# Bifurcation from Infinity of the Schrödinger Equation via Invariant
Manifolds

Chunqiu Li ††E-mail<EMAIL_ADDRESS>Jintao Wang ††Corresponding author.
E-mail<EMAIL_ADDRESS>

Department of Mathematics, Wenzhou University,
Wenzhou, Zhejiang Province, 325035, P. R. China

###### Abstract

This paper is concerned with the bifurcation from infinity of the nonlinear
Schrödinger equation

$-\Delta u+V(x)u=\lambda u+f(x,u),\hskip 11.38092ptx\in\mathbb{R}^{N}.$

We treat this problem in the framework of dynamical systems by considering the
corresponding parabolic equation on unbounded domains. Firstly, we establish a
global invariant manifold for the parabolic equation on $\mathbb{R}^{N}$.
Then, we restrict the parabolic equation to this invariant manifold, which
generates a system of finite dimension. Finally, we use the Conley index
theory and the shape theory of attractors to establish some new results on
bifurcations from infinity and multiplicity of solutions of the Schrödinger
equation under an appropriate Landesman-Lazer type condition.

Keywords: Conley index; Bifurcation from infinity; Invariant manifold;
Schrödinger equation; Parabolic equation

MSC2010: 37B30, 35B32, 35K55, 58J55

##  1 Introduction

In this paper we are concerned with the following nonlinear Schrödinger
equation

$-\Delta u+V(x)u=\lambda u+f(x,u),\hskip 11.38092ptx\in\mathbb{R}^{N},$ (1.1)

where the potential $V\in L^{\infty}(\mathbb{R}^{N})$, $N\geqslant 1$,
$\lambda\in\mathbb{R}$ is the bifurcation parameter, and $f$ is bounded and
satisfies the following Landesman-Lazer type condition

$\liminf\limits_{s\rightarrow+\infty}f(x,s)\geqslant\overline{f}>0,\hskip
22.76228pt\limsup\limits_{s\rightarrow-\infty}f(x,s)\leqslant-\underline{f}<0$
(1.2)

uniformly with respect to $x\in\mathbb{R}^{N}$ (where $\overline{f}$ and
$\underline{f}$ are independent of $x$). We are basically interested in the
bifurcation from infinity and multiplicity of solutions of the equation.

The bifurcation and multiplicity of elliptic equations near resonance have
aroused much interest in the past decades. This topic can be traced back to
the earlier work of Mawhin and Schmitt [19], where the authors studied the
problem of elliptic equations of the following form

$-\Delta u=\lambda u+f(x,u)$ (1.3)

on a bounded domain, where $\lambda\in\mathbb{R}$ and $f$ satisfies an
appropriate Landesman-Lazer type condition. It was shown that if $\mu_{k}$ is
an eigenvalue of odd multiplicity, then (1.3) has at least two distinct
solutions for $\lambda$ on one side of $\mu_{k}$ but close to $\mu_{k}$, and
at least one solution for $\lambda$ on the other side. Later Schmitt and Wang
[30] extended this result to the case when $\lambda$ crosses each eigenvalue
$\mu_{k}$ on bifurcations from infinity for operator equations to cover the
case of even multiplicity. Recently, Li, Li and Zhang [15] presented a
dynamical argument for Schmitt and Wang’s result on (1.3). The interested
reader is referred to [1, 4, 5, 8, 9, 17, 20, 22, 23, 29, 32], etc., on
bifurcations from infinity and multiplicity for elliptic equations under
various boundary conditions.

The bifurcation from infinity of the equation (1.1) on unbounded domains is
also widely investigated; see [3, 14, 10, 11, 27, 28] and the references
therein. However, comparing to the case of bounded domains, the problem on
unbounded domains is more difficult. The main reason is that the spectrum of
the operator $A:=-\Delta+V(x)$ is not discrete in general, and may be quite
complicated, which depends on the potential. Under some additional conditions
on the potential $V$ and $f$, Kryszewski and Szulkin [14], using the degree
theory and a variational approach, considered the asymptotic linear
Schrödinger equation (1.1). They showed that if $\lambda_{0}$ is an isolated
eigenvalue for the linearization at infinity, then there exists a sequence
$(u_{n},\lambda_{n})$ of solutions of (1.1) such that

$\|u_{n}\|\rightarrow\infty\hskip 11.38092pt\mbox{and \hskip
11.38092pt$\lambda_{n}\rightarrow\lambda_{0}$,\hskip 22.76228ptas \hskip
11.38092pt$n\rightarrow\infty$,}$

which extends the results of Stuart [27]. Later, by using degree theory and
Conley index theory, Ćwiszewski and Kryszewski [3] studied the bifurcation
from infinity of elliptic equations on $\mathbb{R}^{N}$, and proved that if
the bifurcation parameter is an eigenvalue of the Hamiltonian, then the
bifurcation from infinity also occurs under some suitable conditions.
Motivated by these works mentioned above, in this paper we further study the
bifurcation from infinity and multiplicity of the Schröding equation (1.1)
near an isolated eigenvalue $\lambda_{*}$ of the operator $A=\Delta+V(x)$. We
will show that under some appropriate conditions on $f$ and the potential $V$,
there exists $\theta>0$ such that for each
$\lambda\in[\lambda_{*}-\theta,\lambda_{*})$, (1.1) has at least three
distinct solutions $e_{\lambda}^{1},e_{\lambda}^{2}$ and $e_{\lambda}^{3}$
with

$\|e_{\lambda}^{i}\|\rightarrow\infty,\hskip 11.38092pt\mbox{as}\hskip
11.38092pt\lambda\rightarrow\lambda_{*}^{-},\hskip 11.38092pti=1,2,$ (1.4)

whereas $e_{\lambda}^{3}$ remains bounded on
$[\lambda_{*}-\theta,\lambda_{*})$. The “dual” version of our results also
holds true if, instead of (1.2), we assume that

$\limsup\limits_{s\rightarrow+\infty}f(x,s)\leqslant-\overline{f}<0,\hskip
22.76228pt\liminf\limits_{s\rightarrow-\infty}f(x,s)\geqslant\underline{f}>0$
(1.5)

uniformly with respect to $x\in\mathbb{R}^{N}$.

Our method in this work is arranged as follows. Instead of transforming (1.1)
into an operator equation and using the topological degree or variational
methods, we regard the problem as the stationary one of the following
parabolic equation

$u_{t}-\Delta u+V(x)u=\lambda u+f(x,u),\hskip 11.38092ptx\in\mathbb{R}^{N}.$
(1.6)

Specifically, let $\Phi_{\lambda}$ be the semiflow generated by (1.6) and
$\lambda_{*}$ be an isolated eigenvalue of the operator $A=-\Delta+V(x)$.
Assume that the Lipschitz constant $L_{f}$ of $f$ satisfies

$F_{\mu}L_{f}<1$

for $\mu\in(\frac{1}{4}\beta,\frac{3}{4}\beta)$ fixed, where $F_{\mu}$ and
$\beta>0$ are given by (4.4) and (3.2), respectively. We first present an
existence result on a global invariant manifold ${\mathcal{M}}$ of the
equation (1.6) for $\lambda$ near $\lambda_{*}$. Then the equation (1.6) can
be restricted to this invariant manifold ${\mathcal{M}}$, and therefore it is
reduced to a system of finite dimension. This allows us to apply the Conley
index theory to the reduced system. By virtue of the shape theory of
attractors [13], it can be shown that there exists $\theta>0$ such that for
each $\lambda\in[\lambda_{*}-\theta,\lambda_{*})$, the system (1.6) can
bifurcate from infinity a compact invariant set
${\mathcal{S}}_{\lambda}^{\infty}$, which takes the shape of a sphere
$\mathbb{S}^{m-1}$, where $m$ denotes the algebraic multiplicity of
$\lambda_{*}$. Based on this fact, one can further verify that
${\mathcal{S}}_{\lambda}^{\infty}$ contains at least two distinct equilibria
$e_{\lambda}^{1}$ and $e_{\lambda}^{2}$ of $\Phi_{\lambda}$, which are
actually solutions of (1.1) satisfying (1.4).

Besides, using the Landesman-Lazer type condition (1.2), we show a fundamental
result on $f$ (see Lemma 4.1), which plays a crucial role in establishing our
main results for (1.1). Then we prove that (1.1) has a solution
$e_{\lambda}^{3}$ for $\lambda$ belonging to a two-sided neighborhood
$\Lambda$ of $\lambda_{*}$, which remains bounded on $\Lambda$. We remark that
this lemma extends the result established in [15, 16] (see [15, Lemma 5.2] or
[16, Lemma 6.7]) to the case of unbounded domains, and can be seen as a
nontrivial extension. This is because it is very hard to verify this result by
directly using the techniques given in [15, 16]. To avoid the difficulty
caused by unbounded domains, we establish this lemma on some finite
dimensional subspace of the phase space, which is inspired by the finite-
dimension property of the invariant manifold.

It is worth mentioning that we treat the problem (1.1) in the framework of
dynamical systems, and deal with it via the approach of a pure dynamical
nature, which is different from those in the literature. Moreover, the method
that we restrict the system (1.6) to the global invariant manifold can help us
avoid the verification of the compactness for the semiflow $\Phi_{\lambda}$.
In general, it is not easy to verify that the semiflow is asymptotically
compact for a parabolic equation on unbounded domains.

This work is organized as follows. Section 2 is concerned with some
preliminaries. In Section 3 we first establish a global invariant manifold of
the equation (1.6) on $\mathbb{R}^{N}$, and then we reduce (1.6) to this
invariant manifold. In Section 4, we present a detailed discussion on
dynamical behaviors of the reduced system, and establish our main results on
bifurcations from infinity and multiplicity of solutions of the Schrödinger
equation (1.1).

##  2 Preliminaries

In this section we first make some preliminaries.

###  2.1 Local semiflows

Let $X$ be a complete metric space.

A local semiflow $\Phi$ on $X$ is a continuous mapping from an open set
$\mathcal{D}(\Phi)\subset\mathbb{R}_{+}\times X$ to $X$ that satisfies the
following properties

  1. (1)

For every $x\in X$, there is $0<T_{x}\leqslant\infty$, such that

$(t,x)\in\mathcal{D}(\Phi)\Longleftrightarrow t\in[0,T_{x});$

  2. (2)

$\Phi(0,\cdot)={\rm id}_{X}$, and

$\Phi(t+s,x)=\Phi(t,\Phi(s,x))$

for any $x\in X$ and $t,s\in\mathbb{R}_{+}$ with $t+s\leqslant T_{x}$.

Here, the number $T_{x}$ is called the escape time of $\Phi(t,x)$. For
simplicity, we rewrite $\Phi(t,x)$ as $\Phi(t)x$.

Given an interval $J\subset\mathbb{R}$. A continuous mapping
$\gamma:J\rightarrow X$ is called a trajectory (or solution) of $\Phi$ on $J$
if

$\gamma(t)=\Phi(t-s)\gamma(s),\hskip 11.38092pt\forall t,s\in J,\,t\geqslant
s.$

We call a trajectory $\gamma$ on $\mathbb{R}$ a full trajectory. The orbit of
a full trajectory $\gamma$ is the set

${\rm orb}(\gamma)=\\{\gamma(t):t\in\mathbb{R}\\},$

which is simply called a full orbit.

The $\omega$-limit set $\omega(\gamma)$ and $\omega^{*}$-limit set
$\omega^{*}(\gamma)$ of a full trajectory $\gamma$ are defined, respectively,
by

$\begin{array}[]{ll}\omega(\gamma)=\\{y\in X:\,\,\,\,\mbox{there exists
}t_{n}\rightarrow\infty\mbox{ such that }\gamma(t_{n})\rightarrow
y\\},\end{array}$ $\begin{array}[]{ll}\omega^{*}(\gamma)=\\{y\in
X:\,\,\,\,\mbox{there exists }t_{n}\rightarrow-\infty\mbox{ such that
}\gamma(t_{n})\rightarrow y\\}.\end{array}$

Let $N$ be a subset of $X$. We say that $\Phi$ does not explode in $N$, if
$\Phi([0,T_{x}))x\subset N$ implies that $T_{x}=\infty$.

######  Definition 2.1.

[24] A set $N\subset X$ is said to be admissible, if for every sequences
$x_{n}\in N$ and $t_{n}\rightarrow\infty$ satisfying
$\Phi([0,t_{n}])x_{n}\subset N$ for all $n$, then the sequence
$\Phi(t_{n})x_{n}$ has a convergent subsequence.

A set $N$ is said to be strongly admissible, if it is admissible and moreover,
$\Phi$ does not explode in $N$.

######  Definition 2.2.

$\Phi$ is said to be asymptotically compact on $X$, if every bounded set
$B\subset X$ is strongly admissible.

A set $S\subset X$ is said to be positively invariant (resp., invariant), if
$\Phi(t)S\subset S$ (resp., $\Phi(t)S=S$) for any $t\geqslant 0$.

A compact invariant set ${\mathcal{A}}\subset X$ is called an attractor of
$\Phi$, if it attracts a neighborhood $U$ of itself, that is,

$\lim_{t\rightarrow\infty}{\rm d}_{H}\left(\Phi(t)U,{\mathcal{A}}\right)=0.$

###  2.2 Sectorial operators and spectral sets

Let $X$ be a Banach space. A closed densely defined linear operator
$A:D(A)\subset X\rightarrow X$ is called a sectorial operator, if there are
real numbers $\phi\in(0,\pi/2),a\in\mathbb{R}$ and $M\geqslant 1$ such that
the sector

$S_{a,\phi}=\\{\lambda:\phi\leqslant|{\rm arg}(\lambda-a)|\leqslant\pi,\hskip
11.38092pt\lambda\neq a\\}$

is contained in the resolvent set of $A,$ and

$\|(\lambda I-A)^{-1}\|\leqslant M/|\lambda-a|$

for each $\lambda\in S_{a,\phi}$, where $I$ denotes the identity on $X.$

Let $A$ be a sectorial operator on $X$. By $\sigma(A)$ we denote the spectrum
of $A.$ If $\min_{z\in\sigma(A)}{\rm Re}\,z>0,$ then one can define the
fractional powers of $A$ as follows: for each $\alpha>0,$

$A^{-\alpha}=\frac{1}{\Gamma(\alpha)}\int^{\infty}_{0}t^{\alpha-1}{\rm
e}^{-At}{\rm d}t,$

where $\begin{array}[]{l}\Gamma(s)=\int^{\infty}_{0}t^{s-1}{\rm e}^{-t}{\rm
d}t\end{array}$ is the Gamma function. Let $A^{\alpha}$ be the inverse of
$A^{-\alpha}$ with $D(A^{\alpha})=R(A^{-\alpha}).$

Let $\alpha\geqslant 0$. Denote $X^{\alpha}=D(A_{1}^{\alpha})$ and equip
$X^{\alpha}$ with the norm $\|\cdot\|_{\alpha}$ defined by

$\|u\|_{\alpha}=\|A^{\alpha}_{1}u\|,\hskip 11.38092ptu\in X^{\alpha},$

where $A_{1}=A+aI$, and $a$ is a real number such that
$\min_{z\in\sigma(A_{1})}{\rm Re}z>0$. It is easy to see that $X^{\alpha}$ is
a Banach space, which is called the fractional power of $X$. For more notions
and results on sectorial operators, the interested reader is referred to Henry
[12, Chapter I] for details.

Now let us briefly recall some notions on spectral sets; see [14] and [26],
etc., for details.

Let $X,Y$ be (real) Banach spaces. A closed densely defined linear operator
$L:D(L)\subset X\rightarrow Y$ is called a Fredholm operator if the range
$R(L)$ is closed and $\dim N(L)<\infty,{\rm codim}\,R(L)<\infty,$ where $N(L)$
is the kernel of $L$.

######  Definition 2.3.

Let $E$ be a real Hilbert space and $L:D(L)\subset E\rightarrow E$ be a self-
adjoint linear operator. The essential spectrum $\sigma_{e}(L)$ of $L$ is
defined by the set

$\\{\lambda\in\mathbb{C}:L-\lambda I\,\,\mbox{ is not a Fredholm
operator}\\},$

######  Remark 2.1.

One may immediately deduce from the definition that
$\sigma_{e}(L)\subset\sigma(L)$ and that $\sigma(L)\setminus\sigma_{e}(L)$
consists of isolated eigenvalues of finite multiplicity; see also [14].

###  2.3 Conley index

For the readers’ convenience, we finally recall the definition of Conley
index; see [7, 18] and [24], etc., for details.

Let $\Phi$ be a local semiflow on $X$. From now on we will always assume
$\Phi$ is asymptotically compact.

A compact invariant set $S$ is said to be isolated, if there exists a
neighborhood $N$ of $S$ such that $S$ is the maximal compact invariant set of
$\Phi$ in $\overline{N}$. Correspondingly, the set $N$ is called an isolating
neighborhood of $S$.

Let $B$ be a bounded closed subset of $X$. $x\in\partial B$ is called a strict
ingress (resp., strict egress, bounce-off) point of $B$, if for each
trajectory $\gamma:[-\tau,s]\rightarrow X$ with $\gamma(0)=x$, where
$\tau\geqslant 0$, $s>0$, the following conditions are satisfied: (1) there
exists $0<\varepsilon<s$ such that

$\gamma(t)\in\mbox{int}B\,\,\,(\mbox{resp., }\,\gamma(t)\not\in
B,\,\,\,\mbox{resp., }\,\gamma(t)\not\in B),\hskip
22.76228pt\forall\,t\in(0,\varepsilon);$

(2) if $\tau>0$, then there is a number $\delta\in(0,\tau)$ such that

$\gamma(t)\not\in B\,\,\,(\mbox{resp.,
}\,\gamma(t)\in\mbox{int}B,\,\,\,\mbox{resp., }\,\gamma(t)\not\in B),\hskip
22.76228pt\forall\,t\in(-\delta,0).$

By $B^{i}$ (resp., $B^{e}$, $B^{b}$), we denote the set of all strict ingress
(resp., strict egress, bounce-off) points of the closed set $B$, and write
$B^{-}=B^{e}\cup B^{b}$.

A set $B\subset X$ is called an isolating block [24] if $B^{-}$ is closed and
$\partial B=B^{i}\cup B^{-}.$ Let $N,E$ be two closed subsets of $X$. $E$ is
called an exit set of $N$, if the following properties hold:

  1. (1)

$E$ is N-positively invariant, that is, for each $x\in E$ and $t\geqslant 0$,

$\Phi([0,t])x\subset N\Longrightarrow\Phi([0,t])x\subset E;$

  2. (2)

for every $x\in N,$ if $\Phi(t_{1})x\not\in N$ for some $t_{1}>0$, then there
is $t_{0}\in[0,t_{1}]$ such that $\Phi(t_{0})\in E$.

Let $S$ be a compact isolated invariant set of $\Phi$. A pair of bounded
closed subsets $(N,E)$ is said to be an index pair of $S$, if (1) $N\setminus
E$ is an isolating neighborhood of $S;$ (2) $E$ is an exit set of $N$.

By [24], one can deduce that if $B$ is a bounded isolating block, then
$(B,B^{-})$ is an index pair of the maximal compact invariant set $S$ in $B$.

######  Definition 2.4.

Let $(N,E)$ be an index pair of $S$. The homotopy Conley index of $S$ is
defined to be the homotopy type $[(N/E,[E])]$ of the pointed space
$(N/E,[E])$, denoted by $h(\Phi,S).$

##  3 Invariant manifolds of nonlinear evolution equations

In this section, we establish an existence result on global invariant
manifolds of the following nonlinear evolution equation

$u_{t}+Au=\lambda u+f(u)$ (3.1)

on a Banach space $X$, where $A$ is a sectorial operator on $X$,
$\lambda\in\mathbb{R}$, and $f$ is a globally Lipschitz mapping from
$X^{\alpha}$ to $X$ for some $0\leqslant\alpha<1$.

Let $\|\cdot\|$ and $\|\cdot\|_{\alpha}$ denote the norms of $X$ and
$X^{\alpha}$, respectively.

###  3.1 Prelimiaries

Denote $\sigma(A)$ the spectrum of $A$ and

${\rm Re}\,\sigma(A):=\\{{\rm Re}\,z:z\in\sigma(A)\\}.$

Suppose $\lambda_{0}$ is an isolated eigenvalue of $A$ with finite
multiplicity, and that the spectrum $\sigma(A)$ has a decomposition
$\sigma(A)=\sigma^{1}\cup\sigma^{2}\cup\sigma^{3}$ with

$\sigma^{1}=\sigma(A)\cap\\{{\rm Re}\lambda\leqslant\beta_{1}\\},\hskip
11.38092pt\sigma^{2}=\\{\lambda_{0}\\},\hskip
11.38092pt\sigma^{3}=\sigma(A)\cap\\{{\rm Re}\lambda\geqslant\beta_{2}\\},$

where $\beta_{1},\beta_{2}$ are real numbers. It can be easily seen that

${\rm Re}\,\sigma(A)\cap(\beta_{1},\beta_{2})=\\{{\rm Re}\lambda_{0}\\}.$

Correspondingly, the space $X$ can be decomposed as

$X=X^{1}\oplus X^{2}\oplus X^{3},$

and the space $X^{2}$ is finite dimensional. In addition, we assume the space
$X^{1}$ is also finite dimensional. Denote

$P_{i}:X\rightarrow X^{i},\,i\in\\{1,2,3\\}$

the projection from $X$ to $X_{i}$.

Pick a positive number $\beta$ with

$\beta<\min\\{{\rm Re}\lambda_{0}-\beta_{1},\beta_{2}-{\rm Re}\lambda_{0}\\},$
(3.2)

and write $J=({\rm Re}\lambda_{0}-\frac{1}{4}\beta,{\rm
Re}\lambda_{0}+\frac{1}{4}\beta).$ Let $A_{\lambda}=A-\lambda I,\lambda\in J$
and denote $A^{i}=A_{\lambda}|_{X^{i}}$. Then by the basic knowledge on
sectorial operators (see e.g., [12, Theorems 1.5.3, 1.5.4]), one deduces that
there exists $M>0$ (depending on $A$) such that for each $\alpha\in[0,1)$,

$\displaystyle\|\Lambda^{\alpha}{{\rm e}}^{-A^{1}t}\|\leqslant M{\rm
e}^{\frac{3}{4}\beta t},\hskip 11.38092pt\|{\rm e}^{-A^{1}t}\|\leqslant M{\rm
e}^{\frac{3}{4}\beta t},\hskip 22.76228ptt\leqslant 0,$ (3.3)
$\displaystyle\|\Lambda^{\alpha}{\rm e}^{-A^{2}t}\|\leqslant M{\rm
e}^{\frac{1}{4}\beta|t|},\hskip 11.38092pt\|{\rm e}^{-A^{2}t}\|\leqslant M{\rm
e}^{\frac{1}{4}\beta|t|},\hskip 22.76228ptt\in\mathbb{R},$ (3.4)
$\displaystyle\|\Lambda^{\alpha}{\rm
e}^{-A^{3}t}P_{3}\Lambda^{-\alpha}\|\leqslant M{\rm e}^{-\frac{3}{4}\beta
t},\hskip 11.38092pt\|\Lambda^{\alpha}{\rm e}^{-A^{3}t}\|\leqslant
Mt^{-\alpha}{\rm e}^{-\frac{3}{4}\beta t},\hskip 22.76228ptt>0,$ (3.5)

where $\Lambda=A+aI$, and $a$ is a positive number such that ${\rm
Re}\,\sigma(\Lambda)>0$. The first estimates in (3.3) and (3.4) hold true as
the spaces $X^{1}$ and $X^{2}$ are finite dimensional.

For a given $\mu\geqslant 0$, we define a space by

${\mathcal{X}}_{\mu}=\big{\\{}u\in{\mathcal{C}}(\mathbb{R};X^{\alpha}):\sup\limits_{t\in\mathbb{R}}{\rm
e}^{-\mu|t|}\|u(t)\|_{\alpha}<\infty\big{\\}},$ (3.6)

which is equipped with the following norm

$\|u\|_{{\mathcal{X}}_{\mu}}=\sup\limits_{t\in\mathbb{R}}{\rm
e}^{-\mu|t|}\|u(t)\|_{\alpha}<\infty.$

Then it can be easily seen that ${\mathcal{X}}_{\mu}$ is a Banach space.

Now we rewrite the equation (3.1) as

$u_{t}+A_{\lambda}u=f(u).$ (3.7)

It is well-known that the Cauchy problem of (3.7) is well-posed in
$X^{\alpha}$; see e.g., [12, Theorem 3.3.3]. Specifically, for each $u_{0}\in
X^{\alpha},$ there exist $T>0$ and a (unique) continuous function
$u:[0,T)\rightarrow X^{\alpha}$ with $u(0)=u_{0}$ such that the equation (3.7)
is satisfied on $(0,T)$.

Denote $\Phi_{\lambda}$ the semiflow generated by the equation (3.7).

Let us start with a fundamental result on the solution of (3.7).

######  Lemma 3.1.

Let $\mu\in(\frac{1}{4}\beta,\frac{3}{4}\beta)$. Assume
$u\in{\mathcal{X}}_{\mu}.$ Then $u$ is a full solution of (3.7) if and only if
$u$ satisfies

$\displaystyle u(t)=$ $\displaystyle{\rm
e}^{-A^{2}t}P_{2}u(0)+\\!\int^{t}_{0}{\rm e}^{-A^{2}(t-s)}P_{2}f(u(s)){\rm
d}s+\\!\int^{t}_{-\infty}{\rm e}^{-A^{3}(t-s)}P_{3}f(u(s)){\rm d}s$
$\displaystyle-\int^{\infty}_{t}{\rm e}^{-A^{1}(t-s)}P_{1}f(u(s)){\rm d}s.$
(3.8)

###### Proof.

Assume $u\in{\mathcal{X}}_{\mu}$. Let $u$ be a full solution of (3.7). Then

$u(t)=u^{1}(t)+u^{2}(t)+u^{3}(t),\hskip 11.38092ptt\in\mathbb{R},$ (3.9)

where $u^{i}(t)=P_{i}u(t),i\in\\{1,2,3\\}.$ It is easy to see that $u^{i}(t)$
can be expressed, respectively, as

$\displaystyle u^{2}(t)={\rm e}^{-A^{2}t}u^{2}(0)+\int^{t}_{0}{\rm
e}^{-A^{2}(t-s)}P_{2}f(u(s)){\rm d}s,\hskip 11.38092ptt\in\mathbb{R},$ (3.10)
$\displaystyle u^{1}(t)={\rm
e}^{-A^{1}(t-t_{0})}u^{1}(t_{0})+\int^{t}_{t_{0}}{\rm
e}^{-A^{1}(t-s)}P_{1}f(u(s)){\rm d}s,\hskip 11.38092ptt\leqslant t_{0},$
(3.11) $\displaystyle u^{3}(t)={\rm
e}^{-A^{3}(t-t_{0})}u^{3}(t_{0})+\int^{t}_{t_{0}}{\rm
e}^{-A^{3}(t-s)}P_{3}f(u(s)){\rm d}s,\hskip 11.38092ptt\geqslant t_{0}.$
(3.12)

By (3.3) (note that $t\leqslant t_{0}$), one deduces that

$\displaystyle\|{\rm e}^{-A^{1}(t-t_{0})}u^{1}(t_{0})\|_{\alpha}\leqslant$
$\displaystyle\,M{\rm e}^{\frac{3}{4}\beta(t-t_{0})}\|u(t_{0})\|_{\alpha}$
$\displaystyle=$ $\displaystyle\,M{\rm e}^{\frac{3}{4}\beta t}{\rm
e}^{-(\frac{3}{4}\beta-\mu)t_{0}}\left({\rm e}^{-\mu
t_{0}}\|u(t_{0})\|_{\alpha}\right)$ $\displaystyle\leqslant$
$\displaystyle\,M{\rm e}^{\frac{3}{4}\beta t}{\rm
e}^{-(\frac{3}{4}\beta-\mu)t_{0}}\|u\|_{{\mathcal{X}}_{\mu}}\rightarrow
0\hskip 11.38092pt\mbox{as}\hskip 11.38092ptt_{0}\rightarrow\infty.$

Moreover, we infer from (3.5) that

$\displaystyle\|{\rm e}^{-A^{3}(t-t_{0})}u^{3}(t_{0})\|_{\alpha}\leqslant$
$\displaystyle\,M{\rm e}^{-\frac{3}{4}\beta(t-t_{0})}\|u(t_{0})\|_{\alpha}$
$\displaystyle=$ $\displaystyle M\,{\rm e}^{-\frac{3}{4}\beta t}{\rm
e}^{(\frac{3}{4}\beta-\mu)t_{0}}({\rm e}^{\mu t_{0}}\|u(t_{0})\|_{\alpha})$
$\displaystyle\leqslant$ $\displaystyle\,M{\rm e}^{-\frac{3}{4}\beta t}{\rm
e}^{(\frac{3}{4}\beta-\mu)t_{0}}\|u\|_{{\mathcal{X}}_{\mu}}\rightarrow 0\hskip
11.38092pt\mbox{as}\hskip 11.38092ptt_{0}\rightarrow-\infty.$

Letting $t_{0}\rightarrow\infty$ and $t_{0}\rightarrow-\infty$ in (3.11),
(3.12), respectively, we conclude from (3.9)-(3.12) that the integral equation
(3.8) holds. On the other hand, if (3.8) holds, then one can easily check that
$u(t)$ is a full solution of (3.7). ∎

###  3.2 Existence of global invariant manifolds

In this subsection, we state and prove the existence result of global
invariant manifolds for the equation (3.7). For this purpose, we consider the
space ${\mathcal{X}}_{\mu}$ for $\mu\in(\frac{1}{4}\beta,\frac{3}{4}\beta)$
being fixed, which is defined by (3.6).

Henceforth we always assume that the following condition on $f$ holds true.
($\mathbf{A}$) The Lipschitz constant $L_{f}$ of $f$ satisfies

$F_{\mu}L_{f}<1,$

where

$F_{\mu}=M\big{[}\int^{\infty}_{0}{\rm e}^{-(\mu-\frac{1}{4}\beta)s}{\rm
d}s+\int^{\infty}_{0}(1+s^{-\alpha}){\rm e}^{-(\frac{3}{4}\beta-\mu)s}{\rm
d}s\big{]}.$

Set

$X_{i}^{\alpha}=X^{i}\cap X^{\alpha},\hskip
11.38092ptX_{i,j}^{\alpha}=X_{i}^{\alpha}\oplus X_{j}^{\alpha},$

where $i,j=1,2,3,i\neq j$. Then we have the following result.

######  Theorem 3.1.

Let the assumption ($\mathbf{A}$) hold and
$\mu\in(\frac{1}{4}\beta,\frac{3}{4}\beta)$ be fixed. Then there exists a
Lipschitz continuous mapping $\xi_{\lambda}$ from $X^{\alpha}_{2}$ to
$X^{\alpha}_{13}$ for $\lambda\in J$, such that for each $\lambda\in J,$ the
system (3.7) has a global invariant manifold ${\mathcal{M}}_{\lambda}$, which
is represented as

${\mathcal{M}}_{\lambda}=\\{w+\xi_{\lambda}(w):w\in X^{\alpha}_{2}\\}.$

###### Proof.

Let $\lambda\in J$ and $w\in X^{\alpha}_{2}.$ We use the integral equation
(3.8) to define a mapping $G:=G_{\lambda,w}$ on ${\mathcal{X}}_{\mu}$ by

$\displaystyle G(u)(t)=$ $\displaystyle{\rm e}^{-A^{2}t}w+\\!\int^{t}_{0}{\rm
e}^{-A^{2}(t-s)}P_{2}f(u(s)){\rm d}s+\\!\int^{t}_{-\infty}{\rm
e}^{-A^{3}(t-s)}P_{3}f(u(s)){\rm d}s$ $\displaystyle-\int^{\infty}_{t}{\rm
e}^{-A^{1}(t-s)}P_{1}f(u(s)){\rm d}s.$

Let us first verify that
$G:{\mathcal{X}}_{\mu}\rightarrow{\mathcal{X}}_{\mu}.$ Assume
$u\in{\mathcal{X}}_{\mu}.$ Since $f(u)$ is globally Lipschitz continuous,
there exists $C_{f}>0$ such that

$\|f(u)\|\leqslant C_{f}(\|u\|_{\alpha}+1).$

By (3.3)-(3.5), we have

$\displaystyle\|G(u)\|_{\alpha}\leqslant$ $\displaystyle M{\rm
e}^{\frac{1}{4}\beta|t|}\|w\|_{\alpha}+M\int^{t_{2}}_{t_{1}}{\rm
e}^{\frac{1}{4}\beta|t-s|}C_{f}(\|u(s)\|_{\alpha}+1){\rm d}s$
$\displaystyle+M\int_{-\infty}^{t}(t-s)^{-\alpha}{\rm
e}^{-\frac{3}{4}\beta(t-s)}C_{f}(\|u(s)\|_{\alpha}+1){\rm d}s$
$\displaystyle+M\int^{\infty}_{t}{\rm
e}^{\frac{3}{4}\beta(t-s)}C_{f}(\|u(s)\|_{\alpha}+1){\rm d}s,$ (3.13)

where

$t_{1}:=\min\\{t,0\\},\hskip 11.38092ptt_{2}:=\max\\{t,0\\}.$

In what follows we consider the second term on the right-hand side of the
above inequality. If $t\geqslant 0,$ then

$\displaystyle{\rm e}^{-\mu|t|}M\int^{t_{2}}_{t_{1}}{\rm
e}^{\frac{1}{4}\beta|t-s|}C_{f}(\|u(s)\|_{\alpha}+1){\rm d}s$ $\displaystyle=$
$\displaystyle{\rm e}^{-\mu t}M\int^{t}_{0}{\rm
e}^{\frac{1}{4}\beta(t-s)}C_{f}(\|u(s)\|_{\alpha}+1){\rm d}s$ $\displaystyle=$
$\displaystyle MC_{f}\int^{t}_{0}{\rm e}^{-(\mu-\frac{1}{4}\beta)(t-s)}{\rm
e}^{-\mu s}(\|u(s)\|_{\alpha}+1){\rm d}s$ $\displaystyle\leqslant$
$\displaystyle MC_{f}\int^{t}_{0}{\rm
e}^{-(\mu-\frac{1}{4}\beta)(t-s)}\left({\rm e}^{-\mu
s}\|u(s)\|_{\alpha}+1\right){\rm d}s.$ (3.14)

Similarly, for $t\leqslant 0,$ we have

$\displaystyle{\rm e}^{-\mu|t|}M\int^{t_{2}}_{t_{1}}{\rm
e}^{\frac{1}{4}\beta|t-s|}C_{f}(\|u(s)\|_{\alpha}+1){\rm d}s$ $\displaystyle=$
$\displaystyle{\rm e}^{\mu t}M\int^{0}_{t}{\rm
e}^{\frac{1}{4}\beta(s-t)}C_{f}(\|u(s)\|_{\alpha}+1){\rm d}s$ $\displaystyle=$
$\displaystyle MC_{f}\int^{0}_{t}{\rm e}^{-(\mu-\frac{1}{4}\beta)(s-t)}{\rm
e}^{\mu s}(\|u(s)\|_{\alpha}+1){\rm d}s$ $\displaystyle\leqslant$
$\displaystyle MC_{f}\int^{0}_{t}{\rm
e}^{-(\mu-\frac{1}{4}\beta)(s-t)}\left({\rm e}^{\mu
s}\|u(s)\|_{\alpha}+1\right){\rm d}s.$ (3.15)

Thus it follows from (3.2) and (3.2) that for $t\in\mathbb{R}$,

$\displaystyle{\rm e}^{-\mu|t|}M\int^{t_{2}}_{t_{1}}{\rm
e}^{\frac{1}{4}\beta|t-s|}C_{f}(\|u(s)\|_{\alpha}+1){\rm d}s$
$\displaystyle\leqslant$ $\displaystyle MC_{f}\int^{t_{2}}_{t_{1}}{\rm
e}^{-(\mu-\frac{1}{4}\beta)|t-s|}\big{(}{\rm
e}^{-\mu|s|}\|u(s)\|_{\alpha}+1\big{)}{\rm d}s$ $\displaystyle\leqslant$
$\displaystyle MC_{f}\int^{\infty}_{0}{\rm e}^{-(\mu-\frac{1}{4}\beta)s}{\rm
d}s\cdot(\|u\|_{{\mathcal{X}}_{\mu}}+1).$ (3.16)

We observe that

$\displaystyle{\rm e}^{-\mu|t|}={\rm e}^{-\mu|t-s+s|}\leqslant{\rm
e}^{-\mu|s|}{\rm e}^{\mu|t-s|},\hskip 11.38092ptt\in\mathbb{R},$

from which it can be seen that

$\displaystyle{\rm e}^{-\mu|t|}M\int_{-\infty}^{t}(t-s)^{-\alpha}{\rm
e}^{-\frac{3}{4}\beta(t-s)}C_{f}(\|u(s)\|_{\alpha}+1){\rm d}s$
$\displaystyle\leqslant$ $\displaystyle
MC_{f}\int_{-\infty}^{t}(t-s)^{-\alpha}{\rm e}^{\mu|t-s|}{\rm
e}^{-\frac{3}{4}\beta(t-s)}\big{[}{\rm
e}^{-\mu|s|}(\|u(s)\|_{\alpha}+1)\big{]}{\rm d}s$ $\displaystyle\leqslant$
$\displaystyle MC_{f}\int_{-\infty}^{t}(t-s)^{-\alpha}{\rm
e}^{-(\frac{3}{4}\beta-\mu)(t-s)}\left({\rm
e}^{-\mu|s|}\|u(s)\|_{\alpha}+1\right){\rm d}s$ $\displaystyle\leqslant$
$\displaystyle MC_{f}\int^{\infty}_{0}s^{-\alpha}{\rm
e}^{-(\frac{3}{4}\beta-\mu)s}{\rm d}s\cdot(\|u\|_{{\mathcal{X}}_{\mu}}+1),$
(3.17)

and

$\displaystyle{\rm e}^{-\mu|t|}M\int^{\infty}_{t}{\rm
e}^{\frac{3}{4}\beta(t-s)}C_{f}(\|u(s)\|_{\alpha}+1){\rm d}s$
$\displaystyle\leqslant$ $\displaystyle MC_{f}\int^{\infty}_{t}{\rm
e}^{\mu|t-s|}{\rm e}^{\frac{3}{4}\beta(t-s)}{\rm
e}^{-\mu|s|}\left(\|u(s)\|_{\alpha}+1\right){\rm d}s$ $\displaystyle\leqslant$
$\displaystyle MC_{f}\int^{\infty}_{t}{\rm
e}^{(\frac{3}{4}\beta-\mu)(t-s)}\left({\rm
e}^{-\mu|s|}\|u(s)\|_{\alpha}+1\right){\rm d}s$ $\displaystyle\leqslant$
$\displaystyle MC_{f}\int^{\infty}_{0}{\rm e}^{-(\frac{3}{4}\beta-\mu)s}{{\rm
d}}s\cdot(\|u\|_{{\mathcal{X}}_{\mu}}+1).$ (3.18)

Thus one can conclude from (3.2), (3.2), (3.2) and (3.2) that

$\displaystyle{\rm e}^{-\mu|t|}\|G(u)\|_{\alpha}\leqslant$ $\displaystyle
M{\rm e}^{-(\mu-\frac{1}{4}\beta)|t|}\|w\|_{\alpha}+{\rm
e}^{-\mu|t|}M\int^{t_{2}}_{t_{1}}{\rm
e}^{\frac{1}{4}\beta|t-s|}C_{f}(\|u(s)\|_{\alpha}+1){\rm d}s$
$\displaystyle+{\rm e}^{-\mu|t|}M\int_{-\infty}^{t}(t-s)^{-\alpha}{\rm
e}^{-\frac{3}{4}\beta(t-s)}C_{f}(\|u(s)\|_{\alpha}+1){\rm d}s$
$\displaystyle+{\rm e}^{-\mu|t|}M\int^{\infty}_{t}{\rm
e}^{\frac{3}{4}\beta(t-s)}C_{f}(\|u(s)\|_{\alpha}+1){\rm d}s$
$\displaystyle\leqslant$ $\displaystyle
M\|w\|_{\alpha}+MC_{f}\int^{\infty}_{0}{\rm e}^{-(\mu-\frac{1}{4}\beta)s}{\rm
d}s\cdot(\|u\|_{{\mathcal{X}}_{\mu}}+1)$
$\displaystyle+MC_{f}\int^{\infty}_{0}s^{-\alpha}{\rm
e}^{-(\frac{3}{4}\beta-\mu)s}{\rm d}s\cdot(\|u\|_{{\mathcal{X}}_{\mu}}+1)$
$\displaystyle+MC_{f}\int^{\infty}_{0}{\rm e}^{-(\frac{3}{4}\beta-\mu)s}{{\rm
d}}s\cdot(\|u\|_{{\mathcal{X}}_{\mu}}+1)$ $\displaystyle\leqslant$
$\displaystyle
M\|w\|_{\alpha}+F_{\mu}C_{f}(\|u\|_{{\mathcal{X}}_{\mu}}+1)<\infty,\hskip
11.38092pt\forall t\in\mathbb{R}.$

That is $\|Gu\|_{{\mathcal{X}}_{\mu}}<\infty,$ which implies
$Gu\in{\mathcal{X}}_{\mu}.$ Next we show that $G$ is a contraction mapping on
${\mathcal{X}}_{\mu}.$ Indeed, for each $u_{1},u_{2}\in{\mathcal{X}}_{\mu},$
we have

$\displaystyle{\rm e}^{-\mu|t|}\|G(u_{1})-G(u_{2})\|_{\alpha}$
$\displaystyle\leqslant{\rm e}^{-\mu|t|}\|\int^{t}_{0}{\rm
e}^{-A^{2}(t-s)}P_{2}\big{(}f(u_{1})-f(u_{2})\big{)}{\rm d}s\|_{\alpha}$
$\displaystyle\,+{\rm e}^{-\mu|t|}\|\int^{t}_{-\infty}\\!{\rm
e}^{-A^{3}(t-s)}P_{3}\big{(}f(u_{1})-f(u_{2})\big{)}{\rm d}s\|_{\alpha}$
$\displaystyle\,+{\rm e}^{-\mu|t|}\|\int^{\infty}_{t}\\!{\rm
e}^{-A^{1}(t-s)}P_{1}\big{(}f(u_{1})-f(u_{2})\big{)}{\rm d}s\|_{\alpha}.$
(3.19)

Similar to the derivation of (3.2), it can be easily shown that

$\displaystyle{\rm e}^{-\mu|t|}\|\int^{t}_{0}{\rm
e}^{-A^{2}(t-s)}P_{2}\big{(}f(u_{1})-f(u_{2})\big{)}{\rm d}s\|_{\alpha}$
$\displaystyle\leqslant$ $\displaystyle\,ML_{f}\int^{t_{2}}_{t_{1}}{\rm
e}^{-(\mu-\frac{1}{4}\beta)|t-s|}\big{(}{\rm
e}^{-\mu|s|}\|u_{1}-u_{2}\|_{\alpha}\big{)}{\rm d}s.$ (3.20)

By some computations, we obtain that

$\displaystyle{\rm e}^{-\mu|t|}\|\int^{t}_{-\infty}{\rm
e}^{-A^{3}(t-s)}P_{3}\big{(}f(u_{1})-f(u_{2})\big{)}{\rm d}s\|_{\alpha}$
$\displaystyle\leqslant$
$\displaystyle\,ML_{f}\int^{t}_{-\infty}(t-s)^{-\alpha}{\rm
e}^{-(\frac{3}{4}\beta-\mu)(t-s)}\big{(}{\rm
e}^{-\mu|s|}\|u_{1}-u_{2}\|_{\alpha}\big{)}{\rm d}s,$ (3.21)
$\displaystyle{\rm e}^{-\mu|t|}\|\int^{\infty}_{t}{\rm
e}^{-A^{1}(t-s)}P_{1}(f(u_{1})-f(u_{2})){\rm d}s\|_{\alpha}$
$\displaystyle\leqslant$ $\displaystyle\,ML_{f}\int_{t}^{\infty}{\rm
e}^{(\frac{3}{4}\beta-\mu)(t-s)}\big{(}{\rm
e}^{-\mu|s|}\|u_{1}-u_{2}\|_{\alpha}\big{)}{\rm d}s.$ (3.22)

Then it follows from (3.2)-(3.22) that

$\displaystyle{\rm e}^{-\mu|t|}\|Gu_{1}-Gu_{2}\|_{\alpha}$
$\displaystyle\leqslant ML_{f}\int^{t_{2}}_{t_{1}}{\rm
e}^{-(\mu-\frac{1}{4}\beta)|t-s|}\big{(}{\rm
e}^{-\mu|s|}\|u_{1}-u_{2}\|_{\alpha}\big{)}{\rm d}s$
$\displaystyle+ML_{f}\int^{t}_{-\infty}(t-s)^{-\alpha}{\rm
e}^{-(\frac{3}{4}\beta-\mu)(t-s)}\\!\big{(}{\rm
e}^{-\mu|s|}\|u_{1}-u_{2}\|_{\alpha}\big{)}{\rm d}s$
$\displaystyle+ML_{f}\int_{t}^{\infty}{\rm
e}^{(\frac{3}{4}\beta-\mu)(t-s)}\big{(}{\rm
e}^{-\mu|s|}\|u_{1}-u_{2}\|_{\alpha}\big{)}{\rm d}s$ $\displaystyle\leqslant
F_{\mu}L_{f}\|u_{1}-u_{2}\|_{{\mathcal{X}}_{\mu}}.$ (3.23)

Therefore

$\|Gu_{1}-Gu_{2}\|_{{\mathcal{X}}_{\mu}}\leqslant
F_{\mu}L_{f}\|u_{1}-u_{2}\|_{{\mathcal{X}}_{\mu}}.$

That is, $G$ is a contraction mapping on ${\mathcal{X}}_{\mu}$.

Now by virtue of the Banach contraction mapping principle, $G$ has a fixed
point $u_{w}:=u_{\lambda,w}\in{\mathcal{X}}_{\mu},$ which is a full solution
of (3.7) with $P_{2}u_{w}(0)=w$ and satisfies

$\displaystyle u_{w}(t)=$ $\displaystyle{\rm e}^{-A^{2}t}w+\\!\int^{t}_{0}{\rm
e}^{-A^{2}(t-s)}P_{2}f(u_{w}(s)){\rm d}s+\\!\int^{t}_{-\infty}{\rm
e}^{-A^{3}(t-s)}P_{3}f(u_{w}(s)){\rm d}s$ $\displaystyle-\int^{\infty}_{t}{\rm
e}^{-A^{1}(t-s)}P_{1}f(u_{w}(s)){\rm d}s.$ (3.24)

Define

$\Gamma(w)=u_{w}(0),\hskip 11.38092ptw\in X^{\alpha}_{2}.$

We claim that $\Gamma(w)$ is Lipschitz continuous with respect to $w$. Indeed,
let $w_{1},w_{2}\in X_{2}^{\alpha}$. Then by (3.2) and using the same
estimation of (3.2), we obtain that

$\displaystyle{\rm e}^{-\mu|t|}\|u_{w_{1}}(t)-u_{w_{2}}(t)\|_{\alpha}$
$\displaystyle\leqslant$ $\displaystyle\,M{\rm
e}^{-(\mu-\frac{1}{4}\beta)|t|}\|w_{1}-w_{2}\|_{\alpha}+ML_{f}\int^{t_{2}}_{t_{1}}{\rm
e}^{-(\mu-\frac{1}{4}\beta)|t-s|}\big{(}{\rm
e}^{-\mu|s|}\|u_{w_{1}}-u_{w_{2}}\|_{\alpha}\big{)}{\rm d}s$
$\displaystyle+ML_{f}\int^{t}_{-\infty}(t-s)^{-\alpha}{\rm
e}^{-(\frac{3}{4}\beta-\mu)(t-s)}\big{(}{\rm
e}^{-\mu|s|}\|u_{w_{1}}-u_{w_{2}}\|_{\alpha}\big{)}{\rm d}s$
$\displaystyle+ML_{f}\int_{t}^{\infty}{\rm
e}^{(\frac{3}{4}\beta-\mu)(t-s)}\big{(}{\rm
e}^{-\mu|s|}\|u_{w_{1}}-u_{w_{2}}\|_{\alpha}\big{)}{\rm d}s$
$\displaystyle\leqslant$
$\displaystyle\,M\|w_{1}-w_{2}\|_{\alpha}+F_{\mu}L_{f}\|u_{w_{1}}-u_{w_{2}}\|_{{\mathcal{X}}_{\mu}},\hskip
11.38092ptt\in\mathbb{R},$

from which it can be seen that

$\|u_{w_{1}}-u_{w_{2}}\|_{{\mathcal{X}}_{\mu}}\leqslant
M\|w_{1}-w_{2}\|_{\alpha}+F_{\mu}L_{f}\|u_{w_{1}}-u_{w_{2}}\|_{{\mathcal{X}}_{\mu}}.$

Thus

$\displaystyle\|\Gamma(w_{1})-\Gamma(w_{2})\|_{\alpha}$
$\displaystyle=\|u_{w_{1}}(0)-u_{w_{2}}(0)\|_{\alpha}\leqslant\|u_{w_{1}}(t)-u_{w_{2}}(t)\|_{{\mathcal{X}}_{\mu}}$
$\displaystyle\leqslant M(1-F_{\mu}L_{f})^{-1}\|w_{1}-w_{2}\|_{\alpha},$

which shows that $\Gamma(w)$ is Lipschitz continuous in $w$.

Define a mapping $\xi_{\lambda}:X_{2}^{\alpha}\rightarrow X^{\alpha}_{13}$ by

$\displaystyle\xi_{\lambda}(w)=\int^{0}_{-\infty}{\rm
e}^{A^{3}s}P_{3}f(u_{w}(s)){\rm d}s-\int^{\infty}_{0}{\rm
e}^{A^{1}s}P_{1}f(u_{w}(s)){\rm d}s,\hskip 11.38092ptw\in X^{\alpha}_{2}.$
(3.25)

Let $t=0$ in (3.2). Then we have

$\Gamma(w)=u_{w}(0)=w+\xi_{\lambda}(w),\hskip 11.38092ptw\in X^{\alpha}_{2}.$
(3.26)

Clearly, one can deduce from (3.26) that
$\xi_{\lambda}:X^{\alpha}_{2}\rightarrow X^{\alpha}_{13}$ is Lipschitz
continuous in $w.$

Set

${\mathcal{M}}_{\lambda}=\\{w+\xi_{\lambda}(w):w\in X^{\alpha}_{2}\\}.$

Then ${\mathcal{M}}_{\lambda}$ is a global invariant manifold for the semiflow
generated by (3.7). ∎

######  Remark 3.1.

By (3.25), one can easily deduce that if $f$ is bounded, then there exists
$M_{f}>0$ depending on $f$ such that

$\|\xi_{\lambda}(w)\|_{\alpha}\leqslant
MM_{f}\int_{0}^{\infty}(1+s^{-\alpha}){\rm e}^{-\frac{3}{4}\beta s}{\rm
d}s,\hskip 11.38092ptw\in X^{\alpha}_{2},$

that is, $\xi_{\lambda}$ is bounded.

##  4 Bifurcation from infinity of Schrödinger equations

In this section we study the bifurcation from infinity of the Schrödinger
equation (1.1). For this purpose, let us first consider the following
parabolic equation

$\left\\{\begin{array}[]{ll}u_{t}-\Delta u+V(x)u=\lambda u+f(x,u),\hskip
11.38092ptx\in\mathbb{R}^{N};\\\\[4.30554pt] u(0,x)=u_{0}(x),\hskip
11.38092ptx\in\mathbb{R}^{N}.\end{array}\right.$ (4.1)

where $\lambda\in\mathbb{R}$, $V\in L^{\infty}(\mathbb{R}^{N}),N\geqslant 1$,
and $f$ is bounded and satisfies the Landesman-Lazer type condition (1.2). We
will apply the global invariant manifold established in Section 3 to discuss
the dynamic bifurcation from infinity of (4.1).

###  4.1 Mathematical setting

Let $X=L^{2}(\mathbb{R}^{N})$ and $Y=H^{1}(\mathbb{R}^{N})$. Denote
$(\cdot,\cdot)$ and $|\cdot|$ the usual inner product and norm on $X$,
respectively. The norm $\|\cdot\|$ on $Y$ is defined by

$\|u\|=\big{(}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}\mathrm{d}x+\int_{\mathbb{R}^{N}}V(x)|u|^{2}\mathrm{d}x\big{)}^{1/2},\hskip
22.76228ptu\in Y.$

For simplicity, from now on we use $\mathrm{B}_{X}(R)$ and $\mathrm{B}_{Y}(R)$
to denote the balls in $X$ and $Y$, centered at $0$ with radius $R$,
respectively.

In the following we assume that the conditions on the potential $V$ and
nonlinearity $f$ hold true.

  1. ($\mathbf{A1}$)

There exist positive numbers $a_{1},a_{2}$ and $V_{\infty}$ such that

$0<a_{1}\leqslant V(x)\leqslant a_{2},\hskip
11.38092pt0<V_{\infty}:=\sup_{x\in\mathbb{R}^{N}}V(x)=\lim\limits_{|x|\rightarrow\infty}V(x)<\infty.$

  2. ($\mathbf{A2}$)

The nonlinear term $f:\mathbb{R}^{N}\times\mathbb{R}\rightarrow\mathbb{R}$
satisfies

$|f(x,s)-f(x,t)|\leqslant l(x)|s-t|,\hskip 22.76228pt\forall
s,t\in\mathbb{R},\hskip 11.38092ptx\in\mathbb{R}^{N},$

where $l(x)=l_{1}(x)+l_{2}(x)$, $l_{1}\in L^{\infty}(\mathbb{R}^{N})$, and
$l_{2}$ satisfies

$l_{2}\in L^{p}(\mathbb{R}^{N}),\hskip 11.38092pt\mbox{$p\geqslant 2$ if
$N=1$,\, $p>2$ if $N=2$, \,$p\geqslant N$ if $N\geqslant 3$.}$

  3. ($\mathbf{A3}$)

The function $f$ satisfies the Landesman-Lazer type condition (1.3) in Section
1, and

$|f(x,s)|\leqslant g(x),\hskip 22.76228pt\forall s\in\mathbb{R},\hskip
11.38092ptx\in\mathbb{R}^{N}$

for some function $g\in L^{2}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$.

Denote by $A$ the operator $-\Delta+V(x)$. Then under the assumptions on
$V(x),$ the Schrödinger operator
$A=-\Delta+V(x):H^{2}(\mathbb{R}^{N})\rightarrow L^{2}(\mathbb{R}^{N})$ is
selfadjoint and bounded from below. Furthermore, it follows from [2, 21, 35]
that the interval $[V_{\infty},\infty)$ is the essential spectrum of $A$ and
the discrete spectrum of $A$ on the interval $(-\infty,V_{\infty})$ appears.
Namely, for each $a<V_{\infty}$, $\sigma(A)\cap(-\infty,a)$ consists of at
most finitely many eigenvalues of $A$. Hence

$\sigma(A)=\sigma_{p}\cup\sigma_{e},$

where $\sigma_{p}$ consists of isolated eigenvalues with finite multiplicity,
and $\sigma_{e}=[V_{\infty},\infty)$ is the essential spectrum.

Now we convert (4.1) into an abstract form on $Y$

$\displaystyle u_{t}+Au=$ $\displaystyle\lambda u+\tilde{f}(u),$ (4.2)
$\displaystyle u(0)=$ $\displaystyle u_{0},$ (4.3)

where $\tilde{f}(u)$ is the Nemitski operator from $Y$ to $X$ defined by

$\tilde{f}(u)(x)=f(x,u),\hskip 22.76228ptu\in Y.$

It is well known that the Nemitski operator $\tilde{f}$ is well defined.
Moreover, by ($\mathbf{A2}$), one can easily verify that $\tilde{f}$ is
Lipschitz continuous with a chosen Lipschitz constant denoted by $L_{f}$. Note
that $A$ is a sectorial operator. Then the Cauchy problem of (4.2)-(4.3) is
well-posed; see [12, Theorem 3.3.3]. Denote $\Phi_{\lambda}$ the semiflow
generated by (4.2)-(4.3). Then $\Phi_{\lambda}$ is a global semiflow on $Y$.

Let $A_{\lambda_{*}}=A-\lambda_{*}$, where $\lambda_{*}\in\sigma_{p}$. Then
the space $X$ can be decomposed into the orthogonal direct sum of its
subspaces $X^{1},X^{2},X^{3}$ corresponding to the negative, zero and positive
eigenvalues of $A_{\lambda_{*}}$, respectively. Denote
$P_{\sigma}(\sigma\in\\{1,2,3\\})$ the projection from $X$ to $X^{\sigma}.$ It
can be easily seen that $X^{1}$ and $X^{2}$ are finite-dimensional.

Set

$Y^{\sigma}=Y\cap X^{\sigma},\hskip 22.76228pt\sigma\in\\{1,2,3\\}.$

Then from the finite dimensionality of $X^{1}$ and $X^{2}$, one concludes that
$Y^{1}$ and $Y^{2}$ coincide with $X^{1}$ and $X^{2}$, respectively. Also, it
holds that

$Y=Y^{1}\oplus Y^{2}\oplus Y^{3},$

and $m:=\dim Y^{2}\geqslant 1$. Since $\lambda_{*}$ is an isolated eigenvalue
of $A$, one can take the same positive number $\beta$ as that in Section 3.1.
Let $J=(\lambda_{*}-\frac{1}{4}\beta,\lambda_{*}+\frac{1}{4}\beta).$

Henceforth we always assume that the Lipschitz constant $L_{f}$ of $f$
satisfies the following condition. (F) For each fixed
$\mu\in(\frac{1}{4}\beta,\frac{3}{4}\beta)$,

$F_{\mu}L_{f}<1,$

where

$F_{\mu}=M\big{(}\int^{\infty}_{0}{\rm e}^{-(\mu-\frac{1}{4}\beta)s}{\rm
d}s+\int^{\infty}_{0}(1+s^{-\frac{1}{2}}){\rm
e}^{-(\frac{3}{4}\beta-\mu)s}{\rm d}s\big{)},$ (4.4)

and $M$ is a positive number depending on $A$. Then one can immediately
conclude from Theorem 3.1 that the following result holds true.

######  Theorem 4.1.

Let the assumptions ($\mathbf{A1}$)-($\mathbf{A3}$) and ($\mathbf{F}$) hold.
Then there exists a Lipschitz continuous mapping $\xi_{\lambda}$ from $Y^{2}$
to $Y^{13}$ for $\lambda\in J$, such that for each $\lambda\in J,$ the system
(4.2)-(4.3) has a global invariant manifold ${\mathcal{M}}_{\lambda}$, which
is represented as

${\mathcal{M}}_{\lambda}=\\{w+\xi_{\lambda}(w):w\in Y^{2}\\},$

where $Y^{13}=Y^{1}\oplus Y^{3}$.

###  4.2 Preliminaries

Given a function $w$ on $\mathbb{R}^{N}$, we use $w_{\pm}$ to denote the
positive and negative parts of $w$, respectively. Specifically,

$w_{\pm}=\max\\{\pm w(x),0\\},\hskip 22.76228ptx\in\mathbb{R}^{N}.$

Then $w=w_{+}-w_{-}.$

######  Lemma 4.1.

Let the assumption ($\mathbf{A3}$) hold. Then for any $R>0,\varepsilon>0,$
there exists $s_{0}>0$ such that if $s\geqslant s_{0}$,

$\int_{\mathbb{R}^{N}}f(x,v+sw)w{\rm
d}x\geqslant\frac{1}{2}\int_{\mathbb{R}^{N}}(\bar{f}w_{+}+\underline{f}w_{-}){\rm
d}x-\varepsilon.$

for all $v\in{\rm\bar{B}}_{X}(R)$ and $w\in{\rm\bar{B}}_{X^{2}}(1),$ where
$X^{2}$ is a finite-dimensional Banach space.

###### Proof.

Fix a positive integer $n$ sufficiently large. Denote $\Omega_{n}$ a ball in
$\mathbb{R}^{N}$ centered at $0$ with radius $n$. Let

$I=\int_{\Omega_{n}}f(x,v+sw)w{\rm
d}x-\frac{1}{2}\int_{\Omega_{n}}(\bar{f}w_{+}+\underline{f}w_{-}){\rm d}x.$

Noticing that $w=w_{+}-w_{-},$ we can rewrite $I$ as $I=I_{+}-I_{-},$ where

$I_{+}=\int_{\Omega_{n}}(f(x,v+sw)-\bar{f}/2)w_{+}{\rm d}x,\hskip
11.38092ptI_{-}=\int_{\Omega_{n}}(f(x,v+sw)+\underline{f}/2)w_{-}{\rm d}x.$

In what follows we first estimate $I_{+}$ for $v\in{\rm\bar{B}}_{X}(R)$ and
$w\in{\rm\bar{B}}_{X^{2}}(1)$.

Write $\\{|v|\geqslant\sigma\\}=\\{x\in\Omega_{n}:|v|\geqslant\sigma\\}$.
Since

$R^{2}\geqslant\int_{\mathbb{R}^{N}}|v|^{2}{\rm
d}x\geqslant\int_{\Omega_{n}}|v|^{2}{\rm
d}x\geqslant\int_{\\{|v|\geqslant\sigma\\}}|v|^{2}{\rm
d}x\geqslant\sigma^{2}|\\{|v|\geqslant\sigma\\}|,$

one can deduce that $|\\{|v|\geqslant\sigma\\}|\rightarrow 0$ as
$\sigma\rightarrow\infty$ uniformly with respect to $v\in{\rm\bar{B}}_{X}(R).$
Thus we can pick a $\sigma>0$ sufficiently large so that

$|\\{|v|\geqslant\sigma\\}|^{1/2}<\delta:=\varepsilon/\big{(}8f_{\infty}(|\Omega_{n}|+1)\big{)},$

where $f_{\infty}=\sup_{x\in\mathbb{R}^{N},s\in\mathbb{R}}|f(x,s)|$.

For each $v\in{\rm\bar{B}}_{X}(R)$ and $w\in{\rm\bar{B}}_{X^{2}}(1),$ let
$D_{n}=\\{|v|<\sigma\\}\cap\\{w_{+}>\delta\\}.$ Then

$\Omega_{n}=D_{n}\cup\\{|v|\geqslant\sigma\\}\cup\\{w_{+}\leqslant\delta\\}.$

Therefore

$\displaystyle I_{+}$
$\displaystyle\geqslant\int_{D_{n}}(f(x,v+sw)-\bar{f}/2)w_{+}{\rm
d}x-\int_{\\{|v|\geqslant\sigma\\}}|f(x,v+sw)-\bar{f}/2|w_{+}{\rm d}x$
$\displaystyle\hskip
11.38092pt-\int_{\\{w_{+}\leqslant\delta\\}}|f(x,v+sw)-\bar{f}/2|w_{+}{\rm
d}x$ $\displaystyle\geqslant\int_{D_{n}}(f(x,v+sw)-\bar{f}/2)w_{+}{\rm
d}x-2f_{\infty}\big{(}\int_{\\{|v|\geqslant\sigma\\}}w_{+}{\rm
d}x+\int_{\\{w_{+}\leqslant\delta\\}}w_{+}{\rm d}x\big{)}.$

It is easy to see that

$\displaystyle\int_{\\{|v|\geqslant\sigma\\}}w_{+}{\rm
d}x\leqslant\big{(}\int_{\\{|v|\geqslant\sigma\\}}w_{+}^{2}{\rm
d}x\big{)}^{1/2}|\\{|v|\geqslant\sigma\\}|^{1/2}\leqslant|w|\delta\leqslant\delta.$

Note that

$\int_{\\{w_{+}\leqslant\delta\\}}w_{+}{\rm d}x\leqslant|\Omega_{n}|\delta.$

Thereby

$\displaystyle I_{+}$
$\displaystyle\geqslant\int_{D_{n}}(f(x,v+sw)-\bar{f}/2)w_{+}{\rm
d}x-2f_{\infty}(|\Omega_{n}|+1)\delta$
$\displaystyle=\int_{D_{n}}(f(x,v+sw)-\bar{f}/2)w_{+}{\rm
d}x-\frac{\varepsilon}{4}.$ (4.5)

Since $z+s\eta\rightarrow+\infty$ (as $s\rightarrow+\infty$) uniformly with
respect to $z\in[-\sigma,\sigma]$ and $\eta\geqslant\delta,$ there exists
$s_{1}>0$ (independent of $n$) such that if $s>s_{1},$

$f(x,z+s\eta)-\bar{f}/2\geqslant-\frac{\varepsilon}{4|\Omega_{n}|^{1/2}},\hskip
22.76228ptz\in[-\sigma,\sigma],\hskip 11.38092pt\eta\geqslant\delta.$

Let $s>s_{1}.$ Then by the definition of $D_{n}$ (note that $w=w_{+}$ on
$D_{n}$), we deduce that

$\displaystyle\int_{D_{n}}(f(x,v+sw)-\bar{f}/2)w_{+}{\rm d}x$
$\displaystyle\geqslant-\frac{\varepsilon}{4|\Omega_{n}|^{1/2}}\int_{D_{n}}w_{+}{\rm
d}x$
$\displaystyle\geqslant-\frac{\varepsilon}{4|\Omega_{n}|^{1/2}}|D_{n}|^{1/2}(\int_{D_{n}}|w|^{2}{\rm
d}x)^{1/2}\geqslant-\frac{\varepsilon}{4}.$

It follows from (4.2) that if $s>s_{1}$,

$I_{+}\geqslant\int_{D_{n}}(f(x,v+sw)-\bar{f}/2)w_{+}{\rm
d}x-\frac{\varepsilon}{4}\geqslant-\frac{\varepsilon}{2},$ (4.6)

uniformly with respect to the positive integer $n$. Define
$g_{n}(x)=\chi_{n}(x)[(f(x,v+sw)-\bar{f}/2)w_{+}],$ where

$\chi_{n}(x)=\left\\{\begin{array}[]{ll}1,&x\in\Omega_{n};\\\\[4.30554pt]
0,&x\in\mathbb{R}^{N}\backslash\Omega_{n}.\end{array}\right.$

Then

$\lim_{n\rightarrow\infty}g_{n}(x)=\big{(}f(x,v+sw)-\bar{f}/2\big{)}w_{+}.$

Noticing that

$I_{+}=\int_{\Omega_{n}}(f(x,v+sw)-\bar{f}/2)w_{+}{\rm
d}x=\int_{\mathbb{R}^{N}}\chi_{n}(x)[(f(x,v+sw)-\bar{f}/2)w_{+}]{\rm d}x,$

by (4.6) we have

$\int_{\mathbb{R}^{N}}\chi_{n}(x)[(f(x,v+sw)-\bar{f}/2)w_{+}]{\rm
d}x\geqslant-\frac{\varepsilon}{2},$ (4.7)

provided $s>s_{1}$, uniformly with respect to $n$. Since the norm
$\|\cdot\|_{L^{1}(\mathbb{R}^{N})}$ and that of $X=L^{2}(\mathbb{R}^{N})$ are
equivalent on $X^{2}$, one easily sees that

$\sup\\{\|w\|_{L^{1}(\mathbb{R}^{N})}:w\in X^{2},|w|=1\\}<\infty.$

Therefore

$\displaystyle\int_{\mathbb{R}^{N}}|\chi_{n}(x)[(f(x,v+sw)-\bar{f}/2)w_{+}]|{\rm
d}x$ $\displaystyle\leqslant 2f_{\infty}\int_{\mathbb{R}^{N}}|w|{\rm d}x$
$\displaystyle\leqslant 2f_{\infty}\|w\|_{L^{1}(\mathbb{R}^{N})}<\infty.$

Thanks to the Lebesgue dominated convergence theorem and letting
$n\rightarrow\infty$ in (4.7), we conclude that if $s>s_{1}$,

$\int_{\mathbb{R}^{N}}(f(x,v+sw)-\bar{f}/2)w_{+}{\rm
d}x\geqslant-\frac{\varepsilon}{2}.$

Similarly, it can be shown that there exists $s_{2}>0$ (independent of $n$)
such that

$\int_{\mathbb{R}^{N}}(f(x,v+sw)+\bar{f}/2)w_{-}{\rm
d}x\leqslant\frac{\varepsilon}{2},$

as long as $s>s_{2}.$ Set $s_{0}=\max\\{s_{1},s_{2}\\}$. Then if $s>s_{0},$ we
have

$\int_{\mathbb{R}^{N}}(f(x,v+sw)-\bar{f}/2)w_{+}{\rm
d}x-\\!\int_{\mathbb{R}^{N}}(f(x,v+sw)+\bar{f}/2)w_{-}{\rm
d}x\geqslant\\!-\frac{\varepsilon}{2}-\frac{\varepsilon}{2}=-\varepsilon$

for all $v\in{\rm\bar{B}}_{X}(R)$ and $w\in{\rm\bar{B}}_{X^{2}}(1)$. The proof
of the Lemma is complete. ∎

######  Remark 4.1.

In [15], the authors used the Landesman-Lazer type condition to give a
fundamental result on $f$ (see [15, Lemma 5.2]), which plays an important role
in establishing their main results on bifurcations from infinity and
multiplicity of solutions of parabolic equations on bounded domains. Lemma 4.1
significantly extends this result to the case of unbounded domains.

Now we restrict the system (4.2)-(4.3) to the global invariant manifold
${\mathcal{M}}_{\lambda}$ given by Theorem 4.1. Then

$w_{t}+A^{2}w=P_{2}\tilde{f}(w+\xi_{\lambda}(w)),\hskip 11.38092ptw\in Y^{2},$
(4.8)

where $w=P_{2}u$, $A^{2}=A_{\lambda}|_{X^{2}},\lambda\in J$. Denote
$\phi_{\lambda}$ the semiflow generated by (4.8). For each $0<a\leqslant
b\leqslant\infty,$ write

$\mathrm{B}_{2}[a,b]=\\{w\in Y^{2},a\leqslant|w|\leqslant b\\}.$

######  Lemma 4.2.

Let the assumptions ($\mathbf{A1}$)-($\mathbf{A3}$) and (F) hold true. Then
there exist $R_{0}>0,c_{0}>0$ such that the following assertions hold.

  1. (1)

For $\lambda\in[\lambda_{*},\lambda_{*}+\frac{1}{4}\beta)$, if $w(t)$ is a
solution of (4.8) in $\mathrm{B}_{2}[R_{0},\infty]$, we have

$\frac{{\rm d}}{{\rm d}t}|w(t)|^{2}\geqslant c_{0}|w(t)|.$ (4.9)

  2. (2)

If $R>R_{0}$, then there exists $0<\eta\leqslant\frac{1}{4}\beta$ such that
for each $\lambda\in[\lambda_{*}-\eta,\lambda_{*})$, (4.9) holds true for any
solution $w(t)$ of (4.8) in $\mathrm{B}_{2}[R_{0},R]$.

  3. (3)

There exists $\theta_{1}>0$ such that if
$\lambda\in[\lambda_{*}-\theta_{1},\lambda_{*})$, then the equation (4.8) has
a positively invariant set $\mathrm{B}_{2}[r_{\lambda},R_{\lambda}]$ with
$r_{\lambda},R_{\lambda}\rightarrow\infty,$ as
$\lambda\rightarrow\lambda_{*}^{-}$.

###### Proof.

Taking the inner product of (4.8) with $w$ in $X$, we have

$\frac{1}{2}\frac{{\rm d}}{{\rm
d}t}|w|^{2}+\|w\|^{2}=\lambda|w|^{2}+(\tilde{f}(w+\xi_{\lambda}(w)),w).$
(4.10)

Note that $\|w\|^{2}=\lambda_{*}|w|^{2}$. Then it follows from (4.10) that

$\frac{1}{2}\frac{{\rm d}}{{\rm
d}t}|w|^{2}=(\lambda-\lambda_{*})|w|^{2}+(\tilde{f}(w+\xi_{\lambda}(w)),w).$
(4.11)

We first estimate the last term in (4.11). Since the norm
$\|\cdot\|_{L^{1}(\mathbb{R}^{N})}$ of $L^{1}(\mathbb{R}^{N})$ and that of
$X=L^{2}(\mathbb{R}^{N})$ are equivalent in $Y^{2},$ one easily sees that

$\min\\{\|v\|_{L^{1}(\mathbb{R}^{N})}:\,\,v\in Y^{2},\,\,|v|=1\\}:=m>0.$
(4.12)

Choose a positive number $\delta$ with
$\delta<\frac{1}{2}\min\\{\bar{f},\underline{f}\\}$. By virtue of Remark 3.1,
we see that $\|\xi_{\lambda}(w)\|\leqslant R_{1}$ for some $R_{1}>0$. Thanks
to Lemma 4.1, there exists $s_{0}>0$ such that if $s>s_{0}$,

$(\widetilde{f}(h+sv),v)=\int_{\mathbb{R}^{N}}f(x,h+sv)v\,{\rm
d}x\geqslant\frac{1}{2}\int_{\mathbb{R}^{N}}\left(\overline{f}v_{+}+\underline{f}v_{-}\right){\rm
d}x-\frac{1}{2}m\delta$ (4.13)

for all $h\in\overline{\mbox{B}}_{X}(R_{1})$ and
$v\in\overline{\mbox{B}}_{X^{2}}(1)$. Now we rewrite

$w=sv,\hskip 11.38092pt\mbox{where $s=|w|$}.$

Clearly $v\in\partial\mathrm{B}_{X^{2}}(1)$. Hence if $s\geqslant s_{0}$, we
deduce from (4.13) that

$(\tilde{f}(w+\xi_{\lambda}(w)),w)=s(f(x,sv+\xi_{\lambda}(w)),v)\geqslant
s\left(\frac{1}{2}\int_{\mathbb{R}^{N}}\left(\overline{f}v_{+}+\underline{f}v_{-}\right){\rm
d}x-\frac{1}{2}m\delta\right).$

Note that

$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}\left(\overline{f}v_{+}+\underline{f}v_{-}\right){\rm
d}x-\frac{1}{2}m\delta$
$\displaystyle\geqslant\delta\int_{\mathbb{R}^{N}}|v|{\rm
d}x-\frac{1}{2}m\delta$ $\displaystyle\geqslant(\mbox{by
}\eqref{4.12})\geqslant\frac{1}{2}m\delta.$

Thereby

$(\tilde{f}(w+\xi_{\lambda}(w)),w)\geqslant\frac{1}{2}m\delta
s=\frac{1}{2}m\delta|w|.$ (4.14)

It follows from (4.11) and (4.14) that

$\frac{{\rm d}}{{\rm d}t}|w(t)|^{2}\geqslant
2\left(\lambda-\lambda_{*}\right)|w|^{2}+m\delta|w(t)|$ (4.15)

as long as $|w(t)|\geqslant s_{0}$. Now we pick $R_{0}=s_{0}$,
$c_{0}=m\delta/2$. Then if
$\lambda\in[\lambda_{*},\lambda_{*}+\frac{1}{4}\beta)$, we infer from (4.15)
that

$\frac{{\rm d}}{{\rm d}t}|w(t)|^{2}\geqslant m\delta|w(t)|>c_{0}|w(t)|$

at any point $t$ where $|w(t)|\geqslant R_{0}$. Therefore the assertion (1)
holds.

Assume $R>R_{0}$ and let $\lambda<\lambda_{*}$. Take an $\eta>0$ such that
$\eta R^{2}<m\delta s_{0}/4$. Then for each
$\lambda\in[\lambda_{*}-\eta,\lambda_{*})$, if $w(t)$ is a solution of (4.8)
in $\mathrm{B}_{2}[R_{0},R]$, we conclude from (4.15) that

$\displaystyle\frac{{\rm d}}{{\rm d}t}|w(t)|^{2}$
$\displaystyle\geqslant-2|\lambda-\lambda_{*}|\,R^{2}+m\delta|w(t)|$
$\displaystyle\geqslant c_{0}|w(t)|+\left(c_{0}|w(t)|-2\eta R^{2}\right)$
$\displaystyle\geqslant c_{0}|w(t)|+\left(c_{0}s_{0}-2\eta
R^{2}\right)\geqslant c_{0}|w(t)|,$

which justifies the second assertion (2).

Finally, we verify that the assertion (3) holds. Take a sequence
$R_{m}>R_{0}\,(m=1,2,\cdots)$ such that
$R_{0}<R_{1}<R_{2}<\cdots<R_{m}<\cdots$ and $R_{m}\rightarrow\infty$ as
$m\rightarrow\infty$. Then for each fixed $m$, we conclude from assertion (2)
that there exists a sequence $\eta_{m}>0\,(m=1,2,\cdots)$ with

$\eta_{1}>\eta_{2}>\cdots>\eta_{m}>\cdots,\hskip 11.38092pt\eta_{m}\rightarrow
0\,(m\rightarrow\infty),$

such that if $\lambda\in[\lambda_{*}-\eta_{m},\lambda_{*})$, (4.9) holds for
any solution $w(t)$ of (4.8) in $\mathrm{B}_{2}[R_{0},R_{m}]$. Furthermore, by
the choice of $\eta_{m}$ and (4.9), we deduce that if
$\lambda\in[\lambda_{*}-\eta_{m},\lambda_{*})$, the set
$\mathrm{B}_{2}[R_{m},\infty]$ is positively invariant for the semiflow $\phi$
generated by (4.8).

On the other hand, let $\lambda<\lambda_{*}$. As the norm
$\|\cdot\|_{L^{1}(\mathbb{R}^{N})}$ of $L^{1}(\mathbb{R}^{N})$ and that of
$X=L^{2}(\mathbb{R}^{N})$ are equivalent in $Y^{2},$ one has

$\displaystyle\big{(}\widetilde{f}(w+\xi_{\lambda}(w)),w\big{)}$
$\displaystyle=\int_{\mathbb{R}^{N}}f(x,w+\xi_{\lambda}(w))w{\rm d}x\leqslant
f_{\infty}\|w\|_{L^{1}(\mathbb{R}^{N})}\leqslant C_{1}|w|$
$\displaystyle\leqslant\frac{\lambda_{*}-\lambda}{2}|w|^{2}+\frac{1}{2(\lambda_{*}-\lambda)}C_{1}^{2},$
(4.16)

where $C_{1}>0$ depends on $f_{\infty}$. Combining (4.11) and (4.2), it yields

$\frac{{\rm d}}{{\rm
d}t}|w(t)|^{2}\leqslant(\lambda-\lambda_{*})|w|^{2}+\frac{C_{1}^{2}}{\lambda_{*}-\lambda}.$

Hence it follows from the Gronwall’s inequality that

$|w(t)|^{2}\leqslant{\rm e}^{-(\lambda_{*}-\lambda)t}|w(0)|^{2}+\big{(}1-{\rm
e}^{-(\lambda_{*}-\lambda)t}\big{)}\frac{C_{1}^{2}}{(\lambda_{*}-\lambda)^{2}},\hskip
11.38092ptt\geqslant 0.$ (4.17)

Set

$R_{\lambda}=\frac{C_{1}}{\lambda_{*}-\lambda}.$

Then one can deduce from (4.17) that the set $\\{w\in Y^{2},|w|\leqslant
R_{\lambda}\\}$ is positively invariant for $\phi$.

Recalling that $\eta_{m}\rightarrow 0$ as $m\rightarrow\infty$, one can deduce
that

$[\lambda_{*}-\eta_{1},\lambda_{*})=\bigcup_{m\geqslant
1}[\lambda_{*}-\eta_{m},\lambda_{*}-\eta_{m+1}).$

We pick $\theta_{1}=\eta_{1}$. Then for each
$\lambda\in[\lambda_{*}-\theta_{1},\lambda_{*})$, there exists some
$m\geqslant 1$ such that
$\lambda\in[\lambda_{*}-\eta_{m},\lambda_{*}-\eta_{m+1})$. It is easy to see
that

$R_{m}\rightarrow\infty\hskip 22.76228pt\mbox{as \hskip
11.38092pt$\lambda\rightarrow\lambda_{*}^{-}$}.$

Hence if we choose $r_{\lambda}=R_{m}$, then one can conclude from the above
argument that $\mathrm{B}_{2}[r_{\lambda},R_{\lambda}]$ is the desired
positively invariant set of the semiflow $\phi$ for
$\lambda\in[\lambda_{*}-\theta_{1},\lambda_{*})$. The proof of this lemma is
complete. ∎

###  4.3 dynamic bifurcation from infinity

Now we study the bifurcation from infinity of (4.2)-(4.3) near $\lambda_{*}$.

######  Definition 4.1.

[15] We say that the system (4.2)-(4.3) bifurcates from infinity at
$\lambda=\lambda_{*}$ (or, $(\infty,\lambda_{*})$ is a bifurcation point), if
for every $\varepsilon>0$, there exist $\lambda\in\mathbb{R}$ with
$|\lambda-\lambda_{*}|<\varepsilon$ and a bounded full solution
$u_{\lambda}=u_{\lambda}(t)$ of (4.2)-(4.3) such that

$\|u_{\lambda}\|_{\infty}>1/\varepsilon,$

where $\|u_{\lambda}\|_{\infty}=\sup_{t\in\mathbb{R}}\|u_{\lambda}(t)\|$.

Let $U\subset Y^{2}$ and denote $K_{\infty}(\phi_{\lambda},U)$ the union of
all bounded full orbits in $U$. For simplicity, if $U=Y^{2}$, we write
$K_{\infty}(\phi_{\lambda},Y^{2})=K_{\infty}(\phi_{\lambda})$. Given a set
$K\subset Y^{2}\times J$ and $\lambda\in J$, write

$K[\lambda]=\\{x:(x,\lambda)\in Y^{2}\times J\\}.$

$K[\lambda]$ is called the $\lambda$-section of $K$. In the following we first
give a result on bifurcations from infinity of the reduced system (4.8).

######  Theorem 4.2.

Let the assumptions ($\mathbf{A1}$)-($\mathbf{A3}$) and (F) hold true, and
$\lambda_{*}\in\sigma_{p}$. Then $(\infty,\lambda_{*})$ is a bifurcation point
of (4.8). Specifically, there exist a sequence $\lambda_{n}$ with
$\lambda_{n}\rightarrow\lambda_{*}$ $(n\rightarrow\infty)$ and a bounded full
solution $u_{n}=u_{\lambda_{n}}(t)$ of (4.8) such that

$\|u_{n}\|_{\infty}\rightarrow\infty,\hskip 11.38092pt\mbox{as}\hskip
11.38092ptn\rightarrow\infty.$

Moreover, there exists a neighborhood $\Lambda$ of $\lambda_{*}$ such that for
each $\lambda\in\Lambda\setminus\\{\lambda_{*}\\}$, (4.8) has a bounded full
solution $v_{\lambda}$, which remains bounded on $\Lambda$.

###### Proof.

Consider the system

$w_{t}+A^{2}w=P_{2}\tilde{f}(w+\xi_{\lambda}(w)),\hskip 11.38092ptw\in Y^{2},$
(4.18)

where $A^{2}=P_{2}A_{\lambda}=P_{2}(A-\lambda I)$, $\lambda\in
J=(\lambda_{*}-\frac{\beta}{4},\lambda_{*}+\frac{\beta}{4})$. Let us study the
following linear equation

$\frac{{\rm d}w}{{\rm d}t}+A^{2}w=0.$ (4.19)

Pick two numbers $\varepsilon_{1},\varepsilon_{2}$ with
$\varepsilon_{1}\in(\lambda_{*}-\frac{\beta}{4},\lambda_{*})$ and
$\varepsilon_{2}\in(\lambda_{*},\lambda_{*}+\frac{\beta}{4})$, respectively.
Then if $\lambda=\varepsilon_{1},\varepsilon_{2}$, the trivial solution {0} is
an isolated invariant set for the semiflow $\psi_{\lambda}$ generated by
equation (4.19) in $Y^{2}$. By [24] (see Chapter I, Corollary 11.2) we know
that there is a positive integer $p$ such that

$h(\psi_{\varepsilon_{1}},\\{0\\})=\Sigma^{0},\hskip
11.38092pth(\psi_{\varepsilon_{2}},\\{0\\})=\Sigma^{p}.$ (4.20)

($p$ is actually the algebraic multiplicity of the eigenvalue $\lambda_{*}$ of
$A^{2}$.)

Now we consider the nonlinear equation

$\frac{{\rm d}w}{{\rm d}t}+A^{2}w-\nu P_{2}\tilde{f}(w+\xi_{\lambda}(w))=0,$
(4.21)

where $\nu\in[0,1]$ is the homotopy parameter. By appropriately modifying the
argument in the proof of [24, Chapter II, Theorem 5.1] (see also the proof of
[33, Theorem 3.2]), one can verify that for each $\varepsilon>0$ with

$\varepsilon_{1}<\lambda_{*}-\varepsilon<\lambda_{*}+\varepsilon<\varepsilon_{2},$

there exists $R_{\varepsilon}>0$ such that if $u_{\lambda}(t)$ is a bounded
full solution of (4.21) for
$\lambda\in[\varepsilon_{1},\lambda_{*}-\varepsilon]\cup[\lambda_{*}+\varepsilon,\varepsilon_{2}]$
and $\nu\in[0,1]$, then

$\|u_{\lambda}\|_{\infty}<R_{\varepsilon}.$ (4.22)

Denote $\psi_{\lambda}^{\nu}$ the semiflow generated by the equation (4.21).
By the continuation property of the Conley index, we deduce that

$\begin{array}[]{ll}h\left(\phi_{\lambda},K_{\infty}({\phi_{\lambda}})\right)&=h\left(\psi_{\lambda}^{1},K_{\infty}({\psi^{1}_{\lambda}})\right)\\\\[4.30554pt]
&=h\left(\psi_{\lambda}^{0},K_{\infty}({\psi^{0}_{\lambda}})\right)=h(\psi_{\varepsilon_{1}},\\{0\\})=\Sigma^{0}\end{array}$
(4.23)

for $\lambda\in[\varepsilon_{1},\lambda_{*}-\varepsilon]$ and

$\begin{array}[]{ll}h\left(\phi_{\lambda},K_{\infty}({\phi_{\lambda}})\right)&=h\left(\psi_{\lambda}^{1},K_{\infty}({\psi^{1}_{\lambda}})\right)\\\\[4.30554pt]
&=h\left(\psi_{\lambda}^{0},K_{\infty}({\psi^{0}_{\lambda}})\right)=h(\psi_{\varepsilon_{2}},\\{0\\})=\Sigma^{p}\end{array}$
(4.24)

for $\lambda\in[\lambda_{*}+\varepsilon,\varepsilon_{2}]$. Moreover, we infer
from (4.22) that for the $\varepsilon>0$,

$K_{\infty}({\phi_{\lambda}})[\lambda]\subset{\rm
B}_{Y^{2}}(R_{\varepsilon}),\hskip
11.38092pt\lambda\in[\varepsilon_{1},\lambda_{*}-\varepsilon]\cup[\lambda_{*}+\varepsilon,\varepsilon_{2}],$

where ${\rm B}_{Y^{2}}(R_{\varepsilon})$ denotes a ball in $Y^{2}$ centered at
$0$ with radius $R_{\varepsilon}$. Hence, we conclude from [15, Theorem 3.5]
that there is a sequence
$\lambda_{n}\rightarrow\lambda_{*}(n\rightarrow\infty)$ and a sequence of
bounded full solutions $u_{n}=u_{\lambda_{n}}(t)$ of (4.8) for
$\lambda=\lambda_{n}$ such that

$\|u_{n}\|_{\infty}\rightarrow\infty,\hskip 11.38092ptn\rightarrow\infty.$
(4.25)

On the other hand, we infer from Lemma 4.2 that there are $R_{0},c_{0}>0$ such
that for each $\lambda\in[\lambda_{*},\lambda_{*}+\frac{1}{4}\beta)$, if
$w(t)$ is a bounded full solution of (4.8) in $\mathrm{B}_{2}[R_{0},\infty]$,
then (4.9) holds true. Therefore

$K_{\infty}(\phi_{\lambda})\subset\mathrm{B}_{2}[0,R_{0}],\hskip
11.38092pt\lambda\in[\lambda_{*},\lambda_{*}+\frac{1}{4}\beta).$ (4.26)

Since $\varepsilon_{1}\in(\lambda_{*}-\frac{\beta}{4},\lambda_{*})$,
$\varepsilon_{2}\in(\lambda_{*},\lambda_{*}+\frac{\beta}{4})$ and
$\varepsilon$ can be arbitrary, we deduce from (4.23) and (4.24) that

$h(\phi_{\lambda},K_{\infty}(\phi_{\lambda}))=\left\\{\begin{array}[]{ll}\Sigma^{0},\hskip
22.76228pt\lambda\in(\lambda_{*}-\frac{\beta}{4},\lambda_{*});\\\\[4.30554pt]
\Sigma^{p},\hskip
22.76228pt\lambda\in(\lambda_{*},\lambda_{*}+\frac{\beta}{4}).\end{array}\right.$
(4.27)

Thus it follows from (4.26) that

$h(\phi_{\lambda_{*}},K_{\infty}(\phi_{\lambda_{*}}))=\Sigma^{p}.$

Now we pick a bounded isolating neighborhood $N$ of
$K_{\infty}(\phi_{\lambda_{*}})$. Then there exists $\delta>0$ such that $N$
is an isolating neighborhood of $\phi_{\lambda}$ for
$\lambda\in[\lambda_{*}-\delta,\lambda_{*}+\delta]$. Thus

$h(\phi_{\lambda},K_{\lambda})=\Sigma^{p},\hskip
11.38092pt\lambda\in[\lambda_{*}-\delta,\lambda_{*}+\delta].$

where $K_{\lambda}=K_{\infty}(\phi_{\lambda},N)$. Choose a full solution
$v_{\lambda}$ in $K_{\lambda}$ and write
$\Lambda=[\lambda_{*}-\delta,\lambda_{*}+\delta]$. Then
$\|v_{\lambda}\|_{\infty}$ remains bounded on $\Lambda$. ∎

By virtue of Lemma 4.2, we can present our dynamic bifurcation results from
infinity for the system (4.2)-(4.3).

######  Theorem 4.3.

Let the assumptions ($\mathbf{A1}$)-($\mathbf{A3}$) hold. Assume the Lipschitz
constant $L_{f}$ of $f$ satisfies (F). Then there exists
$\theta_{1}\in(0,\frac{1}{4}\beta)$ such that for each
$\lambda\in\Lambda_{1}:=[\lambda_{*}-\theta_{1},\lambda_{*})$, the system
$\Phi_{\lambda}$ has a compact invariant set
${\mathcal{S}}^{\infty}_{\lambda}$, which takes the shape of
$(m-1)$-dimensional topological sphere $\mathbb{S}^{m-1}$, and

$\lim_{\lambda\rightarrow\lambda_{*}^{-}}\min\\{\|w\|:w\in{\mathcal{S}}^{\infty}_{\lambda}\\}=\infty.$
(4.28)

Moreover, there exists a two-sided neighborhood $\Lambda$ of $\lambda_{*}$
such that for each $\lambda\in\Lambda$, $\Phi_{\lambda}$ has a compact
invariant set ${\mathcal{K}}_{\lambda}$, which remains bounded on $\Lambda$.

###### Proof.

Let $\theta_{1}$ be given in Lemma 4.2. By virtue of Lemma 4.2 (3), we see
that for each $\lambda\in\Lambda_{1}$, the semiflow $\phi_{\lambda}$ generated
by (4.8) has a positively invariant set ${\rm B}_{\lambda}:={\rm
B}_{2}[r_{\lambda},R_{\lambda}]$. Note that $Y_{2}$ is finite dimensional.
Then one can conclude from the attractor theory (see e.g., [6, 31]) that the
set

${\mathcal{A}}_{\lambda}^{\infty}=\bigcap_{s\geqslant
0}\overline{\bigcup_{t\geqslant s}\phi_{\lambda}(t){\rm B}_{\lambda}}$

is a global attractor of the semiflow $\phi_{\lambda}$ restricted on ${\rm
B}_{\lambda}$.

By the basic knowledge on algebraic topology, it can be easily seen that ${\rm
B}_{\lambda}$ has the homotopy type of an $(m-1)$-dimensional topological
sphere. Thus ${\rm B}_{\lambda}$ enjoys the same shape of $\mathbb{S}^{m-1}$.
Thanks to the shape theory of attractors (see [13, Theorem 3.6] or [25, 34]),
we conclude that ${\mathcal{A}}_{\lambda}^{\infty}\subset{\rm B}_{\lambda}$
has the shape of $\mathbb{S}^{m-1}$.

Set

${\mathcal{S}}_{\lambda}^{\infty}=\\{w+\xi_{\lambda}(w):w\in{\mathcal{A}}_{\lambda}^{\infty}\\}.$

Then ${\mathcal{S}}_{\lambda}^{\infty}\subset{\mathcal{M}}_{\lambda}$ is a
compact invariant set of the system $\Phi_{\lambda}$ generated by (4.2)-(4.3),
which takes the shape of an $(m-1)$-dimensional sphere. By the construction of
${\rm B}_{\lambda}$, we see that $r_{\lambda},R_{\lambda}\rightarrow\infty$ as
$\lambda\rightarrow\lambda_{*}^{-}$, from which one can conclude that (4.28)
holds true.

Let $\Lambda$ be obtained in Theorem 4.2. Then we infer from the proof of
Theorem 4.2 that for each $\lambda\in\Lambda$, $\phi_{\lambda}$ has a compact
invariant set $K_{\lambda}$, which remains bounded on $\Lambda$. Similarly,
set

${\mathcal{K}}_{\lambda}=\\{w+\xi_{\lambda}(w):w\in K_{\lambda}\\}.$

Clearly, ${\mathcal{K}}_{\lambda}$ is the desired compact invariant set of
$\Phi_{\lambda}$ for $\lambda\in\Lambda$. The proof of the theorem is
complete. ∎

As a consequence of Theorem 4.3, we can obtain our main results on
bifurcations from infinity and multiplicity of the Schrödinger equation (1.1).

######  Theorem 4.4.

Let the assumptions ($\mathbf{A1}$)-($\mathbf{A3}$) hold and
$\lambda_{*}\in\sigma_{p}$. Assume the Lipschitz constant $L_{f}$ of $f$
satisfies (F). Then there exists $\theta\in(0,\frac{1}{4}\beta)$ such that for
each $\lambda\in\Lambda_{2}:=[\lambda_{*}-\theta,\lambda_{*}),$ the
Schrödinger equation (1.1) has at least three distinct nontrival solutions
$e_{\lambda}^{1},e_{\lambda}^{2},e^{3}_{\lambda}$ with

$\|e^{i}_{\lambda}\|\rightarrow\infty,\hskip 11.38092pt\mbox{as
\,\,$\lambda\rightarrow\lambda_{*}^{-}$,\hskip 11.38092pt$i=1,2$,}$

whereas $e_{\lambda}^{3}$ remains bounded on $\Lambda_{2}$.

###### Proof.

Let $\theta_{1}$ and $\Lambda$ be given in Theorem 4.3. Pick a positive number
$\theta$ such that $\theta<\min\\{\theta_{1},\delta\\}$, where $\delta>0$ is
chosen in the proof of Theorem 4.2. By virtue of Theorem 4.3, we deduce that
for each $\lambda\in[\lambda_{*}-\theta,\lambda_{*})$, the system
$\Phi_{\lambda}$ has a compact invariant set
${\mathcal{S}}^{\infty}_{\lambda}$. In what follows we show that
${\mathcal{S}}^{\infty}_{\lambda}$ has at least two distinct nontrivial
stationary solutions $e^{1}_{\lambda}$ and $e^{2}_{\lambda}$, which are
actually the solutions of (1.1).

Observing that ${\mathcal{S}}^{\infty}_{\lambda}$ has the shape of
$\mathbb{S}^{m-1}$ and $m\geqslant 1$, we deduce that
${\mathcal{S}}^{\infty}_{\lambda}$ has at least two distinct points $u$ and
$v$. If $u$ and $v$ are equilibria, then the result holds true. Now we suppose
that $u$ is not an equilibrium. Then there exists a full solution
$\gamma=\gamma(t)$ of $\Phi_{\lambda}$ in ${\mathcal{S}}_{\lambda}^{\infty}$
such that $\gamma(0)=u$. Since $\Phi_{\lambda}$ is a gradient system, we
conclude that

$\omega(\gamma)\cap\omega^{*}(\gamma)=\emptyset.$

Noticing that $\omega(\gamma)$ and $\omega^{*}(\gamma)$ consist of equilibrium
points, one can see that $\Phi_{\lambda}$ has two distinct equilibria in
${\mathcal{S}}^{\infty}_{\lambda}$. Recalling that
${\mathcal{S}}^{\infty}_{\lambda}$ satisfies (4.28), we conclude that the
equation (1.1) has two distinct nontrivial solutions $e^{1}_{\lambda}$ and
$e^{2}_{\lambda}$ with

$\|e^{i}_{\lambda}\|\rightarrow\infty,\hskip 11.38092pt\mbox{as
\,$\lambda\rightarrow\lambda_{*}^{-}$,\hskip 11.38092pt$i=1,2$.}$

Theorem 4.3 also asserts that for each
$\lambda\in\Lambda=[\lambda_{*}-\delta,\lambda_{*}+\delta]$, the system
$\Phi_{\lambda}$ has a compact invariant set ${\mathcal{K}}_{\lambda}$, which
remains bounded on $\Lambda$. Thus $\Phi_{\lambda}$ has at least one
equilibrium $e_{\lambda}^{3}$ in ${\mathcal{K}}_{\lambda}$. Therefore
$e_{\lambda}^{3}$ is the solution of the Schrödinger equation (1.1) fulfilling
the requirements of the theorem for $\lambda\in\Lambda_{2}$. The proof of the
theorem is complete. ∎

######  Remark 4.2.

The “dual” version of all our results holds true if, instead of (1.2), we
assume that (1.5) is fulfilled.

Acknowledgments This work is supported by National Natural Science Foundation
of China (Grant Nos. 11871368, 11801190).

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