I have completed my M.Sc in Physics with specialization in Quantum Field theory. However, I have average CGPA of 3.35/6 in all the subjects, due to health issues. I completed the course in 2011.

I have some work done independently on Non-Hermitian Quantum mechanics and also hold a fellowship to pursue PhD by CSIR,India. It is earned by ranking in a Physics examination which is applied for, by thousands of students for Physics itself at a single time. I got selected to Pune University, through and nationwide entrance test, too. I also have experience of working in an industry for a startup.

However, whenever I apply to a place, the only reply I get is I did not meet their stringent requirements. I do not know what to do and how to convince. I just want to ask if it is this much competitive to get into a PhD program ! Any steps I need to take to improve my chances?

What country are you applying in? Just India? Do you mean that you are accepted into Pune, but they won't actually let you start a PhD? I don't understand.