# On first subharmonic bifurcations in a branch of Stokes waves.

Vladimir Kozlov1 1Department of Mathematics, Linköping University, SE-581 83
Linköping, Sweden

###### Abstract.

Steady surface waves in a two-dimensional channel are considered. We study
bifurcations, which occur on a branch of Stokes water waves starting from a
uniform stream solution. Two types of bifurcations are considered:
bifurcations in the class of Stokes waves (Stokes bifurcation) and
bifurcations in a class of periodic waves with the period $M$ times the period
of the Stokes wave ($M$-subharmonic bifurcation). If we consider the first
Stokes bifurcation point then there are no $M$-subharmonic bifurcations before
this point and there exists $M$-subharmonic bifurcation points after the first
Stokes bifurcation for sufficiently large $M$, which approach the Stokes
bifurcation point when $M\to\infty$. Moreover the set of $M$-subharmonic
bifurcating solutions is a closed connected continuum. We give also a more
detailed description of this connected set in terms of the set of its limit
points, which must contain extreme waves, or overhanging waves, or solitary
waves or waves with stagnation on the bottom, or Stokes bifurcation points
different from the initial one.

##  1\. Introduction

###  1.1. Background

We consider steady surface waves in a two-dimensional channel bounded below by
a flat, rigid bottom and above by a free surface. The main subject of this
work is subharmonic bifurcations on the branches of Stokes waves.

Stokes and solitary waves (regular waves) were the main subject of study up to
1980. In 1980 (see Chen [4] and Saffman [22]) it was discovered numerically
and in 2000 (see [1, 2]) this was supported theoretically for the ir-
rotational case for a flow of infinite depth that there exist new types of
periodic waves with several crests on the period (the Stokes wave has only one
crest). These waves occur as a result of bifurcation on a branch of Stokes
waves when they approach the wave of greatest amplitude.

Starting point of our study is a branch of Stokes waves starting from uniform
stream solution and approaching an extreme wave. This branch is parameterized
by a parameter $t$. One can study bifurcations of branches of Stokes waves of
period $\Lambda(t)$ in one of the following settings:

(i) in the class of $\Lambda(t)$-periodic solutions (Stokes bifurcation);

(ii) in the class of $M\Lambda(t)$-periodic solutions (M-subharmonic
bifurcation);

(iii) in the class of bounded solutions.

In this paper we will deal mainly with (i) and (ii).

Let us denote by $\\{\tilde{t}_{j}\\}$, $j=1,\ldots,\infty$, the Stokes
bifurcation points. The following theorem is proved in [9]

######  Theorem 1.1.

(i) There exists a sequence $(\widehat{t}_{j},M_{j})$, where
$\widehat{t}_{j}\neq\tilde{t}_{k}$ for all $j$ and $k$; $M_{j}$ are integers,
and

$\widehat{t}_{j}\to\infty,\;\;M_{j}\to\infty\;\;\mbox{as $j\to\infty$}.$

Moreover, $\widehat{t}_{j}$ is a $M_{j}$-subharmonic bifurcation point.

(ii) There exists a sequence $(\widehat{t}_{j},M_{j})$, where
$\widehat{t}_{j}\neq\tilde{t}_{k}$ for all $j$ and $k$, the sequence
$\\{\widehat{t}_{j}\\}$ is bounded and

$M_{j}\to\infty\;\;\mbox{as $j\to\infty$}.$

Furthermore, $\widehat{t}_{j}$ is a a $M_{j}$-subharmonic bifurcation point.
The numbers $\widehat{t}_{j}$ are pairwise different in both cases.

The proof of this theorem is based on the fact that when the Stokes waves
approach the extreme wave many new bifurcation points appear. The proof of
this theorem involves a bifurcation theorem for potential operators with non-
zero crossing number, which gives unfortunately no information on the
structure of the set of bifurcating solutions (see [24] for a discussion of
this structure).

Here we study a mechanism of appearance of the first subharmonic bifurcations.
The main result of this paper is the following

######  Theorem 1.2.

Let $t_{0}>0$ be the first Stokes bifurcation point, i.e. the condition (1.1)
be valid. Then there exists $(t_{M},M)$, where $M$ is a large integer and

$t_{M}\to t_{0}\;\;\;\mbox{as}\;\;M\to\infty,$

such that $t_{M}$ is $M$\- subharmonic bifurcation point. There are no
subharmonic bifurcations for $t<t_{0}$.

We prove this theorem by using a bifurcation theorem with odd crossing number,
which leads to existence of a closed connected continuum of
$M\Lambda$-periodic solutions containing the bifurcation point $t_{M}$ (see
Theorems 8.1, 9.1 and 9.2 for more details).

In the study of bifurcations the Frechet derivative plays important role. In
our case it is a self-adjoint operator bounded from below. Certainly the
Frechet derivatives are defined on different spaces in the cases (i)-(iii). In
the study of the Stokes bifurcations this is a usual Frechet derivative
defined on $\Lambda$-periodic, even solutions. If we denote by $\mu_{0}(t)$
and $\mu_{1}(t)$ the first and the second eigenvalue of the Frechet
derivative, then $\mu_{0}(t)<0$ for all $t$ and $\mu_{1}(0)=0$. We consider
the case when

$\displaystyle\mu_{1}(t)>0\;\;\mbox{for small positive $t$ and assume that
there exists $t_{0}>0$}$ $\displaystyle\mbox{ such that $\mu_{1}(t_{0})=0$ and
$\mu_{1}(t)<0$ for small positive $t-t_{0}$}.$ (1.1)

We will call $t_{0}$ the first Stokes bifurcation.

In the case of $M$ -subharmonic bifurcations we have the same expression for
the Frechet derivative but now it is defined on $M\Lambda$-periodic, even
functions and finally in the case of bifurcations in a class of bounded
solutions the Frechet derivative is defined on the even functions defined on
the whole domain without periodicity condition. By introducing quasi-momentum
$\tau$ we can give another equivalent definition of the Frechet derivative in
the cases (ii) and (iii) which is defined on the space of $\Lambda$-periodic
functions but depending on the real parameter $\tau$. We denote by
$\widehat{\mu}_{j}(t,\tau)$, $j=0,\ldots$, the eigenvalues of this problem,
which are numerated according to the increasing order:

$\widehat{\mu}_{0}(t,\tau)\leq\widehat{\mu}_{1}(t,\tau)\leq\cdots$

The essential part of the paper is devoted to a study of properties of these
eigenvalues. The eigenvalue $\widehat{\mu}_{0}(t,\tau)$ is always less than
$0$ for all $t$ and $\tau\in\mathbb{R}$. If the first Stokes bifurcation
occurs at a certain $t=t_{0}$, then it is proved that the kernel of the
Frechet derivative at this point is one-dimensional. Another important
property concerning the Frechet derivative is the inequalities

$\widehat{\mu}_{1}(t,\tau)<0\;\;\mbox{for $t\in(t_{0},t_{0}+\epsilon)$ and
$\tau\in\mathbb{R}$},$

where $\epsilon$ is a small positive number, and

$\widehat{\mu}_{2}(t_{0},\tau)>0\;\;\mbox{for
$\tau\in(0,\tau_{*})$},\;\;\tau_{*}=2\pi/\Lambda.$

These inequalities together with asymptotic properties of $\widehat{\mu}_{2}$
in a neighborhood of the point $(t,\tau)=(t_{0},0)$ allows to perform analysis
of subharmonic bifurcations near the point $t=t_{0}$. $M$-subharmonic
bifurcation points $t_{M}$ are bigger then $t_{0}$ and $t_{M}\to t_{0}$ as
$M\to\infty$. Moreover, we show that the crossing number at $t_{M}$ is odd and
hence there is a closed connected continuum of $M$-subharmonic bifurcations
started from the bifurcation point.

As is known one of the indication of bifurcation is the change of the Morse
index, or changing of the number of negative eigenvalues of the Frechet
derivatives, which depend on the parameter $t$. In the papers [16] and [9] it
was proved that for the branch of Stokes waves approaching the extreme Stokes
wave unlimited number of negative eigenvalues appears, which implies unlimited
number of bifurcations, see [9]. The problem here is that all these
bifurcations can be Stokes bifurcations and the above fact on negative
eigenvalues does not directly imply existence of subharmonic bifurcations.
More advance analysis is required.

Our aim is to study the first Stokes bifurcation and show that there are
always accompanying $M$-subharmonic bifurcations with arbitrary large $M$.
They generates by $\tau$-roots of the equation $\widehat{\mu}_{2}(t,\tau)=0$.
The main body of the paper is devoted to the study of this equation. This
analysis is based on spectral analysis of different spectral problems and
asymptotic analysis of eigenvalues for small $\tau$.

###  1.2. Formulation of the problem

We consider steady surface waves in a two-dimensional channel bounded below by
a flat, rigid bottom and above by a free surface that does not touch the
bottom. The surface tension is neglected and the water motion can be
rotational. In appropriate Cartesian coordinates $(X,Y)$, the bottom coincides
with the $x$-axis and gravity acts in the negative $y$ -direction. We choose
the frame of reference so that the velocity field is time-independent as well
as the free-surface profile which is supposed to be the graph of $Y=\xi(X)$,
$X\in\mathbb{R}$, where $\xi$ is a positive and continuous unknown function.
Thus

${\mathcal{D}}_{\xi}=\\{X\in\mathbb{R},0<Y<\xi(X)\\},\;\;{\mathcal{S}}_{\xi}=\\{X\in\mathbb{R},\;Y=\xi(X)\\}$

is the water domain and the free surface respectively. We will use the stream
function $\Psi$, which is connected with the velocity vector $({\bf u},{\bf
v})$ by ${\bf u}=-\Psi_{Y}$ and ${\bf v}=\Psi_{X}$.

We assume that $\xi$ is a positive, periodic function having period
$\Lambda>0$ and that $\xi$ is even and strongly decreasing on the interval
$(0,\Lambda/2)$. Since the surface tension is neglected, $\Psi$ and $\xi$
satisfy, after a certain scaling, the following free-boundary problem (see for
example [11]):

$\displaystyle\Delta\Psi+\omega(\Psi)=0\;\;\mbox{in ${\mathcal{D}}_{\xi}$},$
$\displaystyle\frac{1}{2}|\nabla\Psi|^{2}+\xi=R\;\;\mbox{on
${\mathcal{S}}_{\xi}$},$ $\displaystyle\Psi=1\;\;\mbox{on
${\mathcal{S}}_{\xi}$},$ $\displaystyle\Psi=0\;\;\mbox{for $Y=0$},$ (1.2)

where $\omega\in C^{1,\alpha}$, $\alpha\in(0,1)$, is a vorticity function and
$R$ is the Bernoulli constant. We assume that $\Psi$ is even,
$\Lambda$-periodic in $x$ and

$\Psi_{Y}>0\;\;\mbox{on $\overline{{\mathcal{D}}_{\xi}}$},$ (1.3)

which means that the flow is unidirectional.

The Frechet derivative for the problem is evaluated for example in [16], [9],
and the corresponding eigenvalue problem for the Frechet derivative has the
form

$\displaystyle\Delta w+\omega^{\prime}(\Psi)w+\mu w=0\;\;\mbox{in
${\mathcal{D}}_{\xi}$},$ $\displaystyle\partial_{\nu}w-\rho w=0\;\;\mbox{on
${\mathcal{S}}_{\xi}$},$ $\displaystyle w=0\;\;\mbox{for $Y=0$},$ (1.4)

where $\nu$ is the unite outward normal to ${\mathcal{S}}_{\xi}$ and

$\rho=\rho(X)=\frac{(1+\Psi_{X}\Psi_{XY}+\Psi_{Y}\Psi_{YY})}{\Psi_{Y}(\Psi_{X}^{2}+\Psi_{Y}^{2})^{1/2}}\Big{|}_{Y=\xi(X)}.$
(1.5)

The function $w$ in (1.2) is supposed also to be even and $\Lambda$-periodic.

Let us introduce several function spaces. Let $\alpha\in(0,1)$ and
$k=0,1,\ldots$. The space $C^{k,\alpha}({\mathcal{D}})$ consists of bounded
functions in ${\mathcal{D}}$ such that the norms
$C^{k,\alpha}(\overline{{\mathcal{D}}_{a,a+1}})$ are uniformly bounded with
respect to $a\in\mathbb{R}$. Here

${\mathcal{D}}_{a,a+1}=\\{(X,Y)\in{\mathcal{D}},\;:\,a<x<a+1\\}.$

The space $C^{k,\alpha}_{0,\Lambda}({\mathcal{D}})$
($C^{k,\alpha}_{0,\Lambda,e}({\mathcal{D}})\big{)}$ consists of
$\Lambda$-periodic ($\Lambda$-periodic and even) functions, which belong to
$C^{k,\alpha}({\mathcal{D}})$ and vanish at $y=0$.

Similarly we define the space $C^{k,\alpha}_{\Lambda}(\mathbb{R})$
($C^{k,\alpha}_{\Lambda,e}(\mathbb{R})$) consisting of functions in
$C^{k,\alpha}(\mathbb{R})$, which are $\Lambda$-periodic ($\Lambda$-periodic
and even).

We will consider a branch of Stokes water waves depending on a parameter
$t\in\mathbb{R}$, i.e.

$\xi=\xi(X,t),\;\;\Psi=\Psi(X,Y;t),\;\;\Lambda=\Lambda(t).$

For each $t$ the function $\xi\in C^{2,\alpha}(\mathbb{R})$ and $\Psi\in
C^{3,\alpha}({\mathcal{D}})$. This branch starts from a uniform stream
solution for $t=0$ and approach an extreme wave when $t\to\infty$. The
dependence on $t$ is analytic in the following sense: the functions

$\xi(X\Lambda(0)/\Lambda(t),t)\,:\,\mathbb{R}\rightarrow
C^{2,\alpha}_{\Lambda(0),e}(\mathbb{R})\;\;\mbox{and}\;\;\Lambda\,:\,\mathbb{R}\rightarrow(0,\infty)$

are analytic with respect to $t$ and the function $\Psi$ can be found from the
problem

$\displaystyle\Delta\Psi+\omega(\Psi)=0\;\;\mbox{in ${\mathcal{D}}_{\xi}$},$
$\displaystyle\Psi=1\;\;\mbox{on ${\mathcal{S}}_{\xi}$},$
$\displaystyle\Psi=0\;\;\mbox{for $Y=0$}.$

Another equivalent description of the analytical property of functions $\xi$
and $\Psi$ is presented in Sect. 5.2.

The main assumption is given in terms of the second eigenvalue of the spectral
problem (1.2). One can show that the first eigenvalue $\mu_{0}(t)$ is always
negative and the second one $\mu_{1}(t)$ is zero for $t=0$. Our main
assumption is that

$\mu_{1}(t)>0\;\;\mbox{for small positive $t$}.$ (1.6)

This property will be discussed in detail in forthcoming paper.

##  2\. Spectral problem (1.2), $\Lambda$-periodic, even functions

Let

$D=D_{\xi}=\\{(X,Y)\,:\,0<X<\Lambda/2,\;0<X<\xi(X)\\}$

and

$S=S_{\xi}=\\{(X,Y)\,:\,0<X<\Lambda/2,\;Y=\xi(X)\\}.$

We introduce the form

$a(u,v)\\!\\!=\\!\\!a_{D}(u,v)\\!\\!=\\!\\!\int_{D}\Big{(}\nabla
u\cdot\nabla\overline{v}-\omega^{\prime}(\Psi)u\overline{v}\Big{)}dXdY-\int_{0}^{\Lambda/2}\rho(X)u\overline{v}dS,\;\;dS=\sqrt{1+\xi^{\prime
2}}dX,$ (2.1)

which is defined on functions from $H^{1}(D)$ vanishing for $Y=0$. This space
will be denoted by $H^{1}_{0}(D)$.

The assumption (1.3) implies

$a_{D}(u,u)>0\;\;\mbox{for nonzero $u\in H^{1}_{0}(D)$ satisfying
$u(X,\xi(X))=0$ for $X\in(0,\Lambda/2)$}.$ (2.2)

Since $\partial_{X}w(0,Y)=\partial_{X}w(\Lambda/2,Y)=0$ in (1.2), the spectral
problem (1.2) admits the following variational formulation. A real number
$\mu$ and a function $\phi\in H^{1}_{0}(D)$ satisfy (1.2), after natural
extension on ${\mathcal{D}}_{\xi}$, if and only if

$a(\phi,v)=\mu(\phi,v)_{L^{2}(D)},\;\;\mbox{for all $v\in H^{1}_{0}(D)$}.$
(2.3)

Here

$(u,v)_{L^{2}(D)}=\int_{D}u\overline{v}dXdY.$

We put also $||u||_{L^{2}(D)}^{2}=(u,u)_{L^{2}(D)}$. We numerate the
eigenvalues accounting their multiplicity as

$\mu_{0}\leq\mu_{1}\leq\cdots,\;\;\mu_{j}\to\infty\;\;\mbox{as}\;\;j\to\infty,$

and denote by $\Phi_{j}$, $j=0,1,\ldots$, corresponding eigenfunctions. They
can be chosen to be orthogonal to each other, real-valued and with the norm

$||\Phi_{j}||_{L^{2}(D)}=1.$

Thus $\\{\Phi_{j}\\}$ forms an orthogonal basis in $L^{2}(D)$.

In what follows we shall use the function

$u_{*}(X,Y):=\Psi_{X}(X,Y),$ (2.4)

which is odd, $\Lambda$-periodic and belongs to $C^{2,\alpha}(\overline{D})$.
It satisfies the boundary value problem

$\displaystyle\Delta u_{*}+\omega^{\prime}(\Psi)u_{*}=0\;\;\mbox{in
$D_{\xi}$},$ $\displaystyle\partial_{\nu}u_{*}-\rho u_{*}=0\;\;\mbox{on
$S_{\xi}$},$ $\displaystyle u_{*}=0\;\;\mbox{for $X=0$},$ $\displaystyle
u_{*}(0,Y)=0\;\;\mbox{for
$Y\in(0,\xi(0))$},\;\;u_{*}(\Lambda/2,Y)=0\;\;\mbox{for
$Y\in(0,\xi(\Lambda/2))$},$ (2.5)

and it has the following properties along branches of Stokes waves (see [10])

$\displaystyle{\rm(i)}\;\;\;\;u_{*}>0\;\;\mbox{for $(X,Y)\in D\cup S$};$
$\displaystyle{\rm(ii)}\;\;\;(u_{*})_{X}(0,Y)>0\;\;\mbox{for
$0<Y\leq\xi(0)$,\;\; $(u_{*})_{X}(\Lambda/2,Y)<0$ for
$0<Y\leq\xi(\Lambda/2)$};$ $\displaystyle\;\;\;\;\;\;\;\;\mbox{ and
\;\;$(u_{*})_{Y}(X,0)>0$\; for $0<X<\Lambda/2$};$
$\displaystyle{\rm(iii)}\;\;(u_{*})_{XY}(0,0)>0,\;\;(u_{*})_{XY}(\Lambda/2,0)<0.$
(2.6)

The above properties of the function $\Psi_{x}$-the vertical component of the
velocity vector, are found interesting application in study of branches of
Stokes waves, see [5], [7], [27], [26] and [10]. It appears that they are also
important in study of subharmonic bifurcations. They are used, in particular,
in the proof of the following assertion.

######  Proposition 2.1.

Let $\xi$ be not identically constant then the eigenvalue $\mu_{0}$ is simple
111This implies in particular that $\mu_{0}<\mu_{1}$.. Moreover $\mu_{0}<0$
and the corresponding eigenfunction ($\Phi_{0}$) does not change sign inside
$D$. If we assume that $\Phi_{0}$ is positive inside $D$ then $\Phi_{0}$ is
also positive for $Y=\xi(X)$ and for $X=0$, $Y\in(0,\xi(0))$ and
$X=\Lambda/2$, $Y\in(0,\xi(0))$, $Y\in(0,\xi(\Lambda/2))$.

###### Proof.

Let $v_{*}$ be the restriction of the function (2.4) to the domain $D$.
Clearly it is positive inside $D$, belongs to $H^{1}_{0}(D)$ and
$a(v_{*},v_{*})=0$. This implies that either $\mu_{0}<0$ or $\mu_{0}=0$ and
the function $v_{*}$ is an eigenfunction corresponding to $\mu_{0}$. The
latest alternative is impossible. Indeed, if $v_{*}$ is an eigenfunction then
$(v_{*})_{X}=0$ for $X=0$ and for $X=\Lambda/2$. So $v_{*}$ has homogeneous
Cauchy data and satisfies an elliptic equation. Hence $v_{*}$ is identically
zero inside $D$. Thus $\mu_{0}<0$.

Since

$\mu_{0}=\min_{w\in H^{1}_{0}(D),||w||_{0}=1}a(w,w)$ (2.7)

the corresponding eigenfunction cannot change sign inside $D$. Indeed, if
$\phi_{0}$ changes sign then we introduce two functions
$v_{\pm}(X,Y)=\max(0,\pm v(X,Y))$. One can verify that
$a(v_{\pm},v_{\pm})=\mu_{0}||v_{\pm}||^{2}_{L^{2}(D)}$. Therefore both
functions $v_{\pm}$ deliver minimum in (2.7) and hence they are also
eigenfunctions corresponding to $\mu_{0}$ but this is impossible because they
are not smooth. So $\Phi_{0}$ is positive or negative inside $D$.

Assume that $\Phi_{0}>0$ in $D$. Then it cannot vanish on the part of the
boundary where $Y>0$. Indeed if $\Phi$ is zero there then the normal
derivative is also zero and this leads to a chenge of sign inside $D$. ∎

The following Green’s formula for the form $a$ will be useful in the proof of
the nest proposition:

$\displaystyle-\int_{D}(\Delta
u+\omega^{\prime}(\Psi)u)\overline{v}dXdY+\int_{0}^{\Lambda/2}(\partial_{\nu}u-\rho(X)u)\overline{v}dS$
$\displaystyle=a(u,v)+\int_{0}^{\xi(0)}u_{X}\overline{v}|_{X=0}dY-\int_{0}^{\xi(\Lambda/2)}u_{X}\overline{v}|_{X=\Lambda/2}dY.$
(2.8)

Here the function $u$ belongs to the space $H^{2}_{0}(D)$ consisting of
functions from $H^{2}(D)$ vanishing for $Y=0$ and $v\in H^{1}_{0}(D)$.

######  Proposition 2.2.

Let $\xi$ be not identically constant. If $\mu_{1}=0$ then $\mu_{1}=0$ is
simple. Moreover the corresponding eigenfunction has exactly two nodal sets
and the nodal line separating these nodal sets has one of its end points on
the curve $S$.

###### Proof.

Denote by ${\mathcal{X}}$ the space of real eigenfunctions corresponding to
the eigenvalue $\mu_{1}=0$. Since all eigenfunctions from ${\mathcal{X}}$
orthogonal to $\Phi_{0}$, each eigenfunction must change sign inside $D$. Let
us show that $\dim{\mathcal{X}}=1$. The proof consists of several steps.

(a) First we show that every nonzero eigenfunction $w\in{\mathcal{X}}$ has
exactly two nodal sets. Assume that there is an eigenfunction
$w\in{\mathcal{X}}$ which has more then two nodal sets, say $Y_{j}$,
$j=1,\ldots,N$, $N>2$. Introduce the functions $u_{j}(X,Y)=w(X,Y)$ for
$(X,Y)\in Y_{j}$ and zero otherwise. Then $u_{j}\in H^{1}_{0}(D)$ and

$a(u_{j},u_{k})=0\;\;\mbox{for all $k,j=1,\ldots,N$}.$ (2.9)

We choose the constant $\alpha$ such that

$(u_{1}-\alpha u_{2},\Phi_{0})_{L^{2}(D)}=0$

and observe that $a(u_{1}-\alpha u_{2},u_{1}-\alpha u_{2})=0$. Since

$\mu_{1}=\min a(w,w)$

where $\min$ is taken over $w\in H^{1}_{0}(D)$ satisfying
$(w,\Phi_{0})_{L^{2}(D)}=0$ and $||w||_{L^{2}(D)}=1$, the minimum is attained
on the function $(u_{1}-\alpha u_{2})/||u_{1}-\alpha u_{2}||_{L^{2}(D)}$. So
we conclude that $u_{1}-\alpha u_{2}$ is an eigenfunction corresponding to the
eigenvalue $\mu_{1}=0$, which is impossible since this function is zero on
$Y_{j}$ for $j>2$.

(b) Second, let us prove that the nodal line is not closed and one of its end
points lies oh $S$. Consider an eigenfunction $w$ with two nodal sets and let
$\gamma$ be the nodal line separating these two nodal sets. If this nodal line
is closed then introduce the function $w_{1}$ which coincides with $w$ inside
the closed nodal line and vanishes outside. Then $a(w_{1},w_{1})=0$ and
$w_{1}=0$ on $S$ which contradicts to the inequality (2.2). If both end points
of $\gamma$ lie outside $S$ then introduce $w_{2}$ which coincides with $w$ on
the nodal set separated from $S$ and vanishes otherwise. Then
$a(w_{2},w_{2})=0$ and $w_{2}=0$ on $S$ which again contradicts to the
inequality (2.2). Thus one of end points lies on $S$.

(c) Now we are in position to prove that $\dim{\mathcal{X}}=1$. If
$\dim{\mathcal{X}}>1$ then there is an eigenfunction, say $w_{*}$ which is
zero at the point $z_{1}=(0,\eta(0))$, which must be one of the end points of
the nodal line separating two nodal sets. By b) another end-point $z_{2}$ of
the nodal line lies on $S$. Denote by $Y_{1}$ the nodal set attached to the
part of $S$ between $z_{1}$ and $z_{2}$. Let also $Y_{2}$ be the remaining
nodal domain. We can assume that $w_{*}<0$ in $Y_{1}$ and $w_{*}>0$ in
$Y_{2}$.

Let $v_{*}$ be the function introduced in Proposition 2.1. Using that both
functions $w_{*}$ and $u_{*}$ satisfy the problem (1.2) but the first one
satisfies $\partial_{X}w_{*}=0$ for $X=0,\Lambda/2$ and the second is subject
to $v_{*}=0$ for the same values of $x$, we have, by using (2),

$\int_{0}^{\xi(0)}\partial_{X}v_{*}(0,Y)w_{*}(0,Y)dY=\int_{0}^{\xi(\Lambda/2)}\partial_{X}v_{*}(\Lambda/2,Y)w_{*}(\Lambda/2,Y)dY.$
(2.10)

Consider the function

$U=w_{*}+\beta v_{*},\;\;\mbox{where $\beta>0$}.$

Since $v_{*}$ is a positive function inside $D$, for small $\beta$ the
function $U$ has also two nodal sets $\widetilde{Y}_{1}$ and
$\widetilde{Y}_{2}$, separated by a nodal line $\widetilde{\gamma}$ with end
points $z_{1}=(0,\eta(0))$ and $\widetilde{z}_{2}\in S$ which is close to
$z_{2}$. We assume that the nodal set $\widetilde{Y}_{1}$ is attached to the
part of $S$ between the points $z_{1}$ and $\widetilde{z}_{2}$. Introduce the
functions

$\displaystyle U_{1}(X,Y)=U(X,Y)\;\;\mbox{if $(X,Y)\in\widetilde{Y}_{1}$ and
zero otherwise},$ $\displaystyle U_{2}(X,Y)=U(X,Y)\;\;\mbox{if
$(X,Y)\in\widetilde{Y}_{2}$ and zero otherwise}.$

We choose the constant $\theta$ such that

$(U_{1}+\theta U_{2},\Phi_{0})_{L^{2}(D)}=0.$ (2.11)

It is clear that $\theta\neq 0$ because of

$a(U_{1},\Phi_{0})=\mu_{0}(U_{1},\Phi_{0})_{L^{2}(D)}\neq 0.$

Here we have used that $\Phi_{0}>0$ inside $D$ by Proposition 2.1. Furthermore
due to (2.10) and the fact that $w_{*}$ is an eigenfunction corresponding to
$\mu_{1}=0$, we have

$a(U_{1}+\theta U_{2},U_{1}+\theta U_{2})=0.$

This together with (2.11) implies that $U_{1}+\theta U_{2}$ is an
eigenfunction corresponding to the eigenvalues $\mu_{1}=0$ of the spectral
problem (2.3). But this contradicts to the smoothness properties of the
function $U_{1}+\theta U_{2}$. (Compare with the argument used in a) and b)).
∎

###  2.1. Auxiliary spectral problems

In this section we introduce and study eigenvalues of several eigenvalue
problems, which will be used in subsequent sections to estimate eigenvalues of
”generalized” eigenvalue problem introduced in the next section. Zeros of
generalized eigenvalues will give us the subharmonic bifurcations in what
follows.

Let us introduce the spaces

$H^{1}_{00*}(D)=\\{u\in H^{1}_{0}(D)\,:\;u=0\;\mbox{for $X=0$}\\},$
$H^{1}_{0*0}(D)=\\{u\in H^{1}_{0}(D)\,:\;u=0\;\mbox{for $X=\Lambda/2$}\\}$

and

$H^{1}_{000}(D)=\\{u\in H^{1}_{0}(D)\,:\;u=0\;\mbox{for $X=0$ and
$X=\Lambda/2$}\\}.$

The following spectral problems will play important role in what follows. Find
$\nu^{0*}$ and $u\in H^{1}_{00*}(D)$ satisfying

$a(u,v)=\nu^{0*}(u,v)_{L^{2}(D)}\;\;\mbox{for all $v\in H^{1}_{00*}(D)$}.$
(2.12)

Find $\nu^{*0}$ and $u\in H^{1}_{0*0}(D)$ satisfying

$a(u,v)=\nu^{*0}(u,v)_{L^{2}(D)}\;\;\mbox{for all $v\in H^{1}_{0*0}(D)$}.$
(2.13)

Find $\nu^{00}$ and $u\in H^{1}_{000}(D)$ satisfying

$a(u,v)=\nu^{00}(u,v)_{L^{2}(D)}\;\;\mbox{for all $v\in H^{1}_{000}(D)$}.$
(2.14)

We denote the eigenvalues of these problems by

$\nu^{0*}_{0}\leq\nu^{0*}_{1}\leq\cdots,$
$\nu^{*0}_{0}\leq\nu^{*0}_{1}\leq\cdots$

and

$\nu^{00}_{0}\leq\nu^{00}_{1}\leq\cdots,$

where the multiplicity is taken into account. We note that the spectral
problem considered in Sect. 2 represents the fourth problem in this list of
spectral problems, where no restrictions for $X=0$ and $X=\Lambda/2$ are
given.

######  Proposition 2.3.

The following properties are valid:

$\nu^{0*}_{0}<0,\;\;\nu^{*0}_{0}<0,\;\;\nu^{00}_{0}=0$ (2.15)

and

$\nu^{0*}_{1}>\mu_{1},\;\;\nu^{*0}_{1}>\mu_{1},\;\;\nu^{00}_{1}>0.$ (2.16)

###### Proof.

The function $v_{*}$ from Proposition 2.1 satisfies the problem (2.14) with
$\nu^{00}=0$ and it is positive inside $D$. Therefore $\nu^{00}$ is the lowest
eigenvalue of (2.14) and we arrive at the last relation in (2.15). Since

$H^{1}_{000}(D)\subset H^{1}_{00*}(D)\;\;\mbox{and}\;\;H^{1}_{000}(D)\subset
H^{1}_{0*0}(D),$ (2.17)

we obtain the first two inequalities in (2.15), compare with the proof of
Proposition 2.1.

Inclusions

$H^{1}_{00*}(D)\subset H^{1}_{0}(D)\;\;\mbox{and}\;\;H^{1}_{0*0}(D)\subset
H^{1}_{0}(D)$ (2.18)

together with (2.17) implies the inequalities (2.16). ∎

##  3\. Generalized eigenvalue problem

To study subharmonic and more general bifurcations a more general eigenvalue
problem, which includes quasi momentum as a parameter, is needed. This section
is devoted to this spectral problem.

As it is known (see, for example [19], [21]) all bounded solutions to the
problem

$\displaystyle\Delta w+\omega^{\prime}(\Psi)w+\mu w=0\;\;\mbox{in
${\mathcal{D}}_{\xi}$},$ $\displaystyle\partial_{\nu}w-\rho w=0\;\;\mbox{on
${\mathcal{S}}_{\xi}$},$ $\displaystyle w=0\;\;\mbox{for $y=0$},$ (3.1)

with real $\mu$, are described by

$w(X,Y)=\sum_{j}a_{j}e^{i\tau_{j}X}w_{j}(X,Y),$

where $\tau_{j}\in[0,\tau_{*})$ and $w_{j}$ is a $\Lambda$-periodic solution
to the problem

$\displaystyle(\partial_{X}+i\tau)^{2}w+\partial_{Y}^{2}w+\omega^{\prime}(\Psi)w+\mu
w=0\;\;\mbox{in ${\mathcal{D}}_{\xi}$},$ $\displaystyle N(\tau)w-\rho
w=0\;\;\mbox{on ${\mathcal{S}}_{\xi}$},$ $\displaystyle w=0\;\;\mbox{for
$Y=0$},$ (3.2)

where

$N(\tau)w=e^{-i\tau X}\partial_{\nu}(e^{i\tau X}w).$

The number $\tau$ is called quasi momentum. It is not assumed in this
consideration that $w$ is even. By [23] the number of such $\tau_{j}$ is
finite for a fixed $\mu$. In this section we prove important estimates for
eigenvalues of the generalized eigenvalue problem.

###  3.1. Variational formulation

The problem (3) is a spectral problem for a self-adjoint operator for every
real $\tau$. To give its variational formulation we introduce

$\Omega=\\{(X,Y)\,:\,-\Lambda/2<X<\Lambda/2,\;0<Y<\xi(X)\\}$

and denote by $H^{1}_{0,p}(\Omega)$ the subspace of all function $u$ in
$H^{1}(\Omega)$, which satisfies $u(-\Lambda/2,Y)=u(\Lambda/2,Y)$ for
$Y\in(0,\xi(\Lambda/2)$ and $u(X,0)=0$ for $X\in(-\Lambda/2,\Lambda/2)$. We
put

${\bf
a}(u,v;\tau)=\int_{\Omega}\Big{(}(\partial_{X}+i\tau)u\overline{(\partial_{X}+i\tau)v}+\partial_{Y}u\partial_{Y}\overline{v}-\omega^{\prime}(\Psi)u\overline{v}\Big{)}dXdY-\int_{-\Lambda/2}^{\Lambda/2}\rho(X)u\overline{v}dS,$
(3.3)

which can be written also as

${\bf a}(u,v;\tau)={\bf a}(u.v)+i\tau{\bf b}(u,v)+\tau^{2}{\bf c}(u,v),$

where

${\bf a}(u,v)=a_{\Omega}(u,v),\;\;{\bf
b}(u,v)=\int_{\Omega}(u\partial_{X}\overline{v}-\partial_{X}u\overline{v})dXdY,\;\;{\bf
c}(u,v)=(u,v)_{L^{2}(\Omega)}.$

Consider the spectral problem: find $w\in H^{1}_{0p}(\Omega)$ and
$\mu\in\mathbb{R}$ satisfying

${\bf a}(w,V;\tau)=\widehat{\mu}(\tau)(\phi,V)_{L^{2}(\Omega)}\;\;\mbox{for
all $V\in H^{1}_{0,p}(\Omega)$}.$ (3.4)

This is a variational formulation of the spectral problem (3). Note that we do
not suppose in this formulation the functions $w$ and $V$ are even or odd,
both of them are only periodic. We will consider the form ${\bf a}$ only for
real $\tau$. In this case the form is symmetric and the spectrum consists of
isolated real eigenvalues of finite multiplicity.

Denote by $\widehat{\mu}_{j}(\tau)$, $\tau\in\mathbb{R}$, the eigenvalues of
the problem (3) (equivalently (3.4)) numerated according to the increasing
order

$\widehat{\mu}_{0}(\tau)\leq\widehat{\mu}_{1}(\tau)\leq\cdots$

Clearly

$\widehat{\mu}_{0}(0)=\mu_{0},\;\;\widehat{\mu}_{1}(0)=\min(\mu_{1},0),\;\;\widehat{\mu}_{2}(0)=\min(\max(\mu_{1},0),\mu_{2},\nu^{00}_{1})$

and

$\widehat{\mu}_{3}(0)>0\;\;\mbox{if $\mu_{2}>0$}.$ (3.5)

Here we use that the first and second eigenvalues of the problem (1.2),
considered for $\Lambda$-periodic odd functions coincide with $\nu^{00}_{0}$
and $\nu^{00}_{1}$ respectively.

There is another variational formulation. We put $w=e^{-i\tau X}\phi$ in (3.4)
and $V=e^{-i\tau X}v$, $v\in H^{1}_{0}(\Omega,\tau)$, where

$H^{1}_{0}(\Omega,\tau)=\\{v\in
H^{1}_{0}(\Omega)\,:\,v(-\Lambda/2,Y)=e^{-i\tau\Lambda}v(\Lambda/2,Y)\\}.$

Then the spectral problem (3.4) is equivalent to: find $\widehat{\mu}(\tau)$
and $\phi\in H^{1}_{0}(\Omega,\tau)$ satisfying

$a(\phi,v)=\widehat{\mu}(\tau)(\phi,v)_{L^{2}(\Omega)}\;\;\mbox{for all $v\in
H^{1}_{0}(\Omega,\tau)$}.$ (3.6)

###  3.2. Estimates of the eigenvalues $\widehat{\mu}_{j}$.

We will need some estimate for the function $\widehat{\mu}_{j}(\tau)$. For
this purpose we introduce two auxiliary spectral problems. Let
$H^{1}_{00}(\Omega)$ consists of functions from $H^{1}(\Omega)$ which vanish
for $Y=0$ and $x=\pm\Lambda/2$.

The first spectral problem is to find $\mu_{D}\in[0,\tau_{*})$ and $\phi\in
H^{1}_{00}(\Omega)$ such that

$a(\phi,v)=\mu_{D}(\phi,v)_{L^{2}(\Omega)}\;\;\mbox{for all $v\in
H^{1}_{00}(\Omega)$}.$ (3.7)

The second one is to find $\mu_{N}\in[0,\tau_{*})$ and $\phi\in
H^{1}_{0}(\Omega)$ such that

$a(\phi,v)=\mu_{N}(\phi,v)_{L^{2}(\Omega)}\;\;\mbox{for all $v\in
H^{1}_{0}(\Omega)$}.$ (3.8)

We denote by $\\{\mu_{Dj}\\}$ and $\\{\mu_{Nj}\\}$, $j=0,1,\ldots$, the
eigenvalues of the problems (3.7) and (3.8) respectively numerated according
to

$\mu_{D0}\leq\mu_{D1}\leq\cdots,\;\;\mu_{N0}\leq\mu_{N1}\leq\cdots,$

where the multiplicity is taken into account. Since

$H^{1}_{00}(\Omega)\subset H^{1}_{0}(\Omega,\tau)\subset H^{1}_{0}(\Omega),$
(3.9)

we conclude

$\mu_{Nj}\leq\widehat{\mu}_{j}(\tau)\leq\mu_{Dj},\;\;j=0,1,\ldots$ (3.10)

Note that both eigenvalues $\mu_{Dj}$ and $\mu_{Nj}$ do not depend on $\tau$.

In the following lemma we show that all inequalities (3.10) are strong

######  Lemma 3.1.

Let $\xi$ be not identically constant. Then the following inequalities hold:

$\mu_{Nj}<\widehat{\mu}_{j}(\tau)<\mu_{Dj},\;\;j=0,1,\ldots$ (3.11)

for $\tau\in(0,\tau_{*})$.

###### Proof.

First, we assume that $n$ satisfies

$\mu_{Dn}>\mu_{D(n-1)}$ (3.12)

and prove the inequalities in right-hand side of (3.11). Let $\Phi_{j}\in
H^{1}_{00}(\Omega)$ be the eigenfunction of (3.7) corresponding to $\mu_{Dj}$,
$j=0,\ldots$, and let $X_{n}$ be the space of linear combinations of
$\\{\Phi_{j}\\}_{j=0}^{n-1}$ and

${\mathcal{X}}_{n}=\\{\Phi\in
H^{1}_{00}(\Omega)\,:\,(\Phi,\Phi_{j})_{L^{2}(\Omega)},\,j=0,\ldots,n-1\\}.$

Then

$\mu_{Dn}=\min_{\Phi\in
X_{n}}\frac{a(\Phi,\Phi)}{||\Phi||_{L^{2}(\Omega)}^{2}}.$

We assume here and in what follows that $\Phi\neq 0$ in similar relations.

Introduce the subspace

${\mathcal{Y}}_{n}={\mathcal{Y}}_{n}(\tau)=\\{\Phi\in
H^{1}_{0}(\Omega,\tau)\,:\,a(\Phi,\Phi_{j})-\mu(\Phi,\Phi_{j})=0,\,j=0,\ldots,n-1\\},$

where we use the short notation $\mu=\mu_{Dn}$. The relation
${\mathcal{X}}_{n}\subset{\mathcal{Y}}_{n}$, follows from

$a(\Phi,\Phi_{j})-\mu(\Phi,\Phi_{j})_{L^{2}(\Omega)}=(\mu_{Dj}-\mu)(\Phi,\Phi_{j})_{L^{2}(\Omega)}=0\;\;\mbox{for
$\Phi\in{\mathcal{X}}_{n}$, $j=0,\ldots,n-1$}.$

To show that the codimension of ${\mathcal{Y}}_{n}$ is $n$, we assume that
there exists

$\Phi_{*}=\sum_{j=0}^{n-1}c_{j}\Phi_{j}$

such that

$a(\Phi,\Phi_{*})-\mu(\Phi,\Phi_{*})_{L^{2}(\Omega)}=0\;\;\mbox{for all
$\Phi\in H^{1}_{0}(\Omega,\tau)$}.$

Due to (3.12)

$a(\Phi_{*},\Phi_{*})-\mu||\Phi_{*}||^{2}_{L^{2}(\Omega)}<0$

if one of coefficient $c_{j}$ is non-zero and hence co-dimension of
${\mathcal{Y}}_{n}$ is $n$. Let us also prove that every $u\in
H^{1}_{0}(\Omega,\tau)$ admits the representation $u=\Phi+\phi$, where
$\Phi\in{\mathcal{Y}}_{n}$ and $\phi\in X_{n}$. Indeed, we choose $c_{j}$ and
$\phi$ according to

$c_{j}(\mu_{Dj}-\mu)||\Phi_{j}||^{2}_{L^{2}(\Omega)}=a(u,\Phi_{j})-\mu(u,\Phi_{j})_{L^{2}(\Omega)},\;\;\phi=\sum_{j=0}^{n-1}c_{j}\Phi_{j}.$

Then

$a(\Phi,\Phi_{j})-\mu(\Phi,\Phi_{j})_{L^{2}(\Omega)}=a(u,\Phi_{j})-\mu(u,\Phi_{j})_{L^{2}(\Omega)}+a(\phi,\Phi_{j})+\mu(\phi,\Phi_{j})_{L^{2}(\Omega)}=0.$

Therefore $\Phi\in{\mathcal{Y}}_{n}$.

By the min-max principle for eigenvalues of (3.6):

$\widehat{\mu}_{n}\leq\min_{\Phi\in{\mathcal{Y}}_{n}}\frac{a(\Phi,\Phi)}{||\Phi||_{L^{2}(\Omega)}^{2}},$
(3.13)

If this inequality is strong then the right inequality in (3.11) is proved.
Assume that

$\widehat{\mu}_{n}=\min_{\Phi\in{\mathcal{Y}}_{n}}\frac{a(\Phi,\Phi)}{||\Phi||_{L^{2}(\Omega)}^{2}}.$
(3.14)

Then the minimum is attained at a certain function
$\widehat{\Phi}\in{\mathcal{Y}}_{n}$ and

$a(\widehat{\Phi},v)=\mu(\widehat{\Phi},v)_{L^{2}(\Omega)}\;\;\mbox{for all
$v\in{\mathcal{Y}}_{n}$}.$ (3.15)

Using the definition of ${\mathcal{Y}}_{n}$, we conclude that

$a(\widehat{\Phi},v+g)-\mu(\widehat{\Phi},v+g)_{L^{2}(\Omega)}=0\;\;\mbox{for
all $g\in X_{n}$}.$

Therefore $\widehat{\Phi}\in{\mathcal{Y}}_{n}$ is an eigenfunction of (3.6)
and (3.7) with $\mu=\mu_{Dn}$ considered in the space $H^{1}_{0}(\Omega,\tau)$
and $H^{1}_{00}(\Omega)$. Considering the functions
$\widehat{\Phi}(X,Y)+\widehat{\Phi}(-X,Y)$ and
$\widehat{\Phi}(X,Y)-\widehat{\Phi}(-X,Y)$, which are even and odd
respectively and using that $\tau\in(0,\tau_{*})$ (this implies that
$\partial_{X}\Phi=0$ for $X=\pm\Lambda/2$), we conclude that
$\widehat{\Phi}=0$, since there is a homogeneous Cauchy data at
$X=\pm\Lambda/2$ for the function $\widehat{\Phi}$. Thus the right inequality
in (3.11) is strong.

Consider the case when (3.12) is not valid. Then we choose $l$ such that

$\mu_{Dn}=\cdot=\mu_{D(n-l)}>\mu_{D(n-l-1)}.$

By just proved, $\widehat{\mu}_{n-l}(\tau)<\mu_{D(n-l)}$, which implies
$\widehat{\mu}_{n}(\tau)<\mu_{D(n)}$.

Let us turn to the left inequality in (3.11). Now, we assume that $n$
satisfies

$\widehat{\mu}_{n}>\widehat{\mu}_{n-1}.$ (3.16)

We represent $H^{1}_{0}(\Omega)$ as

$H^{1}_{0}(\Omega)={\mathcal{X}}+{\mathcal{Y}},$

where ${\mathcal{X}}$ consists of even functions in $H^{1}_{0}(\Omega)$ and
${\mathcal{Y}}$ consists of odd functions in $H^{1}_{0}(\Omega)$.

Let $\Phi_{j}\in H^{1}_{0}(\Omega,\tau)$ be the eigenfunction of (3.7)
corresponding to $\mu_{Dj}$, $j=0,\ldots$, and let now $X_{n}$ be the space of
linear combinations of $\\{\Phi_{j}\\}_{j=0}^{n-1}$ and

${\mathcal{X}}_{n}(\tau)=\\{\Phi\in
H^{1}_{0}(\Omega,\tau)\,:\,(\Phi,\Phi_{j})_{L^{2}(\Omega)},\,j=0,\ldots,n-1\\}.$

Then

$\widehat{\mu}_{n}=\min_{\Phi\in
X_{n}}\frac{a(\Phi,\Phi)}{||\Phi||_{L^{2}(\Omega)}^{2}}.$

Introduce the subspace

${\mathcal{Y}}_{n}=\\{\Phi\in
H^{1}_{0}(\Omega)\,:\,a(\Phi,\Phi_{j})-\mu(\Phi,\Phi_{j})=0,\,j=0,\ldots,n-1\\},$

where we use the short notation $\mu=\mu_{Dn}$. The relation
${\mathcal{X}}_{n}\subset{\mathcal{Y}}_{n}$, follows from

$a(\Phi,\Phi_{j})-\mu(\Phi,\Phi_{j})_{L^{2}(\Omega)}=(\mu_{Dj}-\mu)(\Phi,\Phi_{j})_{L^{2}(\Omega)}=0\;\;\mbox{for
$\Phi\in{\mathcal{X}}_{n}$, $j=0,\ldots,n-1$}.$

In the same way as before one can show that the codimension of
${\mathcal{Y}}_{n}$ is equal to $n$ and that every $u\in
H^{2}_{0}(\Omega,\tau)$ admits the representation $u=\Phi+\phi$, where
$\Phi\in{\mathcal{Y}}_{n}$ and $\phi\in X_{n}$.

By the min-max principle for eigenvalues of (3.6):

$\widehat{\mu}_{n}\leq\min_{\Phi\in{\mathcal{Y}}_{n}}\frac{a(\Phi,\Phi)}{||\Phi||_{L^{2}(\Omega)}^{2}},$
(3.17)

If this inequality is strong then the right inequality in (3.11) is proved.
Assume that

$\widehat{\mu}_{n}=\min_{\Phi\in{\mathcal{Y}}_{n}}\frac{a(\Phi,\Phi)}{||\Phi||_{L^{2}(\Omega)}^{2}}.$
(3.18)

Then the minimum is attained at a certain function
$\widehat{\Phi}\in{\mathcal{Y}}_{n}$ and

$a(\widehat{\Phi},v)=\mu(\widehat{\Phi},v)_{L^{2}(\Omega)}\;\;\mbox{for all
$v\in{\mathcal{Y}}_{n}$}.$

Using the definition of ${\mathcal{Y}}_{n}$, we conclude that

$a(\widehat{\Phi},v+g)-\mu(\widehat{\Phi},v+g)_{L^{2}(\Omega)}=0\;\;\mbox{for
all $g\in X_{n}$}.$

Therefore $\widehat{\Phi}\in{\mathcal{Y}}_{n}$ is an eigenfunction of (3.6)
and (3.7) with $\mu=\widehat{\mu}_{n}$ considered in the space
$H^{1}_{0}(\Omega)$ and $H^{1}_{0}(\Omega,\tau)$. The same argument as before
proves that the right inequality in (3.11) is strong.

∎

The next lemma gives important estimates for $\widehat{\mu}_{j}$, which are
essential for the proof our main theorem.

######  Lemma 3.2.

For all $\tau\in(0,\tau_{*})$

(i)

$\mu_{0}<\widehat{\mu}_{0}(\tau)<\nu^{*0}_{0}\;\;.$ (3.19)

(ii) If $0<\nu^{*0}_{1}$ and $\nu^{0*}_{0}<\mu_{1}$ then

$\nu^{0*}_{0}<\widehat{\mu}_{1}(\tau)<0,$ (3.20)

and

$\mu_{1}<\widehat{\mu}_{2}(\tau)<\nu^{*0}_{1}$ (3.21)

(iii)

$\nu^{0*}_{1}<\widehat{\mu}_{3}(\tau).$ (3.22)

###### Proof.

First let us prove the equalities

$\mu_{D0}=\nu^{*0}_{0},\;\;\mu_{D1}=\nu^{00}_{0},\;\;\mu_{D2}=\nu^{*0}_{1}$
(3.23)

assuming $\nu^{00}_{0}<\nu^{*0}_{1}$ (this is needed for the second and third
relations above).

We represent $H^{1}_{00}(\Omega)$ as

$H^{1}_{00}(\Omega)=\widehat{X}+\widehat{Y},$

where $\widehat{X}$ consists of even functions in $H^{1}_{00}(\Omega)$ and
$\widehat{Y}$ consists of odd functions in $H^{1}_{00}(\Omega)$. Then the
spaces $\widehat{X}$ and $\widehat{Y}$ are invariant for this spectral
problem.

Introduce two spectral problems

$a_{\Omega}(u,v)=\mu(u,v)_{L^{2}(\Omega)}\;\;\mbox{for all
$v\in\widehat{X}$},$ (3.24)

where $u\in\widehat{X}$, and

$a_{\Omega}(u,v)=\mu(u,v)_{L^{2}(\Omega)}\;\;\mbox{for all
$v\in\widehat{Y}$},$ (3.25)

where $u\in\widehat{Y}$. Then the eigenvalues and eigenfunctions of the
problems (3.24) and (3.25) coincide with the eigenvalues and eigenfunctions of
the problem (3.7). Furthermore, the eigenvalues of the problem (3.24)
coincides with the eigenvalues of the problem (2.12) and the eigenvalues of
the problem (3.25) coincides with the eigenvalues of the problem (2.14). This
implies (3.23) and due to Proposition 2.3 and Lemma 3.1 we get the right-hand
inequalities in (3.19)-(3.21).

Let us turn to the estimates (3.19)-(3.21) and (3.22) from below. We start
from proving

$\mu_{N0}=\mu_{0},\;\;\mu_{N1}=\nu^{0*}_{0},\;\;\mu_{N2}=\mu_{1},\;\;\mu_{N3}=\nu^{0*}_{1}$
(3.26)

assuming $\nu^{0*}_{0}<\mu_{1}$ (this is needed only for the second and third
relations above).

We use the representation

$H^{1}_{0}(\Omega)=\tilde{X}+\tilde{Y},$

where $\tilde{X}$ consists of even functions with respect to $X$ in
$H^{1}_{0}(\Omega)$ and $\tilde{Y}$ consists of odd functions in
$H^{1}_{0}(\Omega)$. Introduce two more spectral problem

$a(u,v)=\mu(u,v)_{L^{2}(\Omega)}\;\;\mbox{for all $v\in\tilde{X}$},$ (3.27)

where $u\in\tilde{X}$, and

$a(u,v)=\mu(u,v)_{L^{2}(\Omega)}\;\;\mbox{for all $v\in\tilde{Y}$},$ (3.28)

where $u\in\tilde{Y}$. Then the eigenvalues and eigenfunctions of the problems
(3.27) and (3.28) coincides with of the eigenvalues and eigenfunctions of the
problem (3.8). Furthermore, the eigenvalues of the problem (3.27) coincides
with the eigenvalues of the problem (2.3) and the eigenvalues of the problem
(3.28) coincides with the eigenvalues of the problem (2.13). This implies
(3.26) and due to Proposition 2.3 and Lemma 3.1 we get the right-hand
inequalities in (3.19)-(3.22).

∎

######  Corollary 3.3.

If $\mu_{1}>0$ then

$\widehat{\mu}_{1}(\tau)<0,\;\;\widehat{\mu}_{2}(\tau)>0\;\;\mbox{for
$\tau\in(0,\tau_{*})$
and}\;\;\widehat{\mu}_{2}(0)=\mu_{1}>0,\;\widehat{\mu}_{1}=0.$

Moreover the eigenvalue $\mu=0$ is simple with the eigenfunction $\psi_{x}$.

###  3.3. Asymptotics of eigenvalues $\widehat{\mu}_{1}(\tau)$ and
$\widehat{\mu}_{2}(\tau)$

In this section we assume that $\mu_{1}=0$. According to Proposition 2.2 this
eigenvalue is simple in the space of periodic even function. Since $0$ is
always simple eigenvalue in the space of odd periodic functions, the
multiplicity of the eigenvalue $\mu_{1}$ is two in the space of periodic
functions. One can check that if $\mu$ is an eigenvalue of (3) for a certain
$\tau$ then $\mu$ is also the eigenvalue of the same multiplicity of the same
problem for $-\tau$. Therefore

$\widehat{\mu}_{1}(-\tau)=\widehat{\mu}_{1}(\tau),\;\;\widehat{\mu}_{2}(-\tau)=\widehat{\mu}_{2}(\tau).$

Furthermore due to Lemma 3.2

$\widehat{\mu}_{1}(\tau)<0\;\;\mbox{and}\;\;\widehat{\mu}_{2}(\tau)>0\;\;\mbox{for
$\tau\in(0,\tau_{*})$}.$ (3.29)

Since $\mu(-\tau)$, $\mu(\tau+\tau_{*})$ are eigenvalues if $\mu(\tau)$ is an
eigenvalue we have

$\widehat{\mu}_{1}(\tau_{*}-\tau)=\widehat{\mu}_{1}(\tau)\;\;\mbox{and}\;\;\widehat{\mu}_{2}(\tau_{*}-\tau)=\widehat{\mu}_{2}(\tau).$

Moreover if $\phi(X,Y;\tau)$ is an eigenfunction corresponding to $\mu(\tau)$
then $\phi(-X.Y;-\tau)$ is an eigenfunction corresponding to the eigenvalue
$\mu(-\tau)$.

######  Lemma 3.4.

Assume that $\mu_{1}=0$. Then one of the following two options is valid

(i) There exists an integer $n\geq 1$ such that

$\widehat{\mu}_{1}(\tau)=-\kappa_{n}\tau^{2n-1}+O(\tau^{2n}),\;\;\widehat{\mu}_{2}(\tau)=\kappa_{n}\tau^{2n-1}+O(\tau^{2n})$
(3.30)

for small positive $\tau$, where $\kappa_{n}>0$.

(ii) There exist integers $n,\,m\geq 1$ such that

$\widehat{\mu}_{1}(\tau)=A\tau^{2n}+O(|\tau|^{2n+1}),\;\;\widehat{\mu}_{2}(\tau)=B\tau^{2m}+O(|\tau|^{2m+1})$
(3.31)

for small positive $\tau$, where $A<0$ and $B>0$.

###### Proof.

Since the multiplicity of the eigenvalue $\mu=0$ of the problem (3) has
multiplicity two for $\tau=0$ there are two analytic in $\tau$ branches of
eigenvalues of (3), denote them by $\theta_{1}(\tau)$ and $\theta_{2}(\tau)$
such that $\theta_{1}(0)=\theta_{2}(0)=0$. Since $\theta_{1}(-\tau)$ and
$\theta_{2}(-\tau)$ are also eigenvalues one of the following options is
valid:

(a) $\theta_{1}(\tau)=\theta_{2}(\tau)$ and the function $\theta_{1}$ is even
with respect to $\tau$;

(b) $\theta_{1}(-\tau)=\theta_{2}(\tau)$;

(c) $\theta_{1}(-\tau)=\theta_{1}(\tau)$ and
$\theta_{2}(-\tau)=\theta_{2}(\tau)$.

Observing that the functions $\theta_{j}(\tau)$ coincides with one of
functions $\widehat{\mu}_{j}$, $j=1,2$, for small positive $\tau$, we conclude
that only the options (b) and (c) may occur. This leads to the options (i) and
(ii) in our lemma. The sign of $\kappa_{n}$, $A$ and $B$ follows from (3.29).
∎

The asymptotics of the corresponding eigenfunctions is given in the following

######  Lemma 3.5.

Let $\upsilon_{1}(\tau)$ and $\upsilon_{2}(\tau)$ be eigenfunctions
analytically depending on $\tau$ corresponding to the eigenvalues $\theta_{1}$
and $\theta_{2}$ in the proof of Lemma 3.4. In the case (3.30) the
corresponding eigenfunctions satisfy

$\upsilon_{2}(0)=\phi_{0}+i\beta\psi_{0},\;\;\upsilon_{1}(0)=\phi_{0}-i\beta^{-1}\psi_{0},\;\;\;\mbox{with
$\beta\neq 0$}.$ (3.32)

In the case (3.31) the corresponding eigenfunctions satisfy

$\upsilon_{2}(0)=\phi_{0}\;\;\mbox{and}\;\;\upsilon_{1}(0)=\psi_{0}.$ (3.33)

###### Proof.

Introduce polynomials

$\Phi(\tau)=\phi_{0}+i\tau\phi_{1}+\cdots+(i\tau)^{p}\phi_{p}\;\;\mbox{and}\;\;\Psi(\tau)=\psi_{0}+i\tau\psi_{1}+\cdots+(i\tau)^{p}\psi_{p},$

where $\phi_{j}$, $\psi_{k}$ are real-valued functions from
$H^{1}_{p0}(\Omega)$ such that they are even when $j$ is even and $k$ is odd,
and they are odd when $j$ is odd and $k$ is even. Let us construct the
functions $\phi_{j}$ and $\psi_{k}$, $j,k=1,\ldots,p-1$, satisfying

${\bf a}(\Phi(\tau),v;\tau)=O(|\tau|^{p}),\;\;{\bf
a}(\Psi(\tau),v;\tau)=O(|\tau|^{p})\;\;\mbox{for all $v\in
H^{1}_{p0}(\Omega)$}.$

Equating terms of the same power of $\tau$ in the first relation, we obtain

${\bf a}(\phi_{1},v)+{\bf b}(\phi_{0},v)=0$ (3.34)

and

${\bf a}(\phi_{k+2},v)+{\bf b}(\phi_{k+1},v)+{\bf
c}(\phi_{k},v)=0\;\;\mbox{for $k=0,1,\ldots,p-3$ and $v\in
H^{1}_{0}(\Omega)$}.$ (3.35)

Similar relations hold with $\phi$ replaced by $\psi$. Equations (3.34) and
(3.35) are solvable if ”the right-hand side is orthogonal to the kernel of the
main operator corresponding to the form ${\bf a}$”, i.e.

${\bf b}(\phi_{0},\psi_{0})=0,$ (3.36) ${\bf b}(\phi_{k+1},\phi_{0})+{\bf
c}(\phi_{k},\phi_{0})=0,\;\;\;{\bf b}(\phi_{k+1},\psi_{0})+{\bf
c}(\phi_{k},\psi_{0})=0,$ (3.37)

and

${\bf b}(\psi_{k+1},\phi_{0})+{\bf c}(\psi_{k},\phi_{0})=0,\;\;\;{\bf
b}(\psi_{k+1},\psi_{0})+{\bf c}(\psi_{k},\psi_{0})=0.$ (3.38)

Here we used that the form ${\bf b}$ is anti-symmetric. In this case solutions
are not unique and we choose the solutions orthogonal to the kernel, i.e.

${\bf c}(\phi_{k},\phi_{0})=0,\;\;{\bf c}(\phi_{k},\psi_{0})=0,\;\;{\bf
c}(\psi_{k},\phi_{0})=0,\;\;{\bf c}(\psi_{k},\psi_{0})=0,$ (3.39)

for $k=1,\ldots,p-1$. We note that some terms in relations (3.39) vanish since
some of functions are odd and some of them are even.

(i) Assume that (3.30) is valid and $p=2n-1$. In order to write equations
similar to (3.34) and (3.35) for the next term we introduce the function

$\Phi(\alpha,\beta,\tau)=\alpha\Phi(\tau)+\beta\Psi(\tau),$

where $\alpha$ and $\beta$ are unknown constants. The equation for finding
$\alpha$ and $\beta$ is

${\bf
a}(\Phi(\alpha,\beta,\tau),v;\tau)=\kappa(\alpha\phi_{0}+\beta\psi_{0},v)+O(|\tau|^{p+1})\;\;\mbox{for
all $v\in H^{1}_{p0}(\Omega$)}.$

as the result we obtain

${\bf a}(\alpha\phi_{p}+\beta\psi_{p},v)+{\bf
b}(\alpha\phi_{p-1}+\beta\psi_{p-1},v)+{\bf
c}(\alpha\phi_{p-2}+\beta\psi_{p-2},v)=(-i)^{p}\kappa{\bf
c}(\alpha\phi_{0}+\beta\psi_{0},v)$

for all $v\in H^{1}_{p0}(\Omega)$. This problem is solvable if

${\bf b}(\alpha\phi_{p-1}+\beta\psi_{p-1},\phi_{0})+{\bf
c}(\alpha\phi_{p-2}+\beta\psi_{p-2},\phi_{0})=(-i)^{p}\kappa\alpha$ (3.40)

and

${\bf b}(\alpha\phi_{p-1}+\beta\psi_{p-1},\psi_{0})+{\bf
c}(\alpha\phi_{p-2}+\beta\psi_{p-2},\psi_{0})=(-i)^{p}\kappa\beta,$ (3.41)

From (3.40) and (3.41) it follows

$\beta{\bf b}(\psi_{p-1},\phi_{0})=(-i)^{p}\kappa\alpha$ (3.42)

and

$\alpha{\bf b}(\phi_{p-1},\psi_{0})=(-i)^{p}\kappa\beta,$ (3.43)

which implies

$\kappa^{2}=(-1)^{p}{\bf b}(\phi_{p-1},\psi_{0}){\bf b}(\psi_{p-1},\phi_{0}).$
(3.44)

This implies that the left-hand side in (3.44) is positive, $\kappa$ and
$-\kappa$ satisfy and

$\alpha=1\;\;\mbox{and}\;\;\beta=(-i)^{p}\kappa^{-1}{\bf
b}(\phi_{p-1},\psi_{0}).$

This implies that

$(-1)^{p}{\bf b}(\phi_{p-1},\psi_{0}){\bf b}(\phi_{p-1},\psi_{0})=\alpha^{2}.$

(ii) Let (3.31) be valid. The same calculation as above shows

$\alpha{\bf b}(\phi_{p-1},\phi_{0})=(-1)^{p/2}\kappa\alpha,$

and

$\beta{\bf b}(\psi_{p-1},\psi_{0})=(-1)^{p/2}\kappa\beta.$

So one of quantities ${\bf b}(\phi_{p-1},\phi_{0}$ or ${\bf
b}(\psi_{p-1},\psi_{0})$ must be different from $0$ and we arrive at (3.33).

∎

######  Lemma 3.6.

Let the eigenvalue $\mu=0$ is simple with the eigenfunction $\psi_{x}$. Then

$\mu(\tau)=c\tau^{2}+O(\tau^{4}),$

where $c$ is negative.

###### Proof.

We are looking for the eigenvalue and eigenfunction in the form

$\mu(\tau)=c_{1}\tau+c_{2}\tau^{2}+\cdots\;\;\mbox{and}\;\;u(\tau)=u_{0}+i\tau
u_{1}+(i\tau)^{2}u_{2}+\cdots$

then

${\bf a}(u_{0},w)=0\;\;\mbox{for all $w\in H^{1}_{0}(\Omega)$}.$

Therefore $u_{0}=\psi_{x}$. Furthermore,

$i\tau{\bf a}(u_{1},w)+i\tau{\bf
b}(u_{0},w)=c_{1}(u_{0},w)_{L^{2}(\Omega)}\;\;\mbox{for all $w\in
H^{1}_{0}(\Omega)$}.$

Choosing $w=u_{0}$ we see that $c_{1}=0$. Taking $w=u_{1}$, we get

${\bf a}(u_{1},u_{1})+{\bf b}(u_{0},u_{1})=0\;\;\Rightarrow\;\;{\bf
b}(u_{1},u_{0})={\bf a}(u_{1},u_{1}).$ (3.45)

Equation for $u_{2}$ is

${\bf a}(u_{2},w)+{\bf
b}(u_{1},w)+(u_{0},w)_{L^{2}(\Omega)}=-c_{2}(u_{0},w)_{L^{2}(\Omega)}\;\;\mbox{for
all $w\in H^{1}_{0}(\Omega)$}.$

Taking $w=u_{0}$ we get

${\bf
b}(u_{1},u_{0})+(u_{0},u_{0})_{L^{2}(\Omega)}=-c_{2}(u_{0},u_{0})_{L^{2}(\Omega)}.$

Using (3.45), we arrive at

$c_{2}||u_{0}||^{2}_{L^{2}(\Omega)}=||u_{0}||^{2}_{L^{2}(\Omega)}+{\bf
a}(u_{1},u_{1}),$

which proves positivity of $c_{2}$.

∎

##  4\. Subharmonic spectral problems

Let $M$ be a positive integer. As is known (see for example [21], [19]),
$M\Lambda$-periodic, even solutions to the problem (1.2) are linear
combinations of functions

$e^{i\tau_{j}X}w_{j}(X,Y)+e^{-i\tau_{j}X}w_{j}(-X,Y),$ (4.1)

where only such $\tau=\tau_{j}=j\tau_{*}/M$ and $w=w_{j}$, $j=0,1,\ldots,M-1$,
are taken which solves the problem (3) with $\mu=0$. Therefore the knowledge
of the characteristic exponents $\tau_{j}$ helps to describe $M$-subharmonic
solutions.

The spectral problem (1.2) can be considered on the functions $w$ which are
even and has period $M\Lambda$. We shall call this spectral problem
$M$-subharmonic spectral problem. To give a variational formulation of this
spectral problem we introduce

$D_{M}=\\{(X,Y)\in{\mathcal{D}}\,:\,0<X<M\Lambda/2\\},\;\;S_{M}=\\{(X,Y)\in{\mathcal{S}}\,:\,0<X<M\Lambda/2\\},$

and denote by $H^{1}_{0}(D_{M})$ the space of functions in $H^{1}(D_{M})$
which vanish for $Y=0$. We put

$a_{M}(u,v)=\int_{D_{M}}\Big{(}\nabla
u\cdot\nabla\overline{v}-\omega^{\prime}(\Psi)u\overline{v}\Big{)}dXdY-\int_{0}^{M\Lambda/2}\rho(X)u\overline{v}dS,$

Using that $\partial_{X}w(0,Y)=0$ and $\partial_{X}w(M\Lambda/2,Y)=0$, we
arrive at the following variational formulation for the spectral problem (1.2)
on even $M\Lambda$-periodic functions: find a function $\phi\in
H^{1}_{0}(D_{M})$ and the number $\mu=\mu^{(M)}\in\mathbb{R}$ satisfying

$a_{M}(\phi,v)=\mu^{(M)}(\phi,v)_{L^{2}(D_{M})}\;\;\mbox{for all $v\in
H^{1}_{0}(D_{M})$}.$ (4.2)

As it is known the pair $(\phi,\mu^{(M)})$ solves the problem (4.2) if and
only if $\mu^{(M)}$ is an eigenvalue of the problem (1.2), considered on even,
$M\Lambda$-periodic functions, and $\phi$, after a natural extension, is a
corresponding eigenfunction. We numerate the eigenvalues accounting their
multiplicity as

$\mu^{(M)}_{0}\leq\mu^{(M)}_{1}\leq\cdots,\;\;\mu^{(M)}_{j}\to\infty\;\;\mbox{as}\;\;j\to\infty.$

Denote the number of eigenvalues of the problem (4.2) which are less than zero
by $n_{0}(M)$ and by $n(M)$ if we include zero eigenvalues, i.e.

$\displaystyle n_{0}(M)=\mbox{the number of $j$ such
that}\;\;\mu^{(M)}_{j}<0,$ $\displaystyle n(M)=\mbox{the number of $j$ such
that}\;\;\mu^{(M)}_{j}\leq 0.$

######  Proposition 4.1.

Let $\mu_{1}=0$. Then for every $M=1,2,\ldots$

$n_{0}(M)=M\;\;\mbox{when $M$ is odd and}\;\;n(M)=M+1\;\;\mbox{when $M$ is
even}$ (4.3)

and

$n(M)=M+1\;\;\mbox{when $M$ is odd and}\;\;n(M)=M+2\;\;\mbox{when $M$ is
even}.$ (4.4)

###### Proof.

Using that

$\widehat{\mu}_{j}(\tau)=\widehat{\mu}_{j}(-\tau)=\widehat{\mu}_{j}(\tau_{*}+\tau)\;\;\mbox{for
$j=0,1,2$},$

we conclude that $\widehat{\mu}_{j}(\tau)=\widehat{\mu}_{j}(\tau_{*}-\tau)$.
Due to the description of even, $M\Lambda$ periodic solution given by (4.1),
the number $n(M)$ is equal to the number of indexes $k$ and $m$ such that

$\widehat{\mu}_{0}\Big{(}\frac{k}{M}\Big{)}\leq
0,\;\;\widehat{\mu}_{1}\Big{(}\frac{m}{M}\Big{)}\leq 0,$ (4.5)

where $0\leq k,m\leq M/2$. Therefore

$n(M)=M+1\;\;\mbox{when $M$ is odd and}\;\;n(M)=M+2\;\;\mbox{when $M$ is
even},$

which proves (4.4).

Since the equality in (4.5) is reached only when $m=0$, we get (4.3). ∎

A necessary condition for $M$ subharmonic bifurcation is the absence of zero
eigenvalues of the Frechet derivative corresponding to $M$-subharmonic
problem. The next proposition deals with this problem.

######  Proposition 4.2.

If $\mu_{1}>0$ then the numbers $\mu^{(M)}_{j}$ are different from $0$.

###### Proof.

The proof of this proposition follows directly from Corollary 3.3 and from
description of zero eigenvalues of the Frechet derivative for $M$-subharmonic
problem given at the beginning of this section.

∎

##  5\. Branches of Stokes waves

###  5.1. Uniform stream solution, dispersion equation

The uniform stream solution $\Psi=U(y)$ with the constant depth $\eta=d$ and
$\lambda=1$, satisfies the problem

$\displaystyle U^{{}^{\prime\prime}}+\omega(U)=0\;\;\mbox{on $(0;d)$},$
$\displaystyle U(0)=0,\;\;U(d)=1,$
$\displaystyle\frac{1}{2}U^{\prime}(d)^{2}+d=R.$ (5.1)

Let $s=U^{\prime}(0)$ and $s>s_{0}:=2\max_{\tau\in[0,1]}\Omega(\tau)$, where

$\Omega(\tau)=\int_{0}^{\tau}\omega(p)dp.$

Then the problem (5.1) has a solution $(U,d)$ with a strongly monotone
function $U$ if

${\mathcal{R}}(s):=\frac{1}{2}s^{2}+d(s)-\Omega(1)=R.$ (5.2)

In this case $(U,d)$ is found from the relations

$y=\int_{0}^{U}\frac{d\tau}{\sqrt{s^{2}-2\Omega(\tau)}},\;\;d=d(s)=\int_{0}^{1}\frac{d\tau}{\sqrt{s^{2}-2\Omega(\tau)}}.$
(5.3)

The equation (5.2) is solvable if $R>R_{c}$,

$R_{c}=\min_{s\geq s_{0}}{\mathcal{R}}(s).$ (5.4)

We denote by $s_{c}$ the point where the minimum in (5.4) is attained.

Existence of small amplitude Stokes waves is determined by the dispersion
equation (see, for example, [11]). It is defined as follows. The strong
monotonicity of $U$ guarantees that the problem

$\gamma^{{}^{\prime\prime}}+\omega^{\prime}(U)\gamma-\tau^{2}\gamma=0,\;\;\gamma(0,\tau)=0,\;\;\gamma(d,\tau)=1$
(5.5)

has a unique solution $\gamma=\gamma(y,\tau)$ for each $\tau\in\mathbb{R}$,
which is even with respect to $\tau$ and depends analytically on $\tau$.
Introduce the function

$\sigma(\tau)=\kappa\gamma^{\prime}(d,\tau)-\kappa^{-1}+\omega(1),\;\;\kappa=\Psi^{\prime}(d).$
(5.6)

It depends also analytically on $\tau$ and it is strongly increasing with
respect to $\tau>0$. Moreover it is an even function. The dispersion equation
(see, for example [11]) is the following

$\sigma(\tau)=0.$ (5.7)

It has a positive solution if

$\sigma(0)<0.$ (5.8)

By [11] this is equivalent to $s+d^{\prime}(s)<0$ or what is the same

$1<\int_{0}^{d}\frac{dy}{U^{\prime 2}(y)}.$ (5.9)

The left-hand side here is equal to $1/F^{2}$ where $F$ is the Froude number
(see [28]). Therefore (5.9) means that $F<1$, which is well-known condition
for existence of water waves of small amplitude. Another equivalent
formulation is given by requirement (see, for example [13])

$s\in(s_{0},s_{c})\;\;\mbox{and satisfies (\ref{M6a})}.$ (5.10)

The existence of such $s$ is guaranteed by $R\in(R_{c},R_{0})$. Moreover in
this case the equation ${\mathcal{R}}(s)=R$ has exactly two solutions $s_{-}$
and $s_{+}$, $s_{0}<s_{+}<s_{c}<s_{-}$. The corresponding solutions to (5.1)
are given by (5.3) and we denote them by

$(U_{+},d_{+})\;\;\mbox{and}\;\;(U_{-},d_{-}).$ (5.11)

The function $\sigma$ has the following asymptotic representation

$\sigma(\tau)=\kappa\tau+O(1)\;\;\mbox{for large $\tau$}$

and equation (5.7) has a unique positive root, which will be denoted by
$\tau_{*}$. It is connected with $\Lambda_{0}$ by the relation

$\tau_{*}=\frac{2\pi}{\Lambda_{0}}.$ (5.12)

The function $\gamma(y,\tau)$ is positive in $(0,d]$ for $\tau>\tau_{*}$.

Let

$\rho_{0}=\frac{1+\Psi^{\prime}(d)\Psi^{{}^{\prime\prime}}(d)}{\Psi^{\prime}(d)^{2}}.$
(5.13)

We note that

$\frac{1+\Psi^{\prime}(d)\Psi^{{}^{\prime\prime}}(d)}{\Psi^{\prime}(d)^{2}}=\kappa^{-2}-\frac{\omega(1)}{\kappa}$

and hence another form for (5.6) is

$\sigma(\tau)=\kappa\gamma^{\prime}(d,\tau)-\kappa\rho_{0}.$ (5.14)

The spectral problem (1.2) takes the form

$\displaystyle\Delta w+\omega^{\prime}(U)w+\mu w=0\;\;\mbox{for
$X\in\mathbb{R}$ and $0<Y<d$},$
$\displaystyle\partial_{Y}w-\rho_{0}w=0\;\;\mbox{for $Y=d$},$ $\displaystyle
w=0\;\;\mbox{for $Y=0$},$ (5.15)

where $w$ is even, $\Lambda=2\pi/\tau_{*}$-function. One can verify that
$\mu_{0}<0$ and corresponding eigenfunction does not depend on $X$,
$\mu_{1}=0$ and corresponding eigenfunction is
$\cos(\tau_{*}X)\gamma(Y;\tau_{*})$ and $\mu_{2}>0$. For small $t$,
$\mu_{0}<0$ and $\mu_{2}>0$. We assume that

$\mu_{1}(t)>0\;\;\mbox{for small positive $t$}.$ (5.16)

The validity of this assumption will be discussed in forthcoming paper.

###  5.2. Partial hodograph transform

In what follows we will study branches of Stokes waves
$(\Psi(X,Y;t),\xi(X;t))$ of period $\Lambda(t)$, $t\in\mathbb{R}$, started
from the uniform stream at $t=0$. It is convenient to make the following
change of variables

$x=\lambda X,\;\;y=Y,\;\;\lambda=\frac{\Lambda(0)}{\Lambda(t)}$ (5.17)

in order to deal with the problem with a fixed period. As the result we get

$\displaystyle\Big{(}\lambda^{2}\partial_{x}^{2}+\partial_{y}^{2}\Big{)}\psi+\omega(\psi)=0\;\;\mbox{in
$D_{\eta}$},$
$\displaystyle\frac{1}{2}\Big{(}\lambda^{2}\psi_{x}^{2}+\psi_{y}^{2}\Big{)}+\eta=R\;\;\mbox{on
$B_{\eta}$},$ $\displaystyle\psi=1\;\;\mbox{on $B_{\eta}$},$
$\displaystyle\psi=0\;\;\mbox{for $y=0$},$ (5.18)

where

$\psi(x,y;t)=\Psi(\lambda^{-1}x,y;t)\;\;\mbox{and}\;\;\eta(x;t)=\xi(\lambda^{-1}x;t).$

Here all functions have the same period $\Lambda_{0}:=\Lambda(0)$, $D_{\eta}$
and $B_{\eta}$ are the domain and the free surface after the change of
variables (5.17).

Due to the assumption (1.3) we can introduce the variables

$q=x,\;\;p=\psi.$

Then

$q_{x}=1,\;\;q_{y}=0,\;\;p_{x}=\psi_{x},\;\;p_{y}=\psi_{y},$

and

$\psi_{x}=-\frac{h_{q}}{h_{p}},\;\;\psi_{y}=\frac{1}{h_{p}},\;\;dxdy=h_{p}dqdp.$
(5.19)

System (5.2) in the new variables takes the form

$\displaystyle\Big{(}\frac{1+\lambda^{2}h_{q}^{2}}{2h_{p}^{2}}+\Omega(p)\Big{)}_{p}-\lambda^{2}\Big{(}\frac{h_{q}}{h_{p}}\Big{)}_{q}=0\;\;\mbox{in
$Q$},$
$\displaystyle\frac{1+\lambda^{2}h_{q}^{2}}{2h_{p}^{2}}+h=R\;\;\mbox{for
$p=1$},$ $\displaystyle h=0\;\;\mbox{for $p=0$}.$ (5.20)

Here

$Q=\\{(q,p)\,:\,q\in\mathbb{R}\,,\;\;p\in(0,1)\\}.$

The uniform stream solution corresponding to the solution $U$ of (5.1) is

$H(p)=\int_{0}^{p}\frac{d\tau}{\sqrt{s^{2}-2\Omega(\tau)}},\;\;s=U^{\prime}(0)=H_{p}^{-1}(0).$
(5.21)

One can check that

$H_{pp}-H_{p}^{3}\omega(p)=0$ (5.22)

or equivalently

$\Big{(}\frac{1}{2H_{p}^{2}}\Big{)}_{p}+\omega(p)=0.$ (5.23)

Moreover it satisfies the boundary conditions

$\frac{1}{2H_{p}^{2}(1)}+H(1)=R,\;\;H(0)=0.$ (5.24)

The problem (5.2) has a variational formulation (see [6]) and the potential is
given by

$f(h;\lambda)=\int_{-\Lambda_{0}/2}^{\Lambda_{0}/2}\int_{0}^{1}\Big{(}\frac{1+\lambda^{2}h_{q}^{2}}{2h_{p}^{2}}-h+R-(\Omega(p)-\Omega(1))\Big{)}h_{p}dqdp.$
(5.25)

Then according to [15] there exists a branch of solutions to (5.2)

$h=h(q,p;t):\mathbb{R})\rightarrow
C^{2,\alpha}_{pe}(\overline{Q}),\;\;\lambda=\lambda(t):\mathbb{R}\rightarrow(0,\infty),$
(5.26)

which has a real analytic reparametrization locally around each $t$.

##  6\. Bifurcation analysis

Here we present two equation for finding bifurcation points in $q,p$
variables. One of them is defined by a boundary value problem in a two-
dimensional domain and another one including a Dirichlet-Neumann operator is
defined one a part of one-dimensional boundary. It was proved in Sect. 6.4
that the Frechet derivatives of operators in these two formulations have the
same number of negative eigenvalues and the same dimension of the kernels. So
both of them can be used in analysis of the Morse index and the corresponding
crossing number. The first bifurcation $2$D problem is useful in analysis of
the number of negative eigenvalues and the second one is more convenient for
application of general bifurcation results.

Both formulations can be easily extended for study of subharmonic
bifurcations, see Sect. 6.5 In Sect. 6.3 we give an explicit connection
between the Frechet derivatives in $(x,y)$ and $(q,p)$ variables. This
material is borrowed from [9], where all missing proofs can be found.

###  6.1. First formulation of the bifurcation equation

In order to find bifurcation points and bifuracating solutions we put $h+w$
instead of $h$ in (5.2) and introduce the operators

$\displaystyle{\mathcal{F}}(w;t)=\Big{(}\frac{1+\lambda^{2}(h_{q}+w_{q})^{2}}{2(h_{p}+w_{p})^{2}}\Big{)}_{p}-\Big{(}\frac{1+\lambda^{2}h_{q}^{2}}{2h_{p}^{2}}\Big{)}_{q}$
$\displaystyle-\lambda^{2}\Big{(}\frac{h_{q}+w_{q}}{h_{p}+w_{p}}\Big{)}_{q}+\lambda^{2}\Big{(}\frac{h_{q}}{h_{p}}\Big{)}_{q}$

and

${\mathcal{G}}(w;t)=\frac{1+\lambda^{2}(h_{q}+w_{q})^{2}}{2(h_{p}+w_{p})^{2}}-\frac{1+\lambda^{2}h_{q}^{2}}{2h_{p}^{2}}+w$

acting on $\Lambda_{0}$-periodic, even functions $w$ defined in $Q$. After
some cancelations we get

${\mathcal{F}}=\Big{(}\frac{\lambda^{2}h_{p}^{2}(2h_{q}+w_{q})w_{q}-(2h_{p}+w_{p})(1+\lambda^{2}h_{q}^{2})w_{p}}{2h_{p}^{2}(h_{p}+w_{p})^{2}}\Big{)}_{p}-\lambda^{2}\Big{(}\frac{h_{p}w_{q}-h_{q}w_{p}}{h_{p}(h_{p}+w_{p})}\Big{)}_{q}$
(6.1)

and

${\mathcal{G}}=\frac{\lambda^{2}h_{p}^{2}(2h_{q}+w_{q})w_{q}-(2h_{p}+w_{p})(1+\lambda^{2}h_{q}^{2})w_{p}}{2h_{p}^{2}(h_{p}+w_{p})^{2}}+w.$
(6.2)

Both these functions are well defined for small $w_{p}$. Then the problem for
finding solutions close to $h$ is the following

$\displaystyle{\mathcal{F}}(w;t)=0\;\;\mbox{in $Q$}$
$\displaystyle{\mathcal{G}}(w;t)=0\;\;\mbox{for $p=1$}$ $\displaystyle
w=0\;\;\mbox{for $p=0$}.$ (6.3)

Furthermore, the Frechet derivative (the linear approximation of the functions
${\mathcal{F}}$ and ${\mathcal{G}}$) is the following

$Aw=A(t)w=\Big{(}\frac{\lambda^{2}h_{q}w_{q}}{h_{p}^{2}}-\frac{(1+\lambda^{2}h_{q}^{2})w_{p}}{h_{p}^{3}}\Big{)}_{p}-\lambda^{2}\Big{(}\frac{w_{q}}{h_{p}}-\frac{h_{q}w_{p}}{h_{p}^{2}}\Big{)}_{q}$
(6.4)

and

${\mathcal{N}}w={\mathcal{N}}(t)w=(Nw-w)|_{p=1},$ (6.5)

where

$Nw=N(t)w=\Big{(}-\frac{\lambda^{2}h_{q}w_{q}}{h_{p}^{2}}+\frac{(1+\lambda^{2}h_{q}^{2})w_{p}}{h_{p}^{3}}\Big{)}\Big{|}_{p=1}.$
(6.6)

The eigenvalue problem for the Frechet derivative, which is important for the
analysis of bifurcations of the problem (6.1), is the following

$\displaystyle A(t)w=\mu w\;\;\mbox{in $Q$},$
$\displaystyle{\mathcal{N}}(t)w=0\;\;\mbox{for $p=1$},$ $\displaystyle
w=0\;\;\mbox{for $p=0$}.$ (6.7)

###  6.2. Second formulation of the bifurcation equation

There is another formulation for the bifurcating solutions, which is used the
Dirichlet-Neumann operator. Let us consider the problem

$\displaystyle{\mathcal{F}}(w;t)=0\;\;\mbox{in $Q$},$ $\displaystyle
w=g\;\;\mbox{for $p=1$},$ $\displaystyle w=0\;\;\mbox{for $p=0$}.$ (6.8)

We define the operator ${\mathcal{S}}={\mathcal{S}}(g;h,t)$ by

${\mathcal{S}}(g;t)={\mathcal{G}}(w;t)|_{p=1},$ (6.9)

where $w$ is the solution of the problem (6.2). Then the equation for
bifurcating solutions is

${\mathcal{S}}(g;t)=0.$ (6.10)

Here we note that spectral problem for the the Frechet derivative of the left-
hand side in (6.10) is given by

$\displaystyle A(t)w=0\;\;\mbox{in $Q$},$ $\displaystyle N(t)w-w=\theta
w\;\;\mbox{for $p=1$},$ $\displaystyle w=0\;\;\mbox{for $p=0$}.$ (6.11)

More exactly if we introduce the problem

$\displaystyle A(t)w=0\;\;\mbox{in $Q$},$ $\displaystyle w=g\;\;\mbox{for
$p=1$},$ $\displaystyle w=0\;\;\mbox{for $p=0$},$ (6.12)

then the operator

$Sg=(Nw-w)|_{p=1}$ (6.13)

is the Frechet derivative of the operator (6.10). The corresponding spectral
problem is

$Sg=\theta g.$ (6.14)

To prove solvability of the problem (6.2) we consider first the Dirichlet
problem

$\displaystyle Aw=f,$ $\displaystyle w=g,\;\;\mbox{for $p=1$}$ $\displaystyle
w=0\;\;\mbox{for $p=0$}.$ (6.15)

Let

$a_{1}=\min_{\overline{Q}}h_{p},\;\;a_{2}=||h||_{C^{2,\alpha}(Q)}.$

######  Proposition 6.1.

(Proposition 2.4, [9]) There exists a positive constant $C$ depending on
$a_{1}$ and $a_{2}$ such that if $g\in C^{2,\alpha}_{pe}(\mathbb{R})$ and
$f\in C^{0,\alpha}_{pe}(Q)$, $\alpha\in(0,\gamma]$, then the problem (6.2) has
a unique solution $w\in C^{2,\alpha}_{pe}(Q)$, which satisfies the estimate

$||w||_{C^{2,\alpha}_{pe}(Q)}\leq
C(||f||_{C^{0,\alpha}_{pe}(Q)}+||g||_{C^{2.,\alpha}_{pe}(\mathbb{R})}).$

Next consider the nonlinear problem

$\displaystyle{\mathcal{F}}(w;t)=f\;\;\mbox{in $Q$},$ $\displaystyle
w=g\;\;\mbox{for $p=1$},$ $\displaystyle w=0\;\;\mbox{for $p=0$}.$ (6.16)

It can be considered as a small perturbation of the problem (6.2).

######  Proposition 6.2.

(Proposition 2.5, [9]) There exist positive number $\delta_{*}$ and a Constant
$C$ depending on $a_{1}$ and $a_{2}$ such that if

$||f||_{C^{0,\gamma}_{pe}(Q)}+||g||_{C^{2,\gamma}_{pe}(\mathbb{R})}\leq\delta\leq\delta_{*},$

then there exists a unique solution $w\in C^{2,\gamma}_{pe}(Q)$ such that

$||w||_{C^{2,\gamma}_{pe}(Q)}\leq C\delta.$

###  6.3. Spectral problems (9) and (6.2) in $(x,y)$ variables

Let

$F(q,p;\lambda)=\Gamma(q,h)h_{p}.$ (6.17)

Then

$AF=-\omega^{\prime}(p)\Gamma-\lambda^{2}\Gamma_{xx}-\Gamma_{yy}.$ (6.18)

and

$-NF+F=\sqrt{\psi_{x}^{2}+\psi_{y}^{2}}\rho\Gamma-\lambda^{2}\psi_{x}\Gamma_{x}-\psi_{y}\Gamma_{y},$
(6.19)

where

$\rho=\rho(x;t)=\frac{(1+\lambda^{2}\psi_{x}\psi_{xy}+\psi_{y}\psi_{yy})}{\psi_{y}(\psi_{x}^{2}+\psi_{y}^{2})^{1/2}}\Big{|}_{y=\eta(x;t)}.$

Then the following assertion is proved in [9].

######  Lemma 6.3.

Let $a=a(x,y)$ be positive, continuous, $\Lambda_{0}$-periodic function. If
$\Gamma=\Gamma(x,y)$ satisfies the problem

$\displaystyle(\lambda^{2}\partial_{x}^{2}+\partial_{y}^{2})\Gamma+\omega^{\prime}(\psi)\Gamma+\mu
a\frac{1}{\psi_{y}}\Gamma=0\;\;\mbox{in $D_{\eta}$},$
$\displaystyle(\lambda^{2}\nu_{x}\Gamma_{x}+\nu_{y}\Gamma_{y}-\rho\Gamma)_{p=1}=0\;\;\mbox{on
$B_{\eta}$},$ $\displaystyle\Gamma=0\;\;\mbox{for $x=0$},$ (6.20)

then

$\displaystyle AF=\mu aF,$ (6.21) $\displaystyle NF-F=0\;\;\mbox{for $p=1$},$
$\displaystyle F=0\;\;\mbox{for $p=0$}.$

###  6.4. Comparison of two spectral problems

Here we will compare the nagative spectrum of the following spectral problems

$\displaystyle A(t)u=0\;\;\mbox{in $Q$}$ $\displaystyle N(t)u-u=\theta
au\;\;\mbox{for $p=1$}$ $\displaystyle u=0\;\;\mbox{for $p=0$}.$ (6.22)

and

$\displaystyle A(t)u=\mu bu\;\;\mbox{in $Q$}$ $\displaystyle
N(t)u-u=0\;\;\mbox{for $p=1$}$ $\displaystyle u=0\;\;\mbox{for $p=0$},$ (6.23)

where $a$ and $b$ are continuous, positive, $\Lambda$-periodic functions.

######  Proposition 6.4.

The spectral problems (6.4) and (6.4) have the same number of negative
eigenvalues (accounting their multiplicities). Moreover, if we consider these
spectral problems problems on $M\Lambda$-periodic, even functions they have
the same number of negative eigenvalues also.

###### Proof.

For $a=1$ and $b=1$ this assertion is proved in [9]. In general case the proof
is literally the same.

∎

###  6.5. Equation for subharmonic bifurcations

For positive integer $M=1,2,\ldots$, and $\alpha\in(0,1)$ let us introduce the
subspaces $C_{M,e}^{k,\alpha}(\overline{Q})$ and
$C_{M,e}^{k,\alpha}(\mathbb{R})$ of $C^{k,\alpha}(\overline{Q})$ and
$C^{k,\alpha}(\mathbb{R})$ respectively consisting of even functions of period
$M\Lambda_{0}$. A similar space for the domain ${\mathcal{D}}_{\xi}$ we denote
by $C_{M,e}^{2,\alpha}(\overline{{\mathcal{D}}_{\xi}})$.

Equation (6.10) can be considered also on functions of period $M\Lambda_{0}$.
Having this in mind we write as before $h+w$ instead of $h$ but now we assume
that $w\in C^{2,\alpha}_{M,e}(\overline{Q})$. Now the operator
${\mathcal{F}}(w;h,t)$, ${\mathcal{G}}(w;h,t)$, $A(t)$ and $N(t)$ from
Sect.6.1 are considered on functions from $C^{2,\gamma}_{Me}(\overline{Q})$.
To indicate this difference we will use the notations
${\mathcal{F}}_{M}(w;h,t)$, ${\mathcal{G}}_{M}(w;h,t)$, $A_{M}(t)$ and
$N_{M}(t)$ for corresponding operators. Moreover we can define analogs of the
operators ${\mathcal{S}}$ and $S$ and denote them by ${\mathcal{S}}_{M}$ and
$S_{M}$ respectively. New operators are acting on functions from
$C^{1,\alpha}_{M,e}(\mathbb{R})$. The equation for subharmonic bifurcations is

${\mathcal{S}}_{M}(g;t)=0,$ (6.24)

where

${\mathcal{S}}_{M}(g;t)\,:\,{\mathcal{U}}\rightarrow
C_{M,e}^{1,\alpha}(\mathbb{R}).$ (6.25)

Here ${\mathcal{U}}$ is a neighborhood oh $0$ in the space
$C_{M,e}^{2,\alpha}(\mathbb{R})$. The operator ${\mathcal{S}}_{M}$ is also
potential and for the definition of the potential the integration in (5.25)
must be taken over the interval $(-M\Lambda_{0}/2,M\Lambda_{0}/2)$. The
corresponding eigenvalue problem for the Frechet derivative $S_{M}(t)$ is

$S_{M}(t)g=-\theta g,$ (6.26)

where $S_{M}$ is defined by the same formulas as $S$ but now the functions $g$
and $w$ are $M\Lambda_{0}$-periodic. Clearly,

$S_{M}(t)\,:\,C_{M,e}^{2,\alpha}(\mathbb{R})\rightarrow
C_{M,e}^{1,\alpha}(\mathbb{R}).$ (6.27)

##  7\. Spectral properties of the Frechet derivative, $t$ dependence.

Since the Frechet derivatives now depends on $t\in[0,\infty)$, we introduce
the notations $\mu_{j}(t)$ and $\widehat{\mu}_{j}(t,\tau)$ in order to
indicate the dependence od the eigenvalues on the variable $t$. Certainly we
assume that

$\mu_{0}(t)<\mu_{1}(t)\leq\mu_{2}(t)\leq\cdots\;\;\mbox{and}\;\;\widehat{\mu}_{0}(t,\tau)<\widehat{\mu}_{1}(t,\tau)\leq\widehat{\mu}_{2}(t,\tau)\leq\cdots$

Clearly $\widehat{\mu}_{j}(t,0)=\mu_{j}(t)$. Using Proposition 2.1, we
conclude $\mu_{0}(t)<0$ for all $t$. In what follows we will assume that

$\mu_{1}(t_{0})=0,\;\;\mu_{1}(t)>0\;\;\mbox{for $t\in(0,t_{0})$
and}\;\;\mu_{1}(t)<0\;\;\mbox{for $t\in(t_{0},t_{0}+\varepsilon)$},$

where $\varepsilon$ is a small positive number.

######  Lemma 7.1.

The functions

$\widehat{\mu}_{1}(t,\tau)+\widehat{\mu}_{2}(t,\tau)\;\;\mbox{
and}\;\;\widehat{\mu}_{1}(t,\tau)\widehat{\mu}_{2}(t,\tau)$ (7.1)

are analytic in a neighborhood of the point $t=t_{0}$, $\tau=0$.

###### Proof.

We introduce the domain

${\mathcal{D}}=\\{u\in H^{2}_{0,p}(Q)\,:\,N(t)u-u=0\;\mbox{for $p=1$}\\}$

and consider the operator $A(t,\tau)$ defined on this domain. If $t=t_{0}$ and
$\tau=0$ then the spectrum of this operator in a neighborhood of $(t_{0},0)$
consists of the point $0$ and its multiplicity is two. For $(t,\tau)$ close to
$(t_{0},0)$ we introduce the spectral projector $P(t,\tau)$ of the operator
$A(t,\tau)$ corresponding to the eigenvalues located close to the point $0$.
Its kernel has dimensions $2$ and it depends analytically on $(t,\tau)$ in a
neighborhood of $(t_{0},0)$. The operator $P(t,\tau))A(t,\tau)$ has finite
rank and its trace is invariant with respect to the choice of orthogonal
normalized basis. Therefore the functions (7.1) are analytical with respect to
$t$ and $\tau$ in a neighborhood of $(t_{0},0)$. ∎

######  Corollary 7.2.

The functions $\widehat{\mu}_{1}(t,\tau)$ and $\widehat{\mu}_{2}(t,\tau)$ are
Lipschitz continuous in a neighborhood of $(t_{0},0)$.

###### Proof.

Consider the equation

$\mu^{2}-B(t,\tau)\mu+A(t,\tau)=0,\;\;A=\widehat{\mu}_{1}\widehat{\mu}_{2},\;\;B=\widehat{\mu}_{1}\widehat{\mu}_{2}.$

By Lemma 7.1 the coefficients $A$ an $B$ are analysic with respect to
$(t,\tau)$ in a neighborhood of $(t_{0},0)$. Since $\mu=\widehat{\mu}_{1}$ and
$\mu=\widehat{\mu}_{2}$ are roots of this equation the result follows from
[20], for example.

∎

###  7.1. Generalized spectral problem, $\mu=0$.

Here we study bounded solutions of the problem (3) with $\mu=0$, i.e.

$\displaystyle\Delta w+\omega^{\prime}(\psi)w=0\;\;\mbox{in
${\mathcal{D}}_{\eta}$},$ $\displaystyle\partial_{\nu}w-\rho w=0\;\;\mbox{on
${\mathcal{S}}_{\eta}$},$ $\displaystyle w=0\;\;\mbox{for $y=0$}.$ (7.2)

According to Lemma 6.3 this problem in $q,\,p$ variables has the form

$\displaystyle A(\tau,t)w=0\;\;\mbox{in $Q$}$ $\displaystyle
N(\tau,t)w+w=0\;\;\mbox{for $p=1$}$ $\displaystyle w=0\;\;\mbox{for $p=0$}.$
(7.3)

where

$A(\tau,t)w=e^{-i\tau q}A(t)(e^{i\tau q}w),\;\;N(t,\tau)w=e^{-i\tau
q}N(t)(e^{i\tau q}w)$

We are interested in such real $\tau$ for which the problem (7.1) has non-
trivial solutions when $|\tau|$ and $|t-t_{0}|$ are small.

######  Proposition 7.3.

Let $t_{0}>0$ be the first value of $t$ when $\mu_{1}(t_{0})=0$ and
$\mu_{1}(t)<0$ for $t\in(t_{0},t_{0}+\varepsilon)$, where $\varepsilon$ is a
small positive number. Then there exist a small positive $\delta$ such that
all $\tau$ eigenvalues of (7.1) subject to $|t-t_{0}|<\delta$ and
$|\tau|<\delta$ are described by

$\widehat{\tau}_{k}(s)=\sum_{j=0}^{\infty}a_{kj}s^{j/n_{k}},\;\;k=1,\ldots L,$
(7.4)

where $s=t-t_{0}$. This formula gives all possible small $\tau$-eigenvalues.
Here $L$ is the algebraic multiplicity of the $\tau$ eigenvalue $\tau=0$ when
$t=t_{0}$.

###### Proof.

We can reduce (7.1) to a standard eigenvalue problem by introducing

$v=\frac{(w_{q}+i\tau w)}{h_{p}}-\frac{h_{q}w_{p}}{h_{p}^{2}}.$

Then

$Aw=\Big{(}\frac{\lambda^{2}h_{q}}{h_{p}}v-\frac{w_{p}}{h_{p}^{2}}\Big{)}_{p}-\lambda^{2}(i\tau+\partial_{q})v$

and

$Nw=\frac{\lambda^{2}h_{q}}{h_{p}}v-\frac{w_{p}}{h_{p}^{2}}.$

The problem (7.1) can be written as

$\displaystyle i\tau w=\frac{h_{q}w_{p}}{h_{p}}-h_{p}v-w_{q}$ (7.5)
$\displaystyle i\tau
v=\lambda^{-2}\Big{(}\frac{\lambda^{2}h_{q}}{h_{p}}v-\frac{w_{p}}{h_{p}^{2}}\Big{)}_{p}-\partial_{q}v$

with boundary conditions

$\displaystyle\frac{\lambda^{2}h_{q}}{h_{p}}v-\frac{w_{p}}{h_{p}^{2}}=0\;\;\mbox{for
$p=1$}$ $\displaystyle w=0\;\;\mbox{for $p=0$}.$ (7.6)

We introduce the operator

${\mathcal{A}}\left(\begin{array}[]{ll}w\\\
u\end{array}\right)=\left(\begin{array}[]{ll}\frac{h_{q}w_{p}}{h_{p}}-w_{q}&-h_{p}v\\\
-\lambda^{-2}\Big{(}\frac{w_{p}}{h_{p}^{2}}\Big{)}_{p}-\partial_{q}v&\Big{(}\frac{h_{q}}{h_{p}}v\Big{)}_{p}-\partial_{q}v\end{array}\right)$

Then

${\mathcal{A}}={\mathcal{A}}(t):C^{2,\gamma}\times C^{1,\gamma}\rightarrow
C^{1,\gamma}\times C^{0,\gamma}$

We put

${\mathcal{D}}=\\{(w,v)\in C^{2,\gamma}\times
C^{1,\gamma}\,:\,\mbox{(\ref{M9aa}) is satisfied}\\}.$

Then the operator

${\mathcal{A}}:{\mathcal{D}}\rightarrow C^{1,\gamma}\times C^{0,\gamma}$

is Fredholm with zero index and its spectrum consists of isolated
$\tau$-eigenvalues of finite algebraic multiplicity. Denote by $X_{0}$ the
kernel of this operator for $t=t_{0}$ and by $P(t)$ the spectral projector
corresponding to the eigenvalues of ${\mathcal{A}}(t)$ located near $0$. Let
also $N_{0}=\dim X_{0}$. Then $P(t)$ is the spectral projector of rank $N_{0}$
depending analytically on $t$ in a small neighborhood of $t_{0}$. Since the
characteristic equation for the operator $P(t){\mathcal{A}}(t)$ is well
defined and is invariant with respect of the choice of the basis, the
coefficient of the characteristic equation analytically depends on $t$ in a
neighborhood of $t_{0}$. Therefore the its coefficients analytically depend on
$t$ also. Therefore we obtain a polynomial equation for $\tau$-eigenvalues
located in a small neighborhood of $0$ with coefficients analytically
depending on $t$. According to [25] such equation has roots (7.4) which give
all possible small $\tau$-eigenvalues. ∎

Assume that $t_{0}>0$ is the first Stokes bifurcation point, which means

$\mu_{1}(t_{0})=0,\;\;\mu_{1}(t)>0\;\;\mbox{for $t\in(0,t_{0})$
and}\;\;\mu_{1}(t)<0\;\;\mbox{for $t\in(t_{0},t_{0}+\epsilon)$},$ (7.7)

where $\epsilon$ is a small positive number.

Since $\widehat{\mu}_{0}\leq\nu_{0}^{*0}$ according to (3.19), we have

$\widehat{\mu}_{0}(t,\tau)<0\;\;\mbox{for all $t\geq 0$, $\tau\in\mathbb{R}$}$
(7.8)

by (2.15). Using (3.22) and (3.5), we get

$\widehat{\mu}_{3}(t,\tau)>0\;\;\mbox{for all $t\geq 0$,
$\tau\in\mathbb{R}$},$ (7.9)

provided $\mu_{2}(t)>0$ for $t\in(t_{0},t_{0}+\epsilon)$. Furthermore, by
(3.21) and Proposition 2.3

$\widehat{\mu}_{1}(t,\tau)<0\;\;\mbox{for $t\in(0,t_{0}+\epsilon)$ and
$\tau\in(0,\tau_{*})$},$ (7.10)

where $\epsilon$ is chosen to satisfy $\nu_{1}^{*0}(t)>0$ for
$t\in(t_{0},t_{0}+\epsilon)$. Since $\nu_{1}^{*0}(t)>\mu_{1}(t)$, the
inequality (7.10) is always valid for $t\leq t_{0}$ and its validity for
$t\in(t_{0},t_{0}+\epsilon)$ requires a restriction on $\epsilon$. Therefore
all small $\tau$ eigenvalues of the problem (7.1) in the interval
$(0,\tau_{*})$ for $t\in(t_{0},t_{0}+\epsilon)$ are given by the equation
$\widehat{\mu}_{2}(t,\tau)=0$, where $t\in(t_{0},t_{0}+\epsilon)$. Here we
have used inequality (3.21), which implies

$\widehat{\mu}_{2}(t,\tau)>0\;\;\mbox{for $t\leq t_{0}$ and
$\tau\in(0,\tau_{*})$}.$ (7.11)

Since $\tau$ eigenvalues are roots of the equation
$\widehat{\mu}_{2}(t,\tau)=0$, there are roots among (7.4) with real
coefficients. Moreover if $\tau(t)$ is a real root then $-\tau(t)$ is also a
real root. We numerate all positive real roots by $\widehat{\tau}_{k}(t)$,
$k=1,\ldots,m$,

$0<\widehat{\tau}_{1}(t)<\cdots<\widehat{\tau}_{m}(t),\;\;\mbox{for
$t\in(t_{0},t_{0}+\epsilon)$}.$ (7.12)

We note that since all roots in (7.12) satisfies
$\widehat{\mu}_{2}(t,\tau_{j}(t))=0$, $j=1,\ldots,m$, they are different and
we have strong inequalities in (7.12). Since the leading term of functions
$\widehat{\tau}_{j}$ is positive the sign of the derivativ is positive also in
a small neigborhood of $t_{0}$. So we will assume that $\varepsilon$ is chosen
to satisfy

$\partial_{t}\widehat{\tau}_{j}(t)>0\;\;\mbox{for
$t\in(t_{0},t_{0}+\varepsilon)$}.$ (7.13)

######  Lemma 7.4.

Let (7.7) hold. For each $j=1,\ldots,m$ there exists an integer $n>0$ and
$\epsilon_{1}>0$ such that

$\partial^{k}_{t}\widehat{\mu}_{2}(t,\tau)=0\;\;\mbox{for
$\tau=\widehat{\tau}_{j}(t)$ and for $t\in(t_{0},t_{0}+\epsilon_{1})$ }$
(7.14)

for $k<n$ and

$\partial^{n}_{t}\widehat{\mu}_{2}(t,\tau)\neq 0\;\;\mbox{for
$\tau=\widehat{\tau}_{j}(t)$ and for $t\in(t_{0},t_{0}+\epsilon_{1})$ }.$
(7.15)

###### Proof.

We note that by (7.8) the function $\widehat{\mu}_{2}(t,\tau)$ is analytic
with respect to $\tau$ near a root $\tau=\widehat{\tau}_{j}$. Let
$n=1,2,\ldots$. If the equation

$\partial^{n}_{\tau}\widehat{\mu}_{2}(t,\widehat{\tau}_{j}(t))=0$ (7.16)

has infinitely many roots in the interval $(t_{0},t_{0}+\epsilon)$ then the
equation (7.16) is valid for all $t\in(t_{0},t_{0}+\epsilon)$. If (7.16) holds
for a certain $t\in(t_{0},t_{0}+\epsilon)$ and all $n$ then due to analyticity
of $\widehat{\mu}_{(}t,\tau)$ with respect to $\tau$ we get that
$\widehat{\mu}_{2}(t,\tau)=0$ for all small $\tau$ in a neighborhood of
$\widehat{\tau}_{j}(t)$, but this is not true by [23].

Let us choose the smallest $n$ such that
$\partial_{t}^{n}\widehat{\mu}_{2}(t,\tau)|_{\tau=\widehat{\tau}_{j}(t)}\neq
0$ for certain $t$. Then we can choose $\epsilon_{1}$ for which

$\partial^{n}_{\tau}\widehat{\mu}_{2}(t,\tau)|_{\tau=\widehat{\tau}_{j}(t)}\neq
0\;\;\mbox{for $t\in(t_{0},t_{0}+\epsilon_{1})$}.$ (7.17)

Differentiating $\widehat{\mu}_{2}(t,\widehat{\tau}_{j}(t))=0$ $n$ times and
using (7.13), we arrive at (7.15). ∎

##  8\. Local subharmonic bifurcation theorem

In this section we assume that $\mu_{1}(t_{0})=0$ for certain $t_{0}>0$ and
(7.7) holds.

Let

$\widehat{a}_{1}=\widehat{a}_{1}(t_{0},\epsilon)=\min_{\overline{Q}\times[t_{0}-\epsilon,t_{0}+\epsilon]}h_{p}(p,q;t),\;\;\widehat{a}_{2}=\widehat{a}_{2}(t_{0},\epsilon)=\sup_{[t_{0}-\epsilon,t_{0}+\epsilon]}||h(\cdot,\cdot;t)||_{C^{2,\alpha}(Q)}.$

Denote

${\mathcal{A}}_{M}=\\{g\in
C_{p,M\Lambda_{0}}^{2,\alpha}(\mathbb{R})\;:\;||g||_{C^{2,\alpha}(\mathbb{R})}\leq\delta=\delta(\widehat{a}_{1},\widehat{a}_{2})\\},$

where $M$ is a positive integer. Consider operator ${\mathcal{S}}_{M}$ defined
on the set ${\mathcal{A}}_{M}$ for each $t\in(t_{0}-\epsilon,t_{0}+\epsilon)$,
i.e

${\mathcal{S}}_{M}\,:\,{\mathcal{A}}_{M}\times(t_{0}-\epsilon,t_{0}+\epsilon)\to
C_{p,M\Lambda_{0}}^{1,\alpha}(\mathbb{R}).$

Here we consider the equation (6.24) for subharmonic bifurcations. This
operator satisfies

${\mathcal{S}}_{M}\in
C({\mathcal{A}}_{M}\times(t_{0}-\epsilon,t_{0}+\epsilon),C_{p,M\Lambda_{0}}^{1,\alpha}(\mathbb{R}))$

and

$S_{M}\in
C({\mathcal{A}}_{M}\times(t_{0}-\epsilon,t_{0}+\epsilon),L(C_{p,M\Lambda_{0}}^{2,\alpha}(\mathbb{R}),C_{p,M\Lambda_{0}}^{1,\alpha}(\mathbb{R}))).$

######  Theorem 8.1.

Let $t_{0}>0$ satisfy (7.7). Then there exists an integer $M_{0}$ such that
for every $M>M_{0}$ there exists $\varepsilon_{M}>0$ such that the branch
(5.26) has $M$-subharmonic bifurcation at $t=t_{0}+\varepsilon_{M}$. Moreover
the crossing number at these bifurcation point is $1$ and $\varepsilon_{M}\to
0$ as $M\to\infty$. The closure of the set of nontrivial solutions of

${\mathcal{S}}_{M}(g;t)=0$

near $(0,t_{0}+\varepsilon_{M})$ contains a connected component to which
$(0,t_{0}+\varepsilon_{M})$ belongs.

###### Proof.

We use the bifurcation equation (6.24) in a neighborhood of $t=t_{0}$. First
we observe that $t_{0}$ is a Stokes bifurcation point. We choose an integer
$M$. In order to apply Theorem II,4,4 from [8], it is sufficient to verify the
following spectral properties of the Frechet derivative $S_{M}(t)$:

(i) find $t_{M}>t_{0}$ close to $t_{0}$ such that the kernel of the operator
$S_{M}(t_{M})$ is non-trivial;

(ii) There exists a positive $\varepsilon$ such that the kernel of $S_{M}(t)$
is trivial for $t\in(t_{M},t_{M}-\varepsilon)\cup(t_{M},t_{M}+\varepsilon)$.
Furthermore

$n_{M}(t_{2})-n_{M}(t_{1})=1\;\;\mbox{for cerain
$t_{2}\in(t_{M},t_{M}+\epsilon)$ and $t_{1}\in(t_{M},t_{M}-\varepsilon)$}.$

Due to Proposition 6.4 it is sufficient to verify properties (i) and (ii) for
the branch of spectral problems (6.4) (or (6.3)) defined on $M\Lambda_{0}$
($M\Lambda(t)$) periodic functions with eigenvalues $\mu^{M}$. We choose a
sufficiently large integer $M_{0}$ and a small $\delta_{0}>0$, and suppose
that $M>M_{0}$ and $t_{0}<t<t_{0}+\delta_{0}$. The size of $M_{0}$ and
$\delta_{0}$ will be clarified below.

We may assume that $\delta_{0}\leq\epsilon$ which guarantees the validity of
(7.8)-(7.10). The inequalities (7.8) and (7.9) show that the the only $\tau$
eigenvalues which may contribute to the kernel of the operator $S_{M}(t)$ come
from equations $\widehat{\mu}_{1}(t,\tau)=0$ or $\widehat{\mu}_{2}(t,\tau)=0$.
By (7.10) the equation $\widehat{\mu}_{1}(t,\tau)=0$ has the only root
$\tau=0$ on the interval $[0,\tau_{*})$ with the corresponding eigenfunction
$\psi_{x}$. Since the function $\psi_{x}$ is odd this equations does not
contribute to the kernel also.

Let us consider the equation $\widehat{\mu}_{2}(t,\tau)=0$. We assume that
$\delta_{0}$ is chosen such that the eigenvalue $\mu=0$ of the problem (3) for
$\tau=0$ and $t\in(t_{0},t_{0}+\delta_{0})$ is simple 222We recall that this
problem is considered on $\Lambda$-periodic solution and it is not a spectral
point for $t\in(t_{0},t_{0}+\delta_{0})$. By Lemma 3.6

$\widehat{\mu}_{2}(t,\tau)<0\;\;\mbox{for all $t\in(t_{0},t_{0}+\delta_{0})$
and small $|\tau|$}.$ (8.1)

Applying (3.21) to the eigenvalue $\widehat{\mu}_{2}(t_{0},\tau)$ we get
$\widehat{\mu}_{2}(t_{0},\tau)>0$ for $\tau\in(0,\tau_{*})$. Using Lemma 3.4,
we may assume that $\delta_{0}$ is chosen to guarantee

$\widehat{\mu}_{2}(t,\tau)>0\;\;\mbox{for $t\in(t_{0},t_{0}+\delta_{0})$ and
$\tau\in[\delta_{1},\tau_{*}/2]$},$ (8.2)

where $\delta_{1}$ is a small positive number.

Second, consider the sequence of functions (7.12)

$\widehat{\tau}_{1}(t)<\widehat{\tau}_{2}(t)<\cdots\widehat{\tau}_{m}(t).$

By Lemma 7.4 some of them satisfy (7.15) with the the inequality $>0$ and some
of the with the inequality $<0$. Let $n_{*}$ be the largest index among
$1,\ldots,m$ for which the inequality $<0$ is valid in (7.15) and hence for
indexes $j=n_{*}+1,\ldots,m$ the inequality $>0$ holds in (7.15). Such index
exists otherwise $\widehat{\mu}_{2}(t,\widehat{\tau}_{j})\geq 0$ for
$t\in(t_{0},t_{0}+\epsilon)$, which contradicts to (8.1) and (8.2).

We choose $t_{M}=t_{0}+\varepsilon_{M}$ satisfying the relation

$\widehat{\tau}_{n_{*}}(t_{M})=\tau_{*}/M.$ (8.3)

Our requirement on $M_{0}$ is that the equation (8.3) is solvable for all
$M>M_{0}$.

(i) Let us check (i). The kernel of the operator $A(t)$ consists of functions
(4.1) with $\tau_{j}$ satisfying

$\widehat{\tau}_{k}(t_{M})=\tau_{j}:=j\tau_{*}/M\;\;\mbox{for certain
$k=n_{*}+1,\ldots,m$}.$ (8.4)

If we take $t\neq t_{M}$ in a small neighborhood of $t_{M}$ then the kernel
will be trivial since all functions $\widehat{\tau}_{j}$ are strongly
increasing.

(ii) Since the functions $\widehat{\tau}_{k}$, $k=n_{*}+1,\ldots,m$, satisfy
(7.15) with the inequality $>0$ they do not contribute to the changing of the
Morse index at $t_{M}$. Since the function $\widehat{\tau}_{n_{*}}$ satisfies
(7.15) with $<0$ instead of $\neq 0$ the eigenvalue
$\widehat{\mu}_{2}(t_{M},\tau)$ changes sign at the point
$\tau=\widehat{\tau}_{n_{*}}(t_{M})$. Hence $\widehat{\tau}_{n_{*}}$
contributes to the changing of the Morse index at $t_{M}$ with $+1$. So the
crossing number at $t_{M}$ is $1$. This proves properties (i) and (ii) and the
theorem.

∎

##  9\. Global subharmonic bifurcation theorems

Let

$X=C^{2,\alpha}_{0,M\Lambda,e}(Q),\;\;Y=Y_{1}\times
Y_{2}=C^{0,\alpha}_{0,M\Lambda,e}(Q)\times
C^{1,\alpha}_{M\Lambda}(\mathbb{R}).$

Introduce a subset in $\mathbb{R}\times X$:

${\mathcal{O}}_{\delta}=\\{(t,w)\in\mathbb{R}\times
X\,:\,(h+w)_{p}>\delta\;\mbox{in}\;\overline{Q},\delta<\lambda(t)\\}.$

Then the functional

$\Gamma(w,t)=({\mathcal{F}}(w,t),{\mathcal{G}}(w,t))\,:\,{\mathcal{O}}_{\delta}\rightarrow
Y=Y_{1}\times Y_{2},$

where ${\mathcal{F}}$ and ${\mathcal{G}}$ are defined by (6.24) and (6.26), is
well defined and continuous.

The Frechet derivative at the point $w$ has the form

$A\xi=A(w;t)\xi=\Big{(}\frac{\lambda^{2}(h+w)_{q}\xi_{q}}{(h+w)_{p}^{2}}-\frac{(1+\lambda^{2}(h+w)_{q}^{2})\xi_{p}}{(h+w)_{p}^{3}}\Big{)}_{p}-\lambda^{2}\Big{(}\frac{\xi_{q}}{(h+w)_{p}}-\frac{(h+w)_{q}\xi_{p}}{(h+w)_{p}^{2}}\Big{)}_{q}$
(9.1)

and

${\mathcal{N}}\xi={\mathcal{N}}(w;t)\xi=(N\xi-\xi)|_{p=1},$ (9.2)

where

$N\xi=N(w;t)\xi=\Big{(}-\frac{\lambda^{2}(h+w)_{q}\xi_{q}}{(h+\xi)_{p}^{2}}+\frac{(1+\lambda^{2}(h+w)_{q}^{2})\xi_{p}}{(h+w)_{p}^{3}}\Big{)}\Big{|}_{p=1}.$
(9.3)

The eigenvalue problem for the Frechet derivative, which is important for the
analysis of bifurcations of the problem (6.1), is the following

$\displaystyle A(w;t)\xi=\mu\xi\;\;\mbox{in $Q$},$
$\displaystyle{\mathcal{N}}(w;t)\xi=0\;\;\mbox{for $p=1$},$
$\displaystyle\xi=0\;\;\mbox{for $p=0$}.$ (9.4)

Introduce some sets:

${\mathcal{S}}_{\delta}=\mbox{closure in $\mathbb{R}\times X$
of}\;\;\\{(t,w)\in{\mathcal{O}}_{\delta}\,:\,\Gamma(w,t)=0,\;w\;\;\mbox{is not
identically zero}\\}.$

Let also ${\mathcal{C}}^{\prime}_{\delta}$ be the connected component of
${\mathcal{S}}_{\delta}$ containing the point $(t_{M},0)$ together with the
connected set from Theorem 8.1.

The following theorem is an analog of Theorem 4.2 [5].

######  Theorem 9.1.

Let $\delta>0$. Then either

(i) ${\mathcal{C}}^{\prime}_{\delta}$ is unbounded in $\mathbb{R}\times X$, or

(ii) ${\mathcal{C}}^{\prime}_{\delta}$ contains another trivial point
$(t_{1},0)$ with $t_{1}\neq t_{M}$, or

(iii) ${\mathcal{C}}^{\prime}_{\delta}$ contains a point
$(t,w)\in\partial{\mathcal{O}}_{\delta}$.

In [15] it is proved an estimate $\Lambda(t)\geq c_{0}>0$ for all
$t\in\mathbb{R}$, where the constant $c_{0}$ depends only on $\omega$ and $r$.
This estimate implies

$\lambda(t)\leq\frac{\Lambda(0)}{c_{0}}.$

###  9.1. Proof of Theorem 9.1

The proof of the theorem is actually the same as that of Theorem 4.2 [5].

We have

$\displaystyle{\mathcal{F}}=-\frac{1+\lambda^{2}(h_{q}+w_{q})^{2}}{(h_{p}+w_{p})^{3}}(h_{pp}+w_{pp})-\lambda^{2}\frac{h_{qq}+w_{qq}}{h_{p}+w_{p}}+2\lambda^{2}\frac{(h_{q}+w_{q})(h_{qp}+w_{qp})}{(h_{p}+w_{p})^{2}}$
$\displaystyle=\frac{1}{(h_{p}+w_{p})^{3}}F(w,t),$

where

$F(w,t)=-(1+\lambda^{2}(h_{q}+w_{q})^{2})(h_{pp}+w_{pp})-\lambda^{2}(h_{p}+w_{p})^{2}(h_{qq}+w_{qq})+2\lambda^{2}(h_{q}+w_{q})(h_{p}+w_{p})(h_{qp}+w_{qp})$

and

$\displaystyle{\mathcal{G}}=\frac{1}{(h_{p}+w_{p})^{2}}G(w,t),$

where

$G(w,t)=1+\lambda^{2}(h_{q}+w_{q})^{2}+2(w+h-R)(h_{p}+w_{p})^{2}.$

Since the functions (5.26) solve (5.2) we have

$F(0,t)=0,\;\;G(0,t)=0\;\;\mbox{for all $t$}.$

Therefore the system $\Gamma(w,t)=0$ has the same solutions as the system
$(F,G)=0$ provided $h_{p}+w_{p}>0$ inside $\overline{Q}$. Since the arguments
in the proof of Theorem 4.2 [5] were based only on the periodicity of
functions and ellipticity of the problem $(F,G)=0$ the proof given in [5] can
be used with small changes to prove Theorem 9.1.

###  9.2. Main Theorem on global subharmonic bifurcations

Let

${\mathcal{C}}^{\prime}=\bigcup_{\delta}{\mathcal{C}}^{\prime}_{\delta}$

The sets ${\mathcal{C}}^{\prime}_{\delta}$ increase as $\delta$ decreases.

Let

$\widehat{\psi}(X,Y;t),\;\widehat{\xi}(X;t),\;\Lambda(t)$ (9.5)

be the elements from ${\mathcal{C}}^{\prime}$ in $(X,\,Y)$-coordinates. We
will denote the set pf such functions also by ${\mathcal{C}}^{\prime}$. Let

$\widehat{\Psi}(X,Y;t)=\Psi(X,Y;t)+\widehat{\psi}(X,Y;t),\;\;\Xi(X;t)=\xi(X;t)+\widehat{\xi}(X,Y;t)$
(9.6)

The main result of this section is the following

######  Theorem 9.2.

The following alternatives are valid for the set (9.5):

(i)
$\sup_{{\mathcal{C}}^{\prime}}\sup_{X\in\mathbb{R}}|\Xi^{\prime}(X)|=\infty$,
or

(ii) $\sup_{{\mathcal{C}}^{\prime}}\max_{X\in\mathbb{R}}(R-\Xi(X))=0$, or

(iii)
$\sup_{{\mathcal{C}}^{\prime}}\sup_{X\in\mathbb{R}}|\widehat{\Psi}_{Y}(X,0)|=0$,
or

(iv) there exists $\delta>0$ such that (ii) from Theorem 9.1 is valid.

###  9.3. Proof of Theorem 9.2

The proof will use the following assertion, which is proved in [15],
Proposition 3.2.

######  Proposition 9.3.

Assume that $\omega\in C^{1,\alpha}([0,1])$. Let $\delta>0$ be given as well
as a ball $B$ of radius $\rho>0$ and let $M=\sup_{(X,\xi(X))\in
B}|\xi^{\prime}(X)|$. Then there exist constants $\widehat{\alpha}\in(0,1)$
and $C>0$, depending only on $R$, $\delta$, $\rho$ and $M$ such that any
solution $(\Psi,\xi)\in C^{2,\alpha}(\overline{D_{\xi}})\times
C^{2,\gamma}(\mathbb{R})$ of (1.2) with $\inf_{B}\Psi_{Y}\geq\delta$ satisfies
$||\Psi||_{C^{3,\widehat{\alpha}}(D_{\xi}\cap\frac{1}{2}B)}\leq C$, where
$\frac{1}{2}B$ is a ball with the same centre and radius $\frac{1}{2}\rho$.

The next lemma contains the main step of the proof.

######  Lemma 9.4.

Let $\widehat{\Psi}_{Y}(X,Y;t)\geq 0$ in ${\mathcal{D}}_{\Xi}$ and let there
exist positive constants $C_{1}$, $C_{2}$, $\delta_{1}$ and
$\widehat{\delta}_{1}$ such that

$|\Xi^{\prime}(X;t)|\leq C_{1},\;\;\Lambda(t)\leq
C_{2}\;\;\mbox{and}\;\;R-\Xi(X;t)\geq\delta_{1}\;\;\widehat{\Psi}_{Y}(X,0;t)\geq\widehat{\delta}_{1}$
(9.7)

for functions from ${\mathcal{C}}^{\prime}$. Then there exist $\delta>0$ such
that

$\widehat{\Psi}_{Y}(X,Y;t)\geq\delta\;\;\mbox{for
$(X,Y)\in{\mathcal{D}}_{\Xi}$}.$ (9.8)

###### Proof.

The proof consists of several steps.

1\. (The first estimate of $\widehat{\Psi}_{Y}$.) Differentiating
$\widehat{\Psi}(X,\Xi(X;t);t)=1$ with respect to $X$, we get

$\widehat{\Psi}_{X}+\widehat{\Psi}_{Y}\Xi^{\prime}=0.$

Hence

$|\widehat{\Psi}_{X}|\leq C_{1}|\widehat{\Psi}_{Y}|\;\;\mbox{on
${\mathcal{S}}_{\Xi}$}.$

From the Bernoulli equation, we get

$\delta_{1}\leq R-\Xi(X)\leq\frac{1}{2}(1+C_{1}^{2})\widehat{\Psi}_{Y}^{2}.$

Therefore

$\widehat{\Psi}_{Y}^{2}\geq\delta_{2}^{2}:=\frac{2\delta_{1}}{(1+C_{1}^{2})}.$

To estimate the function $\widehat{\Psi}_{Y}$ on the whole domain
${\mathcal{D}}_{\Xi}$ we consider the function

$U=\widehat{\Psi}_{Y}-a\sinh(\beta Y),$

where $a$ and $\beta$ are positive constants. Then

$-\Delta U-\omega(\widehat{\Psi})U=a(\beta^{2}+\omega^{\prime})\sinh(\beta
Y)\geq 0\;\;\mbox{if $\beta^{2}\geq\max_{p\in[0,1]}|\omega^{\prime}(p)|$}.$

Furthermore, $U(X,0)=0$ and

$U(X,\Xi(X))\geq\delta_{2}-a\sinh(\beta d_{-}).$

We choose $a$ to satisfy

$\delta_{2}-a\sinh(\beta d_{-})=0.$

By strong maximum principle we get $U\geq 0$ inside ${\mathcal{D}}_{\Xi}$.
Thus

$\widehat{\Psi}_{Y}(X,Y)\geq a\sinh(\beta Y)\;\;\mbox{in
${\mathcal{D}}_{\Xi}$}.$ (9.9)

2\. (The estimate of $\widehat{\Psi}$). Let $d_{-}$ be defined by (5.11). Then
by Theorem 1.1 from [14],

$\xi(X)\geq d_{-}\;\;\mbox{for all $X\in\mathbb{R}$ and for all solutions to
(\ref{K2a}) from ${\mathcal{C}}^{\prime}$}.$

We split the domain ${\mathcal{D}}_{\Xi}$ as follows

${\mathcal{D}}_{\Xi}=\widehat{\mathcal{D}}_{\Xi}\bigcup{\mathcal{Q}}_{d_{-}},$

where

$\widehat{\mathcal{D}}_{\Xi}=\\{(X,Y)\,:\,X\in\mathbb{R},\,d_{-}\leq
Y<\Xi(X)\\},\;\;{\mathcal{Q}}_{d_{-}}=\mathbb{R}\times(0,d_{-}).$

Using estimate (9.9) together with Proposition 9.3, we get

$||\widehat{\Psi}||_{C^{3,\widehat{\alpha}}(\overline{\widehat{\mathcal{D}}_{\Xi}})}\leq
C$ (9.10)

Since $\widehat{\Psi}(\cdot,d_{-})\in C^{3,\widehat{\alpha}}(\mathbb{R})$ we
have that $\widehat{\Psi}\in
C^{3,\widehat{\alpha}}(\overline{{\mathcal{Q}}_{d_{-}}})$.

3\. (The estimate of $\Psi_{Y}$ from below on ${\mathcal{D}}_{\Xi}$) Let us
show that $\widehat{\Psi}_{Y}\geq\delta_{3}$ in ${\mathcal{Q}}_{d_{-}}$, where
$\delta_{3}$ is a positive constant. Indeed, consider all solutions of

$\Delta U+\omega^{\prime}(\widehat{\Psi})U=0\;\;\mbox{in
${\mathcal{Q}}_{d_{-}}$}$

satisfying (9.10), $U(X,d_{-})\geq\delta_{4}$ and $U\geq 0$ inside
${\mathcal{Q}}_{d_{-}}$ and

$U(X,d_{-})\geq\delta_{2}\;\;\mbox{and}\;\;U(X,0)\geq\widehat{\delta}_{1}.$
(9.11)

Since the functions from this set cannot be zero inside
${\mathcal{Q}}_{d_{-}}$ we get that all functions must be positive in
$\overline{\mathcal{Q}}_{d_{-}}$. Since the function $\widehat{\Psi}_{Y}$ also
belongs to this set we get the inequality (9.8) on ${\mathcal{Q}}_{d_{-}}$.
This together with (9.9) gives the proof of (9.8) on ${\mathcal{D}}_{\Xi}$. ∎

Proof of Theorem Asuume that the alternatives (i)-(iii) from Theorem 9.2 are
not valid. Then the bifurcating solutions satisfy (9.8) and (9.10). This
implies boundedness of solutions and that they belong to
${\mathcal{C}}^{\prime}_{\delta}$ for certain positive delta. So we can apply
Theorem 9.1 and it can be verified that only the alternative (ii) in Theorem
9.1 can be applied here. This proves the theorem.

######  Remark 9.5.

In [15] it was shown that under the condition $R<d_{0}$, where

$d_{0}=d(s_{0})=\int_{0}^{1}\frac{d\tau}{\sqrt{s_{0}^{2}-2\Omega(\tau)}},$

the condition (iii) in Theorem 9.2 can be omitted,

Acknowledgements. The author was supported by the Swedish Research Council
(VR), 2017-03837. The author expresses also his appreciation to a anonymous
reviewer for many useful comments and corrections.

Data availability statement Data sharing not applicable to this article as no
datasets were generated or analysed during the current study.

##  10\. Conflict of interest

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

##  11\. References

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