Why doesn't the USA Republican Party (GOP) bother to run for mayor in some cities?

JamesK's answer is great.

I have run as a candidate and here are some other reasons a party may not run:

• Money is limited, especially at local level. Generally money has to be raised at a local level. There may be higher priorities for those resources
• Person-power is limited. There may be higher priorities for those resources
• Someone can be seen as a 'serial candidate' if they run and lose too many times. This is less of an issue if the candidate holds office already (e.g. a local councillor nominating as mayor)
• Being a candidate is crazy hard. Does someone really want to put all that effort into something that is unlikely to succeed
• There may be strategic reasons. Especially with first-past-the-post voting a far left or far right group may drop out to support a centre left or centre right group that is likely to win. Outside the US multiparty is common and at local level issue groups (Save the Trees Party etc) can also act strategically to achieve their goals

In the cases where they have won, it's a dead-end job.

It's been mentioned in other answers that voters in big cities often skew to the liberal end of the political spectrum. A Republican candidate pushing conservative policies has little chance of being elected by such a constituency. However, moderate (e.g. some aspects of Rudolph Giuliani) and liberal (e.g. Michael Bloomberg) Republicans can be competitive in big cities against Democrats, and have actually been elected mayors.

But they have a problem if they seek a higher office, such as governor or president. Higher offices expose the mayor to a more conservative electorate. The moderate/liberal track record needed as mayor now becomes an impediment. They are not welcome by other Republicans, nor does switching parties help.

Giuliani and Bloomberg both campaigned for President, and failed to gain support from either Republicans or Democrats.

James' answer is correct in regards to the specific cases of Portland and Atlanta, but I think you'll find, in the more general case, that it's not at all unusual for either the Democratic party or the Republican party to not run a candidate in lots of different state and local-level elections in the U.S.

There are a few factors that make this especially likely to happen:

• A reasonably-popular incumbent is running for re-election.
• The electorate of the district in question skews heavily toward one major party or the other.
• The office in question simply isn't an especially political one (e.g. school board, property assessor, road supervisor, local district judge, sheriff, etc.)
• The position in question isn't a particularly powerful or prestigious one (doesn't apply to mayor of, say, Atlanta, but might apply to mayor of some tiny rural town in the middle of nowhere.)

Even in positions as powerful as member of the United States House of Representatives, it's not uncommon for a popular incumbent candidate in a district that skews heavily toward one party to run either completely unopposed or only nominally opposed.

When this happens in a more powerful/prestigious position, such as member of Congress, member of state legislature, mayor of medium/large city, etc., it's usually a matter of an electorate that skews heavily toward one party or the other. The party of a reasonably-popular incumbent has no desire to run anyone against them in a primary and the other party can't find anyone who wants to run in a general election they are sure to lose and for which they will receive essentially no financial support. Not many politicians want to waste their time and money being a sacrificial lamb in an election in which they have no chance.

This effect is certainly not unique to the Republican party, though it probably happens more often in the Republican party in the specific case of mayor of a large city simply because large cities tend to have heavily Democratic-leaning electorates. In the case of major of small or medium-sized towns, which tend to skew heavily toward Republican electorates, the trend is likely the opposite with the Democratic party being much more likely not to run a candidate.