关于$\mathbb｛R｝^｛N｝中的Kirchhoff型方程$

IF 1.5 3区数学 Q1 MATHEMATICS

Advances in Differential Equations 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

Advances in Differential Equations MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

1.90

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Advances in Differential Equations will publish carefully selected, longer research papers on mathematical aspects of differential equations and on applications of the mathematical theory to issues arising in the sciences and in engineering. Papers submitted to this journal should be correct, new and non-trivial. Emphasis will be placed on papers that are judged to be specially timely, and of interest to a substantial number of mathematicians working in this area.

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