Is there a way to find an RSA private key after it is encrypted with the corresponding public key? Does there exist an algorithm to recover the private key in this special case?
If an appropriate padding mechanism is used then you cannot directly encrypt the private key with the public key as the input size would be too large. If hybrid encryption is used (RSA + AES) then the answer is no.
You could possibly raise (modulo $n$) the private exponent to the power of the public exponent. So then you would need to retrieve $d$ from $d ^ e \mod n$. I don't see how you could retrieve $d$ from that, even if you take into account how $d$ and $n$ are calculated.
Note that you could not do this if you would require the CRT parameters instead of the private exponent $d$.