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I did a search for there is no any on Google (books, Ngrams). So I think the wording there is no any is grammatical. But I can't see a difference, if any, between two phrases indicated in the title. Are they interchangeable? Is the any redundant here?
There is no any case study based on this.
There is no case study based on this
"There is no any case study based on this" is ungrammatical in standard English.
"There is no case study based on this" is fine.
"There is not any case study based on this" means "There is no case study of any kind based on this."
The two grammatical phrases are not interchangeable.
On Page 1 of your Google Book search, only two of the ten search results (Numbers 1 and 4) are unqualified examples of grammatical usage.
The very first return on the google book search that you did demonstrates that this exact construction there is no any is ungrammatical in standard English.
It is an example from the book The teacher's grammar of English, and the context is talking about grammatical errors that leaners make.
The errors shown in (44) were found in a review by a Korean graduate student. The student clearly is having difficulty producing negative sentences with nonreferential there.
[44] c *Nevertheless, there is no any vocabulary or idiom lists, and no any grammatical explanation or excerises.
(44c) is an example of an error. Note that it is traditional to put an asterisk (*) before an ungrammatical sentence in books about English grammar or usage, as the book has done.
Now let's look at the Google Book search that gives the example you ask about. It just happens to demonstrate, four times, the exact error mentioned in the textbook above.
This return is by highly educated Victor I Chang in the $225 book Delivery and Adoption of Cloud Computing Services in Contemporary Organizations. He has a Faculty homepage at University of Southampton and a penchant for writing in the Royal We.
On this page (96) there are four uses of There is no any... They are all ungrammatical from the point of view of standard English.
For example:
The proposed method is a conceptual idea. It offers a generic recommendation. There is no any case study or demonstration based on this.
This is ungrammatical from the point of view of standard English.
By contrast, the most natural and grammatical way to say this is
The proposed method is a conceptual idea. It offers a generic recommendation. There is no case study or demonstration based on this.
or in the plural
The proposed method is a conceptual idea. It offers a generic recommendation. There are no case studies or demonstrations based on this.
If you want to use "not any," it would work more naturally in the plural:
There are not any case studies...
You can also say
There is not any case study or demonstration based on this.
"There is not any case study" means "There is no case study of any kind."
The "there is not any" construction gets plenty of use.
Note: On Page 174 of their work, Mr Chang also writes, ungrammatically: "There is no much difference whether it has been accessed using a mobile device or a standalone computer" (sic). And they have other grammatical errors in the parts of the book I checked.
Besides the result from The teacher's grammar of English, only Return 4 is definitely grammatical. In two results (Returns 3, 7) on the first page of the Google Book search
there are no any...
appears in a mathematical or scientific or logic formula. I am not a mathematician or a scientist or a logician, and I do not write formulas (formulae). Therefore I don't know if this is considered "grammatical" to the worldwide members of the scientific/mathematical community. It does not seem grammatical to me.
In regular writing, even in scientific journals and technical reports, the phrase is ungrammatical (again: from the point of view of standard English).
Other returns from the first page of the search show ungrammatical/nonstandard uses (Returns 5, 9, 10) or misprints of various sorts (Return 6, 8).
The third return is the first example appearing in a formula. It comes from Proceedings of the Sixth Asian Logic Conference: Beijing, China, 20-24 May 1996 in a chapter called "Default Logic and it's Variants: A Semantical View," by M. Zhang.
Notice that there is an error in the chapter's title (it's should be its), so I am not sure how much we can trust this author to demonstrate standard usage.
There is no any subset D' of D such that...
Logicians might consider it acceptable usage in such a formula. To me, it is not grammatical, for the same reason it is not grammatical in any English.
Compare this to the fourth return (Outline of a Phenomenology of Right):
there is no "any given" individual member
which is grammatical. Because "any given" is in quotes, the phrase is not "there is no any." This usage is similar to:
There is no "any given Sunday" on which I watch American football.
Note: "any given Sunday" is a U.S. idiom about American football.
The fifth return (Handbook of the Sociology of Health, Illness, and Healing: A Blueprint for ...) is absolutely ungrammatical and contains another nonstandard usage (get recovered from):
Since there is no any treatment that one can receive and get recovered from AIDS, then it is better to use the condoms which can protect our lives for some days before death.
On further inspection, one can see that the whole section (the four paragraphs) contains very many errors, from the point of view of standard grammar.
The summary of Return 6 (London Medical and Surgical Journal, Том 2) is a misprint. When you click on the return link, you get
If there is not any pain or inflammation...
Return 7 is a usage in a mathematical formula (Scientia Magna, International Book Series, Vol. 3, No. 1) in a chapter written by Q. Wu:
Conjecture: There is no any positive integer n...
See comments about Return 3.
Return 8 (A Complete Collection Of State-Trials And Proceedings For High ..., Том 4) shows that the scan of Page 584 has been cut off vertically along the right side. So what looks like
..., but there is no
followed by
any
starting a new line is not how the text actually reads.
In Return 9, Modern Bamboo Structures: Proceedings of the First International Conference by Yan Xiao, Masafumi Inoue, Shyam K. Paudel
In Japan, nearly all the bamboo forests are not maintained any longer...Especially since there is no any longer a large demand for bamboo material...
is especially ungrammatical.
Return 10 (I am not intelligent) contains very many errors from the point of view of standard English, including
Ir's just because there is no any obvious enemy in front of him to fight back.
Other nonstandard usages in this paragraph alone include:
a. The subject will find out what he face are the invisible enemies.
b. If the subject can't do that it's the another way to drive people mad.
You can find out more about I am Not Intelligent at the publisher's website. It appears to be about mental illness and schizophrenia in India, so this might explain the nonstandard English.
There is no any
is ungrammatical in the standard Englishes of the USA, Canada, the UK, Ireland, Australia, New Zealand, the Philippines, and other places that accept such English(es) as standard. This is shown in the very first return, a book about teaching English (and a very good one, I might add, that I have consulted many times--and I recommend it to the advanced learner).
There is not any case study
means "There is no case study of any kind."
It is similar to, but not always interchangeable with
There is no case study
Two returns (3 and 7) use "there is no any" in a formula, and I do not speak the English of scientific formulae. It would be considered ungrammatical outside of a formula.
Finally: A great number of the results on Pages 2 and 3 of your search are ungrammatical and are written by authors whose background seems not to be that of a native speaker of the Englishes I know about. Many of these may be (native) speakers of Indian English, and I do not know what is standard in that variety of English.
