\documentclass[twoside]{article}
\usepackage{amssymb} % font used for R in Real numbers
\pagestyle{myheadings}
\markboth{\hfil Spike solutions concentrating at local minima \hfil EJDE--2000/32}
{EJDE--2000/32\hfil Gregory S. Spradlin \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~32, pp.~1--14. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
 An elliptic equation with spike solutions concentrating at local 
minima of the \\ Laplacian of the potential 
\thanks{ {\em Mathematics Subject Classifications:} 35J50.
\hfil\break\indent
{\em Key words and phrases:} Nonlinear Schr\"odinger Equation, 
variational methods, \hfil\break\indent
 singularly perturbed elliptic equation,
   mountain-pass theorem, concentration compactness, \hfil\break\indent
 degenerate critical points.
\hfil\break\indent
\copyright 2000 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted February 4, 2000. Published May 2, 2000.} }
\date{}
%
\author{ Gregory S. Spradlin }
\maketitle

\begin{abstract} 
We consider the equation $-\epsilon^2 \Delta u + V(z)u = f(u)$
which arises in the study of nonlinear Schr\"odinger equations.
We seek solutions that are positive on ${\mathbb R}^N$ and that vanish at
infinity.  Under the assumption that $f$ satisfies super-linear and 
sub-critical growth conditions, we show that for 
 small $\epsilon$ there exist solutions that concentrate 
near local minima of $V$.  The local minima may occur in 
unbounded components, as long as the Laplacian of $V$ achieves a strict 
local minimum along such a component.  Our proofs employ 
variational mountain-pass and concentration compactness arguments.  
A penalization technique developed by Felmer and del~Pino is used 
to handle the lack of compactness and the absence of the 
Palais-Smale condition in the variational framework.  
\end{abstract}


\renewcommand{\theequation}{\thesection.\arabic{equation}} 
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

This paper concerns the equation 
\begin{equation} \label{1.0}
-\epsilon^2 \Delta u + V(z)u = f(u)   
\end{equation}
on ${\mathbb R}^N$  with $N \geq 1$, where $f(u)$ is a ``superlinear'' 
type function such as $f(u) = u^p$, $p > 1$. 
Such an equation arises when searching for standing wave solutions of the 
nonlinear  Schr\"odinger equation (see \cite{f1}).
For small positive $\epsilon$, we seek ``ground states,'' 
that is, positive solutions 
$u$ with $u(z) \to 0$ as $|z| \to \infty$. 
Floer and Weinstein (\cite{f4}) examined the case $N=1$, $f(u) = u^3$ 
and found that for small $\epsilon$, a ground state $u_\epsilon$ exists which 
concentrates near a non-degenerate 
critical point of $V$.  
Similar results for $N > 1$ were obtained by Oh in \cite{o1}-\cite{o3}. 
In \cite{f1}, del~Pino and Felmer found that 
if $V$ has a strict local minimum, then
for small $\epsilon$, (\ref{1.0}) has a ground 
state concentrating near that minimum.  
A strict local minimum occurs when there 
exists a bounded, open set $\Lambda \subset {\mathbb R}^N$ with
$\inf_\Lambda V < \inf_{\partial \Lambda} V$.
They 
extended their results in \cite{f2} to the more 
general case where $V$ has a ``topologically 
stable'' critical point, that is, a critical point 
obtained via a topological 
linking argument (see \cite{f2} for a precise 
formulation).  Such a critical point 
persists under small perturbations of $V$. Examples are a strict 
local extremum and a saddle point.
This very strong result 
is notable because the critical points 
of $V$ in question need not be 
non-degenerate or even isolated.  Similar results have been obtained by
 Li \cite{l1}, and 
earlier work of Rabinowitz \cite{r1} is also interesting. 
The recent results of \cite{a1} and \cite{l2} also permit $V$ to have 
degenerate critical points. 

A common feature of all the papers above is that $V$ must have a 
non-degenerate, or at least topologically stable, 
set of critical points.  Therefore 
it is natural to try to remove this requirement.  
That we must assume some conditions on $V$ is 
shown by Wang's counterexample \cite{w1} - if $V$ is nondecreasing 
and nonconstant in one variable
(e.g. $V(x_1, x_2, x_3) = 2 + \tan^{-1}(x_1)$), then 
no ground states exist.  In \cite{s1} the 
author showed that ground states to (\ref{1.0}) exist under 
the assumption that $V$ is almost
periodic, together with another mild assumption.  
Those assumptions did not guarantee
that $V$ had a topologically stable critical point.  


    Aside from periodicity or 
recurrence properties of $V$, another approach is to impose conditions on the 
derivatives of $V$.  That is the approach taken here.  
We will assume that $V$ has 
a (perhaps unbounded) component of local minima, along which $\Delta V$ achieves
a strict local minimum.  More specifically, assume $f$ satisfies the following:

\begin{description}
\item{(F1)} $f \in C^1({\mathbb R}^+, {\mathbb R})$ 
\item{(F2)} $f'(0) = 0 = f(0)$.
\item{(F3)} $\lim_{q \to \infty} f(q)/q^s = 0$ for some 
$s > 1$, with $s < (N+2)/(N-2)$ if $N \geq 3$.
\item{(F4)} For some $\theta > 2$, 
$0 < \theta F(q) \leq f(q)q$
for all $q > 0$, where $F(\xi) \equiv \int_0^\xi f(t)\,dt$.
\item{(F5)} The function $q \mapsto f(q)/q$ is increasing
on $(0, \infty)$.
\end{description}

Assumptions  (F1)-(F5) are the same 
as in \cite{f1} and are satisfied by $f(q) = q^{s}$, 
for example, if $1 < s < (N+2)/(N-2)$.  
Assume that $V$ satisfies the following:

\begin{description}
\item{(V1)} $V \in C^2({\mathbb R}^n,{\mathbb R})$
\item{(V2)} $D^\alpha V$ is bounded 
    and Lipschitz continuous for $|\alpha| = 2$.
\item{(V3)} $0 < V_- \equiv \inf_{{\mathbb R}^N} V <
                \sup_{{\mathbb R}^N} V \equiv V^+ < \infty$
\item{(V4)} There exists a bounded, nonempty open set 
        $\Lambda \subset {\mathbb R}^N$ and 
    a point $z_0 \in \Lambda$ with 
    $V(z_0) = \inf_\Lambda V \equiv V_0$,
    and 
$$  \Delta_0 \equiv 
        \inf\{\Delta V(z) \mid z \in \Lambda,
                        \ V(z) = V_0\} <
    \inf\{\Delta V(z) \mid z \in \partial \Lambda,
                        \ V(z) = V_0\} $$
\end{description}

\paragraph{Note:} A special case of (V4) occurs when 
$\Lambda$ is bounded and 
$V(z_0) < \inf_{\partial \Lambda} V$; this case 
is treated, under weaker hypotheses, 
in \cite{f1}.  A specific example of (V4) is if $N = 2$ and $V$ 
satisfies (V1)-(V4), with 
$V(z_1, z_2) = 1 + (z_1^2 - z_2)^2$ 
for $z_1^2 + z_2^2 \leq 1$.  Then 
$\Delta V(z_1, z_1^2) = 8z_1^2 + 2$ 
for $z_1^2 + z_2^2 \leq 1$, so we may 
take $\Lambda = B_1(0,0) \subset {\mathbb R}^2$ and $z_0 = (0,0)$.  
Then $V$ has a component
of local minima that includes the 
parabolic arc 
$\{z_2 = z_1^2\} \cap B_1(0,0)$, along which 
$\Delta V$ has a minimum of 
$2$ at $(0,0)$, with $\Delta V > 2$ at the two endpoints of the arc.

    We prove the following:


\begin{theorem} \label{thm1.1} Let $V$ and $f$ satisfy
{\rm (V1)-(V4)} and {\rm (F1)-(F5)}.
Then there exists $\epsilon_0 > 0$ such that if 
$\epsilon \leq \epsilon_0$, then 
(1.0) has a positive solution $u_\epsilon$ with $u_\epsilon(z) \to 0$ as 
$|z| \to \infty$.    
$u_\epsilon$ has exactly one local maximum (hence, global maximum) 
point $z_\epsilon \in \Lambda$, where $\Lambda$ is as in {\rm (V4)}.
There exist 
$\alpha, \beta > 0$ with 
$u_\epsilon (z) \leq 
\alpha \exp(- {\beta \over \epsilon} |z-z_\epsilon|)$
for $\epsilon \leq \epsilon_0$.  
Furthermore, with $V_0$ and $\Delta_0$ as in 
(V4), $V(z_\epsilon) \to V_0$ and $\Delta V(z_\epsilon) \to \Delta_0$ as 
$\epsilon \to 0$.
\end{theorem}


    For small $\epsilon$, $u_\epsilon$ resembles a ``spike,'' which is 
sharper for smaller $\epsilon$.  The spike concentrates near a local 
minimum of $V$ where $\Delta V$ has a strict local minimum.
    The proof of Theorem~1.1 employs 
the techniques of \cite{f1}, with 
some refinements necessary because $V$ does not 
necessarily achieve a {\it strict} local minimum.  Section~2 introduces the penalization scheme developed by Felmer and del~Pino, and continues with the beginning of the proof of Theorem~1.1.  These beginning arguments are taken practically verbatim from \cite{f1}, but are included, since the 
machinery of the penalization technique is used in the remainder of the proof.  The reader is invited to 
consult \cite{f1} for more complete proofs.
Section~3 contains the completion of the proof, which is original.  This part contains delicate computations 
involving $\Delta V$.

\section{The penalization scheme}   \setcounter{equation}{0}

    Extend $f$ to the negative reals by defining 
$f(q) = 0$ for $q < 0$.  Let
$F$ be the primitive of $f$, that is, 
$F(q) = \int_0^q f(t)\,dt$.  Define the functional
$I_\epsilon$ on $W^{1,2}({\mathbb R}^N)$ by 
$$  I_\epsilon(u) = 
    \int_{{\mathbb R}^N} {1 \over 2}
           (\epsilon^2 |\nabla u|^2 + V(z)u^2) - F(u)\,dz.  
$$%                    
$I_\epsilon$ is a $C^1$ functional, and there is a 
one-to-one correspondence between
positive critical points of $I_\epsilon$ and ground 
states of (\ref{1.0}).  It is well known that 
$I_\epsilon$ and similar functionals in related problems 
fail the Palais-Smale condition. That is, 
a ``Palais-Smale sequence,'' defined as a sequence 
$(u_m)$ with $I_\epsilon(u_m)$ convergent
and $I'_\epsilon(u_m) \to 0$ as $m \to \infty$, need not 
have a convergent subsequence.
To get around this difficulty, we formulate 
a ``penalized'' problem, with a 
corresponding ``penalized'' functional satisfying
the Palais-Smale condition, by 
altering $f$ outside of $\Lambda$.

    Let $\theta$ be as in (F4).  Choose $k$ 
so $k > \theta/(\theta - 2)$.  
Let $V_-$ be as in (V3) and 
$a > 0$ be the value at which $f(a)/a = V_-/k$.  
Define ${\tilde f}$ by
\begin{equation}\label{2.1}
  {\tilde f}(s) = \cases{f(s)          &$s \leq a$;\cr
             s{V_- / k}     &$s > a$,\cr}   
\end{equation}                        
$g( \cdot \, , s) = \chi_\Lambda f(s)  + 
(1 - \chi_\Lambda){\tilde f}(s)$, and
$G(z, \xi) = \int_0^\xi g(z, \tau)\,d\tau$.  
Although not continuous, 
$g$ is a Carath\'eodory function.  For $\epsilon > 0$, 
define the penalized 
functional $J_\epsilon$ on $W^{1,2}({\mathbb R}^N)$ by 
\begin{equation}\label{2.2}
J_\epsilon(u) =  \int_{{\mathbb R}^N} {1 \over 2} (\epsilon^2 |\nabla u|^2 + V(z)u^2)
                 - G(z,u)\,dz.   
\end{equation}
A positive critical point of $J_\epsilon$ is 
a weak solution of the ``penalized equation''
\begin{equation} \label{2.3}
  -\epsilon^2 \Delta u + V(z)u = g(z,u),         
\end{equation}                            
that is, a $C^1$ function satisfying (\ref{2.3}) wherever $g$ is 
continuous.
It is proven in \cite{f1} that $J_\epsilon$ satisfies all 
the hypotheses of the Mountain Pass 
Theorem of Ambrosetti and Rabinowitz (\cite{a2}), including 
the Palais-Smale condition.
Therefore $J_\epsilon$ has a critical point $u_\epsilon$, with 
the mountain pass
critical level 
$c(\epsilon) = J_\epsilon(u_\epsilon)$.  $c(\epsilon)$ is defined by the following
minimax:
let the set of paths 
$\Gamma_\epsilon = \{\gamma \in C([0,1], W^{1,2}({\mathbb R}^N)) \mid 
        \gamma(0) = 0,\ J_\epsilon(\gamma(1)) < 0\}$, and 
$$  c(\epsilon) = \inf_{\gamma \in \Gamma_\epsilon}
        \max_{\theta \in [0,1]} 
              J_\epsilon(\gamma(\theta)).
$$ 
As shown in (\cite{f1}), because of (F4), $c(\epsilon)$ can 
be characterized more simply as
$$  c(\epsilon) = \inf_{u \in W^{1,2}({\mathbb R}^N) 
    \setminus \{0\} }
    \sup_{\tau > 0} J_\epsilon(\tau u).
$$
The functions $g(z,q)$ and $f(q)$ agree whenever $z \in \Lambda$ 
or $q < a$.  Therefore if 
$u$ is a weak solution of (\ref{2.3}) with
$u < a$ on $\Lambda^{\bf C} \equiv {\mathbb R}^N \setminus \Lambda$, 
then 
$u$ solves (\ref{1.0}).  Our plan is to find a positive 
critical point 
$u_\epsilon$ of $J_\epsilon$, which is a weak solution of (\ref{2.3}),
then show that $u_\epsilon(z) < a$ for all 
$z \in \Lambda^{\bf C}$.

    For $\epsilon > 0$, let $u_\epsilon$ be a critical point 
of $J_\epsilon$ with 
$J_\epsilon(u_\epsilon) = c(\epsilon)$.  
Maximum principle arguments show that $u_\epsilon$ must be positive.  
Define the ``limiting 
functional'' $I_0$ by 
\begin{equation} \label{2.6}
  I_0 (u) = \int_{{\mathbb R}^N} {1 \over 2} ( |\nabla u|^2 + 
               V_0 u^2) - F(u)  
\end{equation}
and
\begin{equation} \label{2.7}
 \underline {c}  = \inf_{u \in W^{1,2}({\mathbb R}^N)
                \setminus \{0\} }
        \sup_{\tau > 0}  I_0(\tau u).   
\end{equation}  
The equation corresponding to (\ref{2.6}) is 
\begin{equation} \label{2.8}
   -\Delta u + V_0 u = f(u)      
\end{equation}

    We will prove Theorem~1.1
 by proving the following proposition:

\begin{proposition} \label{pr2.9} Let $\epsilon > 0$.  
If  $u_\epsilon$ is a positive solution of 
(\ref{2.3}) satisfying $J_\epsilon(u_\epsilon) = c(\epsilon)$,
then \begin{description} 
\item{\rm (i)} $  \lim_{\epsilon \to 0} 
        \max_{z \in \partial \Lambda} u_\epsilon = 0$.
\item{\rm (ii)} For all $\epsilon$ sufficiently small, $u_\epsilon$ has 
only one local maximum 
point in $\Lambda$ (call it $z_\epsilon$), with
$  \lim_{\epsilon \to 0} V(z_\epsilon) = V_0$   
                    
\item{\rm (iii)} $\lim_{\epsilon \to 0} \Delta V(z_\epsilon) =\Delta_0$.  
\end{description}                          
\end{proposition}   


\paragraph{Proof of Theorem~\ref{thm1.1}:} Assuming Proposition~\ref{pr2.9},
there exists $\epsilon_0 > 0$ such that for $\epsilon < \epsilon_0$, 
$u_\epsilon < a$ on $\partial \Lambda$.  In \cite{f1} it is 
shown that if we multiply
(\ref{2.3}) by $(u_\epsilon  - a)_+$ and integrate by parts,
it follows that 
$u_\epsilon  < a$ on $\Lambda^{\bf C}$,
so $u_\epsilon $ solves (\ref{1.0}).  By the 
definition of $a$ in (\ref{2.1}), and the 
maximum principle, $u_\epsilon $ has no local maxima 
outside of 
$\Lambda$, so $u_\epsilon $ has exactly one local 
maximum point $z_\epsilon $, 
which occurs in $\Lambda$.         

    Define $v_\epsilon $ by translating $u_\epsilon $ from $z_\epsilon $
to zero and 
dilating it by $\epsilon $, that is,
$$  v_\epsilon  (z) = u_\epsilon (z_\epsilon  + \epsilon  z).
$$ % 2.10
Then $v_\epsilon $ is a weak ($C^1$) solution of
 the ``translated and dilated'' equation
$$  -\Delta v_\epsilon  + 
          V(z_\epsilon  + \epsilon  z)v_\epsilon  = 
        g(z_\epsilon  + \epsilon  z, v_\epsilon ).
$$ % 2.11 
Let $\epsilon _j \to 0$.  Along a subsequence (called $(z_{\epsilon _j})$), 
$z_{\epsilon _j} \to {\bar z}  \in {\overline{\Lambda}}$, 
with $V({\bar z} ) = V_0$ 
and $\Delta V({\bar z} ) = \Delta_0$.

    Along a subsequence, $v_{\epsilon _j}$ converges 
locally uniformly 
to a function $v^0$.
Pick $R > 0$ so $v^0 < a$ on ${\mathbb R}^N \setminus B_R(0)$.  
For large enough 
$\epsilon $, $v_\epsilon  < a$ on $\partial B_R(0)$.  
By the maximum principle arguments 
of \cite{f1}, for small $\epsilon $, $v_\epsilon $ decays exponentially, 
uniformly in $\epsilon $. \hfill$\diamondsuit$\smallskip

The proof of Proposition~\ref{pr2.9} will follow if we can prove 
the following statement.

\begin{proposition} \label{pr2.12}
 If $\epsilon _n \to 0$ and 
$(z_n) \subset {\overline{\Lambda}}$
with $u_{\epsilon_n}(z_n) \geq b > 0$, then 
\begin{description}
\item{\rm (i)} $\lim_{n \to \infty} V(z_n) = V_0$. 

\item{\rm (ii)} $\lim_{n \to \infty}\Delta V(z_n) = \Delta_0 $.  
\end{description}
\end{proposition}

It is proven in \cite{f1} that $u_\epsilon$ has exactly one local maximum point 
$z_\epsilon$ for small $\epsilon$.  Since $u_\epsilon$ 
solves (\ref{2.3}), the maximum principle implies that 
$u_\epsilon(z_\epsilon)$ is bounded away from zero.  Thus 
Proposition~\ref{pr2.12} and (V4)
give Proposition~\ref{pr2.9}(ii)-(iii).

    To prove Proposition~\ref{pr2.12}, let  $b$ and $(z_n)$ be as above.  
First we repeat the  argument of \cite{f1} to show that $V(z_n) \to V_0$:
 suppose this does not happen.  Then, along a subsequence, 
$z_n \to {\bar z}  \in {\overline{\Lambda}}$ with 
$V({\bar z} ) > V_0$.
Define $v_n$ by translating $u_{\epsilon_n}$ from $z_n$ 
to $0$ and dilating  
by $\epsilon_n$; that is,
\begin{equation}  \label{2.13}
v_n(z) = u_{\epsilon_n}(z_n + \epsilon_n z).    
\end{equation}
 $v_n$ solves the 
``translated and dilated'' penalized equation
\begin{equation} \label{2.14}
 - \Delta v_n + V(z_n + \epsilon_n z)v_n = g(z_n +\epsilon_n z, v_n) 
\end{equation}
on ${\mathbb R}^N$, with $v_n(z) \to 0$ and $\nabla v_n(z) \to 0$ as 
$|z| \to \infty$.  As shown in \cite{f1}, $(v_n)$ is bounded in 
$W^{1,2}({\mathbb R}^N)$, so by elliptic estimates, $(v_n)$ converges 
locally along a subsequence 
(also denoted $(v_n)$) to $v^0 \in W^{1,2}({\mathbb R}^N)$.  Define 
$\chi_n$ by $\chi_n(z) = \chi_\Lambda (z_n + \epsilon_n z)$,
where $\chi_\Lambda$ is the characteristic function 
of $\Lambda$.  $\chi_n$ 
converges weakly in $L^p$ over compact sets to a function 
$\chi$, for any $p > 1$, with $0 \leq \chi \leq 1$.
Define 
$$  {\bar g}(z,s) = \chi(z)f(s) +  (1- \chi(z)){\tilde f}(s)   
$$
Then $v^0$ satisfies 
\begin{equation} \label{2.16}
 - \Delta v + V({\bar z} )v = {\bar g}(z,v)  
\end{equation}                            
on ${\mathbb R}^N$.  
Define ${\bar G}(z,s) = \int_0^s {\bar g}(z,t)\,dt$.
Associated with (\ref{2.16}) we have the 
limiting functional 
${\bar J}(u) = \int_{{\mathbb R}^N} {1 \over 2}(|\nabla u|^2 + V({\bar z} )u^2) 
            - {\bar G}(z,u)\,dz$.  $v^0$ is a positive 
critical point of ${\bar J}$.

    Define $J_n$ to be the ``translated and dilated'' penalized
functional corresponding to (\ref{2.14}), that is, 
$$ J_n(u) = \int_{{\mathbb R}^N} 
       {1 \over 2}(|\nabla u|^2 + V(z_n + \epsilon_n z)u^2) 
     - G(z_n + \epsilon_n z,u)\,dz.     
$$
Clearly $J_n(v_n) = \epsilon_n^{-N} J_{\epsilon_n}(u_{\epsilon_n})$.  
In \cite{f1} it is proven that 
\begin{equation} \label{2.18}
\lim\inf_{n \to \infty} J_n(v_n) \geq {\bar J}(v^0). 
\end{equation}                        
Also, by letting $w$ be a ground state for (\ref{2.8})
with $I_0(w) = {\underline {c}} $ (the mountain pass value for $I_0$, 
defined in (\ref{2.7}) and 
using $w$ as a test function for $J_n$, it is proven that
${\underline {c}}  \geq \lim\inf_{n \to \infty} J_n(v_n)$.
Thus ${\bar J}(v^0) \leq {\underline {c}} $.  
Therefore, as shown in 
\cite{f1}, $V({\bar z} ) \leq V_0$.  This contradicts our assumption.
Thus $V(z_n) \to V_0$.
   All the above is the same as was proven in \cite{f1}.  Next, 
we must show that $\Delta V(z_n) \to \Delta_0$.
That is the focus of the next section.


\section{The effect of the Laplacian } \setcounter{equation}{0}  

    Proving $\Delta V(z_n) \to \Delta_0$ is a subtle and 
delicate problem.  
Making $\epsilon_n$ approach $0$ is equivalent to dilating 
$V$, which has the effect of making local minima of 
$V$ behave more like global minima.
This assists in finding solutions to (\ref{1.0}).  However,
making $\epsilon_n$ small {\it reduces} the effect of differences in 
$\Delta V$.  For this reason, Theorem~\ref{thm1.1} is 
not only difficult to 
prove, but is not intuitively compelling, either.

    It is known (\cite{g1}) that a ``least energy solution'' of 
(\ref{2.8}), that is, a solution $w$ with 
$I_0(w) = {\underline {c}} $, must be
radially symmetric.  We will need to exploit this fact.
  In order to do this, we will need 
to work with the maximum points of $u_{\epsilon_n}$ instead of merely 
the $(z_n)$ as given in Proposition~\ref{pr2.12}.
 We need the following concentration-compactness result,
which states
that the sequence $(u_{\epsilon_n})$ of ``mountain-pass type 
solutions'' of (\ref{2.3}) does not ``split'':

\begin{lemma} \label{lm3.0}
If $(z_n) \subset {\overline \Lambda}$,
$(y_n) \subset {\mathbb R}^N$, and $b > 0$ with $u_{\epsilon_n}(z_n) > b$ 
and $u_{\epsilon_n}(y_n) > b$ for all $n$, then 
$((z_n - y_n)/\epsilon_n)$ is bounded.
\end{lemma}

\paragraph{Proof:}  define $v_n(z) = u_{\epsilon_n}(z_n + \epsilon_n z)$ 
as in (\ref{2.13}).  
Suppose the lemma is false.  Then, along a subsequence, 
$|y_n - z_n|/\epsilon_n \to \infty$.  
Let $x_n = (y_n - z_n)/\epsilon_n$.
$(\|v_n\|)$ is bounded in $W^{1,2}({\mathbb R}^N)$
and $|x_n| \to \infty$, so we may 
pick a sequence $(R_n) \subset {\mathbb N}$ with 
$R_n \to \infty$,
$|x_n| - R_n \to \infty$, and 
$\|v_n\|_{W^{1,2}
        (    B_{R_n + 1}(0) 
            \setminus
                 B_{R_n - 1}(0)  )
        } \to 0$ 
         as $n \to \infty$.
Define cutoff functions 
$\varphi_n^{1,2} \in C^\infty({\mathbb R}^N, [0,1])$
satisfying $\varphi_1 \equiv 1$ on $B_{R_n - 1}(0)$,
         $\varphi_1 \equiv 0$ on $B_{R_n}(0)^{\bf C}$,
         $\varphi_2 \equiv 1$ on $B_{R_n + 1}(0)^{\bf C}$,
         $\varphi_2 \equiv 0$ on $B_{R_n}(0)$,
and $\|\nabla \varphi_1\|_{L^\infty({\mathbb R}^N)} < 2$,
    $\|\nabla \varphi_2\|_{L^\infty({\mathbb R}^N)} < 2$.
Set $v_n^1 = \varphi_n^1 v_n$ and
    $v_n^2 = \varphi_n^2 v_n$, and 
${\bar v}_n = v_n^1 + v_n^2 = (\varphi_n^1 + \varphi_n^2)v_n$.

    Choose $T_n > 0$ so $J_n(T_n {\bar v}_n) = 0$.  
We claim that 
$T_n$ is well-defined, and bounded in $n$.  Note that the existence of $T_n$ must be checked for the penalized functional
$J_n$, because of the replacement of $F$ with $G$.  
By elliptic estimates, there exists an open 
set $U \subset {\mathbb R}^N$ such that along a subsequence, 
$v_n^1 > b/2$ on $U$ and 
$U \subset (\Lambda - z_n)/\epsilon_n \equiv 
\{z \in {\mathbb R}^N \mid z_n + \epsilon_n z \in \Lambda\}$. 
Let $a$ be as in (\ref{2.1}).  For $t > 2a/b$ and
$z \in U$, $t{\bar v}_n(z) > tb/2 > a$, so
$G(z_n + \epsilon_n z, t{\bar v}_n) = 
        F(t{\bar v}_n) > F(bt/2)$.  Therefore,
for $t > 2a/b$,
\begin{eqnarray*} 
J_n(t{\bar v}_n) 
&=& t^2\int_{{\mathbb R}^N} {1 \over 2}(|\nabla {\bar v}_n|^2 +
   V(z_n + \epsilon_n z){\bar v}_n^2)\,dz
 - \int_{{\mathbb R}^N}  G(z_n + \epsilon_n z,t{\bar v}_n)\,dz \\
&\leq& {t^2 \over 2}(1 + V^+) \|{\bar v}_n\|^2_{W^{1,2}({\mathbb R}^N)}
 -\int_U F(t{\bar v}_n) \\
&\leq& {t^2 \over 2}(1 + V^+) \|{\bar v}_n\|^2_{W^{1,2}({\mathbb R}^N)}
 - \lambda(U) F(tb/2),
\end{eqnarray*}
where $\lambda$ indicates the Lebesgue measure.  
By (F4), there exists $C>0$ such that for $t > 2a/b$, 
$F(tb/2) > Ct^\theta$.  Therefore, for $t > 2a/b$,
\begin{equation} \label{3.2}
  J_n(t{\bar v}_n) \leq {t^2 \over 2}(1 + V^+)
      \|{\bar v}_n\|^2_{W^{1,2}({\mathbb R}^N)} - Ct^\theta.    
\end{equation}
Since $({\bar v}_n)$ is bounded in $W^{1,2}({\mathbb R}^N)$,
 this gives the existence and boundedness of $(T_n)$.

    Since $J_n(T_n {\bar v}_n) = 
            J_n(T_n v_n^1) + J_n(T_n v_n^2) = 0$, 
we may pick $i_n \in \{1, 2\}$ with 
$J_n(T_n v_n^{i_n}) \leq 0$.  
By (F5) and (\ref{2.1}), 
the map $t \mapsto J_n(tv_n^{i_n})$ increases from zero 
at $t = 0$, achieves a positive maximum, then decreases to 
$- \infty$.  We will see more of this in a moment.
Thus there exists a unique
$t_n \in (0, T_n)$ with 
$J_n(t_n v_n^{i_n}) = \max_{t>0} J_n(t v_n^{i_n})$.  
We claim that
$t_n$ and $T_n - t_n$ are both bounded away from zero for 
large $n$: by $(f_1)-(f_4)$ and (\ref{2.1}), 
$J_n(w) \geq 
  {1 \over \theta}
     \min (1, V_-)\|w\|^2_{W^{1,2}({\mathbb R}^N)}- 
        o(\|w\|^2_{W^{1,2}({\mathbb R}^N)})$ uniformly 
in $n$, so $\max_{t>0} J_n(t v_n^{i_n})$ is bounded away from 
zero, 
uniformly in $n$.  It is easy to 
show that $J_n$ is Lipschitz on bounded subsets of 
$W^{1,2}({\mathbb R}^N)$, uniformly in $n$.  Since $(T_n)$ is bounded, this 
implies that  $t_n$ and $T_n - t_n$ are both bounded away from zero for large $n$.

    By definition of $v_n$ as a ``mountain-pass type critical 
point'' of $J_n$, we have 
$$    \max_{t>0} J_n(t v_n^{i_n}) \geq 
            \max_{t>0} J_n(t v_n).      
$$
Using the facts that $\|v_n - {\bar v}_n\|_{W^{1,2}({\mathbb R}^N)} \to 0$
 as $n \to \infty$,  and $(T_n)$ is bounded, we have
\begin{eqnarray}  
    \lim\inf_{n \to \infty} J_n(t_n v_n^{i_n}) &=& 
     \lim\inf_{n \to \infty} \max_{t>0}J_n(t v_n^{i_n})\nonumber\\
& \geq&  \lim\inf_{n \to \infty} \max_{t>0}J_n(t v_n)  \nonumber \\
&=& \lim\inf_{n \to \infty} \max_{t>0}J_n(t {\bar v}_n) \label{3.4} \\
&=& \lim\inf_{n \to \infty} J_n(t_n {\bar v}_n) \nonumber \\
&=& \lim\inf_{n \to \infty}(J_n(t_n v_n^{i_n}) 
    +J_n(t_n v_n^{3 - i_n}) ) \nonumber \\
&\geq& \lim\inf_{n \to \infty}J_n(t_n v_n^{i_n}) 
+\lim\inf_{n \to \infty}J_n(t_n v_n^{3-i_n}). \nonumber
\end{eqnarray}

Now $J_n(T_n v_n^{3-i_n}) = - J_n(T_n v_n^{i_n}) \geq 0$ 
and $t_n < T_n$, so $J_n(t_n v_n^{3-i_n}) \geq 0$.
By (\ref{3.4}), 
$\lim\inf_{n \to \infty}J_n(t_n v_n^{3-i_n}) \leq 0$. 
Therefore $J_n(t_n v_n^{3-i_n}) \to 0$ as $n \to \infty$.  

    Since $J_n(w) \geq {1 \over \theta}
 \min (1, V_-)\|w\|^2_{W^{1,2}({\mathbb R}^N)} - 
            o(\|w\|^2_{W^{1,2}({\mathbb R}^N)})$ uniformly
in $n$, there exists 
$d \in (0, \lim\inf_{n \to \infty} t_n )$ such that 
$\lim\inf_{n \to \infty} J_n(d v_n^{3-i_n}) > 0$.  
Since $d < t_n$ and $J_n(d v_n^{3-i_n}) > J_n(t_n v_n^{3-i_n})$
for large $n$, the map
$t \mapsto J_n(t v_n^{3-i_n})$ achieves a maximum at some 
$t'_n \in (0, t_n)$, and that maximum is bounded away from zero.

    Summarizing the important facts about the mapping
$t \mapsto J_n(t v_n^{3-i_n})$, we have shown that there exists 
$\rho > 0$ such that for large $n$,
\begin{description}
\item{(i)} $0 < t'_n < t_n < T_n$
\item{(ii)} $(T_n)$ is bounded.
\item{(iii)} $(T_n - t_n)$ is bounded away from zero.
\item{(iv)} $J_n(t'_n v_n^{3-i_n}) > \rho > 0$
\item{(v)} $J_n(t_n v_n^{3-i_n}) \to 0$
\item{(vi)} $J_n(T_n v_n^{3-i_n}) \geq 0$
\end{description}
    From (i)-(vi) it is apparent that at some $t^*_n > t'_n$, the mapping 
$t \mapsto J_n(t v_n^{3-i_n})$ is at once decreasing and concave upward.
But this is impossible: let $n \in {\mathbb N}$ and 
$w \in W^{1,2}({\mathbb R}^N) \setminus \{0\}$. 
Define $\psi(t) = J_n(tw)$ for 
$t > 0$.  Then
\begin{eqnarray*}
    \psi'(t) &=& t\int_{{\mathbb R}^N} |\nabla w|^2 + 
                          V(z_n + \epsilon_n z) w^2 \,dz                        -
            \int_{{\mathbb R}^N}  g(z_n + \epsilon_n z, tw)w\,dz \\
&=& t \left[\int_{{\mathbb R}^N} |\nabla w|^2 + 
     V(z_n + \epsilon_n z) w^2 \,dz 
   - \int_{\{w \neq 0\} } {g(z_n + \epsilon_n z, tw) \over{tw} }
    w^2\,dz \right].
\end{eqnarray*}

By (F5) and (\ref{2.1}), 
$t \mapsto g(z_n + \epsilon_n z, tw)/(tw)$ is nondecreasing, so 
if $\psi'(t)$ ever becomes negative, $\psi'$ is increasing for all 
time $t$ after that, and the graph of $\psi$ is concave down.  Therefore
the behavior of  $J_n(t v_n^{3-i_n})$ as described in (i)-(vi)
 is impossible, and  Lemma~\ref{lm3.0} is proven.  \hfill$\diamondsuit$\smallskip 
            
 As mentioned before, it will be advantageous to work with 
the maxima of $(u_{\epsilon_n})$.  Choose $(y_n) \subset {\mathbb R}^N$ with
$$  
u_{\epsilon_n}(y_n) = \max_{{\mathbb R}^N} u_{\epsilon_n}. 
$$
We will prove 
\begin{equation} \label{3.8}
      \Delta V(y_n)   \to \Delta_0.   
\end{equation}
 By Lemma~3.0, $((y_n - z_n)/\epsilon_n)$ is bounded, 
so $y_n - z_n \to 0$.  Thus (\ref{3.8})
gives Proposition~\ref{pr2.12}(ii), completing the proof of 
Theorem~\ref{thm1.1}. \hfill$\diamondsuit$\smallskip

    Along a subsequence, $y_n \to {\bar y} \in {\overline \Lambda}$.  
By Proposition~\ref{pr2.12}(i),
$V({\bar y} ) = V_0$.
Since is not apparent that ${\bar y}  \in \Lambda$, we must proceed 
carefully.  We will redefine the $v_n$'s like in (\ref{2.13}),
by translating $u_{\epsilon_n}$ to $0$ and dilating it.  That is, 
\begin{equation} \label{3.9}
 v_n(z) = u_{\epsilon_n}(y_n +\epsilon_n z). 
\end{equation} 
Then $v_n$ is a positive weak solution, vanishing at infinity, 
of the ``penalized, dilated, and translated'' PDE
$$ -\Delta v + V(y_n + \epsilon_n z)v = g(y_n + \epsilon_n z, v).
$$  % 3.10
Like before, $(v_n)$ converges locally uniformly to a function
$v_0$.  We claim that $v_0$ is actually 
a ground state maximizing at $0$ of the autonomous 
limiting equation (\ref{2.8}).  Proof:  As 
before, define $\chi_n$ by 
$\chi_n(z) = \chi(y_n + \epsilon z)$.
As before, along a subsequence, 
$\chi_n$ converges weakly in 
$L^p$, for any $p > 1$,
 on compact subsets of ${\mathbb R}^N$ to a function $\chi$ with 
$0 \leq \chi \leq 1$.  Define ${\bar g}$ by
$$  {\bar g}(z,s) = \chi(z)f(s) + (1-\chi(z)){\tilde f} (s).
$$ % 3.11
By the argument of Proposition~\ref{pr2.12}, taken from \cite{f1}, 
$(v_n)$ converges locally along a subsequence to
$v_0$, a ground state of
$-\Delta v + V_0 v = {\bar g}(z,v)$.  The functional 
corresponding to this equation is
${\bar J}(u) = 
  \int_{{\mathbb R}^N} {1 \over 2}(|\nabla u|^2 + V_0 u^2) 
                - {\bar G}(z,u)\,dz$,
where ${\bar G}(z,s) = \int_0^s {\bar g}(z,t)\,dt$.
  As before, in (\ref{2.18}),
 ${\underline {c}}  \geq 
        \lim\inf_{n \to \infty}J_n(v_n) \geq 
        {\bar J}(v_0)$, where 
${\underline {c}} $ is from 
(\ref{2.7}). ${\bar J} \geq I_0$, 
where $I_0$ is the \lq\lq autonomous" limiting functional
from (\ref{2.6}), so
$$  {\underline {c}}  \leq \max_{t>0} I_0(t v_0) \leq 
        \max_{t>0} {\bar J}(t v_0) \leq 
            {\underline {c}} ,  $$ % 3.12
and $v_0$ is actually a ground state of (\ref{2.8}). 
\hfill$\diamondsuit$ \medskip

Not only does $(v_n)$ converge locally to $v_0$, but it satisfies the following
lemma.

\begin{lemma} \label{lm3.13} With $(v_n)$ as in 
(\ref{3.9}), for any subsequence of $(v_n)$ there 
is a radially symmetric ground state $v_0$ of (\ref{2.8})
such that $v_n \to v_0$ 
uniformly along a subsequence and the $v_n$'s 
decay exponentially, uniformly in $n$.
\end{lemma}

\paragraph{Proof:}  If one establishes uniform convergence, 
the uniform exponential decay follows readily, using a standard 
maximum principle argument found in \cite{f1}.  Suppose the 
convergence is not uniform.  Then there exist
a subsequence of $(v_n)$ (denoted ($v_n$)) and a sequence
$(x_n) \subset {\mathbb R}^N$ with $|x_n| \to \infty$ and 
$\lim_{n \to \infty} v_n(x_n) > 0$.  Let $d > 0$ 
with  
$d < v_0(0)$ and $d < \lim_{n \to \infty} v_n(x_n)$. 
For large $n$,
$d < v_n(0) = u_{\epsilon_n}(z_n)$ and 
$d < v_n(x_n) = u_{\epsilon_n}(z_n + \epsilon_n x_n)$.  Letting 
$w_n = z_n + \epsilon_n x_n$, we obtain
$((w_n - z_n)/\epsilon_n) = (x_n)$, which is unbounded, violating
Lemma~\ref{lm3.0}.  

    To show $\Delta V(y_n) \to \Delta_0$, 
we again argue 
indirectly.  Suppose otherwise.  Then, along a subsequence, 
$y_n \to {\bar y}  \in {\overline \Lambda}$ with 
\begin{equation} \label{3.14}
\Delta V({\bar y} ) > \Delta_0\,.
\end{equation}      
For 
$x \in {\mathbb R}^N$, define the translation operator
$\tau_x$ by $\tau_x u(z) = u(z-x)$, that is, $\tau_x u$ is 
$u$ translated by $x$.  Assume for convenience, and without 
loss of generality, that 
$$  0 \in \Lambda,\ V(0) = V_0,  \hbox{ and }
        \Delta V(0) = \Delta_0.   $$
We will prove that for large $n$,
\begin{equation} \label{3.16}
 \sup_{t > 0} J_{\epsilon_n}(t\tau_{-y_n/\epsilon_n}u_{\epsilon_n}) < 
                J_{\epsilon_n}(u_{\epsilon_n}) =
        \sup_{t>0}J_{\epsilon_n}(tu_{\epsilon_n}),
\end{equation}
recalling the definition of $J_\epsilon$ in (\ref{2.2}), 
and how $v_n$ is defined from $u_{\epsilon_n}$ in (\ref{3.9}).
That is, translating $t u_{\epsilon_n}$ back to the origin reduces the value of $J_{\epsilon_n}(t v_n)$ because $V$ has lesser concavity at the origin.  This occurs even though 
shrinking $\epsilon$ reduces the difference in concavity.
(\ref{3.16}) contradicts the definition of $u_{\epsilon_n}$. 

    Pick $T > 1$ large enough so that for large $n$,
$J_n(Tv_n) = \epsilon_n^{-N} J_{\epsilon_n}(T u_{\epsilon_n}) < 0$.  
This is possible by the argument of (\ref{3.2}).  
Now (\ref{3.16}) is equivalent to 
$$  \sup_{0 \leq t \leq T} 
        J_{\epsilon_n}(t \tau_{-y_n}u_{\epsilon_n}) 
<  \sup_{0 \leq t \leq T} J_{\epsilon_n}(t u_{\epsilon_n}).   
$$  % 3.17

To prove the above, it will suffice to prove the stronger 
fact that
for large $n$, for all $t \in (0, T)$,
$$  J_{\epsilon_n}(t u_{\epsilon_n}) > 
    J_{\epsilon_n}(t \tau_{-y_n}u_{\epsilon_n}).    
$$ % 3.18
Now, along a subsequence, $v_n \to v_0$ uniformly, so by 
the definition of $v_n$ as a dilation of 
$\tau_{-y_n}u_{\epsilon_n}$ ((\ref{3.9})), $u_{\epsilon_n} \to 0$ uniformly
on ${\mathbb R}^N \setminus \Lambda$ as $n \to \infty$.  Thus for 
large $n$ and $0 \leq t \leq T$, 
the definition of $G$ gives
$G(z, t\tau_{-y_n}u_{\epsilon_n}(z)) = 
F(t\tau_{-y_n}u_{\epsilon_n}(z)$ for all $z \in {\mathbb R}^N$, so
\begin{eqnarray*}
\lefteqn{ J_{\epsilon_n}(t u_{\epsilon_n}) - 
        J_{\epsilon_n}(t \tau_{-y_n}u_{\epsilon_n}) }\\
&=& \int_{{\mathbb R}^N} {1 \over 2} t^2  \left(
 |\nabla u_{\epsilon_n}(z)|^2 + V(z)u_{\epsilon_n}(z)^2\right)
  - G(z, tu_{\epsilon_n}(z))\,dz \\
 &&-\big[ \int_{{\mathbb R}^N}  {1 \over 2} t^2 \left(
     |\nabla \tau_{-y_n}u_{\epsilon_n}(z)|^2 + 
     V(z)\tau_{-y_n}u_{\epsilon_n}(z)^2 \right)
     - F(t\tau_{-y_n}u_{\epsilon_n}(z))\,dz \big] \\
 &\geq& {1 \over 2} t^2 \int_{{\mathbb R}^N} 
   V(z)(u_{\epsilon_n}(z)^2 - u_{\epsilon_n}(z + y_n)^2)\,dz \\
&&  +\int_{{\mathbb R}^N}  F(tu_{\epsilon_n}(z+y_n)-
            F(tu_{\epsilon_n}(z))\,dz \\
&=&{1 \over 2} t^2 \int_{{\mathbb R}^N} (V(z+y_n)- V(z))
      u_{\epsilon_n}(z+y_n)^2\,dz  \\
&=&{1 \over 2} t^2 \epsilon_n^{N}
   \int_{{\mathbb R}^N} (V(y_n + \epsilon_nz) - 
   V(\epsilon_n z)) u_{\epsilon_n}(\epsilon_n z + y_n)^2\,dz \\
&=&{1 \over 2} t^2 \epsilon_n^{N}\int_{{\mathbb R}^N}   
    (V(y_n + \epsilon_nz) - V(\epsilon_n z)) v_n(z)^2\,dz\,.
\end{eqnarray*} 
For $n = 1, 2, \ldots$, define $h_n: {\mathbb R} \to {\mathbb R}$ 
by
$$  h_n(t) = \int_{{\mathbb R}^N} (V(y_n + tz) - V(t z))v_n^2\,dz.    
$$ 
Since $h_n(\epsilon_n) = \int_{{\mathbb R}^N} (V(y_n + \epsilon_n z) - 
        V(\epsilon_n z))v_n^2$,
we must prove that for large $n$, 
\begin{equation}
h_n(\epsilon_n) > 0\,. \label{3.21}
\end{equation} 

    Assume without loss of generality that $\Lambda$ was chosen 
so that there exists $\rho > 0$ with
\begin{equation}
\inf_{N_\rho(\Lambda)} V = V_0, \label{3.22}
\end{equation}
where $N_\rho(\Lambda) = 
    \{x \in {\mathbb R}^N \mid \exists y \in \Lambda 
        \hbox{ with } |y - x| < \rho\}$.
We will prove the following facts about $h_n$:

\begin{lemma} \label{lm3.23} 
 For some $\beta > 0$, for large $n$,
\begin{description}
\item{\rm (i)} $h_n \in C^2({\mathbb R}^+, {\mathbb R})$
\item{\rm (ii)} $h_n(0) \geq 0$ 
\item{\rm (iii)} $|h'_n(0)|^2 \leq o(1)h_n(0)$
\item{\rm (iv)} $h''_n(0) > \beta$
\item{\rm (v)} $h''_n$ is locally Lipschitz on ${\mathbb R}^+$, uniformly 
        in $n$.
\end{description}
\end{lemma}

    Here $o(1) \to 0$ as $n \to \infty$.
Before proving Lemma~\ref{lm3.23}, let us prove how it gives (\ref{3.21}).
By (iv)-(v), there exists $d > 0$ such 
that for large $n$ and $0 \leq t \leq d$, 
$h''_n(t) > \beta/2$.
For $t \in [0, d]$, a Taylor's series expansion shows that 
for large $n$,
\begin{equation} \label{3.24}
h_n(t) \geq h_n(0) + h'_n(0)t + 
            {\beta \over 4}t^2 \equiv l_n(t).   
\end{equation}                          
If $h_n(0) = 0$, then 
by Lemma~\ref{lm3.23}(iii), $h'_n(0) = 0$, so (\ref{3.24}) 
implies that $h_n(t) > 0$ for 
all $t \in (0, d)$, giving (\ref{3.21}) if $n$ is large enough
that $\epsilon_n < d$.  If 
$h_n(0) > 0$, then 
by elementary calculus, $l_n$ attains a minimum value at 
$t = -2h'_n(0)/\beta$, 
and the minimum value is 
$$   \min_{\mathbb R} l_n = l_n(-2h'_n(0)/\beta) = h_n(0) -   
     h'_n(0)^2/\beta \geq (1 - o(1))h_n(0),
$$ % 3.25
where $o(1) \to 0$ as $n \to \infty$.  
For large $n$, if $h_n(0) > 0$ then 
$l_n(t) > 0$ for all $t \in {\mathbb R}$, so $h_n(t) > 0$ 
for all $t \in (0,d)$ for 
large $n$, implying (\ref{3.21}) if $n$ is 
large enough so that $\epsilon_n < d$.  

\paragraph{Proof of Lemma~\ref{lm3.23}} Statement (ii) is trivial, since  
$h_n(0) = (V(y_n) - V_0)\int_{{\mathbb R}^N} v_n^2$, 
and since $z_n \in {\overline \Lambda}$
and $y_n - z_n \to 0$, 
(\ref{3.22}) implies $V(y_n) \geq V_0$ for large $n$.
(i) and (v) follow from Leibniz's Rule, 
$(V_1)-(V_2)$, and the fact that the
$v_n$'s decay exponentially, uniformly in $n$.
For $j = 1, 2$,
$$   h_n^{(j)}(t) =  \int_{{\mathbb R}^N} \sum_{|\alpha| = j}
    (D^\alpha V(y_n + tz) - D^\alpha V(tz))z^\alpha
            v_n(z)^2\,dz.   
$$ % 3.26
Since (V2) holds, $v_n$ decays exponentially, uniformly 
in $n$, $y_n \to {\bar y} $, 
and $v_0$ is radially symmetric, we have
\begin{eqnarray*}
 h''_n(0) &=& \int_{{\mathbb R}^N} \sum_{|\alpha| = 2}
    (D^\alpha V(y_n) - D^\alpha V(0))z^\alpha
            v_n(z)^2\,dz \\
 &\to& \int_{{\mathbb R}^N} \sum_{|\alpha| = 2}
    (D^\alpha V({\bar y} ) - D^\alpha V(0))z^\alpha v_0(z)^2\,dz \\
&=&  \int_{{\mathbb R}^N} \sum_{i=1}^N
    (D^{ii} V({\bar y} ) - D^{ii} V(0))z_i^2 v_0(z)^2\,dz \\
&=&  \int_{{\mathbb R}^N} \sum_{i=1}^N
    (D^{ii} V({\bar y} ) - D^{ii} V(0)){1 \over N} |z|^2 v_0(z)^2\,dz  \\
&=& {1 \over N}(\Delta V({\bar y} ) - \Delta V(0)) 
    \int_{{\mathbb R}^N} |z|^2 v_0(z)^2\,dz > 0
\end{eqnarray*}
by assumption (\ref{3.14}).  Since Lemma~\ref{lm3.23}(v) holds, we have
Lemma~\ref{lm3.23}(iv).

 To prove Lemma~\ref{lm3.23}(iii), we will need the following calculus 
lemma:

\begin{lemma} \label{lm3.28}  Let $U \subset {\mathbb R}^N$ and 
$r > 0$. Let $V \in C^2(N_r(U), {\mathbb R})$ with 
$\inf_{N_r(U)} V \equiv V_0 > - \infty$, 
$|\nabla V|$ bounded on $N_r(U)$, and $D^2 V$ Lipschitz on 
$N_r(U)$.  Then there exists $C > 0$ with 
\begin{equation}
  |\nabla V(z)|^2 \leq C(V(z) - V_0)  \label{3.29}
\end{equation}
for all $z \in U$.
\end{lemma}


\paragraph{Proof:} let $B > 0$ 
with $|D^2 V(z)\xi \cdot \xi| \leq B$ for all 
$\xi \in {\mathbb R}^N$ with $|\xi| = 1$.  Also let $B$ be big enough 
so 
$$      B > |\nabla V(z)|/r 
$$ % 3.30 
for all $z \in U$.  Pick $z \in U$.  If $|\nabla V(z)| = 0$, then 
(\ref{3.29}) is obvious.  Otherwise, let $d = |\nabla V(z)|/B < r$.  
Define $\varphi(t) = V(z - t\nabla V(z)/|\nabla V(z)|)$ 
for $t \in [0, d]$.  $\varphi$ is $C^2$, $\varphi(0) = V(z)$,
and
$\varphi'(0) = -|\nabla V(z)|$.    
By choice of $B$ and the
fact that $B_d(z) \subset N_r(U)$,
$|\varphi''(t)| \leq B$ for all $t \in [0,d]$.
 Taylor's theorem gives
$$    \varphi(d) - \varphi(0) = \varphi'(0)d +
  \varphi''(\xi){d^2 \over 2} \leq -|\nabla V(z)|d + Bd^2/2 = 
                -{|{\nabla V(z)}|^2 \over {2B} }.
$$ % 3.31
Also $\varphi(d) \geq V_0$ because $B_d(z) \subset N_r(U)$.  
Therefore,
$$ {|{\nabla V(z)}|^2 \over {2B} } \leq
   \varphi(0) - \varphi(d) \leq V(z) - V_0\,.
$$ %3.32 
Lemma~\ref{lm3.28} is proven. \hfill$\diamondsuit$ \medskip

    To prove Lemma~\ref{lm3.23}(iii), first note that, by the 
radial symmetry of $v_0$, the uniform exponential decay 
of $v_n$, and the uniform convergence $v_n \to v_0$,
\begin{eqnarray*}
|h'_n(0)| &=& |(\nabla V(y_n) - \nabla V(0)) \cdot
                \int_{{\mathbb R}^N} zv_n^2\,dz | \\
&=&|\nabla V(y_n) \cdot \int_{{\mathbb R}^N} zv_n^2\,dz | \\
&=& |\nabla V(y_n) \cdot \int_{{\mathbb R}^N} zv_0^2\,dz    +                                \ \ 
    \nabla V(y_n) \cdot \int_{{\mathbb R}^N} z(v_n^2 - v_0^2)\,dz| \\
&=& |\nabla V(y_n) \cdot \int_{{\mathbb R}^N} 
                z(v_n^2 - v_0^2)\,dz| \\
&\leq& |\nabla V(y_n)| \, 
            |\int_{{\mathbb R}^N} z(v_n^2 - v_0^2)\,dz| \\
&\leq&  o(1)|\nabla V(y_n)|,
\end{eqnarray*} 
so Lemma~\ref{lm3.28} implies
\begin{eqnarray*}
    |h'_n(0)|^2 &\leq& o(1)|\nabla V(y_n)|^2 \leq
                o(1)(V(y_n)-V_0) \\ 
&\leq& o(1)(V(y_n)-V_0)\int_{{\mathbb R}^N} v_n^2\\
& =&  o(1)h_n(0),
\end{eqnarray*}
since $\int_{{\mathbb R}^N} v_n^2$ is bounded away from zero.  
Lemma~\ref{lm3.23}(iii) is proven.  Thence follow (\ref{3.21}), (\ref{3.8}), 
Proposition~\ref{pr2.12}, and Theorem~\ref{thm1.1}.

\paragraph{Remarks:}
Besides the results cited in the introduction, many important results for 
equations of type (\ref{1.0}) have been found recently.  
For instance, the work in \cite{f1}-\cite{f3} suggests that 
Theorem~\ref{thm1.1} could be strengthened by working 
on a smaller domain than ${\mathbb R}^N$, or by  weakening the hypotheses 
on  $V$.  It is natural to try to extend Theorem~\ref{thm1.1} to cases where 
$V$ is not $C^2$, or to the case where the second derivatives 
of $V$ do not provide a condition like (V4), 
but higher-order derivatives do.

\begin{thebibliography}{19}

\bibitem{a1}  Ambrosetti, A., Badiale, M., Cingolani, S.:
Semiclassical states of nonlinear 
Schr\"odinger equations. Arch.~Rat.~Mech.~Anal. (1997) {\bf 140} no.~3, 285-300.

\bibitem{a2}  Ambrosetti, A., Rabinowitz, P.: 
Dual variational methods in critical point theory and
applications. Journal of Functional Analysis (1973)
{\bf 14}, 349-381. 

\bibitem{f1}  Felmer, P., del~Pino, M.: Local Mountain passes for 
semilinear elliptic problems in unbounded domains. Calc.~Var.~and
PDE (1996) {\bf 4}, 121-137.

\bibitem{f2}   Felmer, P., del~Pino, M.: Semi-classical states for 
nonlinear Schr\"odinger equations.  J.~Funct.~Anal.~(1997)
{\bf 149} no.~1, 245-265. 

\bibitem{f3}  Felmer, P., del~Pino, M.: Multi-peak bound states for 
nonlinear Schr\"odinger equations. Ann.~Inst.~Henri Poincar\'e, 
Analyse non lin\'eaire (1998)
{\bf 15} no.~2, 127-149. 

\bibitem{f4} Floer, A., Weinstein, A.: Nonspreading Wave Packets for the 
Cubic Schr\"odinger Equation with a Bounded Potential, 
J.~Funct.~Anal.~(1986) {\bf 69}, 397-408. 

\bibitem{g1}  Gidas, B., Ni, W.-M., Nirenberg, L.:  
Symmetry of positive 
solutions of nonlinear elliptic equations in ${\mathbb R}^n$.
Math.~Anal.~and Applications, Part A, Advances in Math.
Suppl.~Studies 7A (ed.~L Nachbin), Academic Press (1981),
369-402.

\bibitem{l1} Li, Y.-Y.: On a singularly perturbed elliptic equation.
 Advances in Differential Equations (1997) {\bf 2}, 955-980.

\bibitem{l2} Lu, G., Wei, J.: On Nonlinear Schr\"odinger Equations with
    Totally Degenerate Potentials.  
    C.~R.~Acad.~Sci.~Paris S\'erie I Math.~(1998) {\bf 326} no.~6,
    691-696.

\bibitem{o1} Oh, Y.J.: Existence of semi-classical bound states of 
nonlinear Schr\"odinger equations with potential on the class
$(V_a)$, Comm.~Partial Differ.~Eq. 
(1988) {\bf 13}, 1499-1519. 

\bibitem{o2} Oh, Y.J.: Corrections to Existence of semi-classical bound states of 
nonlinear Schr\"odinger equations with potential on the class
$(V_a)$, Comm.~Partial Differ.~Eq.~(1989) {\bf 14}, 833-834.

\bibitem{o3} Oh, Y.J.: On positive multi-lump bound states nonlinear 
Schr\"odinger equations under multiple well potential,
Comm.~Math.~Phys.~(1990) {\bf 131}, 223-253. 

\bibitem{r1} Rabinowitz, P. H.: On a class of nonlinear
        Schr\"odinger equations. Z.~angew Math.~Phys.~(1992)
        {\bf 43},   270-291. 

\bibitem{s1} Spradlin, G.: A Singularly Perturbed 
Elliptic Partial Differential Equation with an Almost Periodic Term. 
Calc.~Var.~and PDE (1999) {\bf 9}, 207-232.

\bibitem{w1} Wang, X.: On Concentration of positive bound states of 
nonlinear Schr\"odinger equations.  Comm.~Math.~Phys.~(1993)
    {\bf 153}, 229-244.
\end{thebibliography} \medskip


\noindent{\sc Gregory S. Spradlin} \\
 Department of Mathematical Sciences \\
 United States Military Academy \\
 West Point, New York 10996, USA \\
e-mail: gregory-spradlin@usma.edu

\end{document}