Use ElGamal to solve Diffie-Hellman problem Say we are able to decrypt a Elgamal ciphertext $c$ using only the public key. Apparantly it is now possible to solve the Diffie-Hellman problem (given $g^a, g^b$ calculate $g^{ab}$). How?

I know how it works the otherway. When I'm able to solve DH, I found a way to decrypt Elgamal ciphertext. But I can't come up with a good idea for this problem.

I figured since $c=(c_1,c_2)=(v,w)$ maybe $c_1=g^a, c_2=g^b$ such that the plaintext $p=g^{ab}$, but that doesn't work out.

I'm thankful for any tips.
Since you're learning, I won't go ahead and give you the answer; I will try to point you in the right direction.

Consider what Elgamal decryption is: if the private key is $a$ (and hence the public key is $g^a$), and you're given a ciphertext $(b, c)$, the ElGamal decryption is $b^{-a} \cdot c$

Now, assume that we have an Oracle that, given an arbitrary public key $g^x$, and a ciphertext $(g^y, z)$, would return the ElGamal decryption.  Can you see how you could use that by passing it a public key and ciphertext of your choosing (hint: write out what the decryption would look like in that case)?

(Further hint: given $g^x$, it is easy to compute $g^{-x}$, as that's just the multiplicative inverse)