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What is the difference between "there is not" and "there does not" ?

For example:

there does not exist a unique optimal solution for such problems.

versus

there is not exist a unique optimal solution for such problems.

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"There is not exist" is ungrammatical. We negate a present form of verb using the do-support.

See how we negate it "It does not exist".

The same goes to your sentence. There does not exist would be correct.

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