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Weak solutions of semilinear elliptic equation involving Dirac mass
(Elsevier, 2018)
by iterating an increasing sequence {vn}n defined by
v0 = kG[δ0], vn =G[V vpn−1] + kG[δ0], (1.7)
here G[·] is the Green operator defined as
G[f ](x) =
∫
RN
G(x, y)f (y)dy
d G is the Green kernel of −� in RN × RN , where G[δ0] is the fundamental solution...
0 ] + kG[δ0] > v0, d assuming that vn−1(x) ≥ vn−2(x), x ∈RN \ {0}, e deduce that vn =G[V vpn−1] + kG[δ0] ≥G[V vpn−2] + kG[δ0] = vn−1. (2.10) hus {vn}n is an increasing sequence. Move-over, we have that∫ RN vn(−�)ξdx = ∫ RN V v p n−1ξdx + kξ(0), ∀ξ...
0 ] + kG[δ0] > v0, d assuming that vn−1(x) ≥ vn−2(x), x ∈RN \ {0}, e deduce that vn =G[V vpn−1] + kG[δ0] ≥G[V vpn−2] + kG[δ0] = vn−1. (2.10) hus {vn}n is an increasing sequence. Move-over, we have that∫ RN vn(−�)ξdx = ∫ RN V v p n−1ξdx + kξ(0), ∀ξ...
High frequency solutions for the singularly-perturbed one-dimensional nonlinear Schrodinger equation
(SPRINGER, 2006-10)
of the period function).
Next we estimate zk+1n − zkn in terms of the period. We let vn be the solution of the
equation
ε2nv
′′
n − V (ykn)vn(x)) + vpn (x)) = 0,
with initial conditions v′n(ykn) = 0 and vn(ykn) = un(ykn). By our hypothesis on V
we have V (x) � V...