$$\mathbb {R}^{N}$ 中双谐方程正解的存在性

IF 1.2 3区数学 Q1 MATHEMATICS

Annals of Functional Analysis Pub Date : 2024-06-03 DOI:10.1007/s43034-024-00362-9

Wenbo Wang, Jixiang Ma, Jianwen Zhou

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引用次数: 0

摘要

本文研究的是双谐方程 $$\begin{aligned}\Delta ^{2}u=K(x)f(u)\quad \text {in }~\mathbb { R}^{N}.\end{aligned}$$在合适的假设条件下，可以得到正解的存在。这里使用的方法包含积分算子和 Schauder 定点理论。由于$\mathbb {R}^{N}$ 中$\Delta ^{2}u=0$的基本解的形式取决于 N，我们将讨论分为三种情况：（a）$N=2$；（b）$N=4$；（c）$N>2$ but$Nne 4$。在我们的结果中，$\Delta ^{2}$的基本解起着至关重要的作用。

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Existence of positive solutions to the biharmonic equations in $\mathbb {R}^{N}$

This article considers the biharmonic equation

$$\begin{aligned} \Delta ^{2}u=K(x)f(u)\quad \text {in }~\mathbb { R}^{N}. \end{aligned}$$

Under suitable assumptions, the existence of positive solutions is obtained. The methods used here contain the integral operator and the Schauder fixed point theory. Since the form of fundamental solution of $\Delta ^{2}u=0$ in $\mathbb {R}^{N}$ depends on N, we divide our discussions into three cases as (a) $N=2$; (b) $N=4$; (c) $N>2$ but $N\ne 4$. The fundamental solution of $\Delta ^{2}$ plays an essential role in our results.

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来源期刊

Annals of Functional Analysis MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

2.00

自引率

10.00%

发文量

期刊介绍： Annals of Functional Analysis is published by Birkhäuser on behalf of the Tusi Mathematical Research Group. Ann. Funct. Anal. is a peer-reviewed electronic journal publishing papers of high standards with deep results, new ideas, profound impact, and significant implications in all areas of functional analysis and all modern related topics (e.g., operator theory). Ann. Funct. Anal. normally publishes original research papers numbering 18 or fewer pages in the journal’s style. Longer papers may be submitted to the Banach Journal of Mathematical Analysis or Advances in Operator Theory. Ann. Funct. Anal. presents the best paper award yearly. The award in the year n is given to the best paper published in the years n-1 and n-2. The referee committee consists of selected editors of the journal.