scholarly journals On a Min-Max Procedure for the Existence of a Positive Solution for Certain Scalar Field Equations in $\mathbb R^N$

1990 ◽  
pp. 1-15 ◽  
Author(s):  
Abbas Bahri ◽  
YanYan Li
Keyword(s):  
Scalar Field ◽  
Positive Solution ◽  
Field Equations ◽  
Scalar Field Equations

scholarly journals Quantum corrections to slow-roll inflation: scalar and tensor modes

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Jens O. Andersen ◽  
Magdalena Eriksson ◽  
Anders Tranberg
Keyword(s):  
Scalar Field ◽  
Quantum Corrections ◽  
Tree Level ◽  
Field Equations ◽  
Slow Roll ◽  
Scalar Field Equations ◽  
Slow Rolling ◽  
Different Sources ◽  
Suitable Potential ◽  
Quadratic Order

Abstract Inflation is often described through the dynamics of a scalar field, slow-rolling in a suitable potential. Ultimately, this inflaton must be identified with the expectation value of a quantum field, evolving in a quantum effective potential. The shape of this potential is determined by the underlying tree-level potential, dressed by quantum corrections from the scalar field itself and the metric perturbations. Following [1], we compute the effective scalar field equations and the corrected Friedmann equations to quadratic order in both scalar field, scalar metric and tensor perturbations. We identify the quantum corrections from different sources at leading order in slow-roll, and estimate their magnitude in benchmark models of inflation. We comment on the implications of non-minimal coupling to gravity in this context.

scholarly journals COLLAPSING SCALAR FIELD WITH KINEMATIC SELF-SIMILARITY OF THE SECOND KIND IN (2+1) GRAVITY

2005 ◽  
Vol 14 (06) ◽  
pp. 1049-1061 ◽  
Author(s):  
R. CHAN ◽  
M. F. A. DA SILVA ◽  
J. F. VILLAS DA ROCHA ◽  
ANZHONG WANG
Keyword(s):  
Black Holes ◽  
Scalar Field ◽  
Gravitational Collapse ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Field Equations ◽  
Self Similarity ◽  
Global Properties ◽  
Scalar Field Equations

All the (2+1)-dimensional circularly symmetric solutions with kinematic self-similarity of the second kind to the Einstein-massless-scalar field equations are found and their local and global properties are studied. It is found that some of them represent gravitational collapse of a massless scalar field, in which black holes are always formed.

The existence of a positive solution to asymptotically linear scalar field equations

2000 ◽  
Vol 130 (1) ◽  
pp. 81-105 ◽  
Author(s):  
G. Li ◽  
H.-S. Zhou
Keyword(s):  
Positive Solution ◽  
Nonlinear Term ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Field Equations ◽  
Usual Condition ◽  
Image Position ◽  
Scalar Field Equations ◽  
Constraint Method ◽  
Linear Scalar

We consider the following elliptic equation: where m > 0, f(x, u)/u tends to a positive constant as u → + ∞. Here, the nonlinear term f(x, u) does not satisfy the usual condition, that is, for some θ > 0, which is important in using the mountain pass theorem. The aim of this paper is to discuss how to use the mountain pass theorem to show the existence of a positive solution to the present problem when we lose the above condition. Furthermore, if f(x, u) ≡ f(u), we also prove that the above problem has a ground state by using the artificial constraint method.

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