Hi,
I think that in the sentence below "exist" should be used. This is how I interpret the sentence: Many different techniques exist there to solve this problem.
There exists many different techniques to solve this problem.
It is very unnatural besides the mismatch of subject and verb. This is natural English: There are many different techniques that can be employed to solve this problem. There exist many solutions to this problem.
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It is very unnatural besides the mismatch of subject and verb.
This is natural English:
There are many different techniques that can be employed to solve this problem.
There exist many solutions to this problem.