Morse Thoery of Saddle Point Reduction with Applications

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please see the attached PDF.

Comments for author File: Comments.pdf

Author Response

Thanks for your comments.

Reviewer 2 Report

Comments and Suggestions for Authors

The authors study the relationship between the Morse index and the critical groups before and after the saddle point reduction which is motivated by the study of multiple solutions for second-order Hamiltonian systems. Then, they study the existence of periodic solutions in second-order Hamiltonian systems as an application of their results.

I think the proofs have sufficient details and the organization of the paper is good. The paper would be of interest for those working in this direction.

Author Response

Thanks for your comments.

Reviewer 3 Report

Comments and Suggestions for Authors

In general, the writing and presentation of the content is deficient; the summary does not establish the problem being addressed, the methodology followed, nor the results obtained.

 

The introduction does not clearly establish the problem to be addressed or the hypothesis from which the research is developed, the references used are not recent, and it does not clearly establish the state of the art and what is the contribution of the article.

 

“For these problems, the variational method indicates that the solutions to equation (1) correspond to the critical points of a functional on a Hilbert space” The authors do not support this statement

 

“Thus, finding solutions to (1) translates into finding the critical points of a functional” Is it a result of the authors or, how is it justified?

 

Numerous theories have been developed to achieve this objective, including critical

point theory, degree theory, and Morse index theory. However, if the Palais–Smale (PS)

condition does not hold, these methods become significantly more challenging. To address

this issue, saddle point reduction is a viable approach. References are missing

 

The document must be reorganized, in section one the main result is presented, and in the next section the preliminaries that support the result are presented.

 

The same thing happens in section 3, after an example a second result is presented and after that the preliminaries of the same, in addition it is not established in the wording that motivates this second result or what is the hypothesis that leads to it.

 

It is necessary to establish conclusions and final comments

 

In the current form in which the document is presented, the relevance and contribution of the work cannot be defined.

Comments on the Quality of English Language

Minor editing of English language required

Author Response

Thanks for your comments, and we provided detailed responses to your comments in the cover letter. 

Author Response File: Author Response.pdf