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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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 1110-757X 1687-0042 |