A Concentration Phenomenon for p-Laplacian Equation
It is proved that if the bounded function of coefficient Qn in the following equation -div {|∇u|p-2∇u}+V(x)|u|p-2u=Qn(x)|u|q-2u, u(x)=0 as x∈∂Ω. u(x)⟶0 as |x|⟶∞ is positive in a region contained in Ω and negative outside the region, the sets {Qn>0} shrink to a point x0∈Ω as n→∞, and then...
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Wiley
2014-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2014/148902 |
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| author | Yansheng Zhong |
| author_facet | Yansheng Zhong |
| author_sort | Yansheng Zhong |
| collection | DOAJ |
| description | It is proved that if the bounded function of coefficient Qn in the following equation -div {|∇u|p-2∇u}+V(x)|u|p-2u=Qn(x)|u|q-2u, u(x)=0 as x∈∂Ω. u(x)⟶0 as |x|⟶∞ is positive in a region contained in Ω and negative outside the region, the sets {Qn>0} shrink to a point x0∈Ω as n→∞, and then the sequence un generated by the nontrivial solution of the same equation, corresponding to Qn, will concentrate at x0 with respect to W01,p(Ω) and certain Ls(Ω)-norms. In addition, if the sets {Qn>0} shrink to finite points, the corresponding ground states {un} only concentrate at one of these points. These conclusions extend the results proved in the work of Ackermann and Szulkin (2013) for case p=2. |
| format | Article |
| id | doaj-art-265b6037b94f40be8dc159a86e418252 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-265b6037b94f40be8dc159a86e4182522025-02-03T06:15:12ZengWileyJournal of Applied Mathematics1110-757X1687-00422014-01-01201410.1155/2014/148902148902A Concentration Phenomenon for p-Laplacian EquationYansheng Zhong0 Department of Mathematics, Fujian Normal University, Fuzhou 350117, ChinaIt is proved that if the bounded function of coefficient Qn in the following equation -div {|∇u|p-2∇u}+V(x)|u|p-2u=Qn(x)|u|q-2u, u(x)=0 as x∈∂Ω. u(x)⟶0 as |x|⟶∞ is positive in a region contained in Ω and negative outside the region, the sets {Qn>0} shrink to a point x0∈Ω as n→∞, and then the sequence un generated by the nontrivial solution of the same equation, corresponding to Qn, will concentrate at x0 with respect to W01,p(Ω) and certain Ls(Ω)-norms. In addition, if the sets {Qn>0} shrink to finite points, the corresponding ground states {un} only concentrate at one of these points. These conclusions extend the results proved in the work of Ackermann and Szulkin (2013) for case p=2.http://dx.doi.org/10.1155/2014/148902 |
| spellingShingle | Yansheng Zhong A Concentration Phenomenon for p-Laplacian Equation Journal of Applied Mathematics |
| title | A Concentration Phenomenon for p-Laplacian Equation |
| title_full | A Concentration Phenomenon for p-Laplacian Equation |
| title_fullStr | A Concentration Phenomenon for p-Laplacian Equation |
| title_full_unstemmed | A Concentration Phenomenon for p-Laplacian Equation |
| title_short | A Concentration Phenomenon for p-Laplacian Equation |
| title_sort | concentration phenomenon for p laplacian equation |
| url | http://dx.doi.org/10.1155/2014/148902 |
| work_keys_str_mv | AT yanshengzhong aconcentrationphenomenonforplaplacianequation AT yanshengzhong concentrationphenomenonforplaplacianequation |