55,033 research outputs found
Positive solutions to indefinite Neumann problems when the weight has positive average
We deal with positive solutions for the Neumann boundary value problem
associated with the scalar second order ODE where is positive on and is an indefinite weight. Complementary to previous
investigations in the case , we provide existence results
for a suitable class of weights having (small) positive mean, when
at infinity. Our proof relies on a shooting argument for a suitable equivalent
planar system of the type with
a continuous function defined on the whole real line.Comment: 17 pages, 3 figure
Multibump solutions of a class of second-order discrete Hamiltonian systems
For a class of second-order discrete Hamiltonian systems
, we investigate the existence of
homoclinic orbits by applying variational method, where and
are periodic functions. Further, we show that there exist either uncountable
many homoclinic orbits or multibump solutions under certain conditions
Multiple positive solutions to elliptic boundary blow-up problems
We prove the existence of multiple positive radial solutions to the
sign-indefinite elliptic boundary blow-up problem where is a function superlinear at zero and at infinity,
and are the positive/negative part, respectively, of a sign-changing
function and is a large parameter. In particular, we show how the
number of solutions is affected by the nodal behavior of the weight function
. The proof is based on a careful shooting-type argument for the equivalent
singular ODE problem. As a further application of this technique, the existence
of multiple positive radial homoclinic solutions to is also considered
Quantum Horizons of the Standard Model Landscape
The long-distance effective field theory of our Universe--the Standard Model
coupled to gravity--has a unique 4D vacuum, but we show that it also has a
landscape of lower-dimensional vacua, with the potential for moduli arising
from vacuum and Casimir energies. For minimal Majorana neutrino masses, we find
a near-continuous infinity of AdS3xS1 vacua, with circumference ~20 microns and
AdS3 length 4x10^25 m. By AdS/CFT, there is a CFT2 of central charge c~10^90
which contains the Standard Model (and beyond) coupled to quantum gravity in
this vacuum. Physics in these vacua is the same as in ours for energies between
10^-1 eV and 10^48 GeV, so this CFT2 also describes all the physics of our
vacuum in this energy range. We show that it is possible to realize
quantum-stabilized AdS vacua as near-horizon regions of new kinds of quantum
extremal black objects in the higher-dimensional space--near critical black
strings in 4D, near-critical black holes in 3D. The violation of the
null-energy condition by the Casimir energy is crucial for these horizons to
exist, as has already been realized for analogous non-extremal 3D black holes
by Emparan, Fabbri and Kaloper. The new extremal 3D black holes are
particularly interesting--they are (meta)stable with an entropy independent of
hbar and G_N, so a microscopic counting of the entropy may be possible in the
G_N->0 limit. Our results suggest that it should be possible to realize the
larger landscape of AdS vacua in string theory as near-horizon geometries of
new extremal black brane solutions.Comment: 44 pages, 9 figure
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