# Standing Waves for Schrödinger Equations with Kato-Rellich potentials

Aleksander Ćwiszewski  Faculty of Mathematics and Computer Science
Nicolaus Copernicus University
87-100 Toruń, Poland<EMAIL_ADDRESS>and  Piotr Kokocki  Faculty of
Mathematics and Computer Science
Nicolaus Copernicus University
87-100 Toruń, Poland<EMAIL_ADDRESS>
###### Abstract.

We show the existence of standing waves for the nonlinear Schrödinger equation
with Kato-Rellich type potential. We consider both resonant with the
nonlinearity satisfying one of Landesman-Lazer type or sign conditions and
non-resonant case where the linearization at infinity has zero kernel. The
approach relies on the geometric and topological analysis of the parabolic
semiflow associated to the involved elliptic problem. Tail estimates
techniques and spectral theory of unbounded linear operators are used to
exploit subtle compactness properties necessary for use of the Conley index
theory due to Rybakowski. The obtained results extend those by Prizzi [17] for
the non-resonant case, resolve the existence problem at resonance and complete
those from [4] where the bifurcation of stationary solutions from infinity was
studied.

##  1\. Introduction

In the time-dependent nonlinear Schrödinger type equation

$i\psi_{t}=-\Delta\psi+V(x)\psi+g(x,|\psi|)\frac{\psi}{|\psi|},\quad
x\in\mathbb{R}^{N},\ t>0,$

where $V:\mathbb{R}^{N}\to\mathbb{R}$ is a potential and
$g:\mathbb{R}^{N}\times\mathbb{R}\to\mathbb{C}$ is a nonlinear map, searching
for a standing wave solution having the form $\psi(x,t)=e^{-i\lambda t}u(x)$
with $\lambda\in\mathbb{R}$, leads to the following elliptic problem

$None$ $-\Delta u+V(x)u-\lambda u=f(x,u(x)),\quad x\in\mathbb{R}^{N},$

where $f:\mathbb{R}^{N}\times\mathbb{R}\to\mathbb{R}$ is a nonlinear map.
Many authors have extensively studied the equation $(E)_{\lambda}$ by the use
of various methods. In [17] the existence of solutions by Conley index was
obtained in the non-resonant case with Kato-Rellich type $V$. Bifurcation
theory involving topological degree methods was used in [7] for bounded $V$ in
order to obtain the existence of large solutions near eigenvalues, i.e. the
so-called bifurcation from infinity. For the same bifurcation problems with
bounded potential $V$ variational methods were applied in [14], whereas in
[24], an argument relying on topological degree were employed. Conley index
techniques were used in [4] to study bifurcations from infinity for
$(E)_{\lambda}$ with potentials of Kato-Rellich type, while in [2], the same
approach were utilized in the case of bounded potentials $V$.
Throughout the paper we assume that $V$ is a Kato-Rellich type potential, that
is,

$None$ $\begin{gathered}V=V_{\infty}+V_{0},\ \text{where}\ V_{\infty}\in
L^{\infty}(\mathbb{R}^{N})\ \text{and}\ V_{0}\in L^{p}(\mathbb{R}^{N})\
\text{with $p$ satisfying}\\\ p\geqslant 2\ \text{ if }\ 1\leqslant N\leqslant
3\ \ \text{ and }\ \ p>N/2\ \text{ if }\ N\geqslant 4.\end{gathered}\right.$

We also assume that the map $f:\mathbb{R}^{N}\times\mathbb{R}\to\mathbb{R}$
satisfies the conditions:
$(f1)$ the map $f(\,\cdot\,,u)$ is Lebesgue measurable, for all
$u\in\mathbb{R}$, and $c(x):=f(x,0)$ is an element of $L^{2}(\mathbb{R}^{N})$,
$(f2)$ there is a function $l:\mathbb{R}^{N}\to\mathbb{R}$ such that
$|f(x,u)-f(x,v)|\leqslant l(x)|u-v|\quad\text{for a.e.
}x\in\mathbb{R}^{N}\text{ and all }u,v\in\mathbb{R},$ and the following
(stronger Kato-Rellich type) conditions are satisfied $None$
$\begin{gathered}l=l_{\infty}+l_{0},\ \text{where}\ l_{\infty}\in
L^{\infty}(\mathbb{R}^{N})\ \text{and}\ l_{0}\in L^{r}(\mathbb{R}^{N})\
\text{with $r$ satisfying}\\\ r\geqslant 2\ \ \text{if}\ \ N=1,\ \ r>2\ \
\text{if}\ \ N=2\ \ \text{and}\ \ r\geqslant N\ \text{if }\ N\geqslant
3.\end{gathered}\right.$
In some cases we shall also assume that
$(f3)$ there exists $a:\mathbb{R}^{N}\to\mathbb{R}$ satisfying the condition
$(KR)$ such that $u\cdot f(x,u)\leqslant a(x)|u|^{2}\quad\text{for a.e.
}x\in\mathbb{R}^{N}\text{ and all }u\in\mathbb{R}.$
Our study of the existence of standing waves and the related problem
$(E)_{\lambda}$ strongly relies on the fact that $V$ is a Kato-Rellich type
potential and its consequences for the Schrödinger operator $-\Delta+V$,
including its spectral properties. In this paper we shall consider the
realization of $-\Delta+V$ in $L^{2}(\mathbb{R}^{N})$, which appears to be a
closed operator in $L^{2}(\mathbb{R}^{N})$ with the domain equal to
$H^{2}(\mathbb{R}^{N})$. Moreover, $-\Delta+V$ in $L^{2}(\mathbb{R}^{N})$ is
self-adjoint, its spectrum $\sigma(-\Delta+V)$ is bounded from below and the
essential spectrum $\sigma_{ess}(-\Delta+V)$ is located in the interval
$(\varrho(V_{\infty}),+\infty)$ where the so-called asymptotic bottom is given
by

$\varrho(V_{\infty}):=\lim_{R\to+\infty}\operatorname*{ess\,inf\,}_{|x|>R}V_{\infty}(x).$

The spectrum of $-\Delta+V$ may have a nonempty part below the value
$\varrho(V_{\infty})$, which then contains eigenvalues only. Therefore if
$\varrho(V_{\infty})>\lambda$, then $\sigma(-\Delta+V)\cap(-\infty,\lambda)$
consists of a finite number of eigenvalues (see Section 3 for a more detailed
discussion). This enables us to define the collective multiplicity of
eigenvalues less than $\lambda$ as

$d^{-}(V,\lambda):=\\!\\!\\!\\!\\!\sum_{\mu\in\sigma(-\Delta+V)\cap(-\infty,\lambda)}\\!\\!\dim\mathrm{Ker}\,(-\Delta+V-\mu).$

Let us start with the so-called non-resonant case, i.e. when the
linearizations of $f$ at zero and at infinity give potentials that do not
resonate with the operator $-\Delta+V-\lambda$. We shall start with the
following criterion on the existence of solutions of $(E)_{\lambda}$ and
connecting orbits for the parabolic equations associated with $(E)_{\lambda}$

$None$ $u_{t}=\Delta u-V(x)u+\lambda u+f(x,u),\quad x\in\mathbb{R}^{N},\ t>0.$

######  Theorem 1.1.

Let us assume that $f$ satisfies $(f1)-(f3)$ and there exist potentials
$\alpha,\omega:\mathbb{R}^{N}\to\mathbb{R}$ satisfying (PT) such that

$\lim_{u\to\;0}\frac{f(x,u)}{u}=\alpha(x)\quad\text{and}\quad\lim_{|u|\to+\infty}\frac{f(x,u)}{u}=\omega(x)\quad\text{for
a.e. $x\in\mathbb{R}^{N}$}.$

If
$\lambda<\min\left\\{\varrho(V_{\infty}-a_{\infty}),\varrho(V_{\infty}-\alpha_{\infty}),\varrho(V_{\infty}-\omega_{\infty})\right\\}$
is such that

$\lambda\not\in\sigma(-\Delta+V-\alpha)\cup\sigma(-\Delta+V-\omega)\ \ \text{
and }\ \ d^{-}(V-\alpha,\lambda)\neq d^{-}(V-\omega,\lambda),$

then there exists a nonzero solution $\bar{u}\in H^{2}(\mathbb{R}^{N})$ of
$(E)_{\lambda}$ and a bounded solution $u:\mathbb{R}\to H^{1}(\mathbb{R}^{N})$
of the equation $(P)_{\lambda}$ such that either $\bar{u}$ is in the
$\alpha$-limit set of $u$ and $u(t)\to 0$ in $H^{1}(\mathbb{R}^{N})$ as
$t\to+\infty$, or $\bar{u}$ is in the $\omega$-limit set of $u$ and $u(t)\to
0$ in $H^{1}(\mathbb{R}^{N})$ as $t\to-\infty$. Moreover, the set of all
solutions to $(E)_{\lambda}$ is bounded in $H^{1}(\mathbb{R}^{N})$.

Theorem 1.1 is an extension of the result by Prizzi – see [17, Cor. 3.4]; the
difference is that less restrictive assumptions on $\alpha$ and $\omega$ are
considered, no differentiability of $f$ is assumed and a more general
potential $V$ is possible, i.e. satisfying $(KR)$ instead of $(PT)$.
In the resonant case, i.e. when $-\Delta+V-\lambda$ has nonempty kernel we
shall assume that $f$ satisfies $(f2)$ and is "bounded" in the sense that
there is $m\in L^{2}(\mathbb{R}^{N})$ such that

$None$ $|f(x,u)|\leqslant m(x)\quad\text{ for a.e. $x\in\mathbb{R}^{N}$ and
all $u\in\mathbb{R}$}.$

Here we shall impose additional assumptions on $f$ at infinity, the so-called
Landesman-Lazer type and strong resonance conditions. The Landesman-Lazer type
conditions state that either

$None$ $\left\\{\begin{array}[]{c}\check{f}_{+}(x)\geqslant 0\ \mbox{ and }\
\hat{f}_{-}(x)\leqslant 0\ \mbox{ for a.e. }\ x\in\mathbb{R}^{N},\\\
\mbox{there is a set of positive measure on which none
of}\;\;\check{f}_{+}\mbox{ and }\hat{f}_{-}\mbox{vanishes},\end{array}\right.$

or

$None$ $\left\\{\begin{array}[]{c}\hat{f}_{+}(x)\leqslant 0\ \mbox{ and }\
\check{f}_{-}(x)\geqslant 0\ \mbox{ for a.e. }\ x\in\mathbb{R}^{N},\\\
\mbox{there is a set of positive measure on which none of }\;\hat{f}_{+}\mbox{
and }\check{f}_{-}\mbox{ vanishes,}\end{array}\right.$

where $\hat{f}_{\pm}(x):=\limsup_{s\to\pm\infty}f(x,s)$ and
$\check{f}_{\pm}(x):=\liminf_{s\to\pm\infty}f(x,s)$ for $x\in\mathbb{R}^{N}$.
We remark that the continuation property [8] for the eigenvalues of the
Schrödinger operator implies that the conditions $(LL)_{\pm}$ represent a
particular form of the original resonance assumptions introduced by Landesman
and Lazer in [15] (see [4, Remark 1.3] for more details).

The so-called sign or strong resonance conditions state

$None$ $\left\\{\begin{array}[]{c}f(x,s)s\geqslant 0,\mbox{ for a.e.
}x\in\mathbb{R}^{N}\mbox{ and all }s\in\mathbb{R},\\\ \mbox{there is a set of
positive measure on which }\check{k}_{+}\mbox{ and }\check{k}_{-}\mbox{ are
positive,}\end{array}\right.$

or

$None$ $\left\\{\begin{array}[]{c}f(x,s)s\leqslant 0,\mbox{ for a.e.
}x\in\mathbb{R}^{N}\mbox{ and all }s\in\mathbb{R},\\\ \mbox{there is a set of
positive measure on which }\hat{k}_{+}\mbox{ and }\hat{k}_{-}\mbox{ are
negative,}\end{array}\right.$

where $\check{k}_{\pm}(x)=\liminf_{s\to\pm\infty}f(x,s)s$ and
$\hat{k}_{\pm}(x)=\limsup_{s\to\pm\infty}f(x,s)s$ for $x\in\mathbb{R}^{N}$.
In the above setting we obtain the following existence result in the resonant
case, i.e. when $\lambda$ is in the point spectrum of $-\Delta+V$.

######  Theorem 1.2.

Let us assume that $\lambda$ is an eigenvalue of the operator $-\Delta+V$ such
that $\varrho\big{(}V_{\infty}\big{)}>\lambda$ and that $f$ satisfies $(f1)$,
$(f1)^{\prime}$ and $(f2)$. If one of conditions $(LL)_{+}$, $(LL)_{-}$,
$(SR)_{+}$ or $(SR)_{-}$ is satisfied, then the equation $(E)_{\lambda}$
admits a solution $u\in H^{2}(\mathbb{R}^{N})$ and the set of all solutions to
$(E)_{\lambda}$ is bounded.

In case of the existence of a trivial solution we get the following criterion.

######  Theorem 1.3.

Let $f$ be a map satisfying $(f1)$, $(f1)^{\prime}$, $(f2)$, $(f3)$. Assume
also that $f(x,0)=0$ for a.e. $x\in\mathbb{R}^{N}$ and there exist a potential
$\alpha$ satisfying $(PT)$ such that

$\lim_{u\to\;0}\frac{f(x,u)}{u}=\alpha(x)\quad\text{for a.e.
$x\in\mathbb{R}^{N}$}.$

If
$\lambda<\min\\{\varrho(V_{\infty}),\varrho(V_{\infty}-\alpha_{\infty}),\varrho(V_{\infty}-a_{\infty})\\}$
is an eigenvalue of $-\Delta+V$ such that one of the following conditions is
fulfilled
$(i)$ either $(LL)_{+}$ or $(SR)_{+}$ holds and
$d^{-}(V,\lambda)+\dim\mathrm{Ker}\,(-\Delta+V-\lambda)\neq
d^{-}(V-\alpha,\lambda)$,
$(ii)$ either $(LL)_{-}$ or $(SR)_{-}$ holds and $d^{-}(V,\lambda)\neq
d^{-}(V-\alpha,\lambda)$,
then there exists a nonzero solution $\bar{u}\in H^{2}(\mathbb{R}^{N})$ of
$(E)_{\lambda}$ and a bounded solution $u:\mathbb{R}\to H^{1}(\mathbb{R}^{N})$
of such that either $\bar{u}$ is in the $\alpha$-limit set of $u$ and $u(t)\to
0$ in $H^{1}(\mathbb{R}^{N})$ as $t\to+\infty$, or $\bar{u}$ is in the
$\omega$-limit set of $u$ and $u(t)\to 0$ in $H^{1}(\mathbb{R}^{N})$ as
$t\to-\infty$.

As an immediate consequence we get the following straightforward criterion.

######  Corollary 1.4.

Suppose that $V$, $f$ and $\lambda$ are as in Theorem 1.3 and, additionally,
that $\alpha$ is a constant function equal to $\bar{\alpha}\in\mathbb{R}$. If
one of the following conditions is satisfied
$(i)$ if either $(LL)_{+}$ or $(SR)_{+}$ holds and $\bar{\alpha}<0$,
$(ii)$ if either $(LL)_{-}$ or $(SR)_{-}$ holds and $\bar{\alpha}>0$,
$(iii)$ if either $(LL)_{+}$ or $(SR)_{+}$ holds, $\bar{\alpha}>0$ and
$\sigma(-\Delta+V)\cap(\lambda,\bar{\alpha}+\lambda)\neq\emptyset$,
$(iv)$ if either $(LL)_{-}$ or $(SR)_{-}$ holds, $\bar{\alpha}<0$ and
$\sigma(-\Delta+V)\cap(\bar{\alpha}+\lambda,\lambda)\neq\emptyset$,
then the assertion of Theorem 1.3 holds.

Theorem 1.3 provides the existence of nontrivial solution for $(E)_{\lambda}$.
It is showed in [4] that under the Landesman-Lazer or strong resonance
conditions, if $\lambda=\lambda_{0}$ is an isolated eigenvalue (as in Theorem
1.2 or Theorem 1.3), then $\lambda_{0}$ is an asymptotic bifurcation point (or
a point of bifurcation from infinity) for the parameterized family of
equations $(E)_{\lambda}$. In this context our results show that the large
solutions for the equation $(E)_{\lambda}$, with respect to the
$H^{1}(\mathbb{R}^{N})$ norm, appear also in the case $\lambda=\lambda_{0}$.
Our approach is based on the Rybakowski version of Conley index theory for
semiflows on non-compact spaces – see [18] and [19]. In this respect we follow
the pioneering work [20] as well as [17], [12], [13] or [5], where the
techniques involving irreducible sets to problems on bounded domains were
studied. We also refer the reader to the recent paper [21] where the
irreducibility method were used to find specific full nonconstant solutions
for the partial functional differential equation containing delay terms.

Here we consider the semiflow $\Phi$ on the space $H^{1}(\mathbb{R}^{N})$
generated by the nonlinear equation

(1.1) $\dot{u}(t)=-Au(t)+F(u(t)),\quad t>0,$

with $A:=-\Delta+V-\lambda$ and the Nemytskii operator
$F:H^{1}(\mathbb{R}^{N})\to L^{2}(\mathbb{R}^{N})$ determined by $f$. The
admissibility property of the semiflow $\Phi$ and related homotopies, that are
necessary for application of Conley index, is based on tail estimate
techniques (see e.g. [23] and [17]). In order to compute the Conley indices at
resonance, we shall exploit, after [12] (where bounded domains are studied)
and [4] (bifurcations for problems in $\mathbb{R}^{N}$), the fact that both
Landesman-Lazer and sign conditions have geometric consequences in the phase
space of the parabolic equation (1.1). We show the boundedness of the maximal
invariant set $K_{\infty}(\Phi)$, consisting of all $\bar{u}\in
H^{1}(\mathbb{R}^{N})$ such that there exists a full bounded solution of (1.1)
with $u(0)=\bar{u}$. Next we compute the Conley indices of the set
$K_{\infty}(\Phi)$ and the index of the isolated trivial (zero) solution. This
allows us to deduce the existence of a nonzero full solution of (1.1), which
by the gradient-like structure of the semiflow $\Phi$ gives the existence of
nontrivial solution of $(E)_{\lambda}$.
The paper is organized as follows. Section 2 provides basic facts on Conley-
Rybakowski index theory. In Section 3 we recall basic properties of the
operator $-\Delta+V$ and derive an inequality that we shall use for tail
estimates. Section 4 deals with the definition and properties of the semiflow
$\Phi$ determined by $(P)_{\lambda}$ and its parameterized variants. In
particular tail estimates are used to show the admissibility of bounded
subsets of the phase space $H^{1}(\mathbb{R}^{N})$. Section 5 is devoted to
the proof of Theorem 1.1, which is based on the mentioned Conley index
formulas and the properties of irreducible sets. Finally in Section 6 we shall
use the geometric implications of assumptions $(LL)_{\pm}$ and $(SR)_{\pm}$ in
order to prove Theorems 1.2 and 1.3 as well as Corollary 1.4.

##  2\. Preliminaries on Conley index due to Rybakowski

In this section we will briefly recall a version of Conley index for the non-
compact metric spaces due to Rybakowski (see [18] or [19]). Let
$\Phi\colon[0,+\infty)\times X\to X$ be a semiflow on a complete metric space
$X$. A continuous map $u\colon J\to X$, where $J\subset\mathbb{R}$ is an
interval, is a solution of $\Phi$ if $u(t+s)=\Phi_{t}(u(s))$ for all
$t\geqslant 0$ and $s\in J$ such that $t+s\in J$. Given $N\subset X$, we
define the _invariant part_ of the set $N$ by

$\mathrm{Inv}_{\Phi}(N):=\\{x\in X\ |\ \text{there is a
solution}\;u\colon\mathbb{R}\to N\;\text{such that}\;u(0)=x\\}.$

We say that a set $K\subset X$ is a $\Phi$-invariant or invariant (w.r.t.
$\Phi$) if $\mathrm{Inv}_{\Phi}(K)=K$. If there exists an isolating
neighborhood of $K$, i.e. the set $N\subset X$ such that
$K=\mathrm{Inv}_{\Phi}(N)\subset\mathrm{int}\,N$, then $K$ is called an
isolated invariant set.
A set $N\subset X$ is _$\Phi$ -admissible_ or admissible (w.r.t. $\Phi$) if,
for any sequences $(t_{n})$ in $[0,+\infty)$, $(x_{n})$ in $X$ such that
$t_{n}\to+\infty$ and $\\{\Phi(t,x_{n})\ |\ t\in[0,t_{n}]\\}\subset N$, the
sequence of the end-points $\left(\Phi_{t_{n}}(x_{n})\right)$ has a convergent
subsequence.
Suppose that $\\{\Phi^{(s)}\\}_{s\in\Lambda}$, where $\Lambda$ is a metric
space, is a family of semiflows on $X$ such that the map $[0,+\infty)\times
X\times\Lambda\ni(t,x,s)\mapsto\Phi^{(s)}_{t}(x)$ is continuous. A set
$N\subset X$ is admissible with respect to the family of semiflows
$\\{\Phi^{(s)}\\}_{s\in\Lambda}$ if, for any sequences $(t_{n})$ in
$[0,+\infty)$, $(x_{n})$ in $X$ and $(s_{n})$ such that $t_{n}\to+\infty$,
$s_{n}\to s_{0}$ in $\Lambda$ and $\\{\Phi^{(s_{n})}(t,x_{n})\ |\
t\in[0,t_{n}]\\}\subset N$, the sequence $(\Phi^{(s_{n})}_{t_{n}}(x_{n}))$ has
a convergent subsequence.
Let ${\mathcal{I}}(X)$ be the family of all pairs $(\Phi,K)$ such that $\Phi$
is a semiflow on $X$ and $K\subset X$ is an isolated invariant set having an
admissible isolating neighborhood w.r.t. $\Phi$. Given
$(\Phi,K)\in{\mathcal{I}}(X)$, we can choose a pair of closed sets
$(B,B^{-})$, where $B$ is an isolating block of $K$ relative to $\Phi$ and
$B^{-}$ is its exit set (see [18]). Then the Conley homotopy index $h(\Phi,K)$
of $K$ relative to $\Phi$ is defined as the homotopy type of the quotient
space

$h(\Phi,K):=[(B/B^{-},[B^{-}])],$

provided $B^{-}$ is a nonempty set. If $B^{-}=\emptyset$ then we put
$h(\Phi,K):=[(B\,\dot{\cup}\,\\{a\\},a)]$, where $\\{a\\}$ is the one point
space disjoint with $B$. In particular, $h(\Phi,\emptyset)=\overline{0}$,
where $\overline{0}:=[(\\{a\\},a)]$.

Let us enumerate several important properties of the homotopy index.

  1. (H1)

For any $(\Phi,K)\in{\mathcal{I}}(X)$, if $h(\Phi,K)\neq\overline{0}$, then
$K\neq\emptyset$;

  2. (H2)

if $(\Phi,K_{1}),(\Phi,K_{2})\in{\mathcal{I}}(X)$ and $K_{1}\cap
K_{2}=\emptyset$, then $(\Phi,K_{1}\cup K_{2})\in{\mathcal{I}}(X)$ and
$h(\Phi,K_{1}\cup K_{2})=h(\Phi,K_{1})\vee h(\Phi,K_{2})$;

  3. (H3)

for any $(\Phi_{1},K_{1})\in{\mathcal{I}}(X_{1})$ and
$(\Phi_{2},K_{2})\in{\mathcal{I}}(X_{2})$,
$(\Phi_{1}\times\Phi_{2},K_{1}\times K_{2})\in{\mathcal{I}}(X_{1}\times
X_{2})$ and $h(\Phi_{1}\times\Phi_{2},K_{1}\times
K_{2})=h(\Phi_{1},K_{1})\wedge h(\Phi_{2},K_{2})$;

  4. (H4)

if the family of semiflows $\\{\Phi^{(s)}\\}_{s\in[0,1]}$ is continuous and
there exists an admissible (with respect to this family) $N$ such that
$K_{s}=\mathrm{Inv}_{\Phi^{(s)}}(N)\subset\mathrm{int}\ N$ for $s\in[0,1]$,
then

$h(\Phi^{(0)},K_{0})=h(\Phi^{(1)},K_{1}).$

We recall that, if $u\colon\mathbb{R}\to X$ is a solution of the semiflow
$\Phi$, then the $\omega$-limit set of $u$ is defined by

$\omega(u):=\\{x=\lim_{n\to\infty}u(t_{n})\mid\text{for some }\
t_{n}\to+\infty\\}.$

Similarly, the $\alpha$-limit set of $u$ is defined by

$\alpha(u):=\\{x=\lim_{n\to\infty}u(t_{n})\mid\text{for some }\
t_{n}\to-\infty\\}.$

It is known that if $N\subset X$ is $\Phi$-admissible and $u:\mathbb{R}\to N$
is a solution of $\Phi$, then $\alpha(u)$ and $\omega(u)$ are both nonempty
compact invariant sets.
An isolated invariant with respect to the semiflow $\Phi$ set $K$ is called
irreducible if there are no isolated invariant sets $K_{1}$ and $K_{2}$ such
that $K=K_{1}\cup K_{2}$, $K_{1}\cap K_{2}=\emptyset$ and

$h(\Phi,K_{1})\neq\overline{0}\quad\text{ and }\quad
h(\Phi,K_{2})\neq\overline{0}.$

It is known that if $K$ is connected or $h(\Phi,K)=\overline{0}$ or
$h(\Phi,K)=\Sigma^{k}$, where $k\geqslant 0$ is an integer (1) (1) (1) We
recall that $\Sigma^{k}:=[(S^{k},s_{0})]$ is the homotopy type of the pointed
$k$-dimensional sphere $S^{k}:=\\{x\in\mathbb{R}^{k+1}\ |\ |x|=1\\}$., then
$K$ is irreducible (see [18, Theorem 1.11.6]).

######  Theorem 2.1.

(see [18, Theorem 1.11.5]) Assume that $K_{0}\subset K\subset X$ is a pair of
the isolated invariant sets such that $K$ is irreducible and

$\overline{0}\neq h(\Phi,K_{0})\neq h(\Phi,K)\neq\overline{0}.$

Then there exists a full solution $u:\mathbb{R}\to K$ such that
$u(\mathbb{R})\not\subset K_{0}$ and either $\alpha(u)\subset K_{0}$ or
$\omega(u)\subset K_{0}$.

Recall that a continuous function $\mathcal{E}:X\to\mathbb{R}$ is called a
Liapunov functional for the semiflow $\Phi$ if for any $x\in X$, the function
$t\mapsto\mathcal{E}(\Phi_{t}(x))$ is nonincreasing on $(0,+\infty)$. The
semiflow $\Phi$ is called gradient-like with respect to $\mathcal{E}$ if
$\mathcal{E}$ is a Lyapunov functional for $\Phi$ and, for any nonconstant
solution $u:\mathbb{R}\to X$ of $\Phi$, the function
$t\mapsto\mathcal{E}(\Phi_{t}(x))$ is not constant. It is known that if $\Phi$
is gradient-like with respect to some Lyapunov functional for $\Phi$, then,
for any solution $u:\mathbb{R}\to X$ of $\Phi$ with relatively compact
$u(\mathbb{R})$, the sets $\alpha(u)$ and $\omega(u)$ are nonempty and
disjoint sets containing only equilibria of $\Phi$, i.e. elements $x\in X$
such that $\Phi_{t}(x)=x$ for all $t\geqslant 0$ (see [18, Theorem 2.5.4]).

##  3\. Properties of the Schrödinger operator

Let us consider an operator $A$ on the space $X:=L^{2}(\mathbb{R}^{N})$ given
by the formula

(3.1) $Au:=-\Delta u+Vu-\lambda u,\quad\ u\in D(A):=H^{2}(\mathbb{R}^{N}),$

where $\lambda<\varrho(V_{\infty})$. Clearly $A$ is a perturbation of
$A_{0}:H^{2}(\mathbb{R}^{N})\to L^{2}(\mathbb{R}^{N})$, given by
$A_{0}u:=-\Delta u$, $u\in H^{2}(\mathbb{R}^{N})$, by the multiplication
operator $V:H^{2}(\mathbb{R}^{N})\to L^{2}(\mathbb{R}^{N})$. It is known that
$A_{0}$ is self-adjoint and $-A_{0}$ generates an analytic $C_{0}$-semigroup
of bound linear operators on $X$. Since the potential $V$ is of Kato-Rellich
type we infer that $H^{2}(\mathbb{R}^{N})$ continuously embeds into
$L^{q}(\mathbb{R}^{N})$ for $q=+\infty$ if $N=1,2,3$, for arbitrary
$q\geqslant 2$ if $N=4$ and $q=2N/(N-4)$ if $N\geqslant 5$ (see [1, Corollary
9.13]). This implies that the multiplication operator $V$ is well-defined and
$A_{0}$-bounded with zero $A_{0}$-bound, i.e., for any $\varepsilon>0$, there
exists $C_{\varepsilon}>0$ such that

$\|Vu\|_{L^{2}}\leqslant\varepsilon\|A_{0}u\|_{L^{2}}+C_{\varepsilon}\|u\|_{L^{2}}\quad\text{for
all }\ u\in H^{2}(\mathbb{R}^{N}).$

Therefore, by the Rellich-Kato theorem (see e.g. [11, Th. 13.5]), the
perturbed operator $A=A_{0}+V-\lambda I$ is self-adjoint and, by [6, Th.
2.10], the operator $-A$ generates an analytic $C_{0}$-semigroup
$\\{S(t)\\}_{t\geqslant 0}$. Moreover, it appears that the multiplication
operator $V_{0}:H^{2}(\mathbb{R}^{N})\to L^{2}(\mathbb{R}^{N})$ is
$(A_{0}+V_{\infty})$-compact, which due to the Weyl theorem (see e.g. [11, Th.
14.6]) implies that the essential spectra of $A_{0}+V_{\infty}$ and $A$ are
equal, i.e.

$\sigma_{ess}(-\Delta+V-\lambda)=\sigma_{ess}(-\Delta+V_{\infty}-\lambda).$

By the Persson formula (see e.g. [11, Th. 14.11] or [16]) for the bottom of
essential spectrum we get

$\displaystyle\inf\sigma_{\mathrm{ess}}(-\Delta+V_{\infty})=\sup_{R>0}\
\inf\left\\{(-\Delta\phi+V_{\infty}\phi,\phi)_{L^{2}}\ |\ \phi\in
C^{\infty}_{0}(\mathbb{R}^{N}\setminus D(0,R)),\,\|\phi\|_{L^{2}}=1\right\\},$

where $D(0,R):=\\{x\in\mathbb{R}^{N}\mid|x|\leqslant R\\}$. Therefore the
essential part of the spectrum of the operator $A$ satisfies
$\inf\sigma_{\mathrm{ess}}(A)\geq\varrho(V_{\infty})-\lambda$. Hence the part
of the spectrum $\sigma(A)$ contained in the interval
$(-\infty,\varrho(V_{\infty})-\lambda)$ consists of isolated eigenvalues of
finite multiplicity, that in turn satisfy the estimate

(3.2) $\displaystyle|u(x)|\leqslant Ce^{-\delta|x|}\qquad\text{ for a.e. }\
x\in\mathbb{R}^{N},$

where $\delta=\delta(u)>0$ and $C=C(u)>0$ are constants – see [22, Th. C.3.4].
Since $A$ is self-adjoint, by the spectral theorem, there are closed mutually
orthogonal subspaces $X_{-}$, $X_{0}$, $X_{+}$ of $X=L^{2}(\mathbb{R}^{N})$
such that $X=X_{-}\oplus X_{0}\oplus X_{+}$ and

$\sigma(A|X_{-})=\sigma(A)\cap(-\infty,0),\ \ \ X_{0}=\mathrm{Ker}\,\,A,\ \ \
\sigma(A|X_{+})=\sigma(A)\cap(0,+\infty).$

It is known that $\dim\,X_{-}<\infty$ and $\dim X_{0}<\infty$. We denote by
$Q_{-},P,Q_{+}$, be the orthogonal projections onto $X_{-}$, $X_{0}$ and
$X_{+}$, respectively, and we write $Q:=Q_{+}+Q_{-}$. By the semigroup theory
we know that

$S(t)Q_{\pm}u=Q_{\pm}S(t)u,\quad u\in X,\ t\geq 0,$

and there exist $K,\rho_{+}>0$ such that

(3.3) $\|S(t)u\|_{H^{1}}\leqslant Kt^{-1/2}e^{-\rho_{+}t}\|u\|_{L^{2}}\ \
\mbox{ for all }\ \ u\in X_{+},\ t>0.$

Obviously, the restricted semigroup $\\{e^{-tA}|_{X_{-}}\\}_{t\geqslant 0}$
can be extended to a $C_{0}$-group on $X_{-}$ and there exist $\rho_{-}>0$
such that

(3.4) $\|S(t)u\|_{L^{2}}\leqslant e^{\rho_{-}t}\|u\|_{L^{2}},\quad u\in
X_{-},\ t\leqslant 0.$

The following theorem says that the spectral properties enable us to determine
the Conley index of zero in the linear case.

######  Theorem 3.1.

(see [18, Ch. I, Th. 11.1]) The restriction of the semigroup
$\\{S(t)_{|Y}\\}_{t\geq 0}$ to $Y:=H^{1}(\mathbb{R}^{N})\cap(X_{+}\oplus
X_{-})$ determines the semiflow on the space $Y$ equipped with the norm
$\|\cdot\|_{H^{1}}$ (note that $Y=H^{1}(\mathbb{R}^{N})$ if
$\lambda\not\in\sigma(-\Delta+V)$). Furthermore the set $K_{0}=\\{0\\}$ is the
maximal bounded invariant set of the semiflow,
$(S(t)_{|Y},K_{0})\in{\mathcal{I}}(X)$ and

$h(S(t)_{|Y},K_{0})=\Sigma^{\dim X_{-}}.$

Let us consider the semilinear differential equation

(3.5) $\dot{u}(t)=-Au(t)+g(t),\quad t>0,$

where $g:[0,\infty)\to X$ is a continuous function and define the cut-off map
$\phi_{n}(x):=\phi(|x|^{2}/n^{2})$ for $x\in\mathbb{R}^{n}$, where the profile
$\phi:[0,\infty)\to[0,1]$ is a smooth function such that $\phi(x)=0$ for
$x\in[0,1]$ and $\phi(x)=1$ for $|x|\geq 2$. The following version of the
estimates for the solutions of (3.5) can be obtained along the lines of [4,
Lem. 3.6] and [3, Lem. 5.3].

######  Proposition 3.2.

There exist a sequence $(\beta_{n})$ with $\beta_{n}\to 0^{+}$ as $n\to\infty$
and $n_{0}\geqslant 1$, such that if $u\in
C((t_{0},t_{1}),H^{2}(\mathbb{R}^{N}))\cap
C^{1}((t_{0},t_{1}),L^{2}(\mathbb{R}^{N}))$ is a classical solution of the
equation (3.5), then

(3.6)
$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^{N}}\phi_{n}|u(t)|^{2}\,dx\leqslant-\int_{\mathbb{R}^{N}}(V_{\infty}(x)-\lambda)\phi_{n}|u(t)|^{2}\,dx+\gamma_{n}\|u(t)\|_{H^{1}}^{2}+\langle\phi_{n}u(t),g(t)\rangle_{L^{2}}$

for $t\in(t_{0},t_{1})$ and $n\geq n_{0}$. The above sequence $(\beta_{n})$
and integer $n_{0}\geq 1$ depend only on the potential $V$.

###### Proof.

We begin with the observation that

(3.7)
$\displaystyle\frac{1}{2}\frac{d}{dt}\langle\phi_{n}u(t),u(t)\rangle_{L^{2}}=\langle\phi_{n}u(t),\dot{u}(t)\rangle_{L^{2}}=\langle\phi_{n}u(t),-Au(t)+g(t)\rangle_{L^{2}}$
$\displaystyle\quad=\langle\phi_{n}u(t),\Delta
u(t)\rangle_{L^{2}}\\!+\\!\langle\phi_{n}u(t),-Vu(t)\\!+\\!\lambda
u(t)\rangle_{L^{2}}\\!+\\!\langle\phi_{n}u(t),g(t)\rangle_{L^{2}}$
$\displaystyle\quad=:I_{1}(t)+I_{2}(t)+I_{3}(t)$

for $t\in[t_{0},t_{1}]$. Then we have the following estimates

(3.8) $\displaystyle I_{1}(t)$ $\displaystyle=\langle\phi_{n}u(t),\Delta
u(t)\rangle_{L^{2}}=-\langle\nabla(\phi_{n}u(t)),\nabla u(t)\rangle_{L^{2}}$
$\displaystyle=-\\!\int_{\mathbb{R}^{N}}\phi_{n}(x)|\nabla
u(t)|^{2}\,dx-\frac{2}{n^{2}}\int_{\\{n\leqslant|x|\leqslant
n\sqrt{2}\\}}\\!u(t)\phi^{\prime}\left(\tfrac{|x|^{2}}{n^{2}}\right)\,x\cdot\nabla
u(t)\,dx$ $\displaystyle\leqslant\frac{2\sqrt{2}}{n}L_{\phi}\|\nabla
u(t)\|_{L^{2}}\|u(t)\|_{L^{2}}\leqslant\frac{2\sqrt{2}}{n}L_{\phi}\|u(t)\|_{H^{1}}^{2},$

where $L_{\phi}:=\sup_{t\geq 0}|\phi^{\prime}(t)|$. On the other hand we have

(3.9) $\displaystyle I_{2}(t)$
$\displaystyle=\langle\phi_{n}u(t),-Vu(t)+\lambda
u(t)\rangle_{L^{2}}=-\int_{\mathbb{R}^{N}}(V_{\infty}(x)-\lambda)\phi_{n}|u(t)|^{2}\,dx-\int_{\mathbb{R}^{N}}V_{0}(x)\phi_{n}|u(t)|^{2}\,dx$
$\displaystyle\leqslant-\int_{\mathbb{R}^{N}}(V_{\infty}(x)-\lambda)\phi_{n}|u(t)|^{2}\,dx+\|\phi_{n}V_{0}\|_{L^{p}}\|u(t)\|_{L^{2p/(p-1)}}^{2}$
$\displaystyle\leqslant-\int_{\mathbb{R}^{N}}(V_{\infty}(x)-\lambda)\phi_{n}|u(t)|^{2}\,dx+\|\phi_{n}V_{0}\|_{L^{p}}\|u(t)\|_{H^{1}}^{2},$

where, in the last estimate, we used the Sobolev embedding
$H^{1}(\mathbb{R}^{N})\subset L^{2p/(p-1)}(\mathbb{R}^{N})$, which is valid
for $p>2$ satisfying the condition $(KR)$. Therefore, combining
(LABEL:ineq-k1), (3.8) and (3.9) gives (3.6) with

$\displaystyle\beta_{n}:=\frac{4\sqrt{2}}{n}L_{\phi}+2\|\phi_{n}V_{0}\|_{L^{p}},\quad
n\geq 1.$

Since $V_{0}\in L^{p}(\mathbb{R}^{N})$, we have $\beta_{n}\to 0^{+}$ as
$n\to\infty$ and the proof of the proposition is completed. ∎

##  4\. Semiflow and admissible homotopy

Consider the differential equation

(4.1) $\dot{u}(t)=-Au(t)+F(u(t)),\quad t>0,$

where $A$ is defined by (3.1) and the nonlinear map
$F:H^{1}(\mathbb{R}^{N})\to L^{2}(\mathbb{R}^{N})$ is the Nemytskii operator
determined by $f$, i.e. it is given by

(4.2) $[F(u)](x):=f(x,u(x))\quad\mbox{ for a.e. }\ x\in\mathbb{R}^{N}\text{
and all }u\in H^{1}(\mathbb{R}^{N}).$

######  Remark 4.1.

From the assumptions $(f1)$ and $(f2)$ it follows that $f$ is a Carethéodory
function. Therefore, for any $u\in H^{1}(\mathbb{R}^{N})$, the composition in
(4.2) is a measurable function. On the other hand, by $(f2)$ and the Hölder
inequality, for any $u,v\in H^{1}(\mathbb{R}^{N})$, we have

(4.3) $\displaystyle\|F(u)-F(v)\|_{L^{2}}$
$\displaystyle\leqslant\|(l_{\infty}+l_{0})(u-v)\|_{L^{2}}$
$\displaystyle\leqslant\|l_{\infty}\|_{L^{\infty}}\|u-v\|_{L^{2}}+\|l_{0}\|_{L^{r}}\|u-v\|_{L^{2r/(r-2)}}$
$\displaystyle\leqslant\left(\|l_{\infty}\|_{L^{\infty}}+\|l_{0}\|_{L^{r}}\right)\|u-v\|_{H^{1}},$

where we used the Sobolev embedding $H^{1}(\mathbb{R}^{N})\subset
L^{2r/(r-2)}(\mathbb{R}^{N})$, which holds true as $r\in(2,\infty)$ is
restricted in $(PT)$. This implies that $F$ is a Lipschitz map. Furthermore,
by $(f1)$ and (4.3), we have

(4.4)
$\displaystyle\|F(u)\|_{L^{2}}\leqslant\left(\|l_{\infty}\|_{L^{\infty}}+\|l_{0}\|_{L^{r}}\right)\|u\|_{H^{1}}+\|c\|_{L^{2}}$

for $u\in H^{1}(\mathbb{R}^{N})$, which gives the sublinear growth of $F$.
$\square$

Since the operator $A$ is sectorial and $F$ satisfies (4.3) and (4.4), it
follows (see e.g. [10, Theorem 3.3.3]) that, for any $\bar{u}\in
H^{1}(\mathbb{R}^{N})$ and $t_{0}\in\mathbb{R}$, there exists a unique
continuous function $u=u(\,\cdot\,;\bar{u},t_{0}):[t_{0},\infty)\to
H^{1}(\mathbb{R}^{N})$ such that $u(t_{0})=\bar{u}$, $u\in
C((t_{0},+\infty),H^{2}(\mathbb{R}^{N}))\cap
C^{1}((t_{0},+\infty),L^{2}(\mathbb{R}^{N}))$ and (4.1) holds for all
$t>t_{0}$. Consequently the equation (4.1) determines the semiflow
$\\{\Phi_{t}\\}_{t\geqslant 0}$ on the space $H^{1}(\mathbb{R}^{N})$, given by

$\Phi_{t}(\bar{u}):=u(t;\bar{u},0),\quad\bar{u}\in H^{1}(\mathbb{R}^{N}),\
t\geq 0.$

It appears that the semiflow $\Phi$ is gradient-like.

######  Proposition 4.2.

Let ${\mathcal{E}}:H^{1}(\mathbb{R}^{N})\to\mathbb{R}$ be given by

(4.5) ${\mathcal{E}}(v):=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
v(x)|^{2}+(V(x)-\lambda)|v(x)|^{2}\,dx+\int_{\mathbb{R}^{N}}{\mathcal{F}}(x,v(x))\,dx,\quad
v\in H^{1}(\mathbb{R}^{N}),$

where ${\mathcal{F}}:\mathbb{R}^{N}\times\mathbb{R}\to\mathbb{R}$ is given by
${\mathcal{F}}(x,v):=\int_{0}^{v}f(x,w)\,dw$ for $x\in\mathbb{R}^{N}$ and
$v\in\mathbb{R}$. The function ${\mathcal{E}}$ is a well-defined continuous
Lyaponov functional for the semiflow $\Phi$, i.e. if $u:[t_{0},t_{1})\to
H^{1}(\mathbb{R}^{N})$ is a solution of the semiflow $\Phi$, then

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{E}(u(t))=-\int_{\mathbb{R}^{N}}|\dot{u}(t)|^{2}\,dx\quad\text{for}\
\ t\in(t_{0},t_{1}).$

###### Proof.

If $u:[t_{0},t_{1})\to H^{1}(\mathbb{R}^{N})$ is a solution of (4.1), then
both $\dot{u}$ and $F\circ u$ are continuous functions on $(t_{0},+\infty)$
with values in $L^{2}(\mathbb{R}^{N})$, and consequently $A\circ
u=-\dot{u}+F\circ u$ is continuous, which implies that $u\in
C((t_{0},t_{1}),H^{2}(\mathbb{R}^{N}))$. Fix $t\in(t_{0},t_{1})$ and take any
small $h\neq 0$, then, using the fact that $u\in
C((t_{0},t_{1}),H^{2}(\mathbb{R}^{N}))$, we get

$\displaystyle\frac{1}{h}\big{(}\|\nabla u(t+h)\|_{L^{2}}^{2}-\|\nabla
u(t)\|_{L^{2}}^{2}\big{)}$ $\displaystyle\quad=\langle\nabla
u(t+h),\nabla\big{(}h^{-1}(u(t+h)-u(t))\big{)}\rangle_{L^{2}}+\langle\nabla\big{(}h^{-1}(u(t+h)-u(t))\big{)},\nabla
u(t)\rangle_{L^{2}}$ $\displaystyle\quad=-\langle\Delta
u(t+h),h^{-1}(u(t+h)-u(t))\rangle_{L^{2}}-\langle h^{-1}(u(t+h)-u(t)),\Delta
u(t)\rangle_{L^{2}}$ $\displaystyle\quad\to 2\langle-\Delta
u(t),\dot{u}(t)\rangle_{L^{2}}\ \text{ as }h\to 0.$

In a similar manner

$\frac{d}{dt}\int_{\mathbb{R}^{N}}(V(x)-\lambda)|u(t)|^{2}=2\int_{\mathbb{R}^{N}}(V(x)-\lambda)u(t)\dot{u}(t)\,dx.$

By standard arguments

$\displaystyle\frac{1}{h}\int_{\mathbb{R}^{N}}(\mathcal{F}(x,u(t+h)(x))-\mathcal{F}(x,u(t)(x))\,dx$
$\displaystyle\quad=\frac{1}{h}\int_{\mathbb{R}^{N}}g_{n}(x)\big{(}u(t+h)(x)-u(t)(x)\big{)}\,dx\to\langle
F(u(t)),\dot{u}(t)\rangle_{L^{2}}\ \text{ as }h\to 0^{+},$

where $g_{h}(x):=f\big{(}x,u(t)(x)+\theta_{x}(u(t+h)(x)-u(t)(x))\big{)}$, with
$\theta_{x}\in(0,1)$ from the Lagrange mean value theorem applied for fixed
$x\in\mathbb{R}^{N}$, and we use the fact that $g_{h}\to F(u(t))$ in
$L^{2}(\mathbb{R}^{N})$ as $h\to 0$. The latter follows from the following
inequality implied by $(f2)$

$|g_{h}(x)-f(t,u(t)(x))|\leqslant l(x)|u(t+h)(x)-u(t)(x)|\ \text{ for a.e.
}x\in\mathbb{R}^{N},$

which implies $\|g_{h}-F(u(t))\|_{L^{2}}\leqslant
C(\|l_{\infty}\|_{L^{\infty}}+\|l_{0}\|_{L^{r}})\|u(t+h)-u(t)\|_{H^{1}}$ as in
(4.3).
Combining the above three equalities together we obtain

$(\mathcal{E}\circ u)^{\prime}(t)=\langle-\Delta
u(t)+(V-\lambda)u(t)-F(u(t)),\dot{u}(t)\rangle_{L^{2}}=-\|\dot{u}(t)\|_{L^{2}(\mathbb{R}^{N})}^{2},$

which proves the assertion. ∎

Let us also consider a parameterized by $s\in[0,1]$ family of differential
equations

(4.6) $\dot{u}(t)=-Au(t)+G(u(t),s),\quad t>0,$

where $G:H^{1}(\mathbb{R}^{N})\times[0,1]\to L^{2}(\mathbb{R}^{N})$ is a
continuous map satisfying the following conditions
$(G1)$ the map $G$ is locally Lipschitz, that is, for any $R>0$, there is
$L_{R}>0$ such that $\|G(u,s)-G(v,s)\|_{L^{2}}\leqslant L_{R}\|u-v\|_{H^{1}}$
for any $s\in[0,1]$ and $u,v\in H^{1}(\mathbb{R}^{N})$ with $\|u\|_{H^{1}}$,
$\|v\|_{H^{1}}\leqslant R$;
$(G2)$ the map $G$ has sublinear growth, that is, there is $C>0$ such that
$\|G(u,s)\|_{L^{2}}\leqslant C(\|u\|_{H^{1}}+1)$ for any $s\in[0,1]$ and $u\in
H^{1}(\mathbb{R}^{N})$.
Given $s\in[0,1]$, we denote by
$\Psi^{(s)}=\big{\\{}\Psi_{t}^{(s)}\big{\\}}_{t\geqslant 0}$ the semiflow on
$H^{1}(\mathbb{R}^{N})$ determined by the differential equation (4.6). We
shall use the following continuation property – see e.g. [3, Th. 3.2].

######  Proposition 4.3.

The following assertions hold.
$(i)$ If $\bar{u}_{n}\to\bar{u}_{0}$ in $H^{1}(\mathbb{R}^{N})$ and $s_{n}\to
s_{0}$, then
$\Psi_{t}^{(s_{n})}(\bar{u}_{n})\to\Psi_{t}^{(s_{0})}(\bar{u}_{0})$ in
$H^{1}(\mathbb{R}^{N})$ for all $t\geq 0$.
$(ii)$ If $(\bar{u}_{n})$ is a bounded sequence in $H^{1}(\mathbb{R}^{N})$
such that $\bar{u}_{n}\to\bar{u}_{0}$ in $L^{2}(\mathbb{R}^{N})$ for some
$\bar{u}_{0}\in H^{1}(\mathbb{R}^{N})$, then
$\Psi_{t}^{(s_{n})}(\bar{u}_{n})\to\Psi_{t}^{(s_{0})}(\bar{u}_{0})$ in
$H^{1}(\mathbb{R}^{N})$ uniformly with respect to $t$ in compact subsets of
$(0,\infty)$.

The following so-called tail estimates will be used in order to establish
compactness properties of the family of semiflows
$\\{\Psi^{(s)}\\}_{s\in[0,1]}$ – compare [4, Prop. 3.7 and Cor. 3.9].

######  Proposition 4.4.

Let us assume that there is a potential $a=a_{\infty}+a_{0}$, where
$a_{\infty}\in L^{\infty}(\mathbb{R}^{N})$ and $a_{0}$ is a finite sum of
potentials belonging to $L^{p}(\mathbb{R}^{N})$ with $p$ as in the condition
$(KR)$ (possibly with different exponents $p$ for each component) and $b\in
L^{2}(\mathbb{R}^{N})$ such that, for any $s\in[0,1]$ and $u\in
H^{1}(\mathbb{R}^{N})$, the following inequality is satisfied

(4.7) $u(x)\cdot G(u,s)(x)\leqslant a(x)|u(x)|^{2}+b(x)|u(x)|\ \text{ for a.e.
}\ x\in\mathbb{R}^{N}.$

If $\varrho(V_{\infty}-a_{\infty})>\lambda$ then, for any $R>0$, there exists
$\varepsilon>0$, a sequence $(\gamma_{n})$ with $\gamma_{n}\to 0^{+}$ as
$n\to\infty$ and $n_{0}\geqslant 1$, such that if $u:[t_{0},t_{1}]\to
H^{1}(\mathbb{R}^{N})$ is a solution of the equation (4.6) for some
$s\in[0,1]$ and

$\|u(t)\|_{H^{1}}\leqslant R\quad\text{for all}\ \ t\in[t_{0},t_{1}],$

then, for all $n\geqslant n_{0}$, we have

$\int_{\mathbb{R}^{N}}\phi_{n}|u(t)|^{2}\,dx\leqslant
e^{-2\varepsilon(t-t_{0})}\|u(t_{0})\|_{L^{2}}^{2}+\gamma_{n},\quad
t\in[t_{0},t_{1}].$

The above constant $\varepsilon>0$, the sequence $(\gamma_{n})$ and the
integer $n_{0}\geq 1$ depend only on $R>0$, the dimension $N\geq 1$ and the
potentials $a$, $b$ and $V$.

###### Proof.

We shall carry out the proof with assuming that $a_{0}$ is in
$L^{p}(\mathbb{R}^{N})$ with $p$ like in the condition $(KR)$, an adaptation
of the below estimates to the case where $a_{0}$ is a sum of such functionals
is straightforward.
By Proposition 3.2, there exist a sequence $(\beta_{n})$ with $\beta_{n}\to
0^{+}$ as $n\to\infty$ such that

(4.8)
$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^{N}}\phi_{n}|u(t)|^{2}\,dx\\!\leqslant\\!-\\!\int_{\mathbb{R}^{N}}(V_{\infty}-\lambda)\phi_{n}|u(t)|^{2}\,dx+\beta_{n}\|u(t)\|_{H^{1}}^{2}+\langle\phi_{n}(t),G(u(t),s)\rangle_{L^{2}}$

for $t\in[t_{0},t_{1}]$ and $n\geq 1$. Since
$\varrho(V_{\infty}-a_{\infty})>\lambda$, we can choose $\varepsilon>0$ and
$n_{0}\geq 1$ such that, for all $n\geqslant n_{0}$,

$\displaystyle\phi_{n}(x)(V_{\infty}(x)-a_{\infty}(x)-\lambda)\geq\varepsilon\phi_{n}(x)\quad\text{for
a.a. }x\in\mathbb{R}^{N}.$

By the condition (4.7) we obtain

(4.9)
$\displaystyle\langle\phi_{n}u(t),G(u(t),s)\rangle_{L^{2}}\leqslant\int_{\mathbb{R}^{N}}\phi_{n}a_{\infty}|u(t)|^{2}\,dx+\int_{\mathbb{R}^{N}}\phi_{n}a_{0}|u(t)|^{2}\,dx+\int_{\mathbb{R}^{N}}\phi_{n}b|u(t)|\,dx.$

On the other hand, by the Hölder inequality and Sobolev embedding, we have

(4.10)
$\displaystyle\int_{\mathbb{R}^{N}}\phi_{n}a_{0}|u(t)|^{2}\,dx\leqslant\|\phi_{n}a_{0}\|_{L^{p}}\|u(t)\|_{L^{2p/(p-1)}}^{2}\leqslant\|\phi_{n}a_{0}\|_{L^{p}}\|u(t)\|_{H^{1}}^{2}.$

Combining (4.8), (4.9) and (LABEL:eq-r2b) we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^{N}}\phi_{n}|u(t)|^{2}\,dx$
$\displaystyle\leqslant-\int_{\mathbb{R}^{N}}(V_{\infty}-a_{\infty}-\lambda)\phi_{n}|u(t)|^{2}\,dx+(\beta_{n}+\|\phi_{n}a_{0}\|_{L^{p}})\|u(t)\|_{H^{1}}^{2}+\|\phi_{n}b\|_{L^{2}}\|u(t)\|_{H^{1}}$
$\displaystyle\leqslant-\varepsilon\int_{\mathbb{R}^{N}}\phi_{n}|u(t)|^{2}\,dx+(\beta_{n}+\|\phi_{n}a_{0}\|_{L^{p}})\|u(t)\|_{H^{1}}^{2}+\|\phi_{n}b\|_{L^{2}}\|u(t)\|_{H^{1}}$
$\displaystyle\leqslant-\varepsilon\int_{\mathbb{R}^{N}}\phi_{n}|u(t)|^{2}\,dx+\varepsilon\gamma_{n}$

for $t\in[t_{0},t_{1}]$ and $n\geq n_{0}$, where we define
$\gamma_{n}:=\varepsilon^{-1}R^{2}(\beta_{n}+\|\phi_{n}a_{0}\|_{L^{p}})+R\|\phi_{n}b\|_{L^{2}}$.
Since $a_{0}\in L^{p}(\mathbb{R}^{N})$ and $b\in L^{2}(\mathbb{R}^{n})$, we
have $\gamma_{n}\to 0^{+}$ as $n\to\infty$. Integrating the above inequality,
for any $n\geq n_{0}$, gives

$\displaystyle\int_{\mathbb{R}^{N}}\phi_{n}|u(t)|^{2}\,dx$
$\displaystyle\leqslant
e^{-2\varepsilon(t-t_{0})}\int_{\mathbb{R}^{N}}\phi_{n}|u(t_{0})|^{2}\,dx+\gamma_{n}(1-e^{-2\varepsilon(t-t_{0})})$
$\displaystyle\leqslant
e^{-2\varepsilon(t-t_{0})}\int_{\mathbb{R}^{N}}\phi_{n}|u(t_{0})|^{2}\,dx+\gamma_{n},$

which completes the proof. ∎

As an immediate consequence we obtain the following admissibility result.

######  Corollary 4.5.

Let the assumptions of Proposition 4.4 hold. If
$\rho(V_{\infty}-a_{\infty})>\lambda$, then an arbitrary bounded set $M\subset
H^{1}(\mathbb{R}^{N})$ is admissible with respect to the family of semiflows
$\\{\Psi^{(s)}\\}_{s\in[0,1]}$.

###### Proof.

Let us assume that $(s_{m})$ in $[0,1]$, $(t_{m})$ in $[0,\infty)$ and
$(\bar{u}_{m})$ in $H^{1}(\mathbb{R}^{N})$ are sequences such that $s_{m}\to
s_{0}$ and $t_{m}\to\infty$ and $\\{\Psi^{(s_{m})}_{t}(\bar{u}_{m})\ |\
t\in[0,t_{m}]\\}\subset M$ for $m\geq 1$. We show that the set
$\\{\Psi^{(s_{m})}_{t_{m}}(\bar{u}_{m})\\}$ is relatively compact in
$H^{1}(\mathbb{R}^{N})$. Let us assume that $R>0$ is such that $M\subset
B_{H^{1}}(0,R)$ and let $u_{m}:[0,\infty)\to H^{1}(\mathbb{R}^{N})$ be a
solution of the semiflow $\Psi^{(s_{m})}$ starting at $\bar{u}_{m}$. In view
of Proposition 4.3 (ii) it is enough to check that the set $\\{u_{m}(t_{m})\
|\ n\geq 1\\}$ is relatively compact in $L^{2}(\mathbb{R}^{N})$. To this end
we use Proposition 4.4 to obtain $\varepsilon>0$, a sequence $(\gamma_{n})$
with $\gamma_{n}\to 0^{+}$ as $n\to\infty$ and $n_{0}\geqslant 1$, such that

$\int_{\mathbb{R}^{N}}\phi_{n}|u_{m}(t_{m})|^{2}\,dx\leqslant e^{-2\varepsilon
t_{m}}\|\bar{u}_{m}\|_{L^{2}}^{2}+\gamma_{n}\leqslant e^{-2\varepsilon
t_{m}}R^{2}+\gamma_{n}$

for all $n\geqslant n_{0}$ and $m\geq 1$. This together with the Rellich-
Kondrachov theorem (applied for the balls of radius $n$ in $\mathbb{R}^{N}$)
shows that the set $\\{u_{m}(t_{m})\\}$ is relatively compact in
$L^{2}(\mathbb{R}^{N})$ as desired. ∎

##  5\. The non-resonant case

In this section we shall prove Theorem 1.1. The following extended versions of
the Conley index formulae [17, Th. 3.3 and 3.6] will be crucial.

######  Theorem 5.1.

Suppose $f:\mathbb{R}^{N}\times\mathbb{R}\to\mathbb{R}$ satisfies the
conditions $(f1)$, $(f2)$, $(f3)$ and
$\varrho(V_{\infty}-a_{\infty})>\lambda$.
$(i)$ If there there exists a potential $\alpha$ satisfying $(PT)$ such that
$\lim_{u\to\,0}\frac{f(x,u)}{u}=\alpha(x)\ \mbox{ for a.e. }\
x\in\mathbb{R}^{N},$ $\varrho(V_{\infty}-\alpha_{\infty})>\lambda$ and
$\lambda\not\in\sigma(-\Delta+V-\alpha)$, then $K_{0}:=\\{0\\}$ is an isolated
invariant set with respect to the semiflow $\Phi$ and
$h(\Phi,K_{0})=\Sigma^{d^{-}(V-\alpha,\,\lambda)}.$
$(ii)$ If there exists a potential $\omega$ satisfying $(PT)$ such that
$\lim_{|u|\to\infty}\frac{f(x,u)}{u}=\omega(x)\ \mbox{ for a.e. }\
x\in\mathbb{R}^{N},$ $\varrho(V_{\infty}-\omega_{\infty})>\lambda$ and
$\lambda\not\in\sigma(-\Delta+V-\omega)$, then the set $K_{\infty}(\Phi)$,
consisting of all $\bar{u}\in H^{1}(\mathbb{R}^{N})$ such that $u(0)=\bar{u}$
for some bounded solution $u:\mathbb{R}\to H^{1}(\mathbb{R}^{N})$ of $\Phi$,
is a compact isolated invariant set with respect to $\Phi$ and
$h(\Phi,K_{\infty}(\Phi))=\Sigma^{d^{-}(V-\omega,\,\lambda)}.$

###### Proof.

(i) Consider the map $G:H^{1}(\mathbb{R}^{N})\times[0,1]\to
L^{2}(\mathbb{R}^{N})$ given, for any $u\in H^{1}(\mathbb{R}^{N})$ and
$s\in[0,1]$, by

$[G(u,s)](x):=(1-s)F(u)(x)+s\alpha(x)u(x)\quad\text{for a.e.
}x\in\mathbb{R}^{N},$

where $F$ is given by (4.2). It can be easily seen that $G$ satisfies
conditions $(G1)$ and $(G2)$ from Section 4. Let
$\\{\Psi^{(s)}\\}_{s\in[0,1]}$, be the family of semiflows generated by the
equation (4.6). We claim that there exists $r_{0}>0$ such that the set
$N_{r}:=\\{u\in H^{1}(\mathbb{R}^{N})\ \mid\ \|u\|_{H^{1}}\leqslant r\\}$ has
the property

(5.1) $\mathrm{Inv}_{\Psi^{(s)}}\,N_{r}=\\{0\\}\ \text{ for all
}r\in(0,r_{0}]\ \text{ and }s\in[0,1].$

Indeed, if we suppose to the contrary, then we get a sequence of $(s_{m})$ in
$[0,1]$ such that, for any $m\geqslant 1$, there exists a bounded nontrivial
full solution $u_{m}$ of the semiflow $\Psi^{(s_{m})}$ such that

$\|u_{m}(0)\|_{H^{1}}\geqslant(1-1/2m)\cdot\sup_{t\in\mathbb{R}}\|u_{m}(t)\|_{H^{1}}>0\quad\text{and}\quad\rho_{m}:=\|u_{m}(0)\|_{H^{1}}\to
0^{+}\quad\text{as}\ \ m\to\infty.$

Note that, for each $m\geqslant 1$, $v_{m}:=\rho_{m}^{-1}u_{m}$ solves the
equation

$\dot{v}(t)=-Av(t)+F_{m}(v(t)),\quad t\in\mathbb{R},$

with $F_{m}:H^{1}(\mathbb{R}^{N})\to L^{2}(\mathbb{R}^{N})$ given, for any
$u\in H^{1}(\mathbb{R}^{N})$, by

$[F_{m}(u)](x):=f_{m}(x,u(x))\quad\text{for a.e. }x\in\mathbb{R}^{N}$

where $f_{m}:\mathbb{R}^{N}\times\mathbb{R}\to\mathbb{R}$ is defined by

$f_{m}(x,u):=(1-s_{m})\rho_{m}^{-1}f(x,\rho_{m}u)+s_{m}\alpha(x)u\ \mbox{ for
a.e. }\ x\in\mathbb{R}^{N}\mbox{ and }u\in\mathbb{R}.$

Clearly

(5.2) $\|v_{m}(0)\|_{H^{1}}=1\ \ \text{ and }\ \
\sup_{t\in\mathbb{R}}\|v_{m}(t)\|_{H^{1}}\leqslant(1-1/2m)^{-1}\leqslant 2\
\text{ for all }\ m\geqslant 1.$

On the other hand, by the assumption $(f3)$, for all $m\geqslant 1$, we have

$\displaystyle uf_{m}(x,u)$
$\displaystyle=(1-s_{m})u\rho_{m}^{-1}f(x,\rho_{m}u)+s_{m}\alpha(x)|u|^{2}$
$\displaystyle\leqslant(1-s_{m})a(x)|u|^{2}+s_{m}\alpha(x)|u|^{2}\leqslant\widetilde{a}(x)|u|^{2},$

where $\widetilde{a}=\widetilde{a}_{\infty}+\widetilde{a}_{0}$ with
$\widetilde{a}_{\infty}:=\max\\{a_{\infty},\alpha_{\infty}\\}$ and
$\widetilde{a}_{0}:=a_{0}+\alpha_{0}$. Here $a_{0}\in L^{p}(\mathbb{R}^{N})$
and $\alpha_{0}\in L^{r}(\mathbb{R}^{N})$, where the exponents $p=p(a_{0})>2$
and $r=r(\alpha_{0})\geq 2$ are restricted by the conditions $(KR)$ and
$(PT)$, respectively. Since $\varrho(V_{\infty}-a_{\infty})>\lambda$ and
$\varrho(V_{\infty}-\alpha_{\infty})>\lambda$ we infer that

$\varrho\left(V_{\infty}-\widetilde{a}_{\infty}\right)>\lambda.$

Hence, in view of Proposition 4.4 with $G(u,s)=F_{m}(u)$ for $u\in
H^{1}(\mathbb{R}^{N})$ and $s\in[0,1]$, we see that there exist
$\varepsilon>0$, sequence $(\gamma_{n})$ with $\gamma_{n}\to 0^{+}$ and
$n_{0}\geqslant 1$ (independent of $m$) such that, for all $m\geqslant 1$ and
$n\geqslant n_{0}$, we have

$\int_{\mathbb{R}^{N}\setminus B(0,n)}|v_{m}(t)|^{2}\,dx\leqslant
e^{-2\varepsilon(t-r)}\|v_{m}(r)\|^{2}+\gamma_{n}\leqslant
4e^{-2\varepsilon(t-r)}+\gamma_{n},\quad r<t,$

where in the last inequality we used the estimate (5.2). Fixing
$t\in\mathbb{R}$ and passing with $r\to-\infty$, we get

$\int_{\mathbb{R}^{N}\setminus
B(0,n)}|v_{m}(t)|^{2}\,dx\leqslant\gamma_{n},\quad n\geq n_{0},\ m\geq 1.$

Since the sequence $(v_{m}(t))$ is bounded in $H^{1}(\mathbb{R}^{N})$, by the
Rellich-Kondrachov compact embedding theorem, we infer that the set of
restricted functions $\\{v_{m}(t)|_{B(0,n)}\ |\ m\geq 1\\}$ is relatively
compact in $L^{2}(B(0,n))$ for any $n\geq 1$. Combining this with the fact
that $\gamma_{n}\to 0^{+}$ we see that, for any $t\in\mathbb{R}$, the set
$\\{v_{m}(t)\\}_{m\geqslant 1}$ is relatively compact in
$L^{2}(\mathbb{R}^{N})$. Take any $\tau>0$ and let
$\big{(}v_{m_{k}}(t-\tau)\big{)}$ be a subsequence of $(v_{m}(t-\tau))$ such
that $v_{m_{k}}(t-\tau)\to\bar{v}$ in $L^{2}(\mathbb{R}^{N})$. Using the weak
compactness argument we can suppose that $\bar{v}\in H^{1}(\mathbb{R}^{N})$.
Since, for all $u\in\mathbb{R}$ and a.e. $x\in\mathbb{R}^{N}$,

$f_{m}(x,u)\to\alpha(x)u,\quad m\to\infty,$

we can use [3, Proposition 4.3] taking into consideration (5.2), to see that
$v_{m_{k}}$ converges in $H^{1}(\mathbb{R}^{N})$, uniformly on compact subsets
of $(t-\tau,\infty)$, to the solution $v:[t-\tau,\infty)\to
H^{1}(\mathbb{R}^{N})$ of

(5.3) $\dot{v}(t)=-Av(t)+m_{\alpha}v(t),\quad t>0,$

where $m_{\alpha}:H^{1}(\mathbb{R}^{N})\to L^{2}(\mathbb{R}^{N})$ is given by
$m_{\alpha}(u)(x):=\alpha(x)u(x)$ for a.e. $x\in\mathbb{R}^{N}$ and all $u\in
H^{1}(\mathbb{R}^{N})$. Since $\tau>0$ is arbitrary, we can combine this with
a diagonal type argument to obtain the existence of a bounded solution
$v:\mathbb{R}\to H^{1}(\mathbb{R}^{N})$ of the equation (5.3) such that
$\|v(0)\|_{H^{1}}=1$. If we consider the functional
$\mathcal{E}_{\alpha}:H^{1}(\mathbb{R}^{N})\to\mathbb{R}$, given by

$\mathcal{E}_{\alpha}(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}\big{(}|\nabla
u|^{2}+(V(x)-\alpha(x)-\lambda)|u|^{2}\big{)}\,\,dx,\quad u\in
H^{1}(\mathbb{R}^{N}),$

then, by Proposition 4.2, we have

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{E}_{\alpha}(v(t))=-\int_{\mathbb{R}^{N}}|\dot{v}(t)|^{2}\,dx\
\text{ for all }\ t\in\mathbb{R},$

i.e. the semiflow generated by (5.3) is gradient-like with respect to the
Lyapunov functional $\mathcal{E}_{\alpha}$. Hence, since $v$ is nontrivial and
bounded, there exists a nontrivial equilibrium point in the set
$\alpha(u)\cup\omega(u)$. Then we have
$\mathrm{Ker}\,(-A+m_{\alpha})=\mathrm{Ker}\,(-\Delta+V-\alpha-\lambda)\neq\\{0\\}$,
which is a contradiction proving the existence of $r_{0}>0$ such that (5.1)
holds.
Consequently from (5.1) it follows that
$\mathrm{Inv}_{\Psi^{(s)}}\,N_{r}=\\{0\\}\subset\mathrm{int}\,N_{r}$ for
$s\in[0,1]$ and $r\in(0,r_{0}]$ and furthermore, in view of Proposition 4.3
and Corollary 4.7, the set $N_{r}$ is admissible with respect to the family of
semiflows $\\{\Psi^{(s)}\\}_{s\in[0,1]}$. Hence, we can apply the homotopy
invariance property (H4) to get

(5.4) $h(\Phi,K_{0})=h(\Psi^{(0)},K_{0})=h(\Psi^{(1)},K_{0}).$

Since $\Psi^{(1)}$ is the semiflow generated by (5.3) and
$\mathrm{Ker}\,(-A+m_{\alpha})=\\{0\\}$, we infer from Theorem 3.1, that

$h(\Psi^{(1)},K_{0})=\Sigma^{d^{-}(V-\alpha,\,\lambda)},$

which together with (5.4) ends the proof of the assertion (i).
(ii) In this case we consider the family of semiflows
$\\{\Psi^{(s)}\\}_{s\in[0,1]}$ determined by the equation (4.6), where the map
$G$ is given, for any $u\in H^{1}(\mathbb{R}^{N})$ and $s\in[0,1]$, by the
formula

$[G(u,s)](x):=(1-s)F(u)(x)+s\omega(x)u(x)\quad\text{for a.e.
}x\in\mathbb{R}^{N}.$

We claim that there exists $R>0$ such that if $u$ is a bounded in
$H^{1}(\mathbb{R}^{N})$ solution of the semiflow $\Psi^{(s)}$ for some
$s\in[0,1]$, then $\|u(t)\|_{H^{1}}<R$ for all $t\in\mathbb{R}$. If we suppose
to the contrary, then there is a sequence of $(s_{m})$ in $[0,1]$ such that,
for any $m\geqslant 1$, there exists a bounded solution $u_{m}$ of the
semiflow $\Psi^{(s_{m})}$ such that
$\rho_{m}:=\sup_{t\in\mathbb{R}}\|u_{m}(t)\|_{H^{1}}\to\infty$ and
$\|u_{m}(0)\|_{H^{1}}\geqslant\rho_{m}-1$. Taking $v_{m}:=\rho_{m}^{-1}u_{m}$
for $m\geqslant 1$, we can show in a similar manner as in the proof of (i),
that $\mathrm{Ker}\,(-\Delta+V-\omega-\lambda)\neq\\{0\\}$, which is a
contradiction.
Therefore, if we take $N_{R}:=\\{u\in
H^{1}(\mathbb{R}^{N})\,\mid\,\|u\|_{H^{1}}\leqslant R\\}$, then we have

$K_{\infty}(\Phi)\subset N_{R}\quad\text{and}\quad
K_{s}:=\mathrm{Inv}_{\Psi^{(s)}}\,N_{R}\subset\mathrm{int}\,N_{R}\ \text{ for
all }\ s\in[0,1].$

Similarly as in the point $(i)$, Proposition 4.3 and Corollary 4.7 show that
the set $N_{R}$ is admissible with respect to the family of semiflows
$\\{\Psi^{(s)}\\}_{s\in[0,1]}$. Hence, we can apply the homotopy property (H4)
to get

(5.5) $h(\Phi,K_{\infty}(\Phi))=h(\Psi^{(0)},K_{0})=h(\Psi^{(1)},K_{1}).$

Since $\Psi^{(1)}$ is the semiflow generated by the equation (5.3) with
$\alpha$ replaced by $\omega$ and $\mathrm{Ker}\,(-A+m_{\omega})=\\{0\\}$, we
can again apply Theorem 3.1 to obtain

$h(\Psi^{(1)},K_{1})=\Sigma^{d^{-}(V-\omega,\,\lambda)}.$

Combining the above equation with (5.5) completes the proof of the assertion
(ii). ∎

###### Proof of Theorem 1.1.

From Theorem 5.1 (ii) it follows that $K_{\infty}(\Phi)$ is an isolated
invariant set with respect to $\Phi$ and

$h(\Phi,K_{\infty}(\Phi))=\Sigma^{d^{-}(V-\omega,\,\lambda)}.$

On the other hand, by Theorem (5.1) (i), $K_{0}=\\{0\\}$ is also an isolated
$\Phi$-invariant set and

$h(\Phi,K_{0})=\Sigma^{d^{-}(V-\alpha,\,\lambda)}.$

Since, by assumption, $d^{-}(V-\alpha,\lambda)\neq d^{-}(V-\omega,\lambda)$,
we infer that

$\overline{0}\neq\Sigma^{d^{-}(V-\alpha,\,\lambda)}=h(\Phi,K_{0})\neq
h(\Phi,K_{\infty}(\Phi))=\Sigma^{d^{-}(V-\omega,\,\lambda)}\neq\overline{0}.$

This together with Theorem 2.1, implies the existence of a nontrivial full
bounded solution $u:\mathbb{R}\to H^{1}(\mathbb{R}^{N})$ for the semiflow
$\Phi$ such that either $\alpha(u)\subset K_{0}$ or $\omega(u)\subset K_{0}$.
Recall that, in view of Proposition 4.2, $\Phi$ is gradient-like, which
implies that both $\alpha(u)$ and $\omega(u)$ consists of equilibria. Since
$u$ is nontrivial and bounded and hence either $\alpha(u)$ or $\omega(u)$
contains a nonzero equilibrium point. Then the zero solution is in the
remaining limit set of $u$ and the proof is completed. ∎

##  6\. The resonant case

Throughout the whole section we assume that at least the assumptions of
Theorem 1.2 are satisfied, i.e. $f$ satisfies $(f1)^{\prime}$ and $(f2)$, $V$
is a Kato-Rellich type potential, $\varrho(V_{\infty})>\lambda$ and
$\lambda\in\sigma(-\Delta+V)$. Let us define
$G:H^{1}(\mathbb{R}^{N})\times[0,1]\to L^{2}(\mathbb{R}^{N})$ by

(6.1) $G(u,s):=PF(Pu+sQu)+sQF(Pu+sQu)\quad\text{for}\ \ u\in
H^{1}(\mathbb{R}^{N}),\ s\in[0,1],$

where $P$ and $Q=Q_{+}+Q_{-}$ are as in Section 3.

######  Lemma 6.1.

The mapping $G$ satisfies conditions $(G1)$, $(G2)$ and there exists $b\in
L^{2}(\mathbb{R}^{N})$ such that, for all $u\in H^{1}(\mathbb{R}^{N})$,

(6.2) $|G(u,s)(x)|\leqslant b(x)\ \text{ for a.e. }x\in\mathbb{R}^{N}.$

###### Proof.

Indeed, for $(G1)$ note that, by (4.3), there is $L>0$ such that, for all
$u,v\in H^{1}(\mathbb{R}^{N})$ and $s\in[0,1]$,

$\displaystyle\|G(u,s)-G(v,s)\|_{L^{2}}$ $\displaystyle\leqslant
2\|F(Pu+sQu)-F(Pv+sQv)\|_{L^{2}}$ $\displaystyle\leqslant
2L\|P(u-v)+sQ(u-v)\|_{L^{2}}\leqslant 4L\|u-v\|_{L^{2}}.$

Furthermore, by $(f1)^{\prime}$, we have

(6.3) $\displaystyle\|G(u,s)\|_{L^{2}}$ $\displaystyle\leqslant
2\|F(Pu+sQu)\|_{L^{2}}\leqslant 2\|m\|_{L^{2}},\quad u\in
H^{1}(\mathbb{R}^{N}),\ s\in[0,1],$

which implies $(G2)$.
In order to verify the property (6.2) observe that, given $u\in
H^{1}(\mathbb{R}^{N})$ and $s\in[0,1]$, we have

(6.4) $G(u,s)=sF(Pu+sQu)+(1-s)PF(Pu+sQu)\quad\text{ almost everywhere on }\
\mathbb{R}^{N}.$

If we take an orthogonal basis of $X_{0}$ consisting of elements $\varphi_{k}$
for $k=1,\dots,d$, where $d=\dim X_{0}$, then

(6.5) $\displaystyle PF(Pu+sQu)=\sum_{k=1}^{d}e_{k}(u,s)\varphi_{k},\quad u\in
H^{1}(\mathbb{R}^{N}),\ s\in[0,1],$

where

$\displaystyle e_{k}(u,s):=\int_{\mathbb{R}^{N}}F(Pu+sQu)\varphi_{k}\,dx,\quad
1\leqslant k\leqslant d.$

Clearly, by $(f1)^{\prime}$, for all $u\in H^{1}(\mathbb{R}^{N})$ and
$s\in[0,1]$,

$\displaystyle|e_{k}(u,s)|\leqslant\|F(Pu+sQu)\|_{L^{2}}\|\varphi_{k}\|_{L^{2}}\leqslant\|m\|_{L^{2}}\|\varphi_{k}\|_{L^{2}}.$

and consequently, for a.e. $x\in\mathbb{R}^{N}$,

(6.6)
$\displaystyle|PF(Pu+sQu)(x)|\leqslant\sum_{k=1}^{d}|e_{k}(u,s)||\varphi_{k}(x)|\leqslant\|m\|_{L^{2}}\sum_{k=1}^{d}\|\varphi_{k}\|_{L^{2}}|\varphi_{k}(x)|.$

Combining (6.4) and $(f1)^{\prime}$ gives, for all $u\in
H^{1}(\mathbb{R}^{N})$ and a.e. $x\in\mathbb{R}^{N}$,

$\displaystyle|G(u,s)(x)|\leqslant|m(x)|+|PF(Pu+sQu)(x)|.$

and hence, by the inequality (6.6), the condition (6.2) holds with $b\in
L^{2}(\mathbb{R}^{N})$ given by
$b(x):=|m(x)|+\|m\|_{L^{2}}\sum_{k=1}^{d}\|\varphi_{k}\|_{L^{2}}|\varphi_{k}(x)|$
for a.e. $x\in\mathbb{R}^{N}$. ∎

Let us consider the family of semiflows $\\{\Psi^{(s)}\\}_{s\in[0,1]}$,
generated by the equation (4.6) with $G$ given by the formula (6.1). Since
$G(u,1)=F(u)$ for $u\in H^{1}(\mathbb{R}^{N})$, we see that $\Psi^{(1)}=\Phi$
where $\Phi$ is the semiflow defined by $(P)_{\lambda}$. Since $G$ satisfies
(6.2), the conditions (4.7) hold and we can apply Corollary 4.5 to see that
any bounded subset of $H^{1}(\mathbb{R}^{N})$ is admissible with respect to
the family of semiflows $\Psi=\\{\Psi^{(s)}\\}_{s\in[0,1]}$.
Below we obtain further compactness and geometric properties of the family
$\Psi$.

######  Proposition 6.2.

Let $K_{\infty}(\Psi)$ be the set of all $\bar{u}\in H^{1}(\mathbb{R}^{N})$
such that there exist $s\in[0,1]$ and a bounded solution $u:\mathbb{R}\to
H^{1}(\mathbb{R}^{N})$ of $\Psi^{(s)}$ with $u(0)=\bar{u}$. Then the set
$QK_{\infty}(\Psi)$ (the projection of $K_{\infty}(\Psi)$ onto $X_{-}\oplus
X_{+}$) is bounded in $H^{1}(\mathbb{R}^{N})$ and relatively compact in
$L^{2}(\mathbb{R}^{N})$.

###### Proof.

Let $u:\mathbb{R}\to H^{1}(\mathbb{R}^{N})$ be a bounded solution of
$\Psi^{(s)}$ for some $s\in[0,1]$. Then $u$ is also a mild solution of the
equation (4.6) and we have the Duhamel formula

(6.7)
$u(t)=S(t-t^{\prime})u(t^{\prime})+\int_{t^{\prime}}^{t}S(t-\tau)G(u(\tau),s)\,d\tau,\quad
t^{\prime}<t.$

By applying the projection operator $Q_{+}$ we get

$Q_{+}u(t)=S(t-t^{\prime})Q_{+}u(t^{\prime})+\int_{t^{\prime}}^{t}S(t-\tau)Q_{+}G(u(\tau),s)\,\mathrm{d}\tau.$

In view of (3.3) and the boundedness of $u$, by passing to the limit with
$t^{\prime}\to-\infty$, we get

$Q_{+}u(t)=\int_{-\infty}^{t}S(t-\tau)Q_{+}G(u(\tau),s)\,\mathrm{d}\tau,\quad
t\in\mathbb{R},$

where the integral is convergent in $X=L^{2}(\mathbb{R}^{N})$. Therefore, by
(3.3) and (6.3), we have

(6.8) $\displaystyle\|Q_{+}u(t)\|_{H^{1}}$
$\displaystyle\leqslant\int_{-\infty}^{t}K(t-\tau)^{-1/2}e^{-\rho_{+}(t-\tau)}\|Q_{+}G(u(\tau),s)\|_{L^{2}}\,d\tau$
$\displaystyle\leqslant
2K\|m\|_{L^{2}}\int_{0}^{+\infty}z^{-1/2}e^{-\rho_{+}z}\,\mathrm{d}z$

for any $t\in\mathbb{R}$. Let us recall that the restriction of
$\\{S(t)\\}_{t\geqslant 0}$ to the space $X_{-}$ can be extended to a
$C_{0}$-group $\\{S(t)\\}_{t\in\mathbb{R}}$ on this space. Hence, applying
with the projection $Q_{-}$ on both sides of the equation (6.7) and using the
$C_{0}$-group property, we get

$Q_{-}u(t^{\prime})=S(t^{\prime}-t)Q_{-}u(t)-\int_{t^{\prime}}^{t}S(t^{\prime}-\tau)Q_{-}G(u(\tau),s)\,d\tau,\quad
t^{\prime}<t.$

Using (3.4) and passing to the limit with $t\to+\infty$ we obtain

(6.9)
$Q_{-}u(t^{\prime})=-\int_{t^{\prime}}^{+\infty}S(t^{\prime}-\tau)Q_{-}G(u(\tau),s)\,d\tau,\quad
t^{\prime}\in\mathbb{R},$

where the integral is convergent in $X=L^{2}(\mathbb{R}^{N})$. Applying (3.4)
and (6.3) to the equation (6.9) yields

$\|Q_{-}u(t^{\prime})\|_{L^{2}}\leqslant\int_{t^{\prime}}^{+\infty}e^{(t^{\prime}-\tau)\rho_{-}}\|Q_{+}G(u(\tau),s)\|_{L^{2}}\,d\tau\leqslant
2\|m\|_{L^{2}}\int_{t^{\prime}}^{+\infty}e^{(t^{\prime}-\tau)\rho_{-}}\,d\tau\leqslant
2\|m\|_{L^{2}}/\rho_{-}.$

This combined with (6.8) implies the existence of $R_{\infty}>0$ such that

(6.10) $\sup_{t\in\mathbb{R}}\|Qu(t)\|_{H^{1}}<R_{\infty},\quad
t\in\mathbb{R}$

for any full bounded solution $u$ of $\Psi^{(s)}$ with $s\in[0,1]$.
To show the relative compactness, we act on (4.6) with the operator $Q$ and we
write $w(t):=Qu(t)$ for $t\in\mathbb{R}$, to obtain the equation

$\displaystyle\dot{w}(t)=-Aw(t)+QG(u(t),s),\quad t>0.$

By Proposition 3.2, there exist a sequence $(\beta_{n})$ with $\beta_{n}\to
0^{+}$ as $n\to\infty$ and $n_{0}\geqslant 1$, such that if $u$ is a full
solution of the semiflow $\Psi^{(s)}$, where $s\in[0,1]$, then

(6.11)
$\displaystyle\frac{1}{2}\frac{d}{dt}\langle\phi_{n}w(t),w(t)\rangle_{L^{2}}\leqslant-\int_{\mathbb{R}^{N}}(V_{\infty}-\lambda)\phi_{n}|w(t)|^{2}\,dx+\beta_{n}\|w(t)\|_{H^{1}}^{2}+\langle
w(t),\phi_{n}QG(u(t),s)\rangle_{L^{2}}$

for $t\in\mathbb{R}$. From the assumption $\varrho(V_{\infty})>\lambda$, there
are $\varepsilon>0$ and $n_{0}\geq 1$ such that, for all $n\geqslant n_{0}$,

(6.12)
$\displaystyle\phi_{n}(x)(V_{\infty}(x)-\lambda)\geq\varepsilon\phi_{n}(x)\quad\text{for
a.a. }x\in\mathbb{R}^{N}.$

By (6.1) and $(f1)^{\prime}$, we have

(6.13)
$\displaystyle\|\phi_{n}QG(u(t),s)\|_{L^{2}}=\|\phi_{n}QF(Pu(t)+sQu(t))\|_{L^{2}}\leqslant\|\phi_{n}m\|_{L^{2}}+\|\phi_{n}PF(Pu(t)+sQu(t))\|_{L^{2}}.$

Since $m\in L^{2}(\mathbb{R}^{N})$, we have
$\delta^{1}_{n}:=\|\phi_{n}m\|_{L^{2}}\to 0$ as $n\to\infty$. On the other
hand, the image of the map $F$ is contained in the ball
$B_{L^{2}}(0,\|m\|_{L^{2}})$ and hence its projection onto the finite
dimensional space $X_{0}$ is relatively compact in $L^{2}(\mathbb{R}^{N})$.
Therefore, by the Kolmogorov-Riesz theorem (see [9]), there is a sequence
$(\delta^{2}_{n})$, depending on $\|m\|_{L^{2}}$, such that $\delta^{2}_{n}\to
0^{+}$ as $n\to\infty$ and
$\|\phi_{n}PF(Pu(t)+sQu(t))\|_{L^{2}}\leqslant\delta^{2}_{n}$ for
$t\in\mathbb{R}$ and $s\in[0,1]$. Consequently, we get
$\delta_{n}:=\delta^{1}_{n}+\delta^{2}_{n}$ that is independent of $s$, $u$
and $t$ such that

$\displaystyle\langle
w(t),\phi_{n}QG(u(t),s)\rangle_{L^{2}}\leqslant\delta_{n}\|w(t)\|_{H^{1}},\quad
t\in\mathbb{R},$

and $\delta_{n}\to 0^{+}$ as $n\to\infty$. Now we combine this with (6.11) and
(6.12) and use (6.10) to get

$\displaystyle\frac{1}{2}\frac{d}{dt}\langle\phi_{n}w(t),w(t)\rangle_{L^{2}}$
$\displaystyle\leqslant-\varepsilon\int_{\mathbb{R}^{N}}\phi_{n}|w(t)|^{2}\,dx+\beta_{n}R_{\infty}^{2}+\delta_{n}R_{\infty},\quad
t\in\mathbb{R}.$

By integrating the above inequality we obtain, for all $n\geqslant n_{0}$ and
$t,r\in\mathbb{R}$ with $t>r$,

$\displaystyle\int_{\mathbb{R}^{N}}\phi_{n}|w(t)|^{2}\,dx$
$\displaystyle\leqslant
e^{-2\varepsilon(t-r)}\int_{\mathbb{R}^{N}}\phi_{n}|w(r)|^{2}\,dx+\frac{1}{2\varepsilon}(1-e^{-2\varepsilon(t-r)})(\beta_{n}R_{\infty}^{2}+\delta_{n}R_{\infty})$
$\displaystyle\leqslant
e^{-2\varepsilon(t-r)}R^{2}+\frac{1}{2\varepsilon}(\beta_{n}R_{\infty}^{2}+\delta_{n}R_{\infty}).$

Passing with $r\to-\infty$ we see that, for all $t\in\mathbb{R}$ and
$n\geqslant n_{0}$,

$\int_{\mathbb{R}^{N}}\phi_{n}|w(t)|^{2}\,dx\leqslant\frac{1}{2\varepsilon}(\beta_{n}R_{\infty}^{2}+\delta_{n}R_{\infty}),$

which by the Rellich-Kondrachov theorem shows that the set $QK_{\infty}(\Psi)$
is relatively compact in $L^{2}(\mathbb{R}^{N})$. Thus the proof of the
proposition is completed. ∎

From now on, we assume that $f$ satisfies additionally one of the resonance
conditions $(LL)_{\pm}$ or $(SR)_{\pm}$. Both Landesman-Lazer and sign
conditions have geometric implications in the phase space of the parabolic
semiflow $\Phi$.

######  Proposition 6.3.

(see [4, Lemma 5.2]) Let the map $F$ be given by (4.2) and $M\subset
H^{1}(\mathbb{R}^{N})$. If either
(i) $(LL)_{\pm}$ holds and $M$ is bounded in $L^{2}(\mathbb{R}^{N})$; or
(ii) $(SR)_{\pm}$ holds and $M$ is relatively compact in
$L^{2}(\mathbb{R}^{N})$,
then there exists $R_{0}>0$ and $\alpha>0$ such that, for all $v\in X_{0}$
with $\|v\|_{L^{2}}\geqslant R_{0}$ and $w\in M$

$\pm\langle v,F(v+w)\rangle_{L^{2}}>\alpha.$

By Proposition 6.2, there exists $R_{\infty}>0$ such that for any bounded
solution $u$ of $\Psi^{(s)}$ with $s\in[0,1]$,

(6.14) $Qu(t)\in B_{H^{1}}(0,R_{\infty}),\quad\text{ for all }\
t\in\mathbb{R},$

and the set $QK_{\infty}(\Psi)$ is relatively compact in
$L^{2}(\mathbb{R}^{N})$. Therefore, by the use of Proposition 6.3, we can
choose $R_{0}>0$ such that

(6.15) $\displaystyle\pm\langle v,F(v+sw)\rangle_{L^{2}}>\alpha\quad\text{for
\ $s\in[0,1]$, $w\in QK_{\infty}(\Psi)$ \ and \ $v\in X_{0}$ \ with \
$\|v\|_{L^{2}}\geqslant R_{0}$}$

(the sign in (6.15) is the same as in the Landesman-Lazer or sign condition
that is satisfied by $f$). Now let us define

(6.16) $N:=\\{u\in H^{1}(\mathbb{R}^{N})\mid\|Qu\|_{H^{1}}\leqslant
R_{\infty}\ \text{ and }\ \|Pu\|_{L^{2}}\leqslant R_{0}\\}.$

The set $N$ appears to be an isolating neighborhood with respect to the family
$\Psi$ and plays a key rule in the Conley index formula derivation that is the
main result of the section.

######  Theorem 6.4.

Let $\Phi$ be the semiflow generated by (4.1) and $K_{\infty}(\Phi)$ be the
set of all $\bar{u}\in H^{1}(\mathbb{R}^{N})$, for which there exists a
bounded solution $u:\mathbb{R}\to H^{1}(\mathbb{R}^{N})$ of $\Phi$ such that
$u(0)=\bar{u}$. Then $K_{\infty}(\Phi)$ is an isolated invariant set such that

(6.17) $K_{\infty}(\Phi)=\mathrm{Inv}_{\Phi}(N)\subset\mathrm{int}\,N$

where $N$ is given by (6.16) and its Conley index with respect to $\Phi$ is
given by the formula

(6.18) $h(\Phi,K_{\infty}(\Phi))=\left\\{\begin{aligned} &\Sigma^{d}&&\text{
if $(LL)_{+}$ or $(SR)_{+}$ holds},\\\ &\Sigma^{d^{-}}&&\text{ if $(LL)_{-}$
or $(SR)_{-}$ holds},\end{aligned}\right.$

where $d^{-}:=d^{-}(V,\lambda)$ and
$d:=d^{-}+\dim\mathrm{Ker}\,(-\Delta+V-\lambda)$.

###### Proof.

Let $u:\mathbb{R}\to H^{1}(\mathbb{R}^{N})$ be an arbitrary bounded solution
of of $\Psi^{(s)}$ for some $s\in[0,1]$. We show that
$u(R)\subset\mathrm{int}\,N$. Indeed, suppose to the contrary, that, for some
$t_{0}\in\mathbb{R}$, either $\|Qu(t_{0})\|_{L^{2}}\geqslant R_{\infty}$ or
$\|Pu(t_{0})\|_{L^{2}}\geqslant R_{0}$. From (6.14) it follows that
$\|Pu(t_{0})\|_{L^{2}}=R_{0}$ and, by the geometric condition (6.15)

$\pm\left.\frac{d}{dt}\|Pu(t)\|_{L^{2}}^{2}\right|_{t=t_{0}}=\pm 2\langle
Pu(t_{0}),PG(u(t_{0}),s)\rangle_{L^{2}}=\pm 2\langle
Pu(t_{0}),F(Pu(t_{0})+sQu(t_{0}))\rangle_{L^{2}}>2\alpha.$

This in turn implies that either $\|Pu(t)\|_{L^{2}}^{2}\geqslant
R_{0}^{2}+2\alpha(t-t_{0})$ for $t>t_{0}$ (or $\|Pu(t)\|_{L^{2}}^{2}\geqslant
R_{0}^{2}+2\alpha(t_{0}-t)$ for $t<t_{0}$, respectively), which means that $u$
would not be bounded. This gives a contradiction and proves the desired
inclusion. Consequently (6.17) holds. Furthermore, by Corollary 4.5, $N$ is
admissible with respect to $\\{\Psi^{(s)}\\}_{s\in[0,1]}$ and, by the homotopy
invariance of Conley index – see (H4) in Section 2, we get

$h(\Phi,K_{\infty}(\Phi))=h(\Psi^{(1)},K_{1})=h(\Psi^{(0)},K_{0}).$

Now we observe that $\Psi^{(0)}$ is conjugate to the semiflow
$\Phi_{P}\times\Phi_{Q}$, where $\Phi_{P}:X_{0}\to X_{0}$ is generated by
$\dot{v}=PF(v)$, whereas the semiflow $\Phi_{Q}:Y\to Y$, where
$Y=H^{1}(\mathbb{R}^{N})\cap(X_{+}\oplus X_{-})$ is the linear space defined
in Theorem 3.1, is generated by the linear equation $\dot{w}=-A_{|Y}w$ for
$t>0$. In particular

(6.19)
$h(\Psi^{(0)},K_{0})=h(\Phi_{P}\times\Phi_{Q},\mathrm{Inv}_{\Phi_{P}}\,(N_{P})\times\mathrm{Inv}_{\Phi_{Q}}\,(N_{Q})),$

where $N_{P}:=N\cap X_{0}=\\{u\in X_{0}\mid\|u\|_{L^{2}}\leqslant R_{0}\\}$
and $N_{Q}:=N\cap Y=\\{u\in Y\mid\|u\|_{H^{1}}\leqslant R_{\infty}\\}$.

By use of Theorem 3.1, we have $\mathrm{Inv}_{\Phi_{Q}}\,(N_{Q})=\\{0\\}$ and

(6.20) $h(\Phi_{Q},\\{0\\})=\Sigma^{d_{-}}.$

Clearly, if $u_{P}:[\delta_{1},\delta_{2})\to X_{0}$, where $\delta_{1}>0$ and
$\delta_{2}\geq 0$, is a solution of the flow $\Phi_{P}$ and
$u_{P}(0)\in\partial N_{P}$, then $\|u_{P}(0)\|_{L^{2}}=R_{0}$ and, by the use
of (6.15), we have

$\pm\left.\frac{d}{dt}\|u_{P}(t)\|_{L^{2}}^{2}\right|_{t=0}=\pm 2\langle
u_{P}(0),PF(u_{P}(0))\rangle_{L^{2}}=\pm 2\langle
u_{P}(0),F(Pu_{P}(0))\rangle_{L^{2}}>2\alpha.$

This implies that the set $N_{P}$ is an isolating block for $\Phi_{P}$ with
the exist set

(6.21) $N_{P}^{-}=\left\\{\begin{aligned} &\\{u\in
X_{0}\mid\|u\|_{L^{2}}=R_{1}\\}&&\text{ if $(LL)_{+}$ or $(SR)_{+}$ holds},\\\
&\emptyset&&\text{ if $(LL)_{-}$ or $(SR)_{-}$ holds}.\end{aligned}\right.$

Therefore we have

(6.22) $h(\Phi_{P},\mathrm{Inv}_{\Phi_{P}}\,(N_{P}))=\left\\{\begin{aligned}
&\Sigma^{\dim X_{0}}&&\text{ if $(LL)_{+}$ or $(SR)_{+}$ holds},\\\
&\Sigma^{0}&&\text{ if $(LL)_{-}$ or $(SR)_{-}$ holds}.\end{aligned}\right.$

By (6.19) and (6.20) and the multiplication property of the Conley index – see
(H3) in Section 2, we have

$\displaystyle h(\Psi^{(0)},K_{0})$
$\displaystyle=h(\Phi_{P}\times\Phi_{Q},\mathrm{Inv}_{\Phi_{P}}\,(N_{P})\times\\{0\\})$
$\displaystyle=h(\Phi_{P},\mathrm{Inv}_{\Phi_{P}}\,(N_{P}))\wedge
h(\Phi_{Q},\\{0\\})$
$\displaystyle=h(\Phi_{P},\mathrm{Inv}_{\Phi_{P}}\,(N_{P}))\wedge\Sigma^{d^{-}},$

which together with (6.22) yields the required index formula (6.18). ∎

###### Proof of Theorem 1.2.

Note that in view of Theorem 6.4 we have
$h(\Phi,K_{\infty}(\Phi))=\Sigma^{k_{\infty}}$, where

(6.23) $k_{\infty}=\left\\{\begin{aligned}
&d^{-}(V,\lambda)+\dim\,X_{0}&&\text{ if $(LL)_{+}$ or $(SR)_{+}$ holds},\\\
&d^{-}(V,\lambda)&&\text{ if $(LL)_{-}$ or $(SR)_{-}$
holds}.\end{aligned}\right.$

This implies that the Conley index of $K_{\infty}(\Phi)$ is nontrivial and, in
view of the existence property of Conley index – see (H1), we infer that
$K_{\infty}(\Phi)\neq\emptyset$. Hence there exists a full bounded solution
for $\Phi$, which allows us to apply the argument with Lyapunov functional
(4.5) from the proof of Theorem 1.1, to get that the sum
$\alpha(u)\cup\omega(u)$ contains the desired stationary solution. ∎

###### Proof of Theorem 1.3.

By Theorem 5.1 (i) we see that $K_{0}=\\{0\\}$ is an isolated invariant set
with respect to $\Phi$ and $h(\Phi,K_{0})=\Sigma^{d^{-}(V-\alpha,\,\lambda)}$.
In view of Theorem 6.4, $K_{\infty}(\Phi)$ is also an isolated
$\Phi$-invariant set and $h(\Phi,K_{\infty}(\Phi))=\Sigma^{k_{\infty}}$, where
$k_{\infty}$ is given by (6.23). In both cases (i) and (ii) we have

$\overline{0}\neq\Sigma^{d^{-}(V-\alpha,\,\lambda)}=h(\Phi,K_{0})\neq
h(\Phi,K_{\infty}(\Phi))=\Sigma^{k_{\infty}}\neq\overline{0}.$

Therefore, in virtue of Theorem 2.1, there exists a full solution
$u:\mathbb{R}\to H^{1}(\mathbb{R}^{N})$ that is nontrivial and either
$\alpha(u)\subset K_{0}$ or $\omega(u)\subset K_{0}$. Using the Lyapunov
functional (4.5), we derive that either $\alpha(u)$ or $\omega(u)$ contains a
nonzero equilibrium point and the zero solution is the other side limit set of
$u$. Thus the proof of the theorem is completed. ∎

###### Proof of Corollary 1.4.

If (i) holds, then we have
$d^{-}(V-\alpha,\lambda)=d^{-}(V,\bar{\alpha}+\lambda)\leqslant
d^{-}(V,\lambda)<k_{\infty}$, where $k_{\infty}$ is given by (6.23). In case
of (ii) we get
$d^{-}(V-\alpha,\lambda)=d^{-}(V,\bar{\alpha}+\lambda)>d^{-}(V,\lambda)=k_{\infty}$,
since $d^{-}(V,\bar{\alpha}+\lambda)-d^{-}(V,\lambda)\geqslant\dim\,X_{0}$ due
to the fact that $\bar{\alpha}>0$. When (iii) holds, then obviously
$k_{\infty}=d^{-}(V,\lambda)+\dim\,X_{0}<d^{-}(V,\bar{\alpha}+\lambda)=d^{-}(V-\alpha,\lambda)$,
since, by assumption, there is at least one eigenvalue in the interval
$(\lambda,\bar{\alpha}+\lambda)$. Finally, if (iv) holds, then in a similar
way, we obtain
$k_{\infty}=d^{-}(V,\lambda)>d^{-}(V,\bar{\alpha}+\lambda)=d^{-}(V-\alpha,\lambda)$.
In all four cases we use Theorem 1.3 to derive the assertion. ∎

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