本文主要评论的文献是Qi 和 Zou 于 2022 年发表在 SIAM J. Math. Anal. 上的论文“EXACT NUMBER OF POSITIVE SOLUTIONS FOR THE KIRCHHOFF EQUATION”。该论文研究了非线性 Kirchhoff 方程正解的确切数量和表达式，并揭示了非局部项对解的影响。这篇论文的研究成果对于理解Kirchhoff方程的解的性质和结构具有重要意义，并为相关领域的研究提供了理论基础。本文将概述论文的主要结果和方法，分析论文的语言和结构，并具体讨论论文中研究方法和成果的优缺点以及可以带给我们的启发。
This article mainly reviews the paper "EXACT NUMBER OF POSITIVE SOLUTIONS FOR THE KIRCHHOFF EQUATION" published by Qi and Zou in SIAM J. Math. Anal. in 2022. The paper studies the exact number and expression of positive solutions for the nonlinear Kirchhoff equation and reveals the influence of the nonlocal term on the solution. The research findings of this paper are of great significance for understanding the nature and structure of the solutions of the Kirchhoff equation and provide a theoretical basis for research in related fields. This article will summarize the main results and methods of the paper, analyze the language and structure of the paper, and specifically discuss the advantages and disadvantages of the research methods and findings in the paper, as well as the inspiration they can bring to us.

这篇论文中，作者通过变分方法和Lyapunov-Schmidt降低维度方法，分析了具有预定质量规范解的Kirchhoff方程，并给出了正规范解的精确数量和表达式。此外，作者还回答了关于具有固定频率的Kirchhoff方程正解数量的一个未解决问题，并观察到一些新的现象，这些现象与相应的非线性Schrödinger方程的现象完全不同，揭示了非局部项的特殊影响。作者在前言部分以定理的形式列出了下列成果：1. 对于2 < p < 2*的情况，论文建立了质量阈值，并给出了所有正规化解的精确数量和表达式。2. 对于p = 2*的情况，论文建立了非负的正规化解的数量和质量阈值。3. 对于1 ≤ N < 4的情况，论文分别考虑了p = 2 + 8N和2 + 8N < p < 2*的情况，并给出了正规化解的精确数量和表达式。4. 对于p = 2 + 4N的情况，论文建立了质量阈值，并给出了唯一正规范解的表达式。
This paper analyzes the Kirchhoff equation with prescribed quality norm solutions using variational methods and Lyapunov-Schmidt reduction techniques, and provides the exact number and expression of positive norm solutions. In addition, the author answers an unsolved problem regarding the number of positive solutions of the Kirchhoff equation with fixed frequency, and observes some new phenomena that are completely different from those of the corresponding nonlinear Schrödinger equation, revealing the special effect of nonlocal terms. The author lists the following achievements in the introduction in the form of theorems: 1. For the case 2 < p < 2*, the paper establishes a quality threshold and provides the exact number and expression of all regular solutions. 2. For the case p = 2*, the paper establishes the number and quality threshold of non-negative regular solutions. 3. For the case 1 ≤ N < 4, the paper considers the cases p = 2 + 8N and 2 + 8N < p < 2* separately, and provides the exact number and expression of regular solutions. 4. For the case p = 2 + 4N, the paper establishes a quality threshold and provides the expression of the unique positive norm solution.

这篇论文的语言风格严谨、专业、清晰、简洁，体现了作者的专业素养和严谨的治学态度。论文使用了大量的数学符号和公式，并严格遵守数学语言的规范，每个定理和结论都经过了严密的证明，并引用了相关的文献。同时，论文针对具有专业知识背景的读者，使用了专业的术语和概念，给出了能量曲线图，以便深入探讨非线性Kirchhoff方程的解的性质，从而给出清晰的结论。该论文的结构相对简单，论文在引言部分介绍了研究背景和主要采用的研究方法之后，就直接以定理的形式给出主要结论，然后引言部分给出重要的引理以及必要的证明。论文的主体部分就基本上是以定理为中心给出具体的证明，当然，以定理加证明的行文模式这也是数学论文的一大特点。
This paper has a rigorous, professional, clear, and concise writing style, reflecting the author's professional quality and rigorous academic attitude. The paper uses a large number of mathematical symbols and formulas and strictly adheres to the norms of mathematical language. Each theorem and conclusion has been rigorously proven and relevant literature has been cited. At the same time, the paper uses professional terminology and concepts for readers with professional knowledge background, provides energy curve graphs for in-depth discussion of the properties of the solutions of nonlinear Kirchhoff equations, and thus gives clear conclusions. The structure of the paper is relatively simple. After introducing the research background and main research methods in the introduction, the main conclusions are presented in the form of theorems. Then, the introduction provides important lemmas and necessary proofs. The main body of the paper is basically centered around theorems to give specific proofs, of course, the writing pattern of theorems plus proofs is also a major characteristic of mathematical papers.

在这篇论文的引言部分，就介绍了Kirchhoff方程研究的两种主流思路，一种是固定Kirchhoff方程的频率，这样方程只有一个未知元，利用Bernstein和Pohozaev的一些工作，采用Lions提出的泛函抽象框架，我们可以给出某些解的性质，但能得出的结果很有限。另外一种是固定Kirchhoff方程的质量，我们可以得到方程组的形式，但方程的解就需要同时找到两个未知元了。这篇论文就是采用后面一种思路，采用这种思路是近些年比较热门的研究方向。虽然很多学者觉得后一种思路有较大研究空间，但是可靠有效的方法一直还没人找到。这篇论文的作者应该受到了Cazenave研究了固定频率的Schrödinger方程的正归一化解数量的成果的启发，想到了一种伸缩变换的方法，将所研究的Kirchhoff方程和对应的Schrödinger方程建立起联系。这种方法的巧妙之处在于充分利用上已知的结果去探索未知的结果。作者发现Kirchhoff方程和对应的Schrödinger方程之间的桥梁是一个代数方程。我们知道想要研究代数方程的解是要比偏微分方程容易的多。虽然五次以上代数方程没有公式解，但只需借助一些函数单调性的分析，我们就可以确定代数方程解的位置。只要可以确定解的位置，我们就可以很好的分析原来Kirchhoff方程的正规化解的存在性问题了。从这里，我们可以看出这篇论文采用的主要方法是很巧妙的。这种方法避开了复杂和繁琐的推导，以及难以驾驭的高级数学理论，却得到了不错的结果。当然这种方法的局限性也是明显的，一方面这种伸缩变换的技巧并不对所有的Kirchhoff方程都有效。实际上，我通过计算发现大部分Kirchhoff方程都不能采用这种伸缩变换的技巧。另一方面，不是所有的Kirchhoff方程对应的Schrödinger方程都存在很好的研究成果。实际上，非线性Schrödinger方程的解的研究也是一个很困难的问题，大部分的Schrödinger方程都还等着我们继续研究。这样一看，虽然作者采用的方法在他研究的那一个具体方程上十分有效，但是这种方法是缺少普遍性的。
In the introduction of this paper, two mainstream approaches to the study of Kirchhoff equations are introduced. One is to fix the frequency of the Kirchhoff equation, so that the equation has only one unknown variable. By utilizing some work of Bernstein and Pohozaev, and adopting the functional abstract framework proposed by Lions, we can give some properties of certain solutions, but the results obtained are very limited. The other is to fix the mass of the Kirchhoff equation, from which we can obtain the form of the equation system, but the solution of the equation then requires finding two unknown variables at the same time. This paper adopts the latter approach, which is a relatively popular research direction in recent years. Although many scholars believe that the latter approach has a large research space, a reliable and effective method has yet to be found. The authors of this paper should have been inspired by Cazenave's study on the normalizable solution quantity of the fixed-frequency Schrödinger equation, and thought of a scaling transformation method to establish a connection between the Kirchhoff equation under study and the corresponding Schrödinger equation. The cleverness of this method lies in fully utilizing the known results to explore unknown results. The authors found that the bridge between the Kirchhoff equation and the corresponding Schrödinger equation is an algebraic equation. We know that it is much easier to study the solutions of algebraic equations than partial differential equations. Although algebraic equations of degree five or higher do not have formula solutions, we can determine the location of the solutions of the algebraic equation by means of some analysis of the monotonicity of functions. As long as we can determine the location of the solutions, we can analyze the existence problem of the regular solutions of the original Kirchhoff equation very well. From here, we can see that the main method adopted in this paper is very clever. This method avoids complex and tedious derivation and difficult-to-handle advanced mathematical theories, yet achieves good results. Of course, the limitations of this method are also obvious. On the one hand, this scaling transformation technique is not effective for all Kirchhoff equations. In fact, I have found through calculation that most Kirchhoff equations cannot adopt this scaling transformation technique. On the other hand, not all Kirchhoff equations have good research results for the corresponding Schrödinger equations. In fact, the study of the solutions of nonlinear Schrödinger equations is also a very difficult problem, and most Schrödinger equations are still waiting for us to continue to study. In this way, although the method adopted by the author is very effective for the specific equation he studies, this method lacks universality.

论文中，作者还采用了变分法来构造满足质量约束的试探解，并通过分析能量泛函的临界点来证明解的存在性。此外，作者还利用了 Pohozaev 不等式和 Sobolev 不等式等工具来分析解的性质和数量。这些方法和技巧是研究Kirchhoff方程的常规方法，在证明的具体过程中，作者多次采用了这些技巧。不足之处在于，作者给出的证明过程中这类方法做的计算应当放在预备知识部分，以引理的性质展示出来。但作者都夹杂在主要定理的证明之中，给读者阅读造成了一定的压力和混乱。
In the paper, the authors also adopted the variational method to construct a trial solution that satisfies the quality constraints and prove the existence of the solution by analyzing the critical points of the energy functional. In addition, the authors have used tools such as the Pohozaev inequality and Sobolev inequality to analyze the properties and quantity of the solution. These methods and techniques are conventional methods for studying the Kirchhoff equation, and the authors have repeatedly used these techniques in the specific process of proof. The drawback is that the calculations of such methods in the proof given by the author should be placed in the preliminary knowledge section, showing the nature of lemmas. However, they are all mixed in the proof of the main theorem, which creates some pressure and confusion for the readers.

此外，这篇论文也揭示了非局部项对 Kirchhoff 方程解的影响。与非线性 Schrödinger 方程不同，Kirchhoff 方程的正解数量和存在范围与频率阈值密切相关。对应解的存在性和数目相关的结果，这篇论文所给出的也相当不错。当然，不足之处也很明显。论文只考虑了Sobolev次临界情况，而没有研究临界情况或超临界情况。并且论文只考虑了具有特定非线性项的Kirchhoff方程，而没有研究更一般形式的方程。这些都有待进一步的研究。
Additionally, this paper also reveals the influence of non-local terms on the solutions of the Kirchhoff equation. Unlike the nonlinear Schrödinger equation, the number and range of positive solutions of the Kirchhoff equation are closely related to the frequency threshold. The results provided by this paper on the existence and number of corresponding solutions are quite good. Of course, the shortcomings are also obvious. The paper only considers the Sobolev sub-critical case and does not study the critical or super-critical cases. Moreover, the paper only considers the Kirchhoff equation with a specific nonlinear term and does not study more general forms of equations. These issues all require further research.

这篇论文给我们的启示也很多，除了上面提到的对 Sobolev 超临界情况下的 Kirchhoff 方程的研究有很大研究空间以外，我们还可以进一步研究 Kirchhoff 方程解的稳定性、渐近行为和集中现象等问题。此外，我们还可以将论文中的方法推广到其他具有非局部项的偏微分方程，例如非线性 Schrödinger-Poisson 方程等。
This paper also offers many insights. In addition to the aforementioned research space on the Kirchhoff equation under Sobolev supercritical conditions, we can further study issues such as the stability, asymptotic behavior, and concentration phenomena of the solutions of the Kirchhoff equation. Moreover, we can generalize the methods in the paper to other partial differential equations with nonlocal terms, such as the nonlinear Schrödinger-Poisson equation, etc.

总之，这篇论文取得了一些有意义的成果，但也存在一些不足之处。未来可以进一步扩大研究范围，改进证明方法，验证结论的可靠性，并解决一些开放性问题。
In summary, this paper has achieved some meaningful results, but also has some shortcomings. In the future, it can further expand the scope of research, improve the proof methods, verify the reliability of the conclusions, and solve some open questions.