scholarly journals Ground state solutions for quasilinear scalar field equations arising in nonlinear optics

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Alessio Pomponio ◽  
Tatsuya Watanabe
Keyword(s):  
Scalar Field ◽  
Nonlinear Optics ◽  
Ground State ◽  
Field Equations ◽  
Ground State Solutions ◽  
Scalar Field Equations

scholarly journals Ground state solutions for fractional scalar field equations under a general critical nonlinearity

2019 ◽  
Vol 18 (5) ◽  
pp. 2199-2215 ◽  
Author(s):  
Claudianor O. Alves ◽  
◽  
Giovany M. Figueiredo ◽  
Gaetano Siciliano ◽  
◽  
...  
Keyword(s):  
Scalar Field ◽  
Ground State ◽  
Critical Nonlinearity ◽  
Field Equations ◽  
Ground State Solutions ◽  
Scalar Field Equations

scholarly journals Linear instability and nondegeneracy of ground state for combined power-type nonlinear scalar field equations with the Sobolev critical exponent and large frequency parameter

2019 ◽  
Vol 150 (5) ◽  
pp. 2417-2441 ◽  
Author(s):  
Takafumi Akahori ◽  
Slim Ibrahim ◽  
Hiroaki Kikuchi
Keyword(s):  
Scalar Field ◽  
Ground State ◽  
Linear Instability ◽  
Frequency Parameter ◽  
Field Equations ◽  
Power Type ◽  
Nonlinear Scalar Field ◽  
Scalar Field Equations ◽  
Sobolev Critical Exponent ◽  
Large Frequency

AbstractWe consider combined power-type nonlinear scalar field equations with the Sobolev critical exponent. In [3], it was shown that if the frequency parameter is sufficiently small, then the positive ground state is nondegenerate and linearly unstable, together with an application to a study of global dynamics for nonlinear Schrödinger equations. In this paper, we prove the nondegeneracy and linear instability of the ground state frequency for sufficiently large frequency parameters. Moreover, we show that the derivative of the mass of ground state with respect to the frequency is negative.

A BERESTYCKI–LIONS THEOREM REVISITED

2012 ◽  
Vol 14 (05) ◽  
pp. 1250033 ◽  
Author(s):  
JIANJUN ZHANG ◽  
WENMING ZOU
Keyword(s):  
Scalar Field ◽  
Ground State ◽  
Mountain Pass ◽  
Field Equations ◽  
Nonlinear Scalar Field ◽  
Growth Restrictions ◽  
Scalar Field Equations ◽  
Least Energy Solutions ◽  
Least Energy

In 1983, Berestycki and Lions [Nonlinear scalar field equations I. Existence of a ground state, Arch. Ration. Mech. Anal.82 (1983) 313–346] studied the following elliptic problem: [Formula: see text] where N ≥ 3, g is subcritical at infinity. They proved the existence of a ground state under some appropriate growth restrictions on g. In the present paper, we improve this result by showing that under the critical growth assumption on g the problem admits a ground state. In addition we study a mountain pass characterization of the least energy solutions of the problem. Without the assumption of the monotonicity of the function [Formula: see text], we show that the mountain pass value gives the least energy level.

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.

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