RESUMEN
This paper proves the existence of
multiple solutions (μ,u) of
u´´´ (t) -λu'(t) - 2u (t) u' (t) =
μ1
(Du') (t)
+ μ2
sin t,
such that u has period 2π and mean zero. D is a singular integral operator. The equation models steady
gravity waves on the surface of a shallow tank, oscillating near the primary
resonance frequency. It is shown here that for each λ > - 1, the equation has one, two, or three solutions, depending on
the position of μ =
(μ1 , μ2) in a neighbourhood
of zero in R2. This extends previous work, which required that λ be close to -1. The method depends
on the existence of cnoidal solutions when λ > - 1 and μ = 0. The Lyapunov–Schmidt
procedure is used to reduce the problem to a single bifurcation equation, which
is analysed using some results of Hale and Taboas.
