It's often thought, quite wrongly, that empiricism is a philosophy that shows that we can discover through the experimental method what exists, and that only what can be empirically evidenced can exist
I don't know anyone who thinks that...that only what can be empirically evidenced can exist. That doesn't make any sense 
Empiricism is a theory about how knowledge comes to us - it says nothing about what does and doesn't exist, just how we know.
Just because our knowledge is based on experience, and experience is subjective it is a logical fallacy to conclude that therefore all knowledge is subjective.
You are forgetting that there are two components to "experience". The act of experiencing and the thing that's being experienced.
I am typing on my laptop. I am experiencing that & my experience is subjective. The laptop itself is not subjective - it's an objective thing. It's existence does not, in any way, rely on my subjective experience of it. It's either there as a thing, or it is not.
Descartes was right in that it's impossible to truly know with 100% certainty that the laptop (or anything else) is truly there - I could be imagining/dreaming it.
Therefore, empiricism - which is all about evidence & testing our knowledge against natural world observations & data - is the only reliable method we have that at least attempts to sort out for us the subjective from the objective.
It certainly does not lead to the "problem of solipsism" - although it depends what you mean by that.
That we can only be sure that we're thinking and nothing else? That's true, as I said above.
That my mind is the only mind there is? That might be true, but empiricism doesn't inevitably lead to that. Not sure why you think it does.
I have to go out now.