Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains

Title:

Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains

Creator:

Suzuki, Toshiyuki

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:ed2afff1-71ac-47a4-b861-ce1dd7772bec
uuid:ed2afff1-71ac-47a4-b861-ce1dd7772bec
doi:10.21136/MB.2014.143851

Subject:

energy method, nonlinear Schrödinger equation, inverse-square potential, and Hardy-Poincaré inequality

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

Nonlinear Schrödinger equations (NLS)a with strongly singular potential a|x| −2 on a bounded domain Ω are considered. If Ω = R N and a > −(N − 2)2 /4, then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here a = −(N − 2)2 /4 is excluded because D(P 1/2 a(N) ) is not equal to H 1 (R N ), where Pa(N) := −∆ − (N − 2)2 /(4|x| 2 ) is nonnegative and selfadjoint in L 2 (R N ). On the other hand, if Ω is a smooth and bounded domain with 0 ∈ Ω, the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua (2000). Hence we can see that H 1 0 (Ω) ⊂ D(P 1/2 a(N) ) ⊂ H s (Ω) (s < 1). Therefore we can construct global weak solutions to (NLS)a on Ω by the energy methods.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

231-238

Source:

Mathematica bohemica | 2014 Volume:139 | Number:2

Harvested from:

CDK

Metadata only:

false