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|Title:||The Lazer-McKenna conjecture and a free boundary problem in two dimensions||Contributor(s):||Dancer, Edward N (author); Yan, Shusen (author)||Publication Date:||2008||DOI:||10.1112/jlms/jdn045||Handle Link:||https://hdl.handle.net/1959.11/2976||Abstract:||We prove that certain super-linear elliptic equations in two dimensions have many solutions when the diffusion is small. We find these solutions by constructing solutions with many sharp peaks. In three or more dimensions, this has already been proved by the authors in 'Comm. Partial Differential Equations' 30 (2005) 1331-1358. However, in two dimensions, the problem is much more difficult because there is no limit problem in the whole space. Therefore, the proof is quite different, though still a reduction argument. A direct consequence of this result is that we give a positive answer to the Lazer-McKenna conjecture for some typical nonlinearities in two dimensions.||Publication Type:||Journal Article||Source of Publication:||Journal of London Mathematical Society, 78(3), p. 639-662||Publisher:||Oxford University Press||Place of Publication:||Oxford, United Kingdom||ISSN:||0024-6107||Field of Research (FOR):||010506 Statistical Mechanics, Physical Combinatorics and Mathematical Aspects of Condensed Matter||Socio-Economic Outcome Codes:||970101 Expanding Knowledge in the Mathematical Sciences||Peer Reviewed:||Yes||HERDC Category Description:||C1 Refereed Article in a Scholarly Journal||Statistics to Oct 2018:||Visitors: 156
|Appears in Collections:||Journal Article|
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