It should be 'Many people have problems.'

There is much trouble for many people. Some people have many troubles. So much confusion for so many students who want to learn English!

It should be "MANY people HAVE problems." Most of the time in everyday English, "people" is the plural form of "person." "People" = pessoas, "person" = pessoa. Since it is plural, it takes the third-person plural verb form. "The people have spoken." (Not "has") "The people of Brazil speak Portuguese. (Not "speaks.") (It gets complicated because there is a second use, in which "people" means "povo," and the rules are different. However, it is not common in everyday spoken English). In your example "people" is working as a countable noun, we talk about "many" people, not "much" people. We make a distinction between "many" and "much." I guess they are both "muitos" in Portuguese, but they are different in English. "How many apples did you buy?" "I bought five apples." "How much sand did you buy?" "I bought two tons."