关于$\mathbb｛R｝^｛N｝中的Kirchhoff型方程$

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2019-08-04 DOI:10.57262/ade027-0304-97

Juntao Sun, Tsung‐fang Wu

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引用次数: 0

摘要

考虑如下的非线性Kirchhoff型方程\begin{equation*} \\ begin{array}{ll} -\left(a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+b\right) \Delta u+u=f(x)\left\vert u\right\vert ^{p-2}u & \text{in}\mathbb{R}^{N}， \\ u\in H^{1}(\mathbb{R}^{N})， & \end{array}% \right。结束\{方程*}% N \组的1美元,a, b > 0,剩下2 < p < \敏\ \ {4,2 ^ {\ ast} \右\}美元($ 2 ^ {\ ast} = \ infty为N = 1美元2 $和$ ^ {\ ast} = 2 N / (N - 2)对N \组3美元)和函数f \美元在C (\ mathbb {R} ^ {N}) \帽L ^ {\ infty} (\ mathbb {R} ^ {N})美元。区别于已有的文献结果，我们更感兴趣的是与上述问题相关的能量泛函的几何性质。进一步证明了正解的不存在性、存在性、唯一性和多重性依赖于参数a和维数N。特别地，我们得出$1\leq N\leq4$存在一个唯一正解，而$N\geq5$允许至少两个正解。

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On the Kirchhoff type equations in $\mathbb{R}^{N}$

Consider a nonlinear Kirchhoff type equation as follows \begin{equation*} \left\{ \begin{array}{ll} -\left( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+b\right) \Delta u+u=f(x)\left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R}^{N}, \\ u\in H^{1}(\mathbb{R}^{N}), & \end{array}% \right. \end{equation*}% where $N\geq 1,a,b>0,2

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.

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