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DC Field | Value | Language |
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dc.contributor.author | Winter, M | - |
dc.contributor.author | Wei, J | - |
dc.coverage.spatial | 29 | en |
dc.date.accessioned | 2007-01-15T13:14:40Z | - |
dc.date.available | 2007-01-15T13:14:40Z | - |
dc.date.issued | 2005 | - |
dc.identifier.citation | Winter, M. and Wei, J. (2005) 'Symmetry of Nodal Solutions for Singularly Perturbed Elliptic Problems on a Ball', Indiana University Mathematics Journal, 54(3), pp. 707-742. doi:10.1512/iumj.2005.54.2546. | en |
dc.identifier.uri | http://bura.brunel.ac.uk/handle/2438/523 | - |
dc.description.abstract | In [40], it was shown that the following singularly perturbed Dirichlet problem \ep^2 \Delta u - u+ |u|^{p-1} u=0, \ \mbox{in} \ \Om,\] \[ u=0 \ \mbox{on} \ \partial \Om has a nodal solution u_\ep which has the least energy among all nodal solutions. Moreover, it is shown that u_\ep has exactly one local maximum point P_1^\ep with a positive value and one local minimum point P_2^\ep with a negative value and, as \ep \to 0, \varphi (P_1^\ep, P_2^\ep) \to \max_{ (P_1, P_2) \in \Om \times \Om } \varphi (P_1, P_2), where \varphi (P_1, P_2)= \min (\frac{|P_1-P_2}{2}, d(P_1, \partial \Om), d(P_2, \partial \Om)). The following question naturally arises: where is the {\bf nodal surface} \{ u_\ep (x)=0 \}? In this paper, we give an answer in the case of the unit ball \Om=B_1 (0). In particular, we show that for \epsilon sufficiently small, P_1^\ep, P_2^\ep and the origin must lie on a line. Without loss of generality, we may assume that this line is the x_1-axis. Then u_\ep must be even in x_j, j=2, ..., N, and odd in x_1. As a consequence, we show that \{ u_\ep (x)=0 \} = \{ x \in B_1 (0) | x_1=0 \}. Our proof is divided into two steps: first, by using the method of moving planes, we show that P_1^\ep, P_2^\ep and the origin must lie on the x_1-axis and u_\ep must be even in x_j, j=2, ..., N. Then, using the Liapunov-Schmidt reduction method, we prove the uniqueness of u_\ep (which implies the odd symmetry of u_\ep in x_1). Similar results are also proved for the problem with Neumann boundary conditions. | en |
dc.format.extent | 274446 bytes | - |
dc.format.mimetype | application/pdf | - |
dc.language.iso | en | - |
dc.publisher | Indiana Univ Math J | en |
dc.subject | Symmetry of nodal solutions, | en |
dc.subject | singular perturbation problems, method of moving planes, Liapunov-Schmidt reduction | en |
dc.title | Symmetry of Nodal Solutions for Singularly Perturbed Elliptic Problems on a Ball | en |
dc.type | Research Paper | en |
dc.identifier.doi | https://doi.org/10.1512/iumj.2005.54.2546 | - |
Appears in Collections: | Dept of Mathematics Research Papers Mathematical Sciences |
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