scholarly journals Behaviors of solutions to a scalar-field equation involving the critical Sobolev exponent with the Robin condition

2005 ◽  
Vol 14 (1) ◽  
pp. 117-134
Author(s):  
Y. Kabeya
Keyword(s):  
Field Equation ◽  
Scalar Field ◽  
Critical Sobolev Exponent ◽  
Scalar Field Equation ◽  
Robin Condition ◽  
Sobolev Exponent

Quasilinear Scalar Field Equations Involving Critical Sobolev Exponents and Potentials Vanishing at Infinity

2018 ◽  
Vol 61 (3) ◽  
pp. 705-733 ◽  
Author(s):  
Athanasios N. Lyberopoulos
Keyword(s):  
Scalar Field ◽  
Weak Solutions ◽  
Bound States ◽  
Critical Sobolev Exponent ◽  
Field Equations ◽  
Critical Sobolev Exponents ◽  
Sobolev Exponent ◽  
Image Position ◽  
Scalar Field Equations

AbstractWe are concerned with the existence of positive weak solutions, as well as the existence of bound states (i.e. solutions inW1,p(ℝN)), for quasilinear scalar field equations of the form$$ - \Delta _pu + V(x) \vert u \vert ^{p - 2}u = K(x) \vert u \vert ^{q - 2}u + \vert u \vert ^{p^ * - 2}u,\qquad x \in {\open R}^N,$$where Δpu: =div(|∇u|p−2∇u), 1 <p<N,p*: =Np/(N−p) is the critical Sobolev exponent,q∈ (p, p*), whileV(·) andK(·) are non-negative continuous potentials that may decay to zero as |x| → ∞ but are free from any integrability or symmetry assumptions.

scholarly journals THE REAL SCALAR FIELD EQUATION FOR NARIAI BLACK HOLE IN THE 5D SCHWARZSCHILD–DE SITTER BLACK STRING SPACE

2007 ◽  
Vol 22 (24) ◽  
pp. 4451-4465 ◽  
Author(s):  
MOLIN LIU ◽  
HONGYA LIU ◽  
CHUNXIAO WANG ◽  
YONGLI PING
Keyword(s):  
Black Hole ◽  
Field Equation ◽  
Scalar Field ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
De Sitter ◽  
Black String ◽  
Transmission Coefficients ◽  
Scalar Field Equation ◽  
Continuous Potential

The Nariai black hole, whose two horizons are lying close to each other, is an extreme and important case in the research of black hole. In this paper we study the evolution of a massless scalar field scattered around in 5D Schwarzschild–de Sitter black string space. Using the method shown by Brevik and Simonsen (2001) we solve the scalar field equation as a boundary value problem, where real boundary condition is employed. Then with convenient replacement of the 5D continuous potential by square barrier, the reflection and transmission coefficients (R, T) are obtained. At last, we also compare the coefficients with the usual 4D counterpart.

scholarly journals Scalar field equation in the presence of signature change

1993 ◽  
Vol 48 (6) ◽  
pp. 2587-2590 ◽  
Author(s):  
Tevian Dray ◽  
Corinne A. Manogue ◽  
Robin W. Tucker
Keyword(s):  
Field Equation ◽  
Scalar Field ◽  
Scalar Field Equation ◽  
Signature Change

scholarly journals Anti-screening of the Galileon force around a disk center hole

2019 ◽  
Vol 34 (02) ◽  
pp. 1950013 ◽  
Author(s):  
Hiromu Ogawa ◽  
Takashi Hiramatsu ◽  
Tsutomu Kobayashi
Keyword(s):  
Field Equation ◽  
Scalar Field ◽  
Modified Gravity ◽  
Disk Center ◽  
Academic Interest ◽  
Screening Effect ◽  
Matter Source ◽  
Fifth Force ◽  
Scalar Field Equation

The Vainshtein mechanism is known as an efficient way of screening the fifth force around a matter source in modified gravity. This has been verified mainly in highly symmetric matter configurations. To study how the Vainshtein mechanism works in a less symmetric setup, we numerically solve the scalar field equation around a disk with a hole at its center in the cubic Galileon theory. We find, surprisingly, that the Galileon force is enhanced, rather than suppressed, in the vicinity of the hole. This anti-screening effect is larger for a thinner, less massive disk with a smaller hole. At this stage, our setup is only of academic interest and its astrophysical consequences are unclear, but this result implies that the Vainshtein screening mechanism around less symmetric matter configurations are quite nontrivial.

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