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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Wiley
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 |
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| author | Peter Zhidkov |
| author_facet | Peter Zhidkov |
| author_sort | Peter Zhidkov |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-79911a4551804cc79a8bcb32ee8b5c39 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2009-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-79911a4551804cc79a8bcb32ee8b5c392025-02-03T05:52:18ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252009-01-01200910.1155/2009/243048243048On the Existence, Uniqueness, and Basis Properties of Radial Eigenfunctions of a Semilinear Second-Order Elliptic Equation in a BallPeter Zhidkov0Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, RussiaWe 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.http://dx.doi.org/10.1155/2009/243048 |
| spellingShingle | Peter Zhidkov On the Existence, Uniqueness, and Basis Properties of Radial Eigenfunctions of a Semilinear Second-Order Elliptic Equation in a Ball International Journal of Mathematics and Mathematical Sciences |
| title | On the Existence, Uniqueness, and Basis Properties of Radial Eigenfunctions of a Semilinear Second-Order Elliptic Equation in a Ball |
| title_full | On the Existence, Uniqueness, and Basis Properties of Radial Eigenfunctions of a Semilinear Second-Order Elliptic Equation in a Ball |
| title_fullStr | On the Existence, Uniqueness, and Basis Properties of Radial Eigenfunctions of a Semilinear Second-Order Elliptic Equation in a Ball |
| title_full_unstemmed | On the Existence, Uniqueness, and Basis Properties of Radial Eigenfunctions of a Semilinear Second-Order Elliptic Equation in a Ball |
| title_short | On the Existence, Uniqueness, and Basis Properties of Radial Eigenfunctions of a Semilinear Second-Order Elliptic Equation in a Ball |
| title_sort | on the existence uniqueness and basis properties of radial eigenfunctions of a semilinear second order elliptic equation in a ball |
| url | http://dx.doi.org/10.1155/2009/243048 |
| work_keys_str_mv | AT peterzhidkov ontheexistenceuniquenessandbasispropertiesofradialeigenfunctionsofasemilinearsecondorderellipticequationinaball |