When a person has a long-term health problem, you say that they "suffer from" that problem:

suffer from depression

suffer from insomnia (not being able to sleep)

suffer from lower back pain

"Obesity" means being very, very overweight. "Obesity" is a technical term that is used by doctors, but normal people also use it when talking about fatness as a general problem:

Obesity is a serious problem in the U.S.