I have a pain in my back around the upper left shoulder due to frequent coughing. I have taken antibiotics for 1 week but it doesn’t help. Is my back pain the result of a serious health problem?

There are two likely causes for the pain in your back:
 
1.    Frequent coughing can lead to muscle strain and this may not settle until your cough resolves.
 
2.    If you have an infection in your lung tissue, this can of course, lead a cough, as well as inflammation inside your chest wall which manifests as pain.
 
The way to treat your pain is to address the cough. If the antibiotics don't help resolve your symptoms, go back to your doctor to let him or her know as you may need additional treatment.

In addition, further tests may be necessary to eliminate other causes for your cough and pain.
 
I hope that this is helpful to you.