own parties. They are Franklin Pierce, Millard Fillmore, John Tyler, Andrew Johnson and Chester A. Arthur who all did not run for reelection. It's worth noting that only Pierce was elected, the rest were ascended to the presidency after their predecessor either died in office or was assassinated.