On the Existence, Uniqueness, and Basis Properties of Radial Eigenfunctions of a Semilinear Second-Order Elliptic Equation in a Ball
We consider the following eigenvalue problem: −Δ𝑢+𝑓(𝑢)=𝜆𝑢, 𝑢=𝑢(𝑥), 𝑥∈𝐵={𝑥∈ℝ3∶|𝑥|<1}, 𝑢(0)=𝑝>0, 𝑢||𝑥|=1=0, where 𝑝 is an arbitrary fixed parameter and 𝑓 is an odd smooth function. First, we prove that for each integer 𝑛≥0 there exists a radially symmetric eigenfunction 𝑢𝑛 which possesses prec...
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| Format: | Article |
| Language: | English |
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Hindawi Limited
2009-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2009/243048 |
| Summary: | We consider the following eigenvalue problem: −Δ𝑢+𝑓(𝑢)=𝜆𝑢, 𝑢=𝑢(𝑥), 𝑥∈𝐵={𝑥∈ℝ3∶|𝑥|<1}, 𝑢(0)=𝑝>0, 𝑢||𝑥|=1=0, where 𝑝 is an arbitrary fixed parameter and 𝑓 is an odd smooth function. First, we prove that for each integer 𝑛≥0 there exists a radially symmetric eigenfunction 𝑢𝑛 which possesses precisely 𝑛 zeros being regarded as a function of 𝑟=|𝑥|∈[0,1). For 𝑝>0 sufficiently small, such an eigenfunction is unique for each 𝑛. Then, we prove that if 𝑝>0 is sufficiently small, then an arbitrary sequence of radial eigenfunctions {𝑢𝑛}𝑛=0,1,2,…, where for each 𝑛 the 𝑛th eigenfunction 𝑢𝑛 possesses precisely 𝑛 zeros in [0,1), is a basis in 𝐿𝑟2(𝐵) (𝐿𝑟2(𝐵) is the subspace of 𝐿2(𝐵) that
consists of radial functions from 𝐿2(𝐵). In addition, in the latter case, the sequence {𝑢𝑛/‖𝑢𝑛‖𝐿2(𝐵)}𝑛=0,1,2,… is a Bari basis in the same space. |
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| ISSN: | 0161-1712 1687-0425 |