\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 90, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/90\hfil Self-similar solutions]
{Self-similar solutions with compactly supported profile of some
nonlinear Schr\"odinger equations}

\author[P. B\'egout, J. I. D\'iaz \hfil EJDE-2014/90\hfilneg]
{Pascal B\'egout, Jes\'us Ildefonso D\'iaz}  % in alphabetical order

\address{Pascal B\'egout \newline
Institut de Math\'ematiques de Toulouse \& TSE \\
Universit\'e Toulouse I Capitole \\	
Manufacture des Tabacs \\
21, All\'ee de Brienne,
31015 Toulouse Cedex 6, France}
\email{Pascal.Begout@math.cnrs.fr}

\address{Jes\'us Ildefonso D\'iaz \newline
Departamento de Matem\'atica Aplicada\\
Instituto de Matem\'atica Interdisciplinar\\
Universidad Complutense de Madrid,
Plaza de las Ciencias, 3, 28040 Madrid, Spain}
\email{diaz.racefyn@insde.es}

\thanks{Submitted December 9, 2013. Published April 2, 2014.}
\subjclass[2000]{35B99, 35A01, 35A02, 35B65, 35J60}
\keywords{Nonlinear self-similar Schr\"odinger equation; compact support;
\hfill\break\indent  energy method}

\begin{abstract}
``Sharp localized'' solutions (i.e. with compact support for each given time $t$)
 of a singular nonlinear type Schr\"odinger equation in the whole space
 $\mathbb{R}^N$ are constructed here under the assumption that they have
 a self-similar structure. It requires the assumption that the external
 forcing term satisfies that 
 $\mathbf{f}(t,x)=t^{-(\mathbf{p}-2)/2}\mathbf{F}(t^{-1/2}x)$
 for some complex exponent $\mathbf{p}$ and for some profile function
 $\mathbf{F}$ which is assumed to be with compact support in
 $\mathbb{R}^N$. We show the existence of solutions of the form
 $\mathbf{u}(t,x)=t^{\mathbf{p}/2}\mathbf{U}(t^{-1/2}x)$, with a profile
 $\mathbf{U}$,  which also has compact support in $\mathbb{R}^N$. 
 The proof of the  localization of the support of the profile $\mathbf{U}$ 
 uses some suitable energy  method applied to the stationary problem satisfied
 by $\mathbf{U}$ after some unknown transformation.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction and main result}
\label{intro}

This article deals with the study of \emph{sharp localized} solutions of the
 nonlinear type Schr\"odinger equation in the whole space $\mathbb{R}^N$,
\begin{equation} \label{nls}
\mathbf{i}\frac{\partial\mathbf{u}}{\partial t}+\Delta\mathbf{u}
=\mathbf{a}|\mathbf{u}|^{-(1-m)}\mathbf{u}+\mathbf{f}(t,x),
\end{equation}
under the fundamental assumption $m\in (0,1)$ and for different choices
of the complex coefficient $\mathbf{a}$. Here we use the notation of
bold symbols for complex symbols, $\mathbf{i}^2=-1$ and
$\Delta=\sum_{j=1}^N\frac{\partial^2}{\partial x^2_j}$ for the Laplacian
in the variables $x$.

By the term \emph{sharp localized solutions} we understand solutions
 which go beyond the so called \emph{localized solutions} considered earlier
 by many authors. For instance, most of the \emph{localized type solutions}
in the previous literature must vanish at infinity in an asymptotic way:
$|\mathbf{u}(t,x)|\to 0$ as $|x|\to \infty$. They have been intensively studied
 mostly when some other structure property is added to the solution.
It is the case of the special solutions which receive also other names such
as \emph{standing waves, travelling waves, solitons}, etc.

Here we are interested on solutions which have a sharper decay when
$|x|$ approaches infinity in the sense that we will require the support
 of the function $\mathbf{u}(t,\cdot)$ to be a compact set of $\mathbb{R}^N$,
for any $t\geqslant0$.

We recall that equations of the type \eqref{nls} arise in many different contexts:
Nonlinear Optics, Quantum Mechanics, Hydrodynamics, etc., and that, for instance,
in Quantum Mechanics the main interest concerns the case in which
$\operatorname{Re}(\mathbf{a})>0$, $\operatorname{Im}(\mathbf{a})=0$ (here and in which follows
 $\operatorname{Re}(\mathbf{a})$ is the real part of the complex number $\mathbf{a}$ and
$\operatorname{Im}(\mathbf{a})$ is its imaginary part) and that in Nonlinear
Optics the $t$ does not represent time but the main scalar variable
which appears in the propagation of the wave guide direction
(see \cite[p.7]{ak},  \cite[p.517]{MR2169020}). Sometimes equations
of the type \eqref{nls} are named as Gross-Pitaevski{\u\i} type of equations
in honor of two famous papers by those authors in 1961 (\cite{MR0128907}
and \cite{pit}). For some physical details and many references,
 we refer the reader to the general presentations made in the books
\cite{MR2040621,MR2002047,MR2000f:35139}.

In most of the papers on equations of the type \eqref{nls},
it is assumed that $m=3$ (the so called \emph{cubic case}).
Nevertheless there are applications in which the general case $m>0$ is of 
interest.  For instance, it is the case of the so called 
\emph{non-Kerr type equations}
arising in the study of \emph{optical solitons} (see, e.g., \cite[p.14]{ak},
and following).

The case $m\in (0,1)$ has been studied before by other authors but under
different points of view: some explicit self-similar solutions (the so called
\emph{algebraic solitons}) can be found in \cite{MR2042347}
(see also \cite[p.33]{ak}). We also mention here the series of interesting
papers by Rosenau and co-authors (\cite{PhysRevLett.101.264101,MR2756172})
in which \emph{sharp localized} solutions are also considered with other type
of statements and methods.

We also mention that the case $\operatorname{Re}(\mathbf{a})>0$
(which corresponds to the dissipative case, also called defocusing or repulsive
case, when $\operatorname{Im}(\mathbf{a})=0)$ must be well distinguished of the
so called attractive problem (or also focusing case) in which it is assumed
that $\operatorname{Re}(\mathbf{a})<0$ (and $\operatorname{Im}(\mathbf{a})=0)$.
See, e.g., \cite{MR2040621,MR2002047,MR2000f:35139} and their references).

The case of complex potentials with certain types of singularities,
i.e. corresponding to the choice $\operatorname{Im}(\mathbf{a})\ne0$,
has been previously considered by several authors, and arises in many
different situations (see, for instance,
\cite{MR80i:35135,MR2765425,MR2000k:35256,MR1828819} and the references therein).

Here we assume that the datum $\mathbf{f}$ is not zero and represents some other
physical magnitude which may arise in the possible coupling with some
different phenomenon: see the different chapters of Part IV of the
book \cite{MR2000f:35139}, the interaction phenomena between long waves
and short waves (\cite{MR0463715,MR2339808,urr} and their references), etc.

Obviously, the property of the compactness of the support of
$\mathbf{u}(t,\cdot)$ requires the assumption that ``the support'' of the datum
function $\mathbf{f}(t,\cdot)$ is a compact set of $\mathbb{R}^N$,
for a.e. $t>0$. Because of that, the qualitative property we consider
in this paper can be understood as a ``finite speed of propagation
property'' typical of linear wave equations. We point out that our
treatment is very different than other ``propagation properties''
studied previously in the literature for Schr\"odinger equations which
are formulated in terms of the spectrum of the solutions. See, e.g.,
the so called Anderson localization (\cite{MR0129917}), \cite{MR797050}, etc.

One of the main reasons of the study of \emph{sharp localized}
 solutions arises from the fact that, if we assume for the moment
$\mathbf{f}\equiv\mathbf{0}$, then
\[
\frac{\partial}{\partial t}|\mathbf{u}|^2+\operatorname{div}\mathbf{J
}=2\operatorname{Im}(\mathbf{a})|\mathbf{u}|^{m+1},
\]
where
\[
\mathbf{J}:=\big(\mathbf{u}\overline{\nabla\mathbf{u}}
-\overline{\mathbf{u}}\nabla\mathbf{u}\big)
=-2\operatorname{Re}(\mathbf{i} \overline{\mathbf{u}} \nabla\mathbf{u}),
\]
($\overline{\mathbf{u}}$ denotes the conjugate of the complex function $\mathbf{u}$)
 and so we get (at least formally) that
\[
\frac12\frac{\mathrm{d}}{\mathrm{d} t}\int_{\mathbb{R}^N}|\mathbf{u}(t,x)|^2\mathrm{d} x
=\operatorname{Im}(\mathbf{a})\int_{\mathbb{R}^N}|\mathbf{u}(t,x)|^{m+1}\mathrm{d} x.
\]
Note that if $\operatorname{Im}(\mathbf{a})\neq0$ then there is no mass conservation.
 For instance, this is the case studied by \cite{MR2765425} where they prove
that actually the solution vanishes after a finite time, once that $m\in(0,1)$.
More generally, it is easy to see that the two following conservation laws hold,
once $a\in\mathbb{R}$ and $\mathbf{f}\equiv\mathbf{0}$:
if $\mathbf{u}(t)\in\mathbf{H^1}(\mathbb{R}^N)\cap\mathbf{L^{m+1}}(\mathbb{R}^N)$
then we have the mass conservation
$\frac{\mathrm{d}}{\mathrm{d} t}\|\mathbf{u}(t)\|_{\mathbf{L^2}(\mathbb{R}^N)}^2=0$;
 moreover, if $\mathbf{u}(t)\in\mathbf{H^2}(\mathbb{R}^N)\cap\mathbf{L^{2m}}
(\mathbb{R}^N)$ then $\mathbf{u}(t)\in\mathbf{L^{m+1}}(\mathbb{R}^N)$
and we have conservation of energy $\frac{\mathrm{d}}{\mathrm{d} t}E\big(\mathbf{u}(t)\big)=0$, where
\[
E\big(\mathbf{u}(t)\big)=\frac12\|\nabla\mathbf{u}(t)\|_{\mathbf{L^2}(\mathbb{R}^N)}^2
+\frac{a}{m+1}\|\mathbf{u}(t)\|_{\mathbf{L^{m+1}}(\mathbb{R}^N)}^{m+1}.
\]
Indeed, in the first case, $\Delta\mathbf{u}(t)\in\mathbf{H^{-1}}(\mathbb{R}^N)$ and
$|\mathbf{u}(t)|^{-(1-m)}\mathbf{u}(t)\in\mathbf{L^\frac{m+1}{m}}(\mathbb{R}^N)$.
It follows from the equation \eqref{nls} that
$\frac{\partial\mathbf{u}(t)}{\partial t}\in\mathbf{H^{-1}}(\mathbb{R}^N)
 +\mathbf{L^\frac{m+1}{m}}(\mathbb{R}^N)$ and since
$\left(\mathbf{H^1}(\mathbb{R}^N)\cap\mathbf{L^{m+1}}(\mathbb{R}^N)\right)^\star
=\mathbf{H^{-1}}(\mathbb{R}^N)+\mathbf{L^\frac{m+1}{m}}(\mathbb{R}^N)$,
it follows that we may take the duality product of equation~\eqref{nls}
with $\mathbf{i}\mathbf{u}(t)$, from which the mass conservation follows. In the same way,
since $\mathbf{u}(t)\in\mathbf{L^2}(\mathbb{R}^N)\cap\mathbf{L^{2m}}(\mathbb{R}^N)$ and
$0<m<1$, we get that $\mathbf{u}(t)\in\mathbf{L^{m+1}}(\mathbb{R}^N)$.
We also easily have that $\Delta\mathbf{u}(t)\in\mathbf{L^2}(\mathbb{R}^N)$ and
$|\mathbf{u}(t)|^{-(1-m)}\mathbf{u}(t)\in\mathbf{L^2}(\mathbb{R}^N)$. It follows
from the equation \eqref{nls} that
$\frac{\partial\mathbf{u}(t)}{\partial t}\in\mathbf{L^2}(\mathbb{R}^N)$ and so we may
take the duality product of equation~\eqref{nls} with
$\frac{\partial\mathbf{u}(t)}{\partial t}$, from which the conservation of energy follows.

Like in the pioneering study by Schr\"odinger, the condition
$\operatorname{Im}(\mathbf{a})=0$ implies that $|\mathbf{u}|^2$ represents a probability density,
and so the study of
\emph{sharp localized solutions} becomes very relevant
(recall the Heisenberg Uncertainty Principle). As we will show here
(sequel of previous papers by the authors, \cite{MR2214595,MR2876246}),
if $m\in(0,1)$, under suitable conditions on the coefficient $\mathbf{a}$
(for instance for
$\operatorname{Re}(\mathbf{a})>0$ and $\operatorname{Im}(\mathbf{a})=0)$, it is possible
to get some estimates on the support of solutions $\mathbf{u}(t,x)$ showing that
the probability $|\mathbf{u}(t,x)|^2$ to localize a particle is zero outside of a
compact set of $\mathbb{R}^N$.

The natural structure for searching self-similar solutions is based on the
transformation $\lambda\longmapsto\mathbf{u_\lambda}$, where for $\lambda>0$,
$\mathbf{p}\in\mathbb{C}$ and $\mathbf{u}\in\mathbf{C}\big((0,\infty);\mathbf{L^1_\mathrm{loc}}
(\mathbb{R}^N)\big)$, we define
\begin{equation} \label{ula}
\mathbf{u_\lambda}(t,x)=\lambda^{-\mathbf{p}}\mathbf{u}(\lambda^2t,\lambda x), \quad
 \forall t>0, \text{ for a.e. } x\in\mathbb{R}^N.
\end{equation}
Recall that since $\mathbf{p}\in\mathbb{C}$,  it follows that
$\lambda^{\mathbf{p}}:=\mathbf{e}^{\mathbf{p}\ln\lambda}
=e^{\operatorname{Re}(\mathbf{p})\ln\lambda}\mathbf{e}^{\mathbf{i}\operatorname{Im}
(\mathbf{p})\ln\lambda}=\lambda^{\operatorname{Re}(\mathbf{p})}\mathbf{e}^{\mathbf{i}\operatorname{Im}
(\mathbf{p})\ln\lambda}$ and that
$|\lambda^{\mathbf{p}}|=\lambda^{\operatorname{Re}(\mathbf{p})}$. Our main assumption on
the datum $\mathbf{f}$ is that
\begin{equation} \label{fla}
\mathbf{f}(t,x)=\lambda^{-(\mathbf{p}-2)}\mathbf{f}(\lambda^2t,\lambda x), \quad
\forall\lambda>0,
\end{equation}
for some $\mathbf{p}\in\mathbb{C}$, for any $t>0$ and almost every $x\in\mathbb{R}^N$,
or equivalently, that
\begin{equation} \label{eqprof}
\mathbf{f}(t,x)=t^{\frac{\mathbf{p}-2}{2}}\mathbf{F}\big(\frac{x}{\sqrt t}\big),
\end{equation}
for any $t>0$ and almost every $x\in\mathbb{R}^N$, where $\mathbf{F}=\mathbf{f}(1)$.
It is easy to build functions $\mathbf{f}$ satisfying \eqref{fla}. Indeed, for any
given function $\mathbf{F}$, we define $\mathbf{f}$ by \eqref{eqprof}.
Then $\mathbf{f}(1)=\mathbf{F}$ and $\mathbf{f}$ satisfies \eqref{fla}. Finally, if we assume
$\operatorname{Re}(\mathbf{p})=\frac2{1-m}$ then a direct calculation show that if
$\mathbf{u}$ is a solution to~\eqref{nls} then for any $\lambda>0$, $\mathbf{u_\lambda}$
is also a solution to~\eqref{nls}, and conversely.

We easily check that if $\mathbf{u}$ satisfies the invariance property
$\mathbf{u}=\mathbf{u_\lambda}$, for any $\lambda>0$, then
\begin{equation} \label{eqprou}
\mathbf{u}(t,x)=t^{\mathbf{p}/2}\mathbf{U}\Big(\frac{x}{\sqrt t}\Big),
\end{equation}
for any $t>0$ and almost every $x\in\mathbb{R}^N$, where $\mathbf{U}=\mathbf{u}(1)$.
Thus, we arrive to the following notion:

\begin{definition} \label{defselsim} \rm
Let $0<m<1$, let $\mathbf{f}\in\mathbf{C}\big((0,\infty);\mathbf{L^2_\mathrm{loc}}
(\mathbb{R}^N)\big)$ satisfies \eqref{fla} and let $\mathbf{p}\in\mathbb{C}$ be such that
$\operatorname{Re}(\mathbf{p})=\frac2{1-m}$. A solution $\mathbf{u}$ of \eqref{nls}
is said to be \emph{self-similar} if
$\mathbf{u}\in\mathbf{C}\big((0,\infty);\mathbf{L^2_\mathrm{loc}}(\mathbb{R}^N)\big)$
and if for any $\lambda>0$, $\mathbf{u_\lambda}=\mathbf{u}$, where $\mathbf{u_\lambda}$
is defined by \eqref{ula}. In this cases, $\mathbf{u}(1)$ is called the
 \emph{profile} of $\mathbf{u}$ and is denoted by $\mathbf{U}$.
\end{definition}

It follows from equation \eqref{nls} and \eqref{eqprou} that $\mathbf{U}$ satisfies
\begin{equation} \label{U}
-\Delta\mathbf{U} + \mathbf{a}|\mathbf{U}|^{-(1-m)}\mathbf{U} - \frac{\mathbf{i}\mathbf{p}}{2}\mathbf{U} + \frac{\mathbf{i}}{2}x.\nabla\mathbf{U} = -\mathbf{F},
\end{equation}
in $\mathscr{D}'(\mathbb{R}^N)$, where $\mathbf{F}=\mathbf{f}(1)$.
Conversely, if $\mathbf{U}\in\mathbf{L^2_\mathrm{loc}}(\mathbb{R}^N)$ verifies \eqref{U},
in $\mathscr{D}'(\mathbb{R}^N)$, then the function $\mathbf{u}$ defined by
\eqref{eqprou} belongs to
$\mathbf{C}\big((0,\infty);\mathbf{L^2_\mathrm{loc}}(\mathbb{R}^N)\big)$
 and is a self-similar solution to~\eqref{nls}, where $\mathbf{f}$ is defined
by \eqref{eqprof} and satisfies \eqref{fla}. It is useful to introduce
the unknown transformation
\begin{equation} \label{gU}
\mathbf{g}(x)=\mathbf{U}(x)\mathbf{e}^{-\mathbf{i}|x|^2/8}.
\end{equation}
Then for any $m\in\mathbb{R}$, $\mathbf{p}\in\mathbb{C}$ and
$\mathbf{U}\in\mathbf{L^2_\mathrm{loc}}(\mathbb{R}^N)$, $\mathbf{U}$ is a
solution to \eqref{U} in $\mathscr{D}'(\mathbb{R}^N)$  if and only if
$\mathbf{g}\in\mathbf{L^2_\mathrm{loc}}(\mathbb{R}^N)$ is a solution to
\begin{equation} \label{g}
-\Delta\mathbf{g} + \mathbf{a}|\mathbf{g}|^{-(1-m)}\mathbf{g} - \mathbf{i}\frac{N+2\mathbf{p}}{4}\mathbf{g} - \frac1{16}|x|^2\mathbf{g}
= -\mathbf{F}\mathbf{e}^{-\mathbf{i}\frac{|\cdot|^2}{8}},
\end{equation}
in $\mathscr{D}'(\mathbb{R}^N)$. It will be convenient to study~\eqref{g} instead
of~\eqref{U}. Indeed, formally, if we multiply \eqref{g} by $\pm\overline{\mathbf{g}}$ or
$\pm\mathbf{i}\overline{\mathbf{g}}$, integrate by parts and take the real part, one obtains
some positive or negative quantities. But the same method applied to
\eqref{U} gives (at least directly) nothing because of the term $\mathbf{i} x.\nabla\mathbf{U}$.

Notice that if $\mathbf{p}\in\mathbb{C}$ is such that $\operatorname{Re}(\mathbf{p})=\frac2{1-m}$
and if $\mathbf{f}\in\mathbf{C}\big((0,\infty);\mathbf{L^2}(\mathbb{R}^N)\big)$
and satisfies \eqref{fla} with $\mathbf{f}(t_0)$ compactly supported for
some $t_0>0$, then it follows from~\eqref{fla} that for any $t>0$,
$\operatorname{supp}\mathbf{f}(t)$ is compact. Moreover, from~\eqref{eqprou}, if $\mathbf{u}$
is a self-similar solution of \eqref{nls} and if $\operatorname{supp}\mathbf{U}$ is compact
then for any $t>0$, $\operatorname{supp}\mathbf{u}(t)$ is compact. As a matter of fact,
it is enough to have that $\mathbf{u}(t_0)$ is compactly supported for
some $t_0>0$ to have that $\mathbf{u}$ satisfies \eqref{thmmain1} below and
$\operatorname{supp}\mathbf{u}(t)$ is compact, for any $t>0$. Indeed, $\mathbf{U}=\mathbf{u}(1)$
satisfies \eqref{U} and by \eqref{eqprou}, $\operatorname{supp}\mathbf{U}$ and $\operatorname{supp}\mathbf{u}(t)$
are compact  for any $t>0$. Let $\mathbf{g}$ be defined by~\eqref{gU}.
Then $\mathbf{g}$ is a solution compactly supported to~\eqref{g} and it follows
the results of Section~\ref{eus} below that
$\mathbf{g}\in\mathbf{H^2_\mathrm{c}}(\mathbb{R}^N)$. By~\eqref{gU},
we obtain that $\mathbf{U}\in\mathbf{H^2_\mathrm{c}}(\mathbb{R}^N)$
and we deduce easily from \eqref{eqprou} that $\mathbf{u}$ satisfies
 \eqref{thmmain1}.

The main result of this paper  reads as follows.

\begin{theorem} \label{thmmain}
Let $0<m<1$, let $\mathbf{a}\in\mathbb{C}$ be such that $\operatorname{Im}(\mathbf{a})\leqslant0$.
If $\operatorname{Re}(\mathbf{a})\leqslant0$ then assume further that
$\operatorname{Im}(\mathbf{a})<0$. Let $\mathbf{p}\in\mathbb{C}$ be such that
$\operatorname{Re}(\mathbf{p})=\frac2{1-m}$ and let
$\mathbf{f}\in\mathbf{C}\big((0,\infty);\mathbf{L^2}(\mathbb{R}^N)\big)$
satisfying \eqref{fla}. Assume also that $\operatorname{supp}\mathbf{f}(1)$ is compact.
\begin{enumerate}
	\item
	\label{thmmaina}
		If $\|\mathbf{f}(1)\|_{\mathbf{L^2}(\mathbb{R}^N)}$ is small enough
then there exists a self-similar solution
		\begin{equation} 		\label{thmmain1}
			\mathbf{u}\in\boldsymbol C\big((0,\infty);\mathbf{H^2}(\mathbb{R}^N)\big)\cap\mathbf{C^1}\big((0,\infty);\mathbf{H^1}(\mathbb{R}^N)\big)\cap\mathbf{C^2}\big((0,\infty);\mathbf{L^2}(\mathbb{R}^N)\big)
		\end{equation}
		to~\eqref{nls} such that for any $t>0$, $\operatorname{supp}\mathbf{u}(t)$ is compact. In particular, $\mathbf{u}$ is a strong solution and verifies~\eqref{nls} for any $t>0$ in
		$\mathbf{L^2}(\mathbb{R}^N)$, and so almost everywhere in $\mathbb{R}^N$.
	\item
	\label{thmmainb}
		Let $R>0$. For any $\varepsilon>0$, there exists $\delta_0=\delta_0(R,\varepsilon,|\mathbf{a}|,|\mathbf{p}|,N,m)>0$ satisfying the following property$:$ if
		$\operatorname{supp}\mathbf{f}(1)\subset\overline B(0,R)$ and if $\|\mathbf{f}(1)\|_{\mathbf{L^2}(\mathbb{R}^N)}\leqslant\delta_0$ then the profile $\mathbf{U}$ of the solution obtained above verifies
		$\operatorname{supp}\mathbf{U}\subset K(\varepsilon)\subset\overline B(0,R+\varepsilon)$, where
		\[
			K(\varepsilon)=\Big\{x\in\mathbb{R}^N;\; \exists y\in\operatorname{supp}\mathbf{f}(1) \text{ such that } |x-y|\leqslant\varepsilon \Big\},
		\]
		which is compact.
	\item
	\label{thmmainc}
		Let $R_0>0$. Assume now further that $\operatorname{Re}(\mathbf{a})>0$,
$\operatorname{Im}(\mathbf{a})=0$ and
		\[
			4\operatorname{Im}(\mathbf{p})+2\sqrt{4\operatorname{Im}^2(\mathbf{p})+2}
\geqslant R_0^2.
		\]
		Then the solution is unique in the set of functions
$\mathbf{C}\big((0,\infty);\mathbf{L^2_\mathrm{c}}(\mathbb{R}^N)\big)$
whose profile $\mathbf{V}$ satisfies
		$\operatorname{supp}\mathbf{V}\subset\overline B(0,R_0)$.
\end{enumerate}
\end{theorem}

In contrast with many other papers on self-similar solutions of equations
dealing with exponents $m>1$ (see
\cite{MR99d:35149,MR99f:35185,MR1745480} and their references),
in this paper we do not prescribe any initial data $\mathbf{u}(0)$ to \eqref{nls}
since we are only interested on any solution $\mathbf{u}(t)$ by an external source
$\mathbf{f}(t)$ compactly supported. Moreover, we point out that if
$\mathbf{u}\in\mathbf{C}\big([0,\infty);\mathbf{L^q}(\mathbb{R}^N)\big)$
is a self-similar solution to~\eqref{nls}, for some $0<q\leqslant\infty$,
then necessarily $\mathbf{u}(0)=\mathbf{0}$. Indeed, with help of~\eqref{eqprou},
we easily show that $\mathbf{U}\in\mathbf{L^q}(\mathbb{R}^N)$ and that for any $t>0$,
$\|\mathbf{u}(t)\|_{\mathbf{L^q}(\mathbb{R}^N)}
=t^{\frac{1}{1-m}+\frac{N}{2q}}\|\mathbf{U}\|_{\mathbf{L^q}(\mathbb{R}^N)}$,
implying necessarily that $\mathbf{u}(0)=\mathbf{0}$. On the other hand, notice that if
$\mathbf{u}\in\mathbf{C}\big([0,\infty);\mathscr{D}'(\mathbb{R}^N)\big)$
is a self-similar solution to~\eqref{nls} then one cannot expect to have
$\mathbf{u}(0)\in\mathbf{L^q}(\mathbb{R}^N)$, unless
$\mathbf{u}(0)=\mathbf{0}$. Indeed, we would have $\mathbf{u_\lambda}(0)=\mathbf{u}(0)$
in $\mathbf{L^q}(\mathbb{R}^N)$ and for any $\lambda>0$,
$\|\mathbf{u}(0)\|_{\mathbf{L^q}(\mathbb{R}^N)}=\lambda^{\frac{2}{1-m}
+\frac{N}{q}}\|\mathbf{u}(0)\|_{\mathbf{L^q}(\mathbb{R}^N)}$
and again we deduce that necessarily $\mathbf{u}(0)=\mathbf{0}$. More generally,
the set of functions $\mathbf{u}$ satisfying the invariance property,
\[
\forall\lambda>0, \text{ for a.e. } x\in\mathbb{R}^N, \quad
\mathbf{u_\lambda}(x):=\lambda^{-\mathbf{p}}\mathbf{u}(\lambda x)=\mathbf{u}(x),
\]
and lying in $\mathbf{L^q}(\mathbb{R}^N)$ is reduced to $\mathbf{0}$.

In the special case of self-similar solution, the above arguments show
that if $\mathbf{f}\equiv\mathbf{0}$, $a\in\mathbb{R}$ and
$\mathbf{u}\in\mathbf{C}\big((0,\infty);\mathbf{L^2_\mathrm{c}}(\mathbb{R}^N)\big)$
then necessarily $\mathbf{u}(t)=0$, for any $t>0$. Indeed, if
$\mathbf{u}\in\mathbf{C}\big((0,\infty);\mathbf{L^2_\mathrm{c}}(\mathbb{R}^N)\big)$
is a self-similar solution to~\eqref{nls} then its profile $\mathbf{U}$
belongs to $\mathbf{L^2}(\mathbb{R}^N)$ and
$\mathbf{u}\in\boldsymbol C^2((0;\infty)\times\mathbb{R}^N)$ (see Section~\ref{eus} below).
So for any $t>0$, we can multiply the above equation by $-\mathbf{i}\overline{\mathbf{u}}(t)$,
integrate by parts over $\mathbb{R}^N$ and take the real part.
We then deduce the mass conservation,
$\frac{\mathrm{d}}{\mathrm{d} t}\|\mathbf{u}(t)\|_{\mathbf{L^2}(\mathbb{R}^N)}^2=0$, which yields with
the above identity,
\begin{gather*}
\|\mathbf{U}\|_{\mathbf{L^2}(\mathbb{R}^N)}
=\|\mathbf{u}(t)\|_{\mathbf{L^2}(\mathbb{R}^N)}=t^{\frac{1}{1-m}
+\frac{N}{4}}\|\mathbf{U}\|_{\mathbf{L^2}(\mathbb{R}^N)},
\end{gather*}
for any $t>0$. Hence the result. As a matter of fact, if $\ell\in\{0,1,2\}$
and if $\mathbf{u}\in\mathbf{C}\big((0,\infty);\mathbf{H^\ell}(\mathbb{R}^N)\big)$
is a self-similar solution
to~\eqref{nls} then one easily deduces from~\eqref{eqprou} that actually
$\lim_{t\searrow0}\|\mathbf{u}(t)\|_{\mathbf{H^\ell}(\mathbb{R}^N)}=0$.

We also mention here that our treatment of sharp localized solutions
has some indirect connections with the study of the ``unique continuation
property''. Indeed, we are showing that this property does not hold when
$m\in(0,1)$, in contrast to the case of linear and other type of nonlinear
Schr\"odinger equations (see, e.g., \cite{MR1980854,urr}).

The paper is organized as follows. In the next section, we introduce some
notation and give  general versions of the main results
(Theorems~\ref{thmsta} and \ref{thmG}). In Section~\ref{eus},
we recall some existence, uniqueness, \emph{a priori} bound and smoothness
 results of solutions to equation~\eqref{g} associated to the evolution
equation \eqref{nls}. Finally, Section \ref{proof} is devoted to the proofs
of the mentioned results, which we carry out by improving some energy
methods presented in \cite{MR2002i:35001}.

\section{Notation and general versions of the main result}
\label{not}

Before stating our main results, we will indicate here some of the notation
used throughout. For $1\leqslant p\leqslant\infty$, $p'$ is the conjugate of
 $p$ defined by $\frac1p+\frac1{p'}=1$. We denote by $\overline\Omega$ 
the closure of a nonempty subset $\Omega\subseteq\mathbb{R}^N$ and by 
$\Omega^\mathrm{c}=\mathbb{R}^N\setminus\Omega$ its complement.
 We note $\omega\Subset\Omega$ to mean that $\overline\omega\subset\Omega$ 
and that $\overline\omega$ is a compact subset of $\mathbb{R}^N$. 
Unless specified, any function lying in a functional space 
$\big(\mathbf{L^p}(\Omega)$,
$\mathbf{W^{m,p}}(\Omega)$, etc\big) is supposed to be a complex-valued 
function ($\mathbf{L^p}(\Omega;\mathbb{C})$, $\mathbf{W^{m,p}}(\Omega;\mathbb{C})$, etc).
For a functional space $\mathbf{E}\subset\mathbf{L^1_\mathrm{loc}}(\Omega;\mathbb{C})$,
 we denote by $\mathbf{E_\mathrm{c}}=\big\{\mathbf{f}\in\mathbf{E};\operatorname{supp}\mathbf{f}\Subset\Omega\big\}$.
For a Banach space $\mathbf{E}$, we denote by $\mathbf{E}^\star$ its topological dual and by
 $\langle\cdot,\cdot\rangle_{\mathbf{E}^\star,\mathbf{E}}\in\mathbb{R}$ the 
$\mathbf{E}^\star-\mathbf{E}$ duality product. In particular, for any 
$\mathbf{T}\in\mathbf{L^{p'}}(\Omega)$ and 
$\boldsymbol{\varphi}\in\mathbf{L^p}(\Omega)$ with $1\leqslant p<\infty$,
$\langle\mathbf{T},\boldsymbol{\varphi}\rangle_{\mathbf{L^{p'}}(\Omega),
\mathbf{L^p}(\Omega)}=\operatorname{Re}\int_\Omega\mathbf{T}(x)
\overline{\boldsymbol{\varphi}(x)}\mathrm{d} x$. For $x_0\in\mathbb{R}^N$ and $r>0$, 
we denote by $B(x_0,r)$ the open ball of $\mathbb{R}^N$ of center $x_0$ 
and radius $r$, by $\mathbb{S}(x_0,r)$ its boundary and by 
$\overline B(x_0,r)$ its closure. As usual, we denote by $C$ auxiliary 
positive constants, and sometimes, for positive parameters $a_1,\ldots,a_n$, 
write $C(a_1,\ldots,a_n)$ to indicate that the constant $C$ continuously 
depends only on $a_1,\ldots,a_n$ (this convention also holds for constants 
which are not denoted by ``$C$'').

Now, we state the precise notion of solution.

\begin{definition} \label{defsols} \rm
Let $\Omega$ be a nonempty bounded open subset of $\mathbb{R}^N$, 
let $(\mathbf{a},\mathbf{b},\mathbf{c})\in\mathbf{\mathbb{C}^3}$, let $0<m\leqslant1$ and let 
$\mathbf{G}\in\mathbf{L^1_\mathrm{loc}}(\Omega)$.
\begin{enumerate}
   \item
   \label{def1}
    We say that $\mathbf{g}$ is a {\it local very weak solution} to
	\begin{equation}
		\label{eq1}
			-\Delta\mathbf{g} + \mathbf{a}|\mathbf{g}|^{-(1-m)}\mathbf{g} + \mathbf{b}\mathbf{g} + \mathbf{c} x.\nabla\mathbf{g} = \mathbf{G},
	\end{equation}
    in $\mathscr{D}'(\Omega)$, if $\mathbf{g}\in\mathbf{L^2_\mathrm{loc}}(\Omega)$ and if
	\begin{equation}
		\label{defsol0}
			\langle\mathbf{g},-\Delta\boldsymbol\varphi\rangle_{\mathscr{D}'(\Omega),\mathscr{D}(\Omega)}
+\langle\mathbf{H}(\mathbf{g}),\boldsymbol\varphi\rangle_{\mathscr{D}'(\Omega),\mathscr{D}(\Omega)}
			=\langle\mathbf{G},\boldsymbol\varphi\rangle_{\mathscr{D}'(\Omega),\mathscr{D}(\Omega)},
	\end{equation}
    for any $\boldsymbol\varphi\in\mathscr{D}(\Omega)$, where
	\begin{equation}
	\label{H1}
		\mathbf{H}(\mathbf{h})=\mathbf{a}|\mathbf{h}|^{-(1-m)}\mathbf{h} + \mathbf{b}\mathbf{h} + \mathbf{c} x.\nabla\mathbf{h},
	\end{equation}
    for any $\mathbf{h}\in\mathbf{L^2_\mathrm{loc}}(\Omega)$.
If, in addition, $\mathbf{g}\in\mathbf{L^2}(\Omega)$ then we say that $\mathbf{g}$ is
a {\it global very weak solution} to~\eqref{eq1}.
   \item
   \label{def2}
    We say that $\mathbf{g}$ is a {\it local weak solution} to~\eqref{eq1} in
$ \mathscr{D}'(\Omega)$, if $\mathbf{g}\in\mathbf{H^1_\mathrm{loc}}(\Omega)$ and if
	\begin{equation}
		\label{defsol1}
			\langle\nabla\mathbf{g},\nabla\boldsymbol\varphi\rangle_{\mathscr{D}'(\Omega),
\mathscr{D}(\Omega)}+\langle\mathbf{H}(\mathbf{g}),\boldsymbol\varphi\rangle_{\mathscr{D}'(\Omega),\mathscr{D}(\Omega)}
			=\langle\mathbf{G},\boldsymbol\varphi\rangle_{\mathscr{D}'(\Omega),\mathscr{D}(\Omega)},
	\end{equation}
    for any $\boldsymbol\varphi\in\mathscr{D}(\Omega)$, where $\mathbf{H}\in\boldsymbol C
\big(\mathbf{L^2_\mathrm{loc}}(\Omega);\mathscr{D}'(\Omega)\big)$ is defined by \eqref{H1}.
   \item
   \label{def3}
    We say that $\mathbf{g}$ is a {\it local weak solution} to
	\begin{equation}
		\label{eq2}
			-\Delta\mathbf{g} + \mathbf{a}|\mathbf{g}|^{-(1-m)}\mathbf{g} + \mathbf{b}\mathbf{g} + \mathbf{c}|x|^2\mathbf{g} = \mathbf{G},
	\end{equation}
    in $\mathscr{D}'(\Omega)$, if $\mathbf{g}\in\mathbf{H^1_\mathrm{loc}}(\Omega)$ and if
$\mathbf{g}$ satisfies \eqref{defsol1}, for any $\boldsymbol\varphi\in\mathscr{D}(\Omega)$, where
	\begin{equation}
	\label{H2}
		\mathbf{H}(\mathbf{h})=\mathbf{a}|\mathbf{h}|^{-(1-m)}\mathbf{h} + \mathbf{b}\mathbf{h} + \mathbf{c}|x|^2\mathbf{h},
	\end{equation}
    for any $\mathbf{h}\in\mathbf{H^1_\mathrm{loc}}(\Omega)$.
   \item
   \label{def4}
    Assume further that $\mathbf{G}\in\mathbf{L^2}(\Omega)$. We say that $\mathbf{g}$
is a {\it global weak solution} to \eqref{eq1} and
	\begin{equation}
		\label{dir}
			 \mathbf{g}_{|\Gamma}=\mathbf{0},
	\end{equation}
    in $\mathbf{L^2}(\Omega)$, if $\mathbf{g}\in\mathbf{H^1_0}(\Omega)$ and if
	\begin{equation}
		\label{defsol2}
			\langle\nabla\mathbf{g},\nabla\mathbf{v}\rangle_{\mathbf{L^2}(\Omega),
\mathbf{L^2}(\Omega)}+\langle\mathbf{H}(\mathbf{g}),\mathbf{v}\rangle_{\mathbf{L^2}(\Omega),
\mathbf{L^2}(\Omega)}
			=\langle\mathbf{G},\mathbf{v}\rangle_{\mathbf{L^2}(\Omega),\mathbf{L^2}(\Omega)},
	\end{equation}
    for any $\mathbf{v}\in\mathbf{H^1_0}(\Omega)$, where $\mathbf{H}\in\boldsymbol
C\big(\mathbf{H^1}(\Omega);\mathbf{L^2}(\Omega)\big)$ is defined by \eqref{H1}.
Note that $\Delta\mathbf{g}\in\mathbf{L^2}(\Omega)$, so
    that equation \eqref{eq1} makes sense in $\mathbf{L^2}(\Omega)$ and almost
everywhere in $\Omega$.
   \item
   \label{def5}
    Assume further that $\mathbf{G}\in\mathbf{L^2}(\Omega)$. We say that $\mathbf{g}$ is a
 {\it global weak solution} to \eqref{eq2} and \eqref{dir}, in
$\mathbf{L^2}(\Omega)$, if
    $\mathbf{g}\in\mathbf{H^1_0}(\Omega)$ and if $\mathbf{g}$ satisfies \eqref{defsol2},
for any $\mathbf{v}\in\mathbf{H^1_0}(\Omega)$, where $\mathbf{H}\in\boldsymbol
C\big(\mathbf{L^2}(\Omega);\mathbf{L^2}(\Omega)\big)$ is
    defined by \eqref{H2}. Note that $\Delta\mathbf{g}\in\mathbf{L^2}(\Omega)$,
so that equation \eqref{eq2} makes sense in $\mathbf{L^2}(\Omega)$ and
almost everywhere in $\Omega$.
\end{enumerate}
\end{definition}


In the above definition, $\Gamma$ denotes the boundary of $\Omega$ 
and $\boldsymbol C(\Omega)=\mathbf{C^0}(\Omega)$ is the space of complex-valued 
functions which are defined and continuous over $\Omega$. 
Obviously, for $k\in\mathbb{N}$, $\mathbf{C^k}(\Omega)$ denotes the space of complex-valued 
functions lying in $\boldsymbol C(\Omega)$ and having all derivatives of 
order lesser or equal than $k$ belonging to $\boldsymbol C(\Omega)$.

\begin{remark} \label{rmkdefsols} \rm
Here are some comments about Definition~\ref{defsols}.
\begin{enumerate}
 \item
  \label{rmkdefsols1}
	Note that in Definition~\ref{defsols}, any global weak solution is a local 
weak and a global very weak solution, and any local weak or global very
	weak solution is a local very weak solution.
 \item
  \label{rmkdefsols2}
	Assume that $\Omega$ has a $C^{0,1}$ boundary. Let $\mathbf{g}\in\mathbf{H^1}(\Omega)$. 
Then boundary condition $\mathbf{g}_{|\Gamma}=0$ makes sense in the sense of
	the trace $\boldsymbol\gamma(\mathbf{g})=\mathbf{0}$. Thus, it is well-known that 
$\mathbf{g}\in\mathbf{H^1_0}(\Omega)$ if and only if $\boldsymbol\gamma(\mathbf{g})=\mathbf{0}$. 
If furthermore $\Omega$ has a $C^1$
	boundary and if $\mathbf{g}\in\boldsymbol C(\overline\Omega)\cap\mathbf{H^1_0}(\Omega)$ 
then for any $x\in\Gamma$, $\mathbf{g}(x)=\mathbf{0}$ (Theorem~9.17, p.288, in \cite{MR2759829}).
	Finally, if $\mathbf{g}\not\in\boldsymbol C(\overline\Omega)$ and $\Omega$ has not a 
$C^{0,1}$ boundary, the condition $\mathbf{g}_{|\Gamma}=\mathbf{0}$ does not
	make sense and, in this case, has to be understood as 
$\mathbf{g}\in\mathbf{H^1_0}(\Omega)$.
 \item
  \label{rmkdefsols3}
	Let $0<m\leqslant1$ and let $\mathbf{z}\in\mathbb{C}\setminus\{\mathbf{0}\}$. 
Since $\left||\mathbf{z}|^{-(1-m)}\mathbf{z}\right|=|\mathbf{z}|^m$, it is understood in 
Definition~\ref{defsols} that
	$\left||\mathbf{z}|^{-(1-m)}\mathbf{z}\right|=0$ when $\mathbf{z}=\mathbf{0}$.
\end{enumerate}
\end{remark}

The main results of this section are the two following theorems implying, 
as a special case, the statement of Theorem~\ref{thmmain}.

\begin{theorem}\label{thmsta}
Let $\Omega\subset B(0,R)$ be a nonempty bounded open subset of $\mathbb{R}^N$, 
let $0<m<1$, let $(\mathbf{a},\mathbf{b},\mathbf{c})\in\mathbb{C}^3$ be such that 
$\operatorname{Im}(\mathbf{a})\leqslant0$,
$\operatorname{Im}(\mathbf{b})<0$ and $\operatorname{Im}(\mathbf{c})\leqslant0$. 
If $\operatorname{Re}(\mathbf{a})\leqslant0$ then assume further that 
$\operatorname{Im}(\mathbf{a})<0$. Then there exist three positive constants $C=C(N,m)$,
$L=L(R,|\mathbf{a}|,|\mathbf{p}|,N,m)$ and $M=M(R,|\mathbf{a}|,|\mathbf{p}|,N,m)$ satisfying the following 
property$:$ let $\mathbf{G}\in\mathbf{L^1_\mathrm{loc}}(\Omega)$, 
let $\mathbf{g}\in\mathbf{H^1_\mathrm{loc}}(\Omega)$ be any local weak solution 
to~\eqref{eq2}, let $x_0\in\Omega$ and let $\rho_0>0$. 
If $\rho_0>\operatorname{dist}(x_0,\Gamma)$ then assume further that
$\mathbf{g}\in\mathbf{H^1_0}(\Omega)$. Assume now that 
$\mathbf{G}_{|\Omega\cap B(x_0,\rho_0)}\equiv\mathbf{0}$. Then 
$\mathbf{g}_{|\Omega\cap B(x_0,\rho_\mathrm{max})}\equiv\mathbf{0}$, where
\begin{equation} \label{thmsta1}
\begin{aligned}
 \rho_\mathrm{max}^\nu
&=\Big(\rho_0^\nu-CM^2\max\{1,\frac{1}{L^2}\}\max\{\rho_0^{\nu-1},1\} \\
&\quad \times\min_{\tau\in(\frac{m+1}{2},1]}\big\{\frac{E(\rho_0)^{\gamma(\tau)}
  \max\{b(\rho_0)^{\mu(\tau)},b(\rho_0)^{\eta(\tau)}\}}{2\tau-(1+m)}\big\}\Big)_+,
\end{aligned}
\end{equation}
where
\begin{gather*}
E(\rho_0)=\|\nabla\mathbf{g}\|_{\mathbf{L^2}(\Omega\cap B(x_0,\rho_0))}^2,	\quad
b(\rho_0)=\|\mathbf{g}\|_{\mathbf{L^{m+1}}(\Omega\cap B(x_0,\rho_0))}^{m+1},	 \\
k=2(1+m)+N(1-m), \quad	\nu=\frac{k}{m+1}>2,
\end{gather*}
and where
\begin{gather*}
\gamma(\tau)=\frac{2\tau-(1+m)}{k}\in(0,1), \quad
\mu(\tau)=\frac{2(1-\tau)}{k}, \quad
\eta(\tau)=\frac{1-m}{1+m}-\gamma(\tau)>0.
\end{gather*}
for any $\tau\in(\frac{m+1}{2},1]$.
\end{theorem}

Here and in what follows, $r_+=\max\{0,r\}$ denotes the positive part of 
the real number $r$.

\begin{remark} \label{rmkthmsta} \rm
If the solution is too ``large'', it may happen that $\rho_\mathrm{max}=0$ 
and so the above result is not consistent. A sufficient condition to 
observe a localizing effect is that the solution is small enough, in a suitable 
sense. We give below a sufficient condition on the data $\mathbf{a}\in\mathbb{C}$, 
$\mathbf{p}\in\mathbb{C}$ and $\mathbf{G}$ to have $\rho_\mathrm{max}>0$.
\end{remark}

\begin{theorem} \label{thmG}
Let $\Omega\subset B(0,R)$ be a nonempty bounded open subset of $\mathbb{R}^N$, 
let $0<m<1$, let $(\mathbf{a},\mathbf{b},\mathbf{c})\in\mathbb{C}^3$ be such that 
$\operatorname{Im}(\mathbf{a})\leqslant0$,
$\operatorname{Im}(\mathbf{b})<0$ and $\operatorname{Im}(\mathbf{c})\leqslant0$. 
If $\operatorname{Re}(\mathbf{a})\leqslant0$ then assume further that 
$\operatorname{Im}(\mathbf{a})<0$. Let $\mathbf{G}\in\mathbf{L^1_\mathrm{loc}}(\Omega)$, 
let $\mathbf{g}\in\mathbf{H^1_\mathrm{loc}}(\Omega)$ be any local weak solution 
to \eqref{eq2}, let $x_0\in\Omega$ and let $\rho_1>0$. 
If $\rho_1>\operatorname{dist}(x_0,\Gamma)$ then assume further that
$\mathbf{g}\in\mathbf{H^1_0}(\Omega)$. Then there exist two positive constants 
$E_\star>0$ and $\varepsilon_\star>0$ satisfying the following property: let
$\rho_0\in(0,\rho_1)$ and assume that 
$\|\nabla\mathbf{g}\|_{\mathbf{L^2}(\Omega\cap B(x_0,\rho_1))}^2<E_\star$ and
\begin{equation} \label{thmG1}
\forall\rho\in(0,\rho_1), \;
\|\mathbf{G}\|_{\mathbf{L^2}(\Omega\cap B(x_0,\rho))}^2
\leqslant\varepsilon_\star(\rho-\rho_0)_+^p,
\end{equation}
where $p=\frac{2(1+m)+N(1-m)}{1-m}$. Then $\mathbf{g}_{|\Omega\cap B(x_0,\rho_0)}\equiv\mathbf{0}$.
In other words (with the notation of Theorem \ref{thmsta}),
$\rho_\mathrm{max}=\rho_0$.
 \end{theorem}

\begin{remark} \label{rmkthmF} \rm
We may estimate $E_\star$ and $\varepsilon_\star$ as
\begin{gather*}
E_\star=E_\star\Big(\|\mathbf{g}\|_{\mathbf{L^{m+1}}(B(x_0,\rho_1))}^{-1},
 \rho_1,\frac{\rho_0}{\rho_1},\frac{L}{M},N,m\Big), \\
\varepsilon_\star=\varepsilon_\star\Big(\|\mathbf{g}\|_{\mathbf{L^{m+1}}(B(x_0,
\rho_1))}^{-1},\frac{\rho_0}{\rho_1},\frac{L}{M},N,m\Big),
\end{gather*}
where $L>0$ and $M>0$ are given by Theorem~\ref{thmsta}. 
The dependence on $1/\delta$ means that if $\delta$ goes to $0$ then 
$E_\star$ and $\varepsilon_\star$ may be very large. 
Note that $p=1/\gamma(1)$, where $\gamma$ is the function defined in
Theorem~\ref{thmsta}.
\end{remark}

\section{Existence, uniqueness and smoothness}
\label{eus}

We recall the following results which are taken from other works by 
the authors \cite[Theorems 2.4, 2.6 and 2.12]{Beg-Di4}. 
Let $\Omega\subset B(0,R)$ be a nonempty bounded open subset of $\mathbb{R}^N$, 
let $0<m<1$ and let $(\mathbf{a},\mathbf{b},\mathbf{c})\in\mathbb{C}^3$  be such that
 $\operatorname{Im}(\mathbf{a})\leqslant0$, $\operatorname{Im}(\mathbf{b})<0$ and 
$\operatorname{Im}(\mathbf{c})\leqslant0$. If $\operatorname{Re}(\mathbf{a})\leqslant0$ 
then assume further that $\operatorname{Im}(\mathbf{a})<0$. 
For any $\mathbf{G}\in\mathbf{L^2}(\Omega)$, there exists at least one global weak solution
$\mathbf{g}\in\mathbf{H^1_0}(\Omega)\cap\mathbf{H^2_\mathrm{loc}}(\Omega)$ to \eqref{eq2}
 and \eqref{dir}. Moreover, if $\Omega$ has a $C^{1,1}$ boundary then 
$\mathbf{g}\in\mathbf{H^2}(\Omega)$. Finally,
\begin{equation} \label{new}
\|\mathbf{g}\|_{\mathbf{H^1}(\Omega)}\leqslant M_0(R^2+1)\|\mathbf{G}\|_{\mathbf{L^2}(\Omega)},
\end{equation}
where $M_0=M_0(|\mathbf{a}|,|\mathbf{b}|,|\mathbf{c}|)$. Finally, if $\mathbf{U}$ belongs
to $\mathbf{L^2_\mathrm{loc}}(\Omega)$ with $\mathbf{U}$ a local very weak
solution to
\[
-\Delta\mathbf{U} + \mathbf{a}|\mathbf{U}|^{-(1-m)}\mathbf{U} + \mathbf{b}\mathbf{U} + \mathbf{i} cx.\nabla\mathbf{U}
= \mathbf{F}, \quad \text{in } \mathscr{D}'(\Omega),
\]
(with any $(\mathbf{a},\mathbf{b},c)\in\mathbb{C}\times\mathbb{C}\times\mathbb{R}$) then
$\mathbf{U}\in\mathbf{H^2_\mathrm{loc}}(\Omega)$. Indeed, by the unknown transformation
described at the beginning of Section~\ref{proof} below, we are brought
back to the study of the smoothness of solutions to equation,
\[
-\Delta\mathbf{g}+\mathbf{a}|\mathbf{g}|^{-(1-m)}\mathbf{g}+(\mathbf{b}-\mathbf{i}\frac{cN}{2})\mathbf{g}-\frac{c^2}{4}|x|^2\mathbf{g}
=\mathbf{F}(x)\mathbf{e}^{-\mathbf{i} c\frac{|x|^2}{4}}, \quad\text{in } \mathscr{D}'(\Omega),
\]
for which the above smoothness result applies. Concerning the uniqueness
of solutions, we have the following result.

\begin{theorem}[Uniqueness] \label{thmuni}
Let $\Omega\subseteq\mathbb{R}^N$ be a nonempty open subset let $0<m<1$, 
let $(a,\mathbf{b},c)\in\mathbb{R}\times\mathbb{C}\times\mathbb{R}$ be such that $a>0$, 
$\operatorname{Re}(\mathbf{b})\geqslant0$ and $c\geqslant0$. 
Then for any $\mathbf{F}\in\mathbf{L^2}(\Omega)$, equation
\[
-\Delta\mathbf{U} - \mathbf{i} a|\mathbf{U}|^{-(1-m)}\mathbf{U} - \mathbf{i}\mathbf{b}\mathbf{U} + \mathbf{i} cx.\nabla\mathbf{U} 
= \mathbf{F}, \quad \text{in } \mathscr{D}'(\Omega),
\]
admits at most one global very weak solution compact with support 
$\mathbf{U}\in\mathbf{L^2_\mathrm{c}}(\Omega)$.
\end{theorem}

\begin{proof}
Let $\mathbf{U_1},\mathbf{U_2}\in\mathbf{L^2_\mathrm{c}}(\Omega)$ 
be two global very weak solutions both compactly supported to the above equation. 
By the results above, one has 
$\mathbf{U_1},\mathbf{U_2}\in\mathbf{H^2_\mathrm{c}}(\Omega)$.
 Setting $\mathbf{g_1}=\mathbf{U_1}\mathbf{e}^{-\mathbf{i} c\frac{|\cdot|^2}{4}}$ and
$\mathbf{g_2}=\mathbf{U_2}\mathbf{e}^{-\mathbf{i} c\frac{|\cdot|^2}{4}}$,
 a straightforward calculation shows that (see also the beginning of 
Section~\ref{proof} below)
$\mathbf{g_1},\mathbf{g_2}\in\mathbf{H^2_\mathrm{c}}(\Omega)$ satisfy
\[
-\Delta\mathbf{g}+\widetilde{\mathbf{a}}|\mathbf{g}|^{-(1-m)}\mathbf{g}+\widetilde{\mathbf{b}}\mathbf{g}+\widetilde c V^2\mathbf{g} 
= \widetilde{\mathbf{F}}, \text{ in } \mathbf{L^2}(\Omega),
\]
where $\widetilde{\mathbf{a}}=-\mathbf{i} a$, $\widetilde{\mathbf{b}}=-\mathbf{i}(\mathbf{b}+\frac{cN}{2})$, 
$\widetilde c=-\frac{c^2}{4}$, $V(x)=|x|$ and 
$\widetilde{\mathbf{F}}=\mathbf{F}\mathbf{e}^{-\mathbf{i} c\frac{|\cdot|^2}{4}}$. Note that,
\begin{gather*}
	\widetilde{\mathbf{a}}\neq0,	\quad	\operatorname{Re}(\widetilde{\mathbf{a}})=0,														\\
	\operatorname{Re}(\widetilde{\mathbf{a}}\,\overline{\widetilde{\mathbf{b}}})
=\operatorname{Re}\Big(a\big(\overline{\mathbf{b}+\frac{cN}{2}}\big)\Big)
=a\operatorname{Re}(\mathbf{b})+\frac12acN\geqslant0,	\\
	\operatorname{Re}\big(\widetilde{\mathbf{a}}\,\overline{\widetilde c}\big)
=\frac{ac^2}{4}\operatorname{Re}(\mathbf{i})=0.
\end{gather*}
Then it follows from (1) of Theorem~2.10 in \cite{Beg-Di4} that
 $\mathbf{g_1}=\mathbf{g_2}$ and hence, $\mathbf{U_1}=\mathbf{U_2}$.
\end{proof}

\begin{remark} \label{moreuni} \rm
Notice that uniqueness for self-similar solution is relied to uniqueness 
for \eqref{g}. Using Theorem~2.10 in \cite{Beg-Di4}, we can show that 
the uniqueness of self-similar solutions to equation~\eqref{nls} holds 
in the class of functions 
$\mathbf{C}\big((0,\infty);\mathbf{L^2_\mathrm{c}}(\mathbb{R}^N)\big)$ 
when, for instance, $\operatorname{Re}(\mathbf{a})=0$ and $\operatorname{Im}(\mathbf{a})<0$ 
(Theorem~\ref{thmuni}). These hypotheses are the same as in \cite{MR2765425}. 
We point out that it seems possible to adapt the uniqueness method of 
\cite[Theorem 2.10]{Beg-Di4} to obtain other criteria of uniqueness.
\end{remark}

\begin{remark} \label{rmkpoi} \rm
In the proof of uniqueness of Theorem~\ref{thmmain}, we will use the 
Poincar\'e's inequality~\eqref{poibt}. This estimate can be improved in 
several ways. For instance, for any $x_0\in\mathbb{R}^N$ and any $R>0$, we have
\begin{equation}\label{rmkpoi1}
\|\mathbf{u}\|_{\mathbf{L^2}(B(x_0,R))}\leqslant
\frac{2R}{\pi}\|\nabla\mathbf{u}\|_{\mathbf{L^2}(B(x_0,R))},
\end{equation}
which is substantially better than \eqref{poibt},
since $2/\pi <1<\sqrt2$. Actually, \eqref{rmkpoi1} holds for any
 $\mathbf{u}\in\mathbf{H^1}\big(B(x_0,R)\big)$ such that
\[
\int_{B(x_0,R)}\mathbf{u}(x)\mathrm{d} x=\mathbf{0},
\]
and $\dfrac{\partial^2\mathbf{u}}{\partial x_j\partial x_k}\in\mathbf{L^\infty}
\big(B(x_0,R)\big)$, for any
$(j,k)\in[\![ 1,N]\!]\times[\![ 1,N]\!]$.
See \cite{MR0117419} for more details.
\end{remark}

\section{Proofs of the localization properties}
\label{proof}

We start by pointing out that if $\Omega\subseteq\mathbb{R}^N$ is 
a nonempty open subset and if $0<m\leqslant1$, we have the following property: let
$\mathbf{U}\in\mathbf{H^1_\mathrm{loc}}(\Omega)$ be a local weak solution to
\[
-\Delta\mathbf{U} + \mathbf{a}|\mathbf{U}|^{-(1-m)}\mathbf{U} + \mathbf{b}\mathbf{U} + \mathbf{i} c x.\nabla\mathbf{U} = \mathbf{F}(x), \quad
\text{in } \mathscr{D}'(\Omega),
\]
for some $(\mathbf{a},\mathbf{b},c)\in\mathbb{C}\times\mathbb{C}\times\mathbb{R}$ and 
$\mathbf{F}\in\mathbf{L^1_\mathrm{loc}}(\Omega)$. Setting 
$\mathbf{g}(x)=\mathbf{U}(x)\mathbf{e}^{-\mathbf{i} c\frac{|x|^2}{4}}$, 
for almost every $x\in\Omega$, it follows that 
$\mathbf{g}\in\mathbf{H^1_\mathrm{loc}}(\Omega)$ is a local weak solution to
\[
-\Delta\mathbf{g}+\mathbf{a}|\mathbf{g}|^{-(1-m)}\mathbf{g}+(\mathbf{b}-\mathbf{i}\frac{cN}{2})\mathbf{g}
-\frac{c^2}{4}|x|^2\mathbf{g}=\mathbf{F}(x)\mathbf{e}^{-\mathbf{i} c\frac{|x|^2}{4}},\quad
 \text{in } \mathscr{D}'(\Omega).
\]
Conversely, if $\mathbf{g}\in\mathbf{H^1_\mathrm{loc}}(\Omega)$ is a local weak solution to
\[
-\Delta\mathbf{g} + \mathbf{a}|\mathbf{g}|^{-(1-m)}\mathbf{g} + \mathbf{b}\mathbf{g} - c^2|x|^2\mathbf{g} = \mathbf{G}(x), \quad
\text{in } \mathscr{D}'(\Omega),
\]
for some $(\mathbf{a},\mathbf{b},c)\in\mathbb{C}\times\mathbb{C}\times\mathbb{R}$ and 
$\mathbf{G}\in\mathbf{L^1_\mathrm{loc}}(\Omega)$, then setting 
$\mathbf{U}(x)=\mathbf{g}(x)\mathbf{e}^{\mathbf{i} c\frac{|x|^2}{2}}$, 
for almost every $x\in\Omega$, it follows that 
$\mathbf{U}\in\mathbf{H^1_\mathrm{loc}}(\Omega)$ is a local weak solution to
\[
-\Delta\mathbf{U} + \mathbf{a}|\mathbf{U}|^{-(1-m)}\mathbf{U}+(\mathbf{b}+\mathbf{i} cN)\mathbf{U}+2\mathbf{i} c x.\nabla\mathbf{U}=
\mathbf{G}(x)\mathbf{e}^{\mathbf{i} c\frac{|x|^2}{2}}, \text{ in } \mathscr{D}'(\Omega).
\]

The proof of Theorems~\ref{thmsta} and \ref{thmG} follows the main structure 
of application of the energy methods introduced to the study of free 
boundary (see, e.g., the general presentation made in the monograph 
\cite{MR2002i:35001}). In both cases, the conclusions follow quite easily 
once it is obtained a general differential inequality for the local 
energy $E(\rho)$ of the type
\begin{equation} \label{etoile}
E(\rho)^\alpha\leqslant C\rho^{-\beta}E'(\rho)+K(\rho-\rho_0)_+^\omega,
\end{equation}
for some positive constants $C$, $\beta$ and $\omega$ with $K=0$,
in case of Theorem~\ref{thmsta} and $K>0$ small enough, in case of
Theorem~\ref{thmG}. The key estimate which leads to desired
local behaviour is that the exponent $\alpha$ arising in \eqref{etoile}
satisfies that $\alpha\in(0,1)$.

Although the main steps to prove \eqref{etoile} follow the same steps already
 indicated in the monograph \cite{MR2002i:35001}, it turns out that the 
concrete case of the systems of scalar equations generated by the 
Schr\"odinger operator does not fulfill the assumptions imposed in
\cite{MR2002i:35001} for the case of systems of nonlinear equations. 
The extension of the method which applied to the system associated to 
the complex Schr\"odinger operator is far to be trivial and it was the 
main object of \cite{MR2876246}. Unfortunately, the extension of the 
method presented in \cite{MR2876246} is not enough to be applied to 
the fundamental equation of the present paper (i.e. \eqref{g} or \eqref{eq2}) 
mainly due to the presence of the source term $-c^2|x|^2g$. 
A sharper version of the energy method, also applicable to a different 
type of nonlinear complex Schr\"odinger type equations 
(for instance containing a Hartree-Fock type nonlocal term), was developed
 in \cite{Beg-Di5}, where the applicability of the energy method was reduced 
to prove a certain local energy balance. Such a local balance will be proved
 here in the following lemma. Thanks to that, the proofs of Theorems~\ref{thmsta}
 and \ref{thmG} are then a corollary of Theorems~2.1 and 2.2 in \cite{Beg-Di5}.

\begin{lemma} \label{lemest}
Let $\Omega\subset B(0,R)$ be a nonempty bounded open subset of $\mathbb{R}^N$, 
let $0<m<1$, let $(\mathbf{a},\mathbf{b},\mathbf{c})\in\mathbb{C}^3$ be such that 
$\operatorname{Im}(\mathbf{a})\leqslant0$,
$\operatorname{Im}(\mathbf{b})<0$ and $\operatorname{Im}(\mathbf{c})\leqslant0$. 
If $\operatorname{Re}(\mathbf{a})\leqslant0$ then assume further that
 $\operatorname{Im}(\mathbf{a})<0$. Let $\mathbf{G}\in\mathbf{L^1_\mathrm{loc}}(\Omega)$ 
and let $\mathbf{g}\in\mathbf{H^1_\mathrm{loc}}(\Omega)$ be any local weak solution
 to \eqref{eq2}. Then there exist two positive constants 
$L=L(R,|\mathbf{a}|,|\mathbf{b}|,|\mathbf{c}|)$ and $M=M(R,|\mathbf{a}|,|\mathbf{b}|,|\mathbf{c}|)$ such that for 
any $x_0\in\Omega$ and any $\rho_\star>0$, if 
$\mathbf{G}_{|\Omega\cap B(x_0,\rho_\star)}\in\mathbf{L^2}
\big(\Omega\cap B(x_0,\rho_\star)\big)$ then we have
\begin{equation}\label{lemest1}
\begin{aligned}
&\|\nabla\mathbf{g}\|_{\mathbf{L^2}(\Omega\cap B(x_0,\rho))}^2
+L\|\mathbf{g}\|_{\mathbf{L^{m+1}}(\Omega\cap B(x_0,\rho))}^{m+1}
+L\|\mathbf{g}\|_{\mathbf{L^2}(\Omega\cap B(x_0,\rho))}^2	\\
&\leqslant M\Big(\big|\int_{\Omega\cap\mathbb{S}(x_0,\rho)}
\mathbf{g}\overline{\nabla\mathbf{g}}.\frac{x-x_0}{|x-x_0|}\mathrm{d}\sigma\big|
+\int_{\Omega\cap B(x_0,\rho)}|\mathbf{G}(x)\mathbf{g}(x)|\mathrm{d} x\Big),
\end{aligned}
\end{equation}
for every $\rho\in[0,\rho_\star)$, where it is additionally assumed
that $\mathbf{g}\in\mathbf{H^1_0}(\Omega)$ if $\rho_\star>\operatorname{dist}(x_0,\Gamma)$.
\end{lemma}

\begin{proof}
Let $x_0\in\Omega$ and let $\rho_\star>0$. Let $\sigma$ be the surface 
measure on a sphere and set for every $\rho\in[0,\rho_*)$,
\begin{gather*}
 I(\rho)=\big|\int_{\Omega\cap\mathbb{S}(x_0,\rho)}
 \mathbf{g}\overline{\nabla\mathbf{g}}.\frac{x-x_0}{|x-x_0|}\mathrm{d}\sigma\big|, \quad 
 J(\rho)=\int_{\Omega\cap B(x_0,\rho)}|\mathbf{G}(x)\mathbf{g}(x)|\mathrm{d} x, \\
 w(\rho)=\int_{\Omega\cap\mathbb{S}(x_0,\rho)}\mathbf{g}\overline{\nabla\mathbf{g}}.
 \frac{x-x_0}{|x-x_0|}\mathrm{d}\sigma, \quad
 I_{\rm Re}(\rho)=\operatorname{Re}\big(w(\rho)\big), \quad
 I_{\rm Im}(\rho)=\operatorname{Im}\big(w(\rho)\big).
\end{gather*}
By taking as test function
 $\mathbf{\widetilde\varphi_n}(x)=\psi_n(|x-x_0|)\mathbf{\widetilde g}(x)$, 
where $\mathbf{\widetilde g}$ is the extension by $0$ of $\mathbf{g}$ on
$\Omega^\mathrm{c}\cap B(x_0,\rho_0)$ and $\psi_n$ is the cut-off function
\[
\forall t\in\mathbb{R},\quad 
\psi_n(t)= \begin{cases}
   1,			& \text{if } |t|\in[0,\rho-\frac{1}{n}],  \\
   n(\rho-|t|),	& \text{if } |t|\in(\rho-\frac{1}{n},\rho),  \\
   0,			& \text{if } |t|\in[\rho,\infty),
 \end{cases}
\]
it can be proved (see \cite[Theorem 3.1]{Beg-Di5}) that 
$I,J,I_{\rm Re},I_{\rm Im}\in C([0,\rho_*);\mathbb{R})$
and, by passing to the limit as $n \to \infty$, that
\begin{equation} \label{prooflemest-2}
\begin{aligned}
&\|\nabla\mathbf{g}\|_{\mathbf{L^2}(\Omega\cap B(x_0,\rho))}^2
+\operatorname{Re}(\mathbf{a})\|\mathbf{g}\|_{\mathbf{L^{m+1}}(\Omega\cap B(x_0,\rho))}^{m+1}
+\operatorname{Re}(\mathbf{b})\|\mathbf{g}\|_{\mathbf{L^2}(\Omega\cap B(x_0,\rho))}^2	\\					\\
&+\operatorname{Re}(\mathbf{c})\||x|\mathbf{g}\|_{\mathbf{L^2}(\Omega\cap B(x_0,\rho))}^2\\
&=I_{\rm Re}(\rho)+\operatorname{Re}
\Big(\int_{\Omega\cap B(x_0,\rho)}\mathbf{G}(x)\overline{\mathbf{g}(x)}\mathrm{d} x\Big),
\end{aligned}
\end{equation}
\begin{equation} \label{prooflemest-1}
\begin{aligned}
&\operatorname{Im}(\mathbf{a})\|\mathbf{g}\|_{\mathbf{L^{m+1}}(\Omega\cap B(x_0,\rho))}^{m+1}
+\operatorname{Im}(\mathbf{b})\|\mathbf{g}\|_{\mathbf{L^2}(\Omega\cap B(x_0,\rho))}^2
+\operatorname{Im}(\mathbf{c})\||x|\mathbf{g}\|_{\mathbf{L^2}(\Omega\cap B(x_0,\rho))}^2	\\				\\
&=I_{\rm Im}(\rho)+\operatorname{Im}
\Big(\int_{\Omega\cap B(x_0,\rho)}\mathbf{G}(x)\overline{\mathbf{g}(x)}\mathrm{d} x\Big),
\end{aligned}
\end{equation}
for any $\rho\in[0,\rho_\star)$. From these estimates, we obtain
\begin{equation} \label{prooflemest1}
\begin{aligned}
&\Big|\|\nabla\mathbf{g}\|_{\mathbf{L^2}(B(x_0,\rho))}^2
 +\operatorname{Re}(\mathbf{a})\|\mathbf{g}\|_{\mathbf{L^{m+1}}(B(x_0,\rho))}^{m+1}
 +\operatorname{Re}(\mathbf{b})\|\mathbf{g}\|_{\mathbf{L^2}(B(x_0,\rho))}^2	\\
&+\operatorname{Re}(\mathbf{c})\||x|\mathbf{g}\|_{\mathbf{L^2}(B(x_0,\rho))}^2\Big|\\
&\leqslant I(\rho)+J(\rho),
\end{aligned}
\end{equation}
\begin{equation}\label{prooflemest2}
\begin{aligned}
&|\operatorname{Im}(\mathbf{a})|\|\mathbf{g}\|_{\mathbf{L^{m+1}}(B(x_0,\rho))}^{m+1}
+|\operatorname{Im}(\mathbf{b})|\|\mathbf{g}\|_{\mathbf{L^2}(B(x_0,\rho))}^2
+|\operatorname{Im}(\mathbf{c})|\||x|\mathbf{g}\|_{\mathbf{L^2}(B(x_0,\rho))}^2\\
&\leqslant I(\rho)+J(\rho),
\end{aligned}
\end{equation}
for any $\rho\in[0,\rho_\star)$. Let $A>1$ to be chosen later.
 We multiply \eqref{prooflemest2} by $A$ and sum the result with
\eqref{prooflemest1}. This leads to
\begin{equation} \label{prooflemest3}
\begin{aligned}
&\|\nabla\mathbf{g}\|_{\mathbf{L^2}(B(x_0,\rho))}^2
 +A_1\|\mathbf{g}\|_{\mathbf{L^{m+1}}(B(x_0,\rho))}^{m+1}
 +A_2\|\mathbf{g}\|_{\mathbf{L^2}(B(x_0,\rho))}^2
+\operatorname{Re}(\mathbf{c})\||x|\mathbf{g}\|_{\mathbf{L^2}(B(x_0,\rho))}^2\\
&\leqslant2A\big(I(\rho)+J(\rho)\big),
\end{aligned}
\end{equation}
where
\begin{gather*}
 A_1	=\begin{cases}
\operatorname{Re}(\mathbf{a}),	&	\text{if } \operatorname{Re}(\mathbf{a})>0, \\
A|\operatorname{Im}(\mathbf{a})|-|\operatorname{Re}(\mathbf{a})|,	&	\text{if }
\operatorname{Re}(\mathbf{a})\leqslant0,
\end{cases} \\
 A_2	= A|\operatorname{Im}(\mathbf{b})|-|\operatorname{Re}(\mathbf{b})|.
\end{gather*}
But \eqref{prooflemest3} yields,
\begin{equation} \label{prooflemest4}
\begin{aligned}
&\|\nabla\mathbf{g}\|_{\mathbf{L^2}(B(x_0,\rho))}^2
 +A_1\|\mathbf{g}\|_{\mathbf{L^{m+1}}(B(x_0,\rho))}^{m+1}
 +\big(A_2-R^2|\operatorname{Re}(\mathbf{c})|\big)\|\mathbf{g}\|_{\mathbf{L^2}(B(x_0,\rho))}^2\\
&\leqslant2A\big(I(\rho)+J(\rho)\big)
\end{aligned}
\end{equation}
We choose $A=A(R,|\mathbf{a}|,|\mathbf{b}|,|\mathbf{c}|)$ large enough to have
$A|\operatorname{Im}(\mathbf{a})|-|\operatorname{Re}(\mathbf{a})|\geqslant1$
(when $\operatorname{Re}(\mathbf{a})\leqslant0)$ and
$A_2-R^2|\operatorname{Re}(\mathbf{c})|\geqslant1$.
Then~\eqref{lemest1} comes from \eqref{prooflemest4} with
$L=\min\big\{A_1,1\big\}$ and $M=2A$. Note that $L=L(R,|\mathbf{a}|,|\mathbf{b}|,|\mathbf{c}|)$ and
$M=M(R,|\mathbf{a}|,|\mathbf{b}|,|\mathbf{c}|)$. This concludes the proof.
\end{proof}

\begin{remark} \label{rmklemest} \rm
When $\rho_\star\leqslant\operatorname{dist}(x_0,\Gamma)$ and 
$\mathbf{G}\in\mathbf{L^2_\mathrm{loc}}(\Omega)$, one may easily obtain 
\eqref{prooflemest-2}--\eqref{prooflemest-1} without the technical 
\cite[Theorem 3.1]{Beg-Di5}. Indeed, it follows from
\cite[Proposition 4.5]{MR2876246} that
$\mathbf{g}\in\mathbf{H^2_\mathrm{loc}}(\Omega)$, so that equation~\eqref{eq2} 
makes sense in $\mathbf{L^2_\mathrm{loc}}(\Omega)$ and almost everywhere 
in $\Omega$. Thus, if
$\rho_\star\leqslant\operatorname{dist}(x_0,\Gamma)$ then 
$\mathbf{g}_{|B(x_0,\rho)}\in\mathbf{H^2}\big(B(x_0,\rho)\big)$ 
and \eqref{prooflemest-2} \big(respectively, \eqref{prooflemest-1}\big) 
is obtained by multiplying \eqref{eq2} by $\overline{\mathbf{g}}$ 
(respectively, by $\overline{\mathbf{i}\mathbf{g}})$, integrating by parts over 
$B(x_0,\rho)$ and taking the real part.
\end{remark}

\begin{proof}[Proof of Theorem~\ref{thmmain}]
Let $R>0$. Let $\varepsilon>0$ and let 
$\mathbf{f}\in\mathbf{C}\big((0,\infty);\mathbf{L^2}(\mathbb{R}^N)\big)$ 
satisfying~\eqref{fla} and $\operatorname{supp}\mathbf{f}(1)\subset\overline B(0,R)$.
Let $M_0$ be the constant in~\eqref{new}. Let $\mathbf{b}=-\mathbf{i}\frac{N+2\mathbf{p}}{4}$, 
$\mathbf{c}=-\frac1{16}$ and $\mathbf{G}=-\mathbf{f}(1)\mathbf{e}^{-\mathbf{i}\frac{|\cdot|^2}{8}}$. 
Note that $\operatorname{Im}(\mathbf{a})\leqslant0$,
$\operatorname{Im}(\mathbf{b})=-\frac{N(1-m)+4}{4(1-m)}<0$ and 
$\operatorname{Im}(\mathbf{c})=0$. In addition, if $\operatorname{Re}(\mathbf{a})\leqslant0$ 
then $\operatorname{Im}(\mathbf{a})<0$. It follows that the existence result of
Section~\ref{eus} applies to equation~\eqref{g}: 
let $\mathbf{g}\in\mathbf{H^1_0}(B(0,2R+2\varepsilon))\cap\mathbf{H^2}(B(0,2R+2\varepsilon))$
be such a solution to \eqref{g} and \eqref{dir}. We apply Theorem~\ref{thmsta} 
with $\rho_0=2\varepsilon$. By \eqref{new}, there exists 
$\delta_0=\delta_0(R,\varepsilon,|\mathbf{a}|,|\mathbf{b}|,|\mathbf{c}|,N,m)>0$ such that if 
$\|\mathbf{f}(1)\|_{\mathbf{L^2}(\mathbb{R}^N)}\leqslant\delta_0$ then 
$\rho_\mathrm{max}\geqslant\varepsilon$. Set $K=\operatorname{supp}\mathbf{f}(1)=\operatorname{supp}\mathbf{G}$.
Let $x_0\in\overline{K(2\varepsilon)^\mathrm{c}}\cap B(0,2R+2\varepsilon)$. Let
$y\in B(x_0,2\varepsilon)$ and let $z\in K$. By definition of $K(2\varepsilon)$, 
$\operatorname{dist}(\overline{K(2\varepsilon)^\mathrm{c}},K)=2\varepsilon$. 
We then have
\[
2\varepsilon=\operatorname{dist}(\overline{K(2\varepsilon)^\mathrm{c}},K)
\le|x_0-z|\le|x_0-y|+|y-z|<2\varepsilon+|y-z|.
\]
It follows that for any $z\in K$, $|y-z|>0$, so that $y\not\in K$. 
This means that $B(x_0,2\varepsilon)\cap K=\emptyset$, for any
$x_0\in\overline{K(2\varepsilon)^\mathrm{c}}\cap B(0,2R+2\varepsilon)$.
 By Theorem~\ref{thmsta} we deduce that for any 
$x_0\in\overline{K(2\varepsilon)^\mathrm{c}}\cap B(0,2R+2\varepsilon)$,
$\mathbf{g}_{\left|B(x_0,\varepsilon)\right.}\equiv\mathbf{0}$. By compactness, 
$\overline{K(\varepsilon)^\mathrm{c}}\cap B(0,2R+2\varepsilon)$ may be 
covered by a finite number of sets
$B(x_0,\varepsilon)\cap B(0,2R+2\varepsilon)$ with 
$x_0\in\overline{K(2\varepsilon)^\mathrm{c}}$. It follows that
 $\mathbf{g}|_{K(\varepsilon)^\mathrm{c}\cap B(0,2R+2\varepsilon)}\equiv\mathbf{0}$. 
This means that $\operatorname{supp}\mathbf{g}\subset K(\varepsilon)\subset B(0,2R+2\varepsilon)$.
 We then extend $\mathbf{g}$ by $\mathbf{0}$ outside of $B(0,2R+2\varepsilon)$. 
Thus, $\mathbf{g}\in\mathbf{H^2_\mathrm{c}}(\mathbb{R}^N)$ is a solution 
to \eqref{g} in $\mathbb{R}^N$. Now, let $\mathbf{U}=\mathbf{g}\mathbf{e}^{\mathbf{i}\frac{|\cdot|^2}{8}}$ 
and let for any $t>0$, $\mathbf{u}(t)=t^{\mathbf{p}/2}\mathbf{U}(\frac{\cdot }{\sqrt t})$.
 It follows that $\operatorname{supp}\mathbf{U}=\operatorname{supp}\mathbf{g}\subset K(\varepsilon)$,
$\mathbf{U}\in\mathbf{H^2_\mathrm{c}}(\mathbb{R}^N)$ and $\mathbf{U}$ 
is a solution to \eqref{U} in $\mathbb{R}^N$. By \eqref{eqprou},
$\mathbf{u}$ verifies \eqref{thmmain1} and is a solution to~\eqref{nls} 
in $(0,\infty)\times\mathbb{R}^N$ with $\mathbf{u}(1)=\mathbf{U}$ compactly supported in
$K(\varepsilon)$. By Definition~\ref{defselsim}, $\mathbf{u}$ is self-similar 
and still by \eqref{eqprou}, $\operatorname{supp}\mathbf{u}(t)$ is compact for any $t>0$. Hence
Properties~\ref{thmmaina} and \ref{thmmainb}. 
It remains to show Property~\ref{thmmainc}. Let $R_0>0$ and assume further 
that $\operatorname{Re}(\mathbf{a})>0$,
$\operatorname{Im}(\mathbf{a})=0$ and 
$0<R_0^2\leqslant4\operatorname{Im}(\mathbf{p})+2\sqrt{4\operatorname{Im}^2(\mathbf{p})+2}$. 
Let $\mathbf{u_1},\mathbf{u_2}\in\mathbf{C}\big((0,\infty);
\mathbf{L^2_\mathrm{c}}(\mathbb{R}^N)\big)$ be two solutions
to \eqref{nls} whose profile $\mathbf{U_1},\mathbf{U_2}$ satisfy 
$\operatorname{supp}\mathbf{U},\operatorname{supp}\mathbf{V}\subset\overline B(0,R_0)$. By Section~\ref{eus},
$\mathbf{U_1},\mathbf{U_2}\in\mathbf{H^2_\mathrm{c}}(\mathbb{R}^N)$. 
For $j\in\{1,2\}$, let 
$\mathbf{g_j}=\mathbf{U_j}\mathbf{e}^{-\mathbf{i}\frac{|\cdot|^2}{8}}$. 
It follows that $\mathbf{g_1}$ and $\mathbf{g_2}$ belong to 
$\mathbf{H^2_\mathrm{c}}(\mathbb{R}^N)$, are compactly supported in 
$\overline B(0,R_0)$ and satisfy the same equation \eqref{g}.
 Let $\mathbf{g}=\mathbf{g_1}-\mathbf{g_2}$ and set for any 
$\mathbf{h}\in\mathbf{L^2_\mathrm{c}}(\mathbb{R}^N)$,
$\mathbf{H}(\mathbf{h})=|\mathbf{h}|^{-(1-m)}\mathbf{h}$. It follows that,
\begin{gather*}
-\Delta\mathbf{g} + a\big(\mathbf{H}(\mathbf{g_1})-\mathbf{H}(\mathbf{g_2})\big) 
- \mathbf{i}\frac{N+2\mathbf{p}}{4}\mathbf{g} - \frac1{16}|x|^2\mathbf{g} = \mathbf{0}, \quad \text{ a.e. in } \mathbb{R}^N.
\end{gather*}
Multiplying this equation by $\overline{\mathbf{g}}$, integrating by parts over 
$\mathbb{R}^N$ and taking the real part, we obtain
\begin{align*}
&\|\nabla\mathbf{g}\|_{\mathbf{L^2}}^2+a\langle\mathbf{H}(\mathbf{g_1})-\mathbf{H}(\mathbf{g_2}),\mathbf{g_1}
-\mathbf{g_2}\rangle_{\mathbf{L^2},\mathbf{L^2}}
-\operatorname{Re}\big(\mathbf{i}\frac{N+2\mathbf{p}}{4}\big)
\|\mathbf{g}\|_{\mathbf{L^2}}^2-\frac{1}{16}\||\cdot|\mathbf{g}\|_{\mathbf{L^2}}^2		\\
&= \|\nabla\mathbf{g}\|_{\mathbf{L^2}}^2+a\langle\mathbf{H}(\mathbf{g_1})-\mathbf{H}(\mathbf{g_2}),
\mathbf{g_1}-\mathbf{g_2}\rangle_{\mathbf{L^2},\mathbf{L^2}}
+\frac12\operatorname{Im}(\mathbf{p})\|\mathbf{g}\|_{\mathbf{L^2}}^2
-\frac{1}{16}\||\cdot|\mathbf{g}\|_{\mathbf{L^2}}^2		\\
&=	 0,
\end{align*}
We recall the following refined Poincar\'e's inequality \cite{BegTor}.
\begin{equation} \label{poibt}
\forall\mathbf{u}\in\mathbf{H^1_0}\big(B(0,R_0)\big), \quad
 \|\mathbf{u}\|_{\mathbf{L^2}(B(0,R_0))}^2\leqslant2R_0^2
\|\nabla\mathbf{u}\|_{\mathbf{L^2}(B(0,R_0))}^2,
\end{equation}
If follows from \eqref{poibt} and \cite[Lemma 9.1]{MR2876246},
that there exists a positive constant $C$ such that
\[
\Big(\frac{1}{2R_0^2}+\frac12\operatorname{Im}(\mathbf{p})-\frac{R_0^2}{16}\Big)
\|\mathbf{g}\|_{\mathbf{L^2}}^2
+Ca\int_\omega\frac{|\mathbf{g_1}(x)-\mathbf{g_2}(x)|^2}{(|\mathbf{g_1}(x)|
+|\mathbf{g_2}(x)|)^{1-m}}\mathrm{d} x
\leqslant 0,
\]
where $\omega=\Big\{x\in\Omega;|\mathbf{g_1}(x)|+|\mathbf{g_2}(x)|>0\Big\}$. But,
\[
\frac{1}{2R_0^2}+\frac12\operatorname{Im}(\mathbf{p})-\frac{R_0^2}{16}
=\frac{1}{16R_0^2}\left(-R_0^4+8\operatorname{Im}(\mathbf{p})R_0^2+8\right)\geqslant0,
\]
when
\[
0\leqslant R_0^2\leqslant4\operatorname{Im}(\mathbf{p})
+2\sqrt{4\operatorname{Im}^2(\mathbf{p})+2}.
\]
It follows that $\mathbf{g_1}=\mathbf{g_2}$ which implies that
$\mathbf{U_1}=\mathbf{U_2}$ and for any $t>0$,
 $\mathbf{u_1}(t)=\mathbf{u_2}(t)$. This completes the proof.
\end{proof}

\subsection*{Acknowledgements}
J. I. D\'iaz was partially supported by  project MTM2011-26119 of
 the DGISPI (Spain) and the Research Group MOMAT (Ref. 910480)
 supported by UCM. He has received also support from the ITN FIRST of the
 Seventh Framework Program of the European Community's
(grant agreement number 238702).


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