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The phrase "there exist many" is a correct and usable phrase in written English.
You can use this phrase to introduce a list of items or to indicate that something is true or exists in multiple forms. For example, "There exist many different approaches to studying a foreign language" or "There exist many paths one can take to pursue their career in engineering".
Exact(60)
Out in the arms there exist many young, bright, hot blue stars, and these are easily resolved.
In the literature there exist many examples of different distances.
Usually there exist many separating hyperplanes for a given linear binary classification problem.
Bui-Duong [11] pointed out that there exist many doubling balls.
There exist many professionals who promote the tenets of Islam while also condemning terrorism.
There exist many elementary means.
There exist many approaches for profile modeling.
There exist many different methods designed to achieve different goals.
There exist many design criteria, which can be numerically optimized.
There exist many designs for unequal probability sampling.
There exist many soil infiltration rate measurement methods.
More suggestions(17)
there exist frequent
there exist considerable
there exist extensive
there exist diverse
there exist innumerable
there arise many
there exist substantial
there exist myriad
there exist lots
there exist multiple
there exist different
there exist many
there exist numerous
there exist few
there exist several
there exist large
there remain many
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com