on the existence, uniqueness, and basis properties of radial eigenfunctions of a semilinear second-order elliptic equation in a ball

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ID: 229098
2009
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Abstract
We consider the following eigenvalue problem: โˆ’ฮ”๐‘ข+๐‘“(๐‘ข)=๐œ†๐‘ข, ๐‘ข=๐‘ข(๐‘ฅ), ๐‘ฅโˆˆ๐ต={๐‘ฅโˆˆโ„3โˆถ|๐‘ฅ|<1}, ๐‘ข(0)=๐‘>0, ๐‘ข||๐‘ฅ|=1=0, where ๐‘ is an arbitrary fixed parameter and ๐‘“ is an odd smooth function. First, we prove that for each integer ๐‘›โ‰ฅ0 there exists a radially symmetric eigenfunction ๐‘ข๐‘› which possesses precisely ๐‘› zeros being regarded as a function of ๐‘Ÿ=|๐‘ฅ|โˆˆ[0,1). For ๐‘>0 sufficiently small, such an eigenfunction is unique for each ๐‘›. Then, we prove that if ๐‘>0 is sufficiently small, then an arbitrary sequence of radial eigenfunctions {๐‘ข๐‘›}๐‘›=0,1,2,โ€ฆ, where for each ๐‘› the ๐‘›th eigenfunction ๐‘ข๐‘› possesses precisely ๐‘› zeros in [0,1), is a basis in ๐ฟ๐‘Ÿ2(๐ต) (๐ฟ๐‘Ÿ2(๐ต) is the subspace of ๐ฟ2(๐ต) that consists of radial functions from ๐ฟ2(๐ต). In addition, in the latter case, the sequence {๐‘ข๐‘›/โ€–๐‘ข๐‘›โ€–๐ฟ2(๐ต)}๐‘›=0,1,2,โ€ฆ is a Bari basis in the same space.
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zhidkov2009internationalon Use this key to autocite in the manuscript while using SciMatic Manuscript Manager or Thesis Manager
Authors Peter Zhidkov
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Journal structural engineering and mechanics
Year 2009
DOI
10.1155/2009/243048
URL
http://dx.doi.org/10.1155/2009/243048
Keywords
mathematics

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