Non-negative solutions of a sublinear elliptic problem

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Julián López-Gómez, Paul H. Rabinowitz, Fabio Zanolin
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引用次数: 0

Abstract

In this paper, the existence of solutions, \((\lambda ,u)\), of the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\lambda u -a(x)|u|^{p-1}u &{} \quad \hbox {in }\Omega ,\\ u=0 &{}\quad \hbox {on}\;\;\partial \Omega , \end{array}\right. \end{aligned}$$

is explored for \(0< p < 1\). When \(p>1\), it is known that there is an unbounded component of such solutions bifurcating from \((\sigma _1, 0)\), where \(\sigma _1\) is the smallest eigenvalue of \(-\Delta \) in \(\Omega \) under Dirichlet boundary conditions on \(\partial \Omega \). These solutions have \(u \in P\), the interior of the positive cone. The continuation argument used when \(p>1\) to keep \(u \in P\) fails if \(0< p < 1\). Nevertheless when \(0< p < 1\), we are still able to show that there is a component of solutions bifurcating from \((\sigma _1, \infty )\), unbounded outside of a neighborhood of \((\sigma _1, \infty )\), and having \(u \gneq 0\). This non-negativity for u cannot be improved as is shown via a detailed analysis of the simplest autonomous one-dimensional version of the problem: its set of non-negative solutions possesses a countable set of components, each of them consisting of positive solutions with a fixed (arbitrary) number of bumps. Finally, the structure of these components is fully described.

Abstract Image

亚线性椭圆问题的非负解法
在本文中,问题$$\begin{aligned}的解((\lambda ,u)\)的存在性-Delta u=\lambda u -a(x)|u|^{p-1}u &{}\quad \hbox {in }\Omega ,\ u=0 &{}\quad \hbox {on}\;\;\partial \Omega , \end{array}\right.\end{aligned}$$是针对(0< p < 1)进行探索的。当(p>1)时,众所周知,这种解有一个无界部分从((sigma _1,0))分叉,其中(sigma _1)是在迪里希特边界条件下(partial \Omega \)中(-\Delta \)的最小特征值。这些解都有\(u \in P\), 正锥的内部。如果\(0< p <1\),当\(p>1\)时用来保持\(u\in P\) 的延续论证就失效了。尽管如此,当\(0< p < 1\) 时,我们仍然能够证明有一部分解是从\((\sigma _1,\infty )\)分叉出来的,在\((\sigma _1,\infty )\)的邻域之外是无界的,并且有\(u gneq 0\).通过对最简单的自主一维问题的详细分析,我们可以发现u的这种非负性是无法改进的:它的非负解集合拥有一组可数的成分,其中每个成分都由具有固定(任意)凹凸数的正解组成。最后,对这些组成部分的结构进行了全面描述。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Accounts of Chemical Research
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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