Does "have a problem with" refer to "being bothered" or "being the cause of a problem"?

Question

The following quote is from the BBC:

The strike has divided opinions in Rio. On social media, many support the sweepers' demands for better salaries. Others say they are opportunists and the situation is a big embarrassment to the city.

But Brazil's culture of littering the streets doesn't help. A video that went viral online shows that even authorities have a problem with that. Rio's mayor, Eduardo Paes, is seen throwing what seemed to be the rest of an apple on the sidewalk - and now promised to impose himself a fine for his wrongdoing. His government recently created a programme to keep Rio's residents from littering the streets.

I'm wondering what the phrase in bold means.

Apparently it means that the fact that many people litter on the street bothers them.
But I think the word "problem" has nothing to do with bothering or being bothered, does it?
I would say it means that authorities themselves stimulate littering on the streets.

I am really interested in hints to help figure these kinds of things out in other contexts by giving some examples saying "they have a problem with . . ." and to write the meaning.

Answer 1

The two conflicting meanings of "have a problem" presented here are:

1) Suffer from a problem/Find something difficult

2) Have an issue with someone/something (You don't like or don't agree with)

The meaning of "have a problem" in the text is clear; the culture of throwing litter is rife, and people are having a problem (finding it difficult) with getting out of this cultural habit (hence the phrase "even the authorities", with the example of the mayor throwing the apple core away).

Here are a couple of unrelated examples to clear up your doubts:

(John enters the classroom and overhears two other students saying negative things about him. He approaches them and says...)

"If you have a problem with me, say it to my face."

John means: "you don't like me/I bother you/we don't agree".

In a different example, we can see a similar but different meaning:

(John has been suffering an addiction for a while now, and it's getting out of control.)

"I have a problem with substance abuse and I can't get out of it".

Clearly, John is "suffering/struggling with" an addiction.

Lastly:

(John is finding 3rd conditionals difficult to form)

"I'm having a problem with this bloody perfect conditional thing. I can't seem to get it right!"

John is "finding it difficult" to do his grammar exercises.

Hopefully, these examples will help you. Always use the context of the sentence and other clues that texts almost always contain to help you choose the right meaning for a phrase you're having a problem with.

Answer 2

In this case I think he means 'are guilty of littering'. 'Littering' being the problem and 'authorities' being the people with it.