This paper presents local and global bifurcation results for radially symmetric solutions of the cubic Helmholtz systemIt is shown that every point along any given branch of radial semitrivial solutions (u 0 , 0, b) or diagonal solutions (u b , u b , b) (for µ = ν) is a bifurcation point. Our analysis is based on a detailed investigation of the oscillatory behavior of solutions at infinity that are shown to decay like 1 |x| as |x| → ∞.
We obtain real-valued, time-periodic and radially symmetric solutions of the cubic Klein–Gordon equation
∂
t
2
U
−
Δ
U
+
m
2
U
=
Γ
(
x
)
U
3
on
R
×
R
3
,
which are weakly localized in space. Various families of such ‘breather’ solutions are shown to bifurcate from any given nontrivial stationary solution. The construction of weakly localized breathers in three space dimensions is, to the author’s knowledge, a new concept and based on the reformulation of the cubic Klein–Gordon equation as a system of coupled nonlinear Helmholtz equations involving suitable conditions on the far field behavior.
This paper considers a pair of coupled nonlinear Helmholtz equationsThe existence of nontrivial strong solutions in W 2,p (R N ) is established using dual variational methods. The focus lies on necessary and sufficient conditions on the parameters deciding whether or not both components of such solutions are nontrivial.
We construct infinitely many real-valued, time-periodic breather solutions of the nonlinear wave equationwith suitable N ≥ 2, p > 2 and localized nonnegative Q. These solutions are obtained from critical points of a dual functional and they are weakly localized in space. Our abstract framework allows to find similar existence results for the Klein-Gordon equation or biharmonic wave equations.
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