# easyaccess: Enhanced SQL command line interpreter for astronomical surveys

Matias Carrasco Kind  National Center for Supercomputing Applications,
University of Illinois at Urbana-Champaign. 1205 W Clark St, Urbana, IL USA
61801  Alex Drlica-Wagner  Fermi National Accelerator Laboratory, P. O. Box
500, Batavia,IL 60510, USA  Audrey Koziol  National Center for Supercomputing
Applications, University of Illinois at Urbana-Champaign. 1205 W Clark St,
Urbana, IL USA 61801  Don Petravick  National Center for Supercomputing
Applications, University of Illinois at Urbana-Champaign. 1205 W Clark St,
Urbana, IL USA 61801

††margin:  DOI: 00.00000/joss.00000 Software • Review • Repository • Archive
Submitted: 00 January 0000
Published: 00 January 0000 License
Authors of papers retain copyright and release the work under a Creative
Commons Attribution 4.0 International License (CC-BY).

## Summary

easyaccess is an enhanced command line interpreter and Python package created
to facilitate access to astronomical catalogs stored in SQL Databases. It
provides a custom interface with custom commands and was specifically designed
to access data from the Dark Energy Survey Oracle database, although it can
easily be extended to another survey or SQL database. The package was
completely written in Python and support customized addition of commands and
functionalities. Visit https://github.com/mgckind/easyaccess to view
installation instructions, tutorials, and the Python source code for
easyaccess.

## The Dark Energy Survey

The Dark Energy Survey (DES) (DES Collaboration 2005; DES Collaboration 2016)
is an international, collaborative effort of over 500 scientists from 26
institutions in seven countries. The primary goals of DES are reveal the
nature of the mysterious dark energy and dark matter by mapping hundreds of
millions of galaxies, detecting thousands of supernovae, and finding patterns
in the large-scale structure of the Universe. Survey operations began on on
August 31, 2013 and will conclude in early 2019. For about 500 nights, DES has
been taking thousands of deep images of southern sky, which are transferred
and processed at the National Center for Supercomputing Applications (NCSA).
The images are processed to produce catalogs of astronomical sources with
hundreds of millions of entries (billions in the case of individual
detections), describing the sources found within the images and other relevant
metadata. A significant subset of the DES data was recently made public (DES
Collaboration 2018) and can be accessed through several mechanisms including
easyaccess and web interfaces that run easyaccess as a backend. This public
release includes information for almost 400 million astrophysical sources and
complementary tables to allow scientific analysis.

### DES users

The first release of easyaccess was on February 17th, 2015 and since then,
over 300 DES Collaborators have used it to access the DES databases (Figure
1). We note that roughly 800 DES accounts exist, but this includes all
database users including those that created accounts before the release of
easyaccess. In August 2018 (version 1.4.4), we added support for the public
DES data release, and since then we have increased the number of public users.

Figure 1: Number of user since first version

## easyaccess

easyaccess is a command line interpreter that is heavily based on termcolor
(Lepa 2018) and the cmd Python core module. It interfaces with cx_Oracle
(Oracle Corp. 2018) to communicate with Oracle, requests (Reitz 2012–2018) for
external URL requests, and other external open source libraries, including
NumPy (Oliphant 2006), pandas (McKinney 2010), fitsio (Sheldon 2018) and h5py
(Collette 2013) to handle and transform array data. Figure 2 shows an example
of the welcome screen as seen by a DES user.

Figure 2: Welcome screenshot

### Features

easyaccess has a variety of features including a history of past commands and
smart tab auto-completion for commands, functions, columns, users, tables, and
paths. Tables can be written directly into comma-separated-value (CSV) or
white-space separated text files, FITS (Wells, Greisen, and Harten 1981)
files, and HDF5 (The HDF Group 1997–2018) files. It provides an iteration
scheme to avoid memory constraints when retrieving large tables. Tables can
also be displayed on the command line and most of the formatting is done using
pandas. Similarly, privileged users can easily upload tables to the database
from any of the file format described above in order to share data with other
users. The uploading mechanism is done chunk-wise, allowing large tables to be
loaded while keeping memory usage low.

In addition, there are a variety of customized functions to search and
describe the tables, search for users and user tables, check quota usage,
check the Oracle execution plan, and soon the ability to run asynchronous jobs
through a dedicated server. There are dozens of other minor features that
allow for a seamless experience while exploring and discovering data within
the hundreds of tables inside the DB.

One can also load SQL queries from a file into the database, or run SQL
queries inside the easyaccess python module in another IDE. Most of the
features are also exposed through a Python API and can be run inside a Jupyter
(Kluyver et al. 2016) notebook or similar tool alongside scientific analyses.

Users can also use easyaccess to submit and request cutouts around specific
positions or objects which are generated from the images. This allows better
integration with other data services for a richer scientific workflow.

### Architecture

We have included a simplified UML diagram describing the architecture and
dependencies of easyaccess (Figure 3). Figure 3 shows only the different
methods for a given class and the name of the file hosting a given class. The
main class, easy_or(), inherits all methods from all different subclasses,
making this model flexible and extendable to other surveys or databases. These
methods are then converted to command line commands and functions that can be
called inside easyaccess. Given that there are some DES specific functions, we
have moved DES methods into a separate class DesActions().

Figure 3: easyaccess architecture diagram

### Installation

To download easyaccess you can clone the source code from GitHub at
https://github.com/mgckind/easyaccess or follow any of the standard
installation channels described below.

  * •

From source

python setup.py install

  * •

conda

conda install easyaccess -c mgckind

  * •

Docker

docker pull mgckind/easyaccess

  * •

pip

pip install easyaccess

## Acknowledgments

The DES Data Management System is supported by the National Science Foundation
under Grant NSF AST 07-15036 and NSF AST 08-13543.

## References

Collette, Andrew. 2013. _Python and Hdf5_. O’Reilly.

DES Collaboration. 2005. “The Dark Energy Survey.” _ArXiv Astrophysics
E-Prints_ , October.

———. 2016. “The Dark Energy Survey: more than dark energy - an overview.”
_MNRAS_ 460 (August): 1270–99. doi:10.1093/mnras/stw641.

———. 2018. “The Dark Energy Survey Data Release 1.” _ArXiv E-Prints_ ,
January.

Kluyver, Thomas, Benjamin Ragan-Kelley, Fernando Pérez, Brian Granger,
Matthias Bussonnier, Jonathan Frederic, Kyle Kelley, et al. 2016. “Jupyter
Notebooks – a Publishing Format for Reproducible Computational Workflows.”
Edited by F. Loizides and B. Schmidt. IOS Press.

Lepa, Konstantin. 2018. “Termcolorr.” _PyPi Repository_.
https://pypi.org/project/termcolor/; PyPi.

McKinney, Wes. 2010. “Data Structures for Statistical Computing in Python.” In
_Proceedings of the 9th Python in Science Conference_ , edited by Stéfan van
der Walt and Jarrod Millman, 51–56.

Oliphant, Travis E. 2006. _A Guide to Numpy_. Trelgol Publishing.

Oracle Corp. 2018. “CxOracle.” _GitHub Repository_.
https://github.com/oracle/python-cx_Oracle; GitHub.

Reitz, Kenneth. 2012–2018. “Requests: HTTP for Humans.” http://docs.python-
requests.org/.

Sheldon, Erin. 2018. “Fitsio.” _GitHub Repository_.
https://github.com/esheldon/fitsio; GitHub.

The HDF Group. 1997–2018. “Hierarchical Data Format, version 5.”

Wells, D. C., E. W. Greisen, and R. H. Harten. 1981. “FITS - a Flexible Image
Transport System” 44 (June): 363.


# Multiplicity and orbital stability of normalized solutions to non-autonomous
Schrödinger equation with mixed nonlinearities

###### Abstract.

This paper studies the multiplicity of normalized solutions to the Schrödinger
equation with mixed nonlinearities

$\begin{cases}-\Delta u=\lambda u+h(\epsilon x)|u|^{q-2}u+\eta|u|^{p-2}u,\quad
x\in\mathbb{R}^{N},\\\ \int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2},\end{cases}$

where $a,\epsilon,\eta>0$, $q$ is $L^{2}$-subcritical, $p$ is
$L^{2}$-supercritical, $\lambda\in\mathbb{R}$ is an unknown parameter that
appears as a Lagrange multiplier, $h$ is a positive and continuous function.
It is proved that the numbers of normalized solutions are at least the numbers
of global maximum points of $h$ when $\epsilon$ is small enough. Moreover, the
orbital stability of the solutions obtained is analyzed as well. In
particular, our results cover the Sobolev critical case $p=2N/(N-2)$.
2020 Mathematics Subject Classification: 35A15; 35J10; 35B33.

###### Key words and phrases:

Normalized solutions; Multiplicity; Orbital stability; Non-autonomous Sobolev
critical Schrödinger equation; Variational methods

*Corresponding author. Email Addresses<EMAIL_ADDRESS><EMAIL_ADDRESS>

Xinfu Lia, Li Xua∗, Meiling Zhub

aSchool of Science, Tianjin University of Commerce, Tianjin 300134, P. R.
China

bCollege of Computer Science and Engineering, Cangzhou Normal University,

Cangzhou, Hebei, 061000, P. R. China

##  1\. Introduction and main results

In this paper, we study the multiplicity and orbital stability of normalized
solutions to the non-autonomous Schrödinger equation with mixed nonlinearities

(1.1) $\begin{cases}-\Delta u=\lambda u+h(\epsilon
x)|u|^{q-2}u+\eta|u|^{p-2}u,\quad x\in\mathbb{R}^{N},\\\
\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2},\end{cases}$

where $N\geq 1,a,\epsilon,\eta>0$,
$2<q<2+\frac{4}{N}<p\left\\{\begin{array}[]{ll}<+\infty,&N=1,2,\\\ \leq
2^{*}:=\frac{2N}{N-2},&N\geq 3,\end{array}\right.$ and $\lambda\in\mathbb{R}$
is an unknown parameter that appears as a Lagrange multiplier. The function
$h$ satisfies the following conditions:
($h_{1}$) $h\in C(\mathbb{R}^{N},\mathbb{R})$ and
$0<h_{0}=\inf_{x\in\mathbb{R}^{N}}h(x)\leq\max_{x\in\mathbb{R}^{N}}=h_{\text{max}}$;
($h_{2}$) $h_{\infty}=\lim_{|x|\to+\infty}h(x)<h_{\text{max}}$;
($h_{3}$) $h^{-1}(h_{\text{max}})=\\{a_{1},a_{2},\cdots,a_{l}\\}$ with
$a_{1}=0$ and $a_{j}\neq a_{i}$ if $i\neq j$.

A solution $u$ to the problem (1.1) corresponds to a critical point of the
functional

(1.2) $E_{\epsilon}(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{1}{q}\int_{\mathbb{R}^{N}}h(\epsilon
x)|u|^{q}dx-\frac{\eta}{p}\int_{\mathbb{R}^{N}}|u|^{p}dx$

restricted to the sphere

$S(a):=\\{u\in H^{1}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}\\}.$

It is well known that $E_{\epsilon}\in
C^{1}(H^{1}(\mathbb{R}^{N}),\mathbb{R})$ and

$E_{\epsilon}^{\prime}(u)\varphi=\int_{\mathbb{R}^{N}}\nabla u\nabla\varphi
dx-\int_{\mathbb{R}^{N}}h(\epsilon x)|u|^{q-2}u\varphi
dx-\eta\int_{\mathbb{R}^{N}}|u|^{p-2}u\varphi dx$

for any $\varphi\in H^{1}(\mathbb{R}^{N})$.

One motivation driving the search for normalized solutions to (1.1) is the
nonlinear Schrödinger equation

(1.3) $i\frac{\partial\psi}{\partial t}+\Delta\psi+g(|\psi|^{2})\psi=0,\
(t,x)\in\mathbb{R}\times\mathbb{R}^{N}.$

Since the mass $\int_{\mathbb{R}^{N}}|\psi|^{2}dx$ is preserved along the flow
associated with (1.3), it is natural to consider it as prescribed. Searching
for standing wave solution $\psi(t,x)=e^{-i\lambda t}u(x)$ of (1.3) leads to
(1.1) for $u$ with $g(|s|^{2})s=h(\epsilon x)|s|^{q-2}s+\eta|s|^{p-2}s$. In
recent decades, the research of finding normalized solutions to Schrödinger
equations has received a special attention. This seems to be particularly
meaningful from the physical point of view, as the $L^{2}$-norm is a preserved
quantity of the evolution and the variational characterization of such
solutions is often a strong help to analyze their orbital stability, see
[Bellazzini, Jeanjean, Luo13, 9, 28, 29] and the references therein.

In the study of normalized solutions to the Schrödinger equation

(1.4) $-\Delta u=\lambda u+|u|^{p-2}u,\quad x\in\mathbb{R}^{N},$

the number $\bar{p}:=2+4/N$, labeled $L^{2}$-critical exponent, is a very
important number, because in the study of (1.4) using variational methods, the
functional

$J(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{N}}|u|^{p}dx,\ u\in
H^{1}(\mathbb{R}^{N})$

is bounded from below on $S(a)$ for the $L^{2}$-subcritical problem, i.e.,
$2<p<2+4/N$. Thus, a solution of (1.4) can be found as a global minimizer of
$J|_{S(a)}$, see [27, 31]. For the purely $L^{2}$-supercritical problem, i.e.,
$2+4/N<p<2^{*}$, $J|_{S(a)}$ is unbounded from below (and from above). Related
to this case, a seminar paper due to Jeanjean [15] exploited the mountain pass
geometry to get a normalized solution, see [3, 4, 8, 12, 14, 18, 19] for more
results. In the purely $L^{2}$-critical case (i.e., $p=2+4/N$), the result is
delicate. Recently, the Schrödinger equation with double power form
nonlinearity $\mu|u|^{q-2}u+|u|^{p-2}u$ has been extensively studied due to
Soave [28, 29], see [2, 16, 17, 20, 30, 33] for more results. The multiplicity
of normalized solutions to the autonomous Schrödinger equation or systems has
also been considered extensively at the last years, see [2, 3, 5, 10, 12, 14,
17, 18, 19, 23, 24]. As for the existence of normalized solutions to the non-
autonomous Schrödinger equation

(1.5) $-\Delta u=\lambda u+f(x,u),\quad x\in\mathbb{R}^{N},$

we refer to [6, 13, 21, 22, 25] and the references therein.

Our study is motivated by Alves [1], where they considered the multiplicity of
normalized solutions to

(1.6) $-\Delta u=\lambda u+h(\epsilon x)f(u),\quad x\in\mathbb{R}^{N}$

with $f$ being $L^{2}$-subcritical. Their arguments depend on the existence of
global minimizer and the relative compactness of any minimizing sequence of
the functional $\tilde{J}|_{S(a)}$ corresponding to the limit problem

(1.7) $-\Delta u=\lambda u+\mu f(u),\quad x\in\mathbb{R}^{N}.$

While in our problem (1.1), the appearance of the $L^{2}$-supercritical term
$\eta|u|^{p-2}u$ makes the functional to the limit problem

(1.8) $-\Delta u=\lambda u+\mu|u|^{q-2}u+\eta|u|^{p-2}u,\quad
x\in\mathbb{R}^{N}$

with $q<2+4/N<p$ is unbounded from below (and from above). But in view of the
studies of [17, 28], we know that the functional in this case has a local
minimizer. So employing the truncated skill used in [2, 26], we can isolate
the local minimizer and obtain the multiplicity of normalized solutions to the
problem (1.1). The application of truncated functions and the appearance of
the Sobolev critical exponent $p=2^{*}$ make more delicate analysis is needed.
Furthermore, we also consider the orbital stability of the solutions obtained
(see Section 5). We should point out that in [2], the authors studied the
multiplicity of normalized solutions to the autonomous Schrödinger equation
(1.8) with $q<2+4/N<p=2^{*}$ in radial symmetry space
$H_{rad}^{1}(\mathbb{R}^{N})$ by using truncated skill and genus theory. Note
that our problem (1.1) is non-autonomous and not radially symmetry, so their
method is not work in our problem.

The main results of this paper are as follows.

######  Theorem 1.1.

Let $N,\epsilon,a,\eta,p,q,h$ be as in (1.1). We further assume that
$h_{\text{max}},a$ and $\eta$ satisfy some condition (i.e.,(2.3) holds for
$p<2^{*}$, (2.3) and (2.6) hold for $p=2^{*}$). Then there exists
$\epsilon_{0}>0$ such that (1.1) admits at least $l$ couples
$(u_{j},\lambda_{j})\in H^{1}(\mathbb{R}^{N})\times\mathbb{R}$ of weak
solutions for $\epsilon\in(0,\epsilon_{0})$ with
$\int_{\mathbb{R}^{N}}|u_{j}|^{2}dx=a^{2}$, $\lambda_{j}<0$ and
$E_{\epsilon}(u_{j})<0$ for $j=1,2,\cdots,l$.

######  Theorem 1.2.

The solutions obtained in Theorem 1.1 is orbitally stable in some sense. To
state this theorem, we need some notations used in the proof of Theorem 1.1,
so we give the details in Section 5. We point out that we give the stability
of $l$ different sets, which is very different from the existing results.

######  Remark 1.3.

In [16, 17, 28, 29], the authors considered the normalized solutions to (1.8)
with $q<2+4/N<p$. They obtained a ground state solution to (1.8) with negative
energy which is local minimizer and orbitally stable and a mountain-pass type
solution with positive energy which is strongly instable. In this paper, the
solutions obtained in Theorem 1.1 are also local minimizers, but we do not
know whether they are ground state solutions. The appearance of the potential
$h$ increases the number of the local minimizer and maintains its stability.

This paper is organized as follows. In Section 2, we define the truncated
functional used in the study. In Section 3, we study the properties of the
truncated autonomous functional. In Section 4, we study the truncated non-
autonomous problem and give the proof of Theorem 1.1. In Section 5, we study
the orbital stability of the solutions obtained in Theorem 1.1.

Notation: For $t\geq 1$, the $L^{t}$-norm of $u\in L^{t}(\mathbb{R}^{N})$ is
denoted by $\|u\|_{t}$. The usual norm of $u\in H^{1}(\mathbb{R}^{N})$ is
denoted by $\|u\|$. $C,C_{1},C_{2},\cdots$ denote any positive constant, whose
value is not relevant and may be change from line to line. $o_{n}(1)$ denotes
a real sequence with $o_{n}(1)\to 0$ as $n\to+\infty$. ‘$\rightarrow$’ denotes
strong convergence and ‘$\rightharpoonup$’ denotes weak convergence.
$B_{r}(x_{0}):=\\{x\in\mathbb{R}^{N}:|x-x_{0}|<r\\}$.

##  2\. Truncated functionals

In the proof of Theorem 1.1, we will adapt for our case a truncated function
found in Peral Alonso ([26], Chapter 2, Theorem 2.4.6).

In what follows, we will consider the functional $E_{\epsilon}$ given by (1.2)
restricts to $S(a)$. By the Sobolev embedding and the Gagliardo-Nirenberg
inequality (see [34])

(2.1) $\begin{cases}\|u\|_{t}\leq C_{N,t}\|u\|_{2}^{1-\gamma_{t}}\|\nabla
u\|_{2}^{\gamma_{t}},\ 2<t\left\\{\begin{array}[]{ll}<\infty,&N=1,2,\\\ \leq
2^{*},&N\geq 3,\end{array}\right.\gamma_{t}:=\frac{N}{2}-\frac{N}{t},\\\
C_{N,2^{*}}=S^{-\frac{1}{2}},\ \text{and}\ S:=\inf_{u\in
D^{1,2}(\mathbb{R}^{N})\setminus\\{0\\}}\frac{\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx}{\left(\int_{\mathbb{R}^{N}}|u|^{2^{*}}dx\right)^{2/2^{*}}},\end{cases}$

we have

(2.2)
$\begin{split}E_{\epsilon}(u)&\geq\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{1}{q}h_{\text{max}}\int_{\mathbb{R}^{N}}|u|^{q}dx-\frac{\eta}{p}\int_{\mathbb{R}^{N}}|u|^{p}dx\\\
&\geq\frac{1}{2}\|\nabla
u\|_{2}^{2}-\frac{1}{q}h_{\text{max}}C_{N,q}^{q}a^{q(1-\gamma_{q})}\|\nabla
u\|_{2}^{q\gamma_{q}}-\frac{\eta}{p}C_{N,p}^{p}a^{p(1-\gamma_{p})}\|\nabla
u\|_{2}^{p\gamma_{p}}\\\ &=g_{a}(\|\nabla u\|_{2})\end{split}$

for any $u\in S(a)$, where

$g_{a}(r):=\frac{1}{2}r^{2}-\frac{1}{q}h_{\text{max}}C_{N,q}^{q}a^{q(1-\gamma_{q})}r^{q\gamma_{q}}-\frac{\eta}{p}C_{N,p}^{p}a^{p(1-\gamma_{p})}r^{p\gamma_{p}},\
r>0.$

Set $g_{a}(r)=r^{2}w_{a}(r)$ with

$w_{a}(r):=\frac{1}{2}-\frac{1}{q}h_{\text{max}}C_{N,q}^{q}a^{q(1-\gamma_{q})}r^{q\gamma_{q}-2}-\frac{\eta}{p}C_{N,p}^{p}a^{p(1-\gamma_{p})}r^{p\gamma_{p}-2},\
r>0.$

Now we study the properties of $w_{a}(r)$. Note that

$t\gamma_{t}\left\\{\begin{array}[]{ll}<2,&2<t<2+4/N,\\\ =2,&t=2+4/N,\\\
>2,&2+4/N<t\leq 2^{*}\end{array}\right.\quad\text{and}\quad\gamma_{2^{*}}=1.$

It is obvious that $\lim_{r\to 0^{+}}w_{a}(r)=-\infty$ and
$\lim_{r\to+\infty}w_{a}(r)=-\infty$. By direct calculations, we obtain that

$w_{a}^{\prime}(r)=-\frac{1}{q}h_{\text{max}}C_{N,q}^{q}a^{q(1-\gamma_{q})}(q\gamma_{q}-2)r^{q\gamma_{q}-3}-\frac{\eta}{p}C_{N,p}^{p}a^{p(1-\gamma_{p})}(p\gamma_{p}-2)r^{p\gamma_{p}-3}.$

Then the equation $w_{a}^{\prime}(r)=0$ has a unique solution

$r_{0}=\left(\frac{(2-q\gamma_{q})\frac{1}{q}h_{\text{max}}C_{N,q}^{q}a^{q(1-\gamma_{q})}}{(p\gamma_{p}-2)\frac{\eta}{p}C_{N,p}^{p}a^{p(1-\gamma_{p})}}\right)^{\frac{1}{p\gamma_{p}-q\gamma_{q}}}$

and the maximum of $w_{a}(r)$ is

$w_{a}(r_{0})=\frac{1}{2}-B\left(h_{\text{max}}a^{q(1-\gamma_{q})}\right)^{\frac{p\gamma_{p}-2}{p\gamma_{p}-q\gamma_{q}}}\left(\eta
a^{p(1-\gamma_{p})}\right)^{\frac{2-q\gamma_{q}}{p\gamma_{p}-q\gamma_{q}}},$

where

$B=\frac{p\gamma_{p}-q\gamma_{q}}{2-q\gamma_{q}}\left(\frac{2-q\gamma_{q}}{p\gamma_{p}-2}\right)^{\frac{p\gamma_{p}-2}{p\gamma_{p}-q\gamma_{q}}}\left(\frac{C_{N,q}^{q}}{q}\right)^{\frac{p\gamma_{p}-2}{p\gamma_{p}-q\gamma_{q}}}\left(\frac{C_{N,p}^{p}}{p}\right)^{\frac{2-q\gamma_{q}}{p\gamma_{p}-q\gamma_{q}}}.$

Thus, if we assume

(2.3)
$\left(h_{\text{max}}a^{q(1-\gamma_{q})}\right)^{\frac{p\gamma_{p}-2}{p\gamma_{p}-q\gamma_{q}}}\left(\eta
a^{p(1-\gamma_{p})}\right)^{\frac{2-q\gamma_{q}}{p\gamma_{p}-q\gamma_{q}}}<\frac{1}{2B},$

then the maximum of $w_{a}(r)$ is positive and $w_{a}(r)$ has exactly two
zeros $0<R_{0}<R_{1}<\infty$, which are also the zeros of $g_{a}(r)$. It is
obvious that $g_{a}(r)$ has the following properties

(2.4) $\begin{cases}&g_{a}(0)=g_{a}(R_{0})=g_{a}(R_{1})=0;\ g_{a}(r)<0\
\text{for}\ r>0\ \text{small};\\\ &\lim_{r\to+\infty}g_{a}(r)=-\infty;\
g_{a}(r)\ \text{has\ exactly\ two\ critical\ points\ }\\\ &r_{1}\in(0,R_{0})\
\text{and}\ r_{2}\in(R_{0},R_{1})\ \text{with}\ g_{a}(r_{1})<0\ \text{and}\
g_{a}(r_{2})>0.\end{cases}$

For $p=2^{*}$, we further assume that
$R_{0}<\eta^{-\frac{N-2}{4}}S^{\frac{N}{4}}$, which is satisfied if we assume

(2.5) $r_{0}<\eta^{-\frac{N-2}{4}}S^{\frac{N}{4}},$

because $R_{0}<r_{0}<R_{1}$. By the expression of $r_{0}$, (2.5) is equivalent
to

(2.6)
$\begin{split}\left(h_{\text{max}}a^{q(1-\gamma_{q})}\right)^{\frac{1}{p\gamma_{p}-q\gamma_{q}}}&\eta^{\frac{N-2}{4}-\frac{1}{p\gamma_{p}-q\gamma_{q}}}\\\
&\leq\left(\frac{(2-q\gamma_{q})C_{N,q}^{q}2^{*}S^{\frac{2^{*}}{2}}}{q(p-2)}\right)^{\frac{-1}{p\gamma_{p}-q\gamma_{q}}}S^{\frac{N}{4}}.\end{split}$

Now fix $\tau:(0,+\infty)\to[0,1]$ as being a non-increasing and $C^{\infty}$
function that satisfies

(2.7) $\tau(x)=\left\\{\begin{array}[]{ll}1,&\text{if}\ x\leq R_{0},\\\
0,&\text{if}\ x\geq R_{1}\end{array}\right.$

and consider the truncated functional

(2.8) $E_{\epsilon,T}(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{1}{q}\int_{\mathbb{R}^{N}}h(\epsilon
x)|u|^{q}dx-\frac{\eta}{p}\tau(\|\nabla
u\|_{2})\int_{\mathbb{R}^{N}}|u|^{p}dx.$

Thus,

(2.9) $\begin{split}E_{\epsilon,T}(u)&\geq\frac{1}{2}\|\nabla
u\|_{2}^{2}-\frac{1}{q}h_{\text{max}}C_{N,q}^{q}a^{q(1-\gamma_{q})}\|\nabla
u\|_{2}^{q\gamma_{q}}\\\ &\qquad-\frac{\eta}{p}\tau(\|\nabla
u\|_{2})C_{N,p}^{p}a^{p(1-\gamma_{p})}\|\nabla u\|_{2}^{p\gamma_{p}}\\\
&=\bar{g}_{a}(\|\nabla u\|_{2})\end{split}$

for any $u\in S(a)$, where

$\bar{g}_{a}(r):=\frac{1}{2}r^{2}-\frac{1}{q}h_{\text{max}}C_{N,q}^{q}a^{q(1-\gamma_{q})}r^{q\gamma_{q}}-\frac{\eta}{p}\tau(r)C_{N,p}^{p}a^{p(1-\gamma_{p})}r^{p\gamma_{p}}.$

It is easy to see that $\bar{g}_{a}(r)$ has the following properties

(2.10) $\begin{cases}&\bar{g}_{a}(r)\equiv{g}_{a}(r)\ \text{for\ all}\
r\in[0,R_{0}];\\\ &\bar{g}_{a}(r)\ \text{is\ positive\ and\ strictly\
increasing\ in\ }(R_{0},+\infty).\end{cases}$

Correspondingly, for any $\mu\in(0,h_{\text{max}}]$, we denote by $J_{\mu},\
J_{\mu,T}:H^{1}(\mathbb{R}^{N})\to\mathbb{R}$ the following functionals

(2.11) $J_{\mu}(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{\mu}{q}\int_{\mathbb{R}^{N}}|u|^{q}dx-\frac{\eta}{p}\int_{\mathbb{R}^{N}}|u|^{p}dx$

and

(2.12) $J_{\mu,T}(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{\mu}{q}\int_{\mathbb{R}^{N}}|u|^{q}dx-\frac{\eta}{p}\tau(\|\nabla
u\|_{2})\int_{\mathbb{R}^{N}}|u|^{p}dx.$

The properties of $J_{\mu,T}$ and $E_{\epsilon,T}$ will be studied in Sections
3 and 4, respectively.

##  3\. The truncated autonomous functional

In this section, we study the properties of the functional $J_{\mu,T}$ defined
in (2.12) restricted to $S(a_{1})$, where $\mu\in(0,h_{\text{max}}]$ and
$a_{1}\in(0,a]$.

######  Lemma 3.1.

Let $N,a,\eta,p,q$ be as in (1.1), (2.3) hold, $\mu\in(0,h_{\text{max}}]$,
$0<a_{1}\leq a$. Then the functional $J_{\mu,T}$ is bounded from below in
$S(a_{1})$.

###### Proof.

By (2.9) and (2.10), for any $u\in S(a_{1})$,

$\begin{split}J_{\mu,T}(u)&\geq J_{\text{max},T}(u)\\\
&\geq\frac{1}{2}\|\nabla
u\|_{2}^{2}-\frac{1}{q}h_{\text{max}}C_{N,q}^{q}a_{1}^{q(1-\gamma_{q})}\|\nabla
u\|_{2}^{q\gamma_{q}}\\\ &\qquad\qquad-\frac{\eta}{p}\tau(\|\nabla
u\|_{2})C_{N,p}^{p}a_{1}^{p(1-\gamma_{p})}\|\nabla u\|_{2}^{p\gamma_{p}}\\\
&\geq\bar{g}_{a}(\|\nabla u\|_{2})\geq\inf_{r\geq
0}\bar{g}_{a}(r)>-\infty.\end{split}$

The proof is complete. ∎

######  Lemma 3.2.

Let $N,a,\eta,p,q$ be as in (1.1), (2.3) hold, $\mu\in(0,h_{\text{max}}]$,
$0<a_{1}\leq a$. $\Upsilon_{\mu,T,a_{1}}:=\inf_{u\in S(a_{1})}J_{\mu,T}(u)<0$.

###### Proof.

Fix $u\in S(a_{1})$. For $t>0$, we define $u_{t}(x)=t^{\frac{N}{2}}u(tx)$.
Then $u_{t}\in S(a_{1})$ for all $t>0$. By $\tau\geq 0$ and $q\gamma_{q}<2$,
we obtain that

$\begin{split}J_{\mu,T}(u_{t})&\leq\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u_{t}|^{2}dx-\frac{\mu}{q}\int_{\mathbb{R}^{N}}|u_{t}|^{q}dx\\\
&=\frac{1}{2}t^{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{\mu}{q}t^{q\gamma_{q}}\int_{\mathbb{R}^{N}}|u|^{q}dx\\\
&<0\end{split}$

for $t>0$ small enough. Thus $\Upsilon_{\mu,T,a_{1}}<0$. The proof is
complete. ∎

######  Lemma 3.3.

Let $N,a,\eta,p,q$ be as in (1.1), (2.3) hold, $\mu\in(0,h_{\text{max}}]$.
Then
(1) $J_{\mu,T}\in C^{1}(H^{1}(\mathbb{R}^{N}),\mathbb{R})$;
(2) Let $a_{1}\in(0,a]$. If $u\in S(a_{1})$ such that $J_{\mu,T}(u)<0$, then
$\|\nabla u\|_{2}<R_{0}$ and $J_{\mu,T}(v)=J_{\mu}(v)$ for all $v$ satisfying
$\|v\|_{2}\leq a$ and being in a small neighborhood of $u$ in
$H^{1}(\mathbb{R}^{N})$.

###### Proof.

(1) is trivial. Now we prove (2). It follows from $J_{\mu,T}(u)<0$ and

$J_{\mu,T}(u)\geq\bar{g}_{a_{1}}(\|\nabla u\|_{2})\geq\bar{g}_{a}(\|\nabla
u\|_{2})$

that $\bar{g}_{a}(\|\nabla u\|_{2})<0$, which implies that $\|\nabla
u\|_{2}<R_{0}$ by (2.10). By (1) and $J_{\mu,T}(u)<0$, we obtain that
$J_{\mu,T}(v)<0$ for all $v$ in a small neighborhood of $u$ in
$H^{1}(\mathbb{R}^{N})$, which combined with $\|v\|_{2}\leq a$ gives that
$\|\nabla v\|_{2}<R_{0}$ and thus $J_{\mu,T}(v)=J_{\mu}(v)$. The proof is
complete. ∎

For any $a_{1}\in(0,a]$, we define

$m_{\mu}(a_{1}):=\inf_{u\in V(a_{1})}J_{\mu}(u),\ V(a_{1}):=\\{u\in
S(a_{1}):\|\nabla u\|_{2}<R_{0}\\}.$

Since $J_{\mu,T}(u)\geq\bar{g}_{a_{1}}(\|\nabla
u\|_{2})\geq\bar{g}_{a}(\|\nabla u\|_{2})$ for any $u\in S(a_{1})$, by Lemmas
3.1-3.3, we obtain that

(3.1) $\Upsilon_{\mu,T,a_{1}}=\inf_{u\in
S(a_{1})}J_{\mu,T}(u)=m_{\mu}(a_{1}).$

In ([16], Lemma 2.6 and Theorem 1.2), the authors obtained that

######  Lemma 3.4.

Let $N,a,\eta,p,q$ be as in (1.1), (2.3) hold, $\mu\in(0,h_{\text{max}}]$.
Then
(1) $a_{1}\in(0,a]\mapsto m_{\mu}(a_{1})$ is continuous;
(2) Let $0<a_{1}<a_{2}\leq a$, then
$\frac{a_{1}^{2}}{a_{2}^{2}}m_{\mu}(a_{2})<m_{\mu}(a_{1})<0$.

Consequently, by (3.1) and Lemma 3.4, we obtain that

######  Lemma 3.5.

Let $N,a,\eta,p,q$ be as in (1.1), (2.3) hold, $\mu\in(0,h_{\text{max}}]$.
Then
(1) $a_{1}\in(0,a]\mapsto\Upsilon_{\mu,T,a_{1}}$ is continuous;
(2) Let $0<a_{1}<a_{2}\leq a$, then
$\frac{a_{1}^{2}}{a_{2}^{2}}\Upsilon_{\mu,T,a_{2}}<\Upsilon_{\mu,T,a_{1}}<0$.

The next compactness lemma is useful in the study of the autonomous problem as
well as in the non-autonomous problem.

######  Lemma 3.6.

Let $N,a,\eta,p,q$ be as in (1.1), (2.3) hold, $\mu\in(0,h_{\text{max}}]$,
$a_{1}\in(0,a]$. $\\{u_{n}\\}\subset S(a_{1})$ be a minimizing sequence with
respect to $\Upsilon_{\mu,T,a_{1}}$. Then, for some subsequence, either
(i) $\\{u_{n}\\}$ is strongly convergent,
or
(ii) There exists $\\{y_{n}\\}\subset\mathbb{R}^{N}$ with $|y_{n}|\to\infty$
such that the sequence $v_{n}(x)=u_{n}(x+y_{n})$ is strongly convergent to a
function $v\in S(a_{1})$ with $J_{\mu,T}(v)=\Upsilon_{\mu,T,a_{1}}$.

###### Proof.

Noting that $\|\nabla u_{n}\|_{2}<R_{0}$ for $n$ large enough, there exists
$u\in H^{1}(\mathbb{R}^{N})$ such that $u_{n}\rightharpoonup u$ in
$H^{1}(\mathbb{R}^{N})$ for some subsequence. Now we consider the following
three possibilities.

(1) If $u\not\equiv 0$ and $\|u\|_{2}=b\neq a_{1}$, we must have
$b\in(0,a_{1})$. Setting $v_{n}=u_{n}-u$, $d_{n}=\|v_{n}\|_{2}$, and by using

$\|u_{n}\|_{2}^{2}=\|v_{n}\|_{2}^{2}+\|u\|_{2}^{2}+o_{n}(1),$

we obtain that $\|v_{n}\|_{2}\to d$, where $a_{1}^{2}=d^{2}+b^{2}$. Noting
that $d_{n}\in(0,a_{1})$ for $n$ large enough, and using the Brézis-Lieb Lemma
(see [35]), Lemma 3.5, $\|\nabla u_{n}\|_{2}^{2}=\|\nabla u\|_{2}^{2}+\|\nabla
v_{n}\|_{2}^{2}+o_{n}(1)$, $\|\nabla
u\|_{2}^{2}\leq\liminf_{n\to+\infty}\|\nabla u_{n}\|_{2}^{2}$, $\tau$ is
continuous and non-increasing, we obtain that

$\begin{split}\Upsilon_{\mu,T,a_{1}}+o_{n}(1)=J_{\mu,T}(u_{n})&=\frac{1}{2}\|\nabla
v_{n}\|_{2}^{2}-\frac{\mu}{q}\|v_{n}\|_{q}^{q}-\frac{\eta}{p}\tau(\|\nabla
u_{n}\|_{2})\|v_{n}\|_{p}^{p}\\\ &\quad+\frac{1}{2}\|\nabla
u\|_{2}^{2}-\frac{\mu}{q}\|u\|_{q}^{q}-\frac{\eta}{p}\tau(\|\nabla
u_{n}\|_{2})\|u\|_{p}^{p}+o_{n}(1)\\\ &\geq
J_{\mu,T}(v_{n})+J_{\mu,T}(u)+o_{n}(1)\\\
&\geq\Upsilon_{\mu,T,d_{n}}+\Upsilon_{\mu,T,b}+o_{n}(1)\\\
&\geq\frac{d_{n}^{2}}{a_{1}^{2}}\Upsilon_{\mu,T,a_{1}}+\Upsilon_{\mu,T,b}+o_{n}(1).\end{split}$

Letting $n\to+\infty$, we find that

$\begin{split}\Upsilon_{\mu,T,a_{1}}&\geq\frac{d^{2}}{a_{1}^{2}}\Upsilon_{\mu,T,a_{1}}+\Upsilon_{\mu,T,b}\\\
&>\frac{d^{2}}{a_{1}^{2}}\Upsilon_{\mu,T,a_{1}}+\frac{b^{2}}{a_{1}^{2}}\Upsilon_{\mu,T,a_{1}}=\Upsilon_{\mu,T,a_{1}},\end{split}$

which is a contradiction. So this possibility can not exist.

(2) If $\|u\|_{2}=a_{1}$, then $u_{n}\to u$ in $L^{2}(\mathbb{R}^{N})$ and
thus $u_{n}\to u$ in $L^{t}(\mathbb{R}^{N})$ for all $t\in(2,2^{*})$.

Case $p<2^{*}$, then

$\begin{split}\Upsilon_{\mu,T,a_{1}}&=\lim_{n\to+\infty}J_{\mu,T}(u_{n})\\\
&=\lim_{n\to+\infty}\left(\frac{1}{2}\|\nabla
u_{n}\|_{2}^{2}-\frac{\mu}{q}\|u_{n}\|_{q}^{q}-\frac{\eta}{p}\tau(\|\nabla
u_{n}\|_{2})\|u_{n}\|_{p}^{p}\right)\\\ &\geq J_{\mu,T}(u).\end{split}$

As $u\in S(a_{1})$, we infer that $J_{\mu,T}(u)=\Upsilon_{\mu,T,a_{1}}$, then
$\|\nabla u_{n}\|_{2}\to\|\nabla u\|_{2}$ and thus $u_{n}\to u$ in
$H^{1}(\mathbb{R}^{N})$, which implies that (i) occurs.

Case $p=2^{*}$, noting that $\|\nabla v_{n}\|_{2}\leq\|\nabla
u_{n}\|_{2}<R_{0}$ for $n$ large enough, and using the Soblev inequality, we
have

(3.2)
$\begin{split}J_{\mu,T}(v_{n})=J_{\mu}(v_{n})&=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
v_{n}|^{2}dx-\frac{\mu}{q}\int_{\mathbb{R}^{N}}|v_{n}|^{q}dx-\frac{\eta}{p}\int_{\mathbb{R}^{N}}|v_{n}|^{p}dx\\\
&\geq\frac{1}{2}\|\nabla
v_{n}\|_{2}^{2}-\frac{\eta}{2^{*}}\frac{1}{S^{\frac{2^{*}}{2}}}\|\nabla
v_{n}\|_{2}^{2^{*}}+o_{n}(1)\\\ &=\|\nabla
v_{n}\|_{2}^{2}\left(\frac{1}{2}-\frac{\eta}{2^{*}}\frac{1}{S^{\frac{2^{*}}{2}}}\|\nabla
v_{n}\|_{2}^{2^{*}-2}\right)+o_{n}(1)\\\ &\geq\|\nabla
v_{n}\|_{2}^{2}\left(\frac{1}{2}-\frac{\eta}{2^{*}}\frac{1}{S^{\frac{2^{*}}{2}}}R_{0}^{2^{*}-2}\right)+o_{n}(1)\\\
&=\|\nabla
v_{n}\|_{2}^{2}\frac{1}{q}h_{\text{max}}C_{N,q}^{q}a^{q(1-\gamma_{q})}R_{0}^{q\gamma_{q}-2}+o_{n}(1),\end{split}$

because
$w_{a}(R_{0})=\frac{1}{2}-\frac{\eta}{2^{*}}\frac{1}{S^{\frac{2^{*}}{2}}}R_{0}^{2^{*}-2}-\frac{1}{q}h_{\text{max}}C_{N,q}^{q}a^{q(1-\gamma_{q})}R_{0}^{q\gamma_{q}-2}=0$.
Now we remember that

(3.3) $\Upsilon_{\mu,T,a_{1}}\leftarrow J_{\mu,T}(u_{n})\geq
J_{\mu,T}(v_{n})+J_{\mu,T}(u)+o_{n}(1).$

Since $u\in S(a_{1})$, we have $J_{\mu,T}(u)\geq\Upsilon_{\mu,T,a_{1}}$, which
combined with (3.2) and (3.3) gives that $\|\nabla v_{n}\|_{2}^{2}\to 0$ and
then $u_{n}\to u$ in $H^{1}(\mathbb{R}^{N})$. This implies that (i) occurs.

(3) If $u\equiv 0$, that is, $u_{n}\rightharpoonup 0$ in
$H^{1}(\mathbb{R}^{N})$. We claim that there exist $R,\beta>0$ and
$\\{y_{n}\\}\subset\mathbb{R}^{N}$ such that

(3.4) $\int_{B_{R}(y_{n})}|u_{n}|^{2}dx\geq\beta,\ \text{for\ all\ }n.$

Indeed, otherwise we must have $u_{n}\to 0$ in $L^{t}(\mathbb{R}^{N})$ for all
$t\in(2,2^{*})$. Thus, for $p<2^{*}$, $J_{\mu,T}(u_{n})\geq\frac{1}{2}\|\nabla
u_{n}\|_{2}^{2}+o_{n}(1)$, which contradicts
$J_{\mu,T}(u_{n})\to\Upsilon_{\mu,T,a_{1}}<0$. For $p=2^{*}$, similarly to
(3.2), we obtain that

$J_{\mu,T}(u_{n})\geq\|\nabla
u_{n}\|_{2}^{2}\frac{1}{q}h_{\text{max}}C_{N,q}^{q}a^{q(1-\gamma_{q})}R_{0}^{q\gamma_{q}-2}+o_{n}(1).$

We also get a contradiction in this case. Hence, in all cases, (3.4) holds and
$|y_{n}|\to+\infty$ obviously. From this, considering
$\bar{u}_{n}(x)=u_{n}(x+y_{n})$, clearly $\\{\bar{u}_{n}\\}\subset S(a_{1})$
and it is also a minimizing sequence with respect to $\Upsilon_{\mu,T,a_{1}}$.
Moreover, there exists $\bar{u}\in H^{1}(\mathbb{R}^{N})\backslash\\{0\\}$
such that $\bar{u}_{n}\rightharpoonup\bar{u}$ in $H^{1}(\mathbb{R}^{N})$.
Following as in the first two possibilities of the proof, we derive that
$\bar{u}_{n}\to\bar{u}$ in $H^{1}(\mathbb{R}^{N})$, which implies that (ii)
occurs. This proves the lemma. ∎

######  Lemma 3.7.

Let $N,a,\eta,p,q$ be as in (1.1), $\mu\in(0,h_{\text{max}}]$,
$a_{1}\in(0,a]$, (2.3) hold. Then $\Upsilon_{\mu,T,a_{1}}$ is attained.

###### Proof.

By Lemmas 3.1-3.3, there exists a bounded minimizing sequence
$\\{u_{n}\\}\subset S(a_{1})$ satisfying
$J_{\mu,T}(u_{n})\to\Upsilon_{\mu,T,a_{1}}$ as $n\to+\infty$. Now, applying
Lemma 3.6, there exists $u\in S(a_{1})$ such that
$J_{\mu,T}(u)=\Upsilon_{\mu,T,a_{1}}$. The proof is complete. ∎

An immediate consequence of Lemma 3.7 is the following corollary.

######  Corollary 3.8.

Let $N,a,\eta,p,q$ be as in (1.1), (2.3) hold. Fix $a_{1}\in(0,a]$ and let
$0<\mu_{1}<\mu_{2}\leq h_{\text{max}}$. Then
$\Upsilon_{\mu_{2},T,a_{1}}<\Upsilon_{\mu_{1},T,a_{1}}$.

###### Proof.

Let $u\in S(a_{1})$ satisfying $J_{\mu_{1},T}(u)=\Upsilon_{\mu_{1},T,a_{1}}$.
Then, $\Upsilon_{\mu_{2},T,a_{1}}\leq
J_{\mu_{2},T}(u)<J_{\mu_{1},T}(u)=\Upsilon_{\mu_{1},T,a_{1}}$. ∎

##  4\. Proof of Theorem 1.1

In this section, we first prove some properties of the functional
$E_{\epsilon,T}$ defined in (2.8) restricted to the sphere $S(a)$, and then
give the proof of Theorem 1.1.

Denote

$J_{\text{max},T}:=J_{h_{\text{max}},T},\
\Upsilon_{\text{max},T,a}:=\Upsilon_{h_{\text{max}},T,a},$

and

$J_{\infty,T}:=J_{h_{\infty},T},\
\Upsilon_{\infty,T,a}:=\Upsilon_{h_{\infty},T,a}.$

It is obvious that $J_{\infty,T}(u)\geq J_{\text{max},T}(u)$ and
$E_{\epsilon,T}(u)\geq J_{\text{max},T}(u)$ for any $u\in S(a)$. By Lemma 3.1,
the definition

$\Gamma_{\epsilon,T,a}:=\inf_{u\in S(a)}E_{\epsilon,T}(u)$

is well defined and $\Gamma_{\epsilon,T,a}\geq\Upsilon_{\text{max},T,a}$.

The next lemma establishes some crucial relations involving the levels
$\Gamma_{\epsilon,T,a}$, $\Upsilon_{\infty,T,a}$ and
$\Upsilon_{\text{max},T,a}$.

######  Lemma 4.1.

Let $N,a,\eta,p,q,h,\epsilon$ be as in (1.1), (2.3) hold. Then

$\limsup_{\epsilon\to
0^{+}}\Gamma_{\epsilon,T,a}\leq\Upsilon_{\text{max},T,a}<\Upsilon_{\infty,T,a}<0.$

###### Proof.

By Lemma 3.7, choose $u\in S(a)$ such that
$J_{\text{max},T}(u)=\Upsilon_{\text{max},T,a}$. Then,

$\begin{split}\Gamma_{\epsilon,T,a}\leq
E_{\epsilon,T}(u)&=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{1}{q}\int_{\mathbb{R}^{N}}h(\epsilon x)|u|^{q}dx\\\
&\qquad-\frac{\eta}{p}\tau(\|\nabla
u\|_{2})\int_{\mathbb{R}^{N}}|u|^{p}dx.\end{split}$

Letting $\epsilon\to 0^{+}$, by the Lebesgue dominated convergence theorem, we
deduce that

$\limsup_{\epsilon\to 0^{+}}\Gamma_{\epsilon,T,a}\leq\limsup_{\epsilon\to
0^{+}}E_{\epsilon,T}(u)=J_{h(0),T}(u)=J_{\text{max},T}(u)=\Upsilon_{\text{max},T,a},$

which combined with Lemma 3.2 and Corollary 3.8 completes the proof. ∎

By Lemma 4.1, there exists $\epsilon_{1}>0$ such that
$\Gamma_{\epsilon,T,a}<\Upsilon_{\infty,T,a}$ for all
$\epsilon\in(0,\epsilon_{1})$. In the following, we always assume that
$\epsilon\in(0,\epsilon_{1})$. Similarly to the proof of Lemma 3.3, we have
the following result, whose proof is omitted.

######  Lemma 4.2.

Let $N,a,\eta,p,q,h,\epsilon$ be as in (1.1), $\epsilon\in(0,\epsilon_{1})$,
(2.3) hold. Then
(1) $E_{\epsilon,T}\in C^{1}(H^{1}(\mathbb{R}^{N}),\mathbb{R})$;
(2) If $u\in S(a)$ such that $E_{\epsilon,T}(u)<0$, then $\|\nabla
u\|_{2}<R_{0}$ and $E_{\epsilon,T}(v)=E_{\epsilon}(v)$ for all $v$ satisfying
$\|v\|_{2}\leq a$ and being in a small neighborhood of $u$ in
$H^{1}(\mathbb{R}^{N})$.

The next two lemmas will be used to prove the $(PS)$ condition for
$E_{\epsilon,T}$ restricts to $S(a)$ at some levels.

######  Lemma 4.3.

Let $N,a,\eta,p,q,h$ be as in (1.1), $\epsilon\in(0,\epsilon_{1})$, (2.3)
hold. Assume $\\{u_{n}\\}\subset S(a)$ such that $E_{\epsilon,T}(u_{n})\to c$
as $n\to+\infty$ with $c<\Upsilon_{\infty,T,a}$. If $u_{n}\rightharpoonup u$
in $H^{1}(\mathbb{R}^{N})$, then $u\not\equiv 0$.

###### Proof.

Assume by contradiction that $u\equiv 0$. Then,

$c+o_{n}(1)=E_{\epsilon,T}(u_{n})=J_{\infty,T}(u_{n})+\frac{1}{q}\int_{\mathbb{R}^{N}}(h_{\infty}-h(\epsilon
x))|u_{n}|^{q}dx.$

By ($h_{2}$), for any given $\delta>0$, there exists $R>0$ such that
$h_{\infty}\geq h(x)-\delta$ for all $|x|\geq R$. Hence,

$\begin{split}c+o_{n}(1)=E_{\epsilon,T}(u_{n})&\geq
J_{\infty,T}(u_{n})+\frac{1}{q}\int_{B_{R/\epsilon}(0)}(h_{\infty}-h(\epsilon
x))|u_{n}|^{q}dx\\\
&\qquad-\frac{\delta}{q}\int_{B^{c}_{R/\epsilon}(0)}|u_{n}|^{q}dx.\end{split}$

Recalling that $\\{u_{n}\\}$ is bounded in $H^{1}(\mathbb{R}^{N})$ and
$u_{n}\to 0$ in $L^{t}(B_{R/\epsilon}(0))$ for all $t\in[1,2^{*})$, it follows
that

$c+o_{n}(1)=E_{\epsilon,T}(u_{n})\geq J_{\infty,T}(u_{n})-\delta C+o_{n}(1)$

for some $C>0$. Since $\delta>0$ is arbitrary, we deduce that
$c\geq\Upsilon_{\infty,T,a}$, which is a contradiction. Thus, $u\not\equiv 0$.
∎

######  Lemma 4.4.

Let $N,a,\eta,p,q,h$ be as in (1.1), $\epsilon\in(0,\epsilon_{1})$, (2.3)
hold. If $p=2^{*}$, we further assume (2.6) hold. Let $\\{u_{n}\\}$ be a
$(PS)_{c}$ sequence of $E_{\epsilon,T}$ restricted to $S(a)$ with
$c<\Upsilon_{\infty,T,a}$ and let $u_{n}\rightharpoonup u_{\epsilon}$ in
$H^{1}(\mathbb{R}^{N})$. If $u_{n}\not\rightarrow u_{\epsilon}$ in
$H^{1}(\mathbb{R}^{N})$, there exists $\beta>0$ independent of
$\epsilon\in(0,\epsilon_{1})$ such that

$\limsup_{n\to+\infty}\|u_{n}-u_{\epsilon}\|_{2}\geq\beta.$

###### Proof.

By Lemma 4.2, we must have $\|\nabla u_{n}\|_{2}<R_{0}$ for $n$ large enough,
and so, $\\{u_{n}\\}$ is also a $(PS)_{c}$ sequence of $E_{\epsilon}$
restricted to $S(a)$. Hence,

$E_{\epsilon}(u_{n})\to c\ \text{and}\
\|E_{\epsilon}|^{\prime}_{S(a)}(u_{n})\|\to 0\ \text{as}\ n\to+\infty.$

Setting the functional $\Psi:H^{1}(\mathbb{R}^{N})\to\mathbb{R}$ given by

$\Psi(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|u|^{2}dx,$

it follows that $S(a)=\Psi^{-1}(a^{2}/2)$. Then, by Willem ([35], Proposition
5.12), there exists $\\{\lambda_{n}\\}\subset\mathbb{R}$ such that

(4.1)
$\|E^{\prime}_{\epsilon}(u_{n})-\lambda_{n}\Psi^{\prime}(u_{n})\|_{H^{-1}(\mathbb{R}^{N})}\to
0\ \text{as}\ n\to+\infty.$

By the boundedness of $\\{u_{n}\\}$ in $H^{1}(\mathbb{R}^{N})$, we know
$\\{\lambda_{n}\\}$ is bounded and thus, for some subsequence, there exists
$\lambda_{\epsilon}$ such that $\lambda_{n}\to\lambda_{\epsilon}$ as
$n\to+\infty$. This together with (4.1) leads to

(4.2)
$E^{\prime}_{\epsilon}(u_{\epsilon})-\lambda_{\epsilon}\Psi^{\prime}(u_{\epsilon})=0\text{\
in \ }H^{-1}(\mathbb{R}^{N})$

and then

(4.3)
$\|E^{\prime}_{\epsilon}(v_{n})-\lambda_{\epsilon}\Psi^{\prime}(v_{n})\|_{H^{-1}(\mathbb{R}^{N})}\to
0\ \text{as}\ n\to+\infty,$

where $v_{n}:=u_{n}-u_{\epsilon}$. By direct calculations, we get that

$\begin{split}\Upsilon_{\infty,T,a}&>\lim_{n\to+\infty}E_{\epsilon}(u_{n})\\\
&=\lim_{n\to+\infty}\left(E_{\epsilon}(u_{n})-\frac{1}{2}E_{\epsilon}^{\prime}(u_{n})u_{n}+\frac{1}{2}\lambda_{n}\|u_{n}\|_{2}^{2}+o_{n}(1)\right)\\\
&=\lim_{n\to+\infty}\left[\left(\frac{1}{2}-\frac{1}{q}\right)\int_{\mathbb{R}^{N}}h(\epsilon
x)|u_{n}|^{q}dx\right.\\\
&\qquad\qquad\qquad\left.+\left(\frac{1}{2}-\frac{1}{p}\right)\eta\int_{\mathbb{R}^{N}}|u_{n}|^{p}dx+\frac{1}{2}\lambda_{n}a^{2}+o_{n}(1)\right]\\\
&\geq\frac{1}{2}\lambda_{\epsilon}a^{2},\end{split}$

which implies that

(4.4) $\lambda_{\epsilon}\leq\frac{2\Upsilon_{\infty,T,a}}{a^{2}}<0\
\text{for\ all\ }\epsilon\in(0,\epsilon_{1}).$

By (4.3), we know

(4.5) $\int_{\mathbb{R}^{N}}|\nabla
v_{n}|^{2}dx-\lambda_{\epsilon}\int_{\mathbb{R}^{N}}|v_{n}|^{2}dx-\int_{\mathbb{R}^{N}}h(\epsilon
x)|v_{n}|^{q}dx-\eta\int_{\mathbb{R}^{N}}|v_{n}|^{p}dx=o_{n}(1),$

which combined with (4.4) gives that

(4.6) $\begin{split}\int_{\mathbb{R}^{N}}|\nabla
v_{n}|^{2}dx&-\frac{2\Upsilon_{\infty,T,a}}{a^{2}}\int_{\mathbb{R}^{N}}|v_{n}|^{2}dx\\\
&\leq
h_{\text{max}}\int_{\mathbb{R}^{N}}|v_{n}|^{q}dx+\eta\int_{\mathbb{R}^{N}}|v_{n}|^{p}dx+o_{n}(1).\end{split}$

If $u_{n}\not\rightarrow u_{\epsilon}$ in $H^{1}(\mathbb{R}^{N})$, that is,
$v_{n}\not\rightarrow 0$ in $H^{1}(\mathbb{R}^{N})$, by (4.6) and the Sobolev
inequality, we deduce that

$\begin{split}\int_{\mathbb{R}^{N}}|\nabla
v_{n}|^{2}dx&-\frac{2\Upsilon_{\infty,T,a}}{a^{2}}\int_{\mathbb{R}^{N}}|v_{n}|^{2}dx\\\
&\leq h_{\text{max}}C_{N,q}^{q}\|v_{n}\|^{q}+\eta
C_{N,p}^{p}\|v_{n}\|^{p}+o_{n}(1).\end{split}$

So there exists $C>0$ independent of $\epsilon$ such that $\|v_{n}\|\geq C$
and then by (4.6)

(4.7)
$\limsup_{n\to+\infty}\left(h_{\text{max}}\int_{\mathbb{R}^{N}}|v_{n}|^{q}dx+\eta\int_{\mathbb{R}^{N}}|v_{n}|^{p}dx\right)\geq
C.$

Case $p<2^{*}$, by (4.7) and the Gagliardo-Nirenberg inequality (2.1), there
exists $\beta>0$ independent of $\epsilon\in(0,\epsilon_{1})$ such that

(4.8) $\limsup_{n\to+\infty}\|v_{n}\|_{2}\geq\beta.$

Case $p=2^{*}$, if

$\limsup_{n\to+\infty}\int_{\mathbb{R}^{N}}|v_{n}|^{q}dx\geq C$

for some $C>0$ independent of $\epsilon$, we obtain (4.8) as well. If

$\liminf_{\epsilon\to
0^{+}}\limsup_{n\to+\infty}\int_{\mathbb{R}^{N}}|v_{n}|^{q}dx=0\ \text{and}\
\liminf_{\epsilon\to 0^{+}}\limsup_{n\to+\infty}\|v_{n}\|_{2}=0,$

by (4.7) we have

$\liminf_{\epsilon\to
0^{+}}\limsup_{n\to+\infty}\int_{\mathbb{R}^{N}}|v_{n}|^{p}dx\geq C,$

and by (4.5) we have

$\liminf_{\epsilon\to 0^{+}}\limsup_{n\to+\infty}\int_{\mathbb{R}^{N}}|\nabla
v_{n}|^{2}dx=\liminf_{\epsilon\to
0^{+}}\limsup_{n\to+\infty}\eta\int_{\mathbb{R}^{N}}|v_{n}|^{p}dx.$

Applying the Sobolev inequality to the above equality, we obtain that

$\liminf_{\epsilon\to 0^{+}}\limsup_{n\to+\infty}\|\nabla
v_{n}\|_{2}^{2}=\liminf_{\epsilon\to
0^{+}}\limsup_{n\to+\infty}\eta\|v_{n}\|_{2^{*}}^{2^{*}}\leq\liminf_{\epsilon\to
0^{+}}\limsup_{n\to+\infty}\eta S^{-2^{*}/2}\|\nabla v_{n}\|_{2}^{2^{*}},$

which implies that

$R_{0}\geq\liminf_{\epsilon\to 0^{+}}\limsup_{n\to+\infty}\|\nabla
v_{n}\|_{2}\geq\eta^{-\frac{N-2}{4}}S^{N/4}.$

That contradicts $R_{0}<\eta^{-\frac{N-2}{4}}S^{N/4}$, that is, assumption
(2.6). So we must have (4.8) for the case $p=2^{*}$. The proof is complete. ∎

Now we give the compactness lemma.

######  Lemma 4.5.

Let $N,a,\eta,p,q,h$ be as in (1.1), $\epsilon\in(0,\epsilon_{1})$, $\beta$ be
as in Lemma 4.4,

$0<\rho_{0}\leq\min\left\\{\Upsilon_{\infty,T,a}-\Upsilon_{\text{max},T,a},\frac{\beta^{2}}{a^{2}}(\Upsilon_{\infty,T,a}-\Upsilon_{\text{max},T,a})\right\\},$

and (2.3) hold. If $p=2^{*}$, we further assume (2.6) hold. Then
$E_{\epsilon,T}$ satisfies the $(PS)_{c}$ condition restricted to $S(a)$ if
$c<\Upsilon_{\text{max},T,a}+\rho_{0}$.

###### Proof.

Let $\\{u_{n}\\}\subset S(a)$ be a $(PS)_{c}$ sequence of $E_{\epsilon,T}$
restricted to $S(a)$. Noting that $c<\Upsilon_{\infty,T,a}<0$, by Lemma 4.2,
$\\{u_{n}\\}$ is bounded in $H^{1}(\mathbb{R}^{N})$. Let $u_{n}\rightharpoonup
u_{\epsilon}$ in $H^{1}(\mathbb{R}^{N})$. By Lemma 4.3,
$u_{\epsilon}\not\equiv 0$. Set $v_{n}:=u_{n}-u_{\epsilon}$. If $u_{n}\to
u_{\epsilon}$ in $H^{1}(\mathbb{R}^{N})$, the proof is complete. If
$u_{n}\not\rightarrow u_{\epsilon}$ in $H^{1}(\mathbb{R}^{N})$ for some
$\epsilon\in(0,\epsilon_{1})$, by Lemma 4.4,

$\limsup_{n\to+\infty}\|v_{n}\|_{2}\geq\beta.$

Set $b=\|u_{\epsilon}\|_{2},\ d_{n}=\|v_{n}\|_{2}$ and suppose that
$\|v_{n}\|_{2}\to d$, then we get $d\geq\beta>0$ and $a^{2}=b^{2}+d^{2}$. From
$d_{n}\in(0,a)$ for $n$ large enough, we have

(4.9) $c+o_{n}(1)=E_{\epsilon,T}(u_{n})\geq
E_{\epsilon,T}(v_{n})+E_{\epsilon,T}(u_{\epsilon})+o_{n}(1).$

Since $v_{n}\rightharpoonup 0$ in $H^{1}(\mathbb{R}^{N})$, similarly to the
proof of Lemma 4.3, we deduce that

(4.10) $E_{\epsilon,T}(v_{n})\geq J_{\infty,T}(v_{n})-\delta C+o_{n}(1)$

for any $\delta>0$, where $C>0$ is a constant independent of $\delta,\
\epsilon$ and $n$. By (4.9) and (4.10), we obtain that

$\begin{split}c+o_{n}(1)=E_{\epsilon,T}(u_{n})&\geq
J_{\infty,T}(v_{n})+E_{\epsilon,T}(u_{\epsilon})-\delta C+o_{n}(1)\\\
&\geq\Upsilon_{\infty,T,d_{n}}+\Upsilon_{\text{max},T,b}-\delta
C+o_{n}(1).\end{split}$

Letting $n\to+\infty$, by Lemma 3.5 and the arbitrariness of $\delta>0$, we
obtain that

$\begin{split}c\geq\Upsilon_{\infty,T,d}+\Upsilon_{\text{max},T,b}&\geq\frac{d^{2}}{a^{2}}\Upsilon_{\infty,T,a}+\frac{b^{2}}{a^{2}}\Upsilon_{\text{max},T,a}\\\
&=\Upsilon_{\text{max},T,a}+\frac{d^{2}}{a^{2}}(\Upsilon_{\infty,T,a}-\Upsilon_{\text{max},T,a})\\\
&\geq\Upsilon_{\text{max},T,a}+\frac{\beta^{2}}{a^{2}}(\Upsilon_{\infty,T,a}-\Upsilon_{\text{max},T,a}),\end{split}$

which contradicts
$c<\Upsilon_{\text{max},T,a}+\frac{\beta^{2}}{a^{2}}(\Upsilon_{\infty,T,a}-\Upsilon_{\text{max},T,a})$.
Thus, we must have $u_{n}\to u_{\epsilon}$ in $H^{1}(\mathbb{R}^{N})$. ∎

In what follows, let us fix $\tilde{\rho},\tilde{r}>0$ satisfying:

$\bullet$
$\overline{B_{\tilde{\rho}}(a_{i})}\cap\overline{B_{\tilde{\rho}}(a_{j})}=\emptyset$
for $i\neq j$ and $i,j\in\\{1,\cdots,l\\}$;

$\bullet$ $\cup_{i=1}^{l}B_{\tilde{\rho}}(a_{i})\subset B_{\tilde{r}}(0)$;

$\bullet$
$K_{\frac{\tilde{\rho}}{2}}=\cup_{i=1}^{l}\overline{B_{\frac{\tilde{\rho}}{2}}(a_{i})}$.

We also set the function
$Q_{\epsilon}:H^{1}(\mathbb{R}^{N})\backslash\\{0\\}\to\mathbb{R}^{N}$ by

$Q_{\epsilon}(u):=\frac{\int_{\mathbb{R}^{N}}\chi(\epsilon
x)|u|^{2}dx}{\int_{\mathbb{R}^{N}}|u|^{2}dx},$

where $\chi:\mathbb{R}^{N}\to\mathbb{R}^{N}$ is given by

$\chi(x):=\left\\{\begin{array}[]{ll}x,&\ \text{if}\ |x|\leq\tilde{r},\\\
\tilde{r}\frac{x}{|x|},&\ \text{if}\ |x|>\tilde{r}.\end{array}\right.$

The next two lemmas will be useful to get important ($PS$) sequences for
$E_{\epsilon,T}$ restricted to $S(a)$.

######  Lemma 4.6.

Let $N,a,\eta,p,q,h$ be as in (1.1), and (2.3) hold. Then there exist
$\epsilon_{2}\in(0,\epsilon_{1}],\rho_{1}\in(0,\rho_{0}]$ such that if
$\epsilon\in(0,\epsilon_{2})$, $u\in S(a)$ and
$E_{\epsilon,T}(u)\leq\Upsilon_{\text{max},T,a}+\rho_{1}$, then

$Q_{\epsilon}(u)\in K_{\frac{\tilde{\rho}}{2}}.$

###### Proof.

If the lemma does not occur, there must be $\rho_{n}\to 0,\epsilon_{n}\to 0$
and $\\{u_{n}\\}\subset S(a)$ such that

(4.11) $E_{\epsilon_{n},T}(u_{n})\leq\Upsilon_{\text{max},T,a}+\rho_{n}\
\text{and}\ Q_{\epsilon_{n}}(u_{n})\not\in K_{\frac{\tilde{\rho}}{2}}.$

Consequently,

$\Upsilon_{\text{max},T,a}\leq J_{\text{max},T}(u_{n})\leq
E_{\epsilon_{n},T}(u_{n})\leq\Upsilon_{\text{max},T,a}+\rho_{n},$

then

$\\{u_{n}\\}\subset S(a)\ \text{and}\
J_{\text{max},T}(u_{n})\to\Upsilon_{\text{max},T,a}.$

According to Lemma 3.6, we have two cases:
(i) $u_{n}\to u$ in $H^{1}(\mathbb{R}^{N})$ for some $u\in S(a)$,
or
(ii) There exists $\\{y_{n}\\}\subset\mathbb{R}^{N}$ with $|y_{n}|\to+\infty$
such that $v_{n}(x)=u_{n}(x+y_{n})$ converges in $H^{1}(\mathbb{R}^{N})$ to
some $v\in S(a)$.

Analysis of (i): By the Lebesgue dominated convergence theorem,

$Q_{\epsilon_{n}}(u_{n})=\frac{\int_{\mathbb{R}^{N}}\chi(\epsilon_{n}x)|u_{n}|^{2}dx}{\int_{\mathbb{R}^{N}}|u_{n}|^{2}dx}\to\frac{\int_{\mathbb{R}^{N}}\chi(0)|u|^{2}dx}{\int_{\mathbb{R}^{N}}|u|^{2}dx}=0\in
K_{\frac{\tilde{\rho}}{2}}.$

From this, $Q_{\epsilon_{n}}(u_{n})\in K_{\frac{\tilde{\rho}}{2}}$ for $n$
large enough, which contradicts $Q_{\epsilon_{n}}(u_{n})\not\in
K_{\frac{\tilde{\rho}}{2}}$ in (4.11).

Analysis of (ii): Now we will study two cases: (I)
$|\epsilon_{n}y_{n}|\to+\infty$ and (II) $\epsilon_{n}y_{n}\to y$ for some
$y\in\mathbb{R}^{N}$.

If (I) holds, the limit $v_{n}\to v$ in $H^{1}(\mathbb{R}^{N})$ provides

$\begin{split}E_{\epsilon_{n},T}(u_{n})&=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
v_{n}|^{2}dx-\frac{1}{q}\int_{\mathbb{R}^{N}}h(\epsilon_{n}x+\epsilon_{n}y_{n})|v_{n}|^{q}dx\\\
&\qquad-\frac{\eta}{p}\tau(\|\nabla
v_{n}\|_{2})\int_{\mathbb{R}^{N}}|v_{n}|^{p}dx\\\ &\to
J_{\infty,T}(v).\end{split}$

Since $E_{\epsilon_{n},T}(u_{n})\leq\Upsilon_{\text{max},T,a}+\rho_{n}$, we
deduce that

$\Upsilon_{\infty,T,a}\leq J_{\infty,T}(v)\leq\Upsilon_{\text{max},T,a},$

which contradicts $\Upsilon_{\infty,T,a}>\Upsilon_{\text{max},T,a}$ in Lemma
4.1.

Now if (II) holds, then

$\begin{split}E_{\epsilon_{n},T}(u_{n})&=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
v_{n}|^{2}dx-\frac{1}{q}\int_{\mathbb{R}^{N}}h(\epsilon_{n}x+\epsilon_{n}y_{n})|v_{n}|^{q}dx\\\
&\qquad-\frac{\eta}{p}\tau(\|\nabla
v_{n}\|_{2})\int_{\mathbb{R}^{N}}|v_{n}|^{p}dx\\\ &\to
J_{h(y),T}(v),\end{split}$

which combined with
$E_{\epsilon_{n},T}(u_{n})\leq\Upsilon_{\text{max},T,a}+\rho_{n}$ gives that

$\Upsilon_{h(y),T,a}\leq J_{h(y),T}(v)\leq\Upsilon_{\text{max},T,a}.$

By Corollary 3.8, we must have $h(y)=h_{\text{max}}$ and $y=a_{i}$ for some
$i=1,2,\cdots,l$. Hence,

$\begin{split}Q_{\epsilon_{n}}(u_{n})=\frac{\int_{\mathbb{R}^{N}}\chi(\epsilon_{n}x)|u_{n}|^{2}dx}{\int_{\mathbb{R}^{N}}|u_{n}|^{2}dx}&=\frac{\int_{\mathbb{R}^{N}}\chi(\epsilon_{n}x+\epsilon_{n}y_{n})|v_{n}|^{2}dx}{\int_{\mathbb{R}^{N}}|v_{n}|^{2}dx}\\\
&\to\frac{\int_{\mathbb{R}^{N}}\chi(y)|v|^{2}dx}{\int_{\mathbb{R}^{N}}|v|^{2}dx}=a_{i}\in
K_{\frac{\tilde{\rho}}{2}},\end{split}$

which implies that $Q_{\epsilon_{n}}(u_{n})\in K_{\frac{\tilde{\rho}}{2}}$ for
$n$ large enough. That contradicts (4.11). The proof is complete. ∎

From now on, we will use the following notations:

$\bullet$ $\theta_{\epsilon}^{i}:=\\{u\in
S(a):|Q_{\epsilon}(u)-a_{i}|\leq\tilde{\rho}\\}$;

$\bullet$ $\partial\theta_{\epsilon}^{i}:=\\{u\in
S(a):|Q_{\epsilon}(u)-a_{i}|=\tilde{\rho}\\}$;

$\bullet$
$\beta_{\epsilon}^{i}:=\inf_{u\in\theta_{\epsilon}^{i}}E_{\epsilon,T}(u)$;

$\bullet$
$\tilde{\beta}_{\epsilon}^{i}:=\inf_{u\in\partial\theta_{\epsilon}^{i}}E_{\epsilon,T}(u)$.

######  Lemma 4.7.

Let $N,a,\eta,p,q,h$ be as in (1.1), (2.3) hold, $\epsilon_{2}$ and $\rho_{1}$
be obtained in Lemma 4.6. Then there exists $\epsilon_{3}\in(0,\epsilon_{2}]$
such that

$\beta_{\epsilon}^{i}<\Upsilon_{\text{max},T,a}+\frac{\rho_{1}}{2}\
\text{and}\ \beta_{\epsilon}^{i}<\tilde{\beta}_{\epsilon}^{i},\ \text{for\
any\ }\epsilon\in(0,\epsilon_{3}).$

###### Proof.

Let $u\in S(a)$ be such that

$J_{\text{max},T}(u)=\Upsilon_{\text{max},T,a}.$

For $1\leq i\leq l$, we define

$\hat{u}_{\epsilon}^{i}(x):=u\left(x-\frac{a_{i}}{\epsilon}\right),\
x\in\mathbb{R}^{N}.$

Then $\hat{u}_{\epsilon}^{i}\in S(a)$ for all $\epsilon>0$ and $1\leq i\leq
l$. Direct calculations give that

$E_{\epsilon,T}(\hat{u}_{\epsilon}^{i})=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{1}{q}\int_{\mathbb{R}^{N}}h(\epsilon
x+a_{i})|u|^{q}dx-\frac{\eta}{p}\tau(\|\nabla
u\|_{2})\int_{\mathbb{R}^{N}}|u|^{p}dx,$

and then

(4.12) $\lim_{\epsilon\to
0^{+}}E_{\epsilon,T}(\hat{u}_{\epsilon}^{i})=J_{h(a_{i}),T}(u)=J_{\text{max},T}(u)=\Upsilon_{\text{max},T,a}.$

Note that

$Q_{\epsilon}(\hat{u}_{\epsilon}^{i})=\frac{\int_{\mathbb{R}^{N}}\chi(\epsilon
x+a_{i})|u|^{2}dx}{\int_{\mathbb{R}^{N}}|u|^{2}dx}\to a_{i}\text{\ as}\
\epsilon\to 0^{+}.$

So $\hat{u}_{\epsilon}^{i}\in\theta_{\epsilon}^{i}$ for $\epsilon$ small
enough, which combined with (4.12) implies that there exists
$\epsilon_{3}\in(0,\epsilon_{2}]$ such that

$\beta_{\epsilon}^{i}<\Upsilon_{\text{max},T,a}+\frac{\rho_{1}}{2}\text{\ for\
any}\ \epsilon\in(0,\epsilon_{3}).$

For any $v\in\partial\theta_{\epsilon}^{i}$, that is, $v\in S(a)$ and
$|Q_{\epsilon}(v)-a_{i}|=\tilde{\rho}$, we obtain that
$|Q_{\epsilon}(v)\not\in K_{\frac{\tilde{\rho}}{2}}$. Thus, by Lemma 4.6,

$E_{\epsilon,T}(v)>\Upsilon_{\text{max},T,a}+\rho_{1},\ \text{for\ all}\
v\in\partial\theta_{\epsilon}^{i}\ \text{and}\ \epsilon\in(0,\epsilon_{3}),$

which implies that

$\tilde{\beta}_{\epsilon}^{i}=\inf_{v\in\partial\theta_{\epsilon}^{i}}E_{\epsilon,T}(v)\geq\Upsilon_{\text{max},T,a}+\rho_{1},\
\text{for\ all}\ \epsilon\in(0,\epsilon_{3}).$

Thus,

$\beta_{\epsilon}^{i}<\tilde{\beta}_{\epsilon}^{i},\ \text{for\ all\
}\epsilon\in(0,\epsilon_{3}).$

∎

Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1. Set $\epsilon_{0}:=\epsilon_{3}$, where $\epsilon_{3}$
is obtained in Lemma 4.7. Let $\epsilon\in(0,\epsilon_{0})$. By Lemma 4.7, for
each $i\in\\{1,2,\cdots,l\\}$, we can use the Ekeland’s variational principle
to find a sequence $\\{u_{n}^{i})\subset\theta_{\epsilon}^{i}$ satisfying

$E_{\epsilon,T}(u_{n}^{i})\to\beta_{\epsilon}^{i}\ \text{and}\
\|E_{\epsilon,T}|_{S(a)}^{\prime}(u_{n}^{i})\|\to 0\ \text{as}\ n\to+\infty,$

that is, $\\{u_{n}^{i}\\}_{n}$ is a $(PS)_{\beta_{\epsilon}^{i}}$ sequence for
$E_{\epsilon,T}$ restricted to $S(a)$. Since
$\beta_{\epsilon}^{i}<\Upsilon_{\text{max},T,a}+\rho_{0}$, it follows from
Lemma 4.5 that there exists $u^{i}$ such that $u_{n}^{i}\to u^{i}$ in
$H^{1}(\mathbb{R}^{N})$. Thus

$u^{i}\in\theta_{\epsilon}^{i},\ E_{\epsilon,T}(u^{i})=\beta_{\epsilon}^{i}\
\text{and}\ E_{\epsilon,T}|_{S(a)}^{\prime}(u^{i})=0.$

As

$Q_{\epsilon}(u^{i})\in\overline{B_{\tilde{\rho}}(a_{i})},\
Q_{\epsilon}(u^{j})\in\overline{B_{\tilde{\rho}}(a_{j})},$

and

$\overline{B_{\tilde{\rho}}(a_{i})}\cap\overline{B_{\tilde{\rho}}(a_{j})}=\emptyset\
\text{for}\ i\neq j,$

we conclude that $u^{i}\not\equiv u^{j}$ for $i\neq j$ while $1\leq i,j\leq
l$. Therefore, $E_{\epsilon,T}$ has at least $l$ nontrivial critical points
for all $\epsilon\in(0,\epsilon_{0})$.

As $E_{\epsilon,T}(u^{i})<0$ for any $i=1,2,\cdots,l$, by Lemma 4.2,
$u^{i}\,(i=1,2,\cdots,l)$ are in fact the critical points of $E_{\epsilon}$ on
$S(a)$ with $E_{\epsilon}(u^{i})<0$ and then there exists
$\lambda_{i}\in\mathbb{R}$ such that

$-\Delta u^{i}=\lambda_{i}u^{i}+h(\epsilon
x)|u^{i}|^{q-2}u^{i}+\eta|u^{i}|^{p-2}u^{i},\quad x\in\mathbb{R}^{N}.$

By using $E_{\epsilon}(u^{i})=\beta_{\epsilon}^{i}<0$ and
$E_{\epsilon}^{\prime}(u^{i})u^{i}=\lambda_{i}a^{2}$, we obtain that

$\frac{1}{2}\lambda_{i}a^{2}=E_{\epsilon}(u^{i})+\left(\frac{1}{q}-\frac{1}{2}\right)\int_{\mathbb{R}^{N}}h(\epsilon
x)|u^{i}|^{q}dx+\left(\frac{1}{p}-\frac{1}{2}\right)\int_{\mathbb{R}^{N}}|u^{i}|^{p}dx,$

which implies that $\lambda_{i}<0$ for $i=1,2,\cdots,l$. The proof is
complete.

##  5\. Orbital stability

In this section we investigate the orbital stability of the solutions obtained
in Theorem 1.1. We first give the definition of orbital stability.

######  Definition 5.1.

A set $\Omega\subset H^{1}(\mathbb{R}^{N})$ is orbitally stable under the flow
associated with the problem

(5.1) $\begin{cases}i\frac{\partial\psi}{\partial t}+\Delta\psi+h(\epsilon
x)|\psi|^{q-2}\psi+\eta|\psi|^{p-2}\psi=0,\ t>0,x\in\mathbb{R}^{N},\\\
\psi(0,x)=u_{0}(x)\end{cases}$

if for any $\theta>0$ there exists $\gamma>0$ such that for any $u_{0}\in
H^{1}(\mathbb{R}^{N})$ satisfying

$\text{dist}_{H^{1}(\mathbb{R}^{N})}(u_{0},\Omega)<\gamma,$

the solution $\psi(t,\cdot)$ of problem (5.1) with $\psi(0,x)=u_{0}$ satisfies

$\sup_{t\in\mathbb{R}^{+}}\text{dist}_{H^{1}(\mathbb{R}^{N})}(\psi(t,\cdot),\Omega)<\theta.$

For any $i=1,2,\cdots,l$, we define

$\begin{split}\Omega_{i}:&=\\{v\in\theta_{\epsilon}^{i}:\
E_{\epsilon,T}|_{S(a)}^{\prime}(v)=0,\
E_{\epsilon,T}(v)=\beta_{\epsilon}^{i}\\}\\\ &=\\{v\in\theta_{\epsilon}^{i}:\
E_{\epsilon}|_{S(a)}^{\prime}(v)=0,\ E_{\epsilon}(v)=\beta_{\epsilon}^{i},\
\|\nabla v\|_{2}\leq R_{0}\\}.\end{split}$

Next we show the stability of the sets $\Omega_{i}\,(i=1,\cdots,l)$ in two
cases $p<2^{*}$ or $p=2^{*}$.

######  Theorem 5.2.

Let $N,a,\eta,q,h,\epsilon_{0}$ be as in Theorem 1.1, $p<2^{*}$,
$\epsilon\in(0,\epsilon_{0})$, (2.3) hold. Then $\Omega_{i}\,(i=1,\cdots,l)$
is orbitally stable under the flow associated with the problem (5.1).

###### Proof.

Letting $r_{1}$ be such that $\bar{g}_{a}(r_{1})=\beta_{\epsilon}^{i}$, and
considering (2.9), the definition of $\Omega_{i}$ and
$\beta_{\epsilon}^{i}<0$, we know that

$\|\nabla v\|_{2}\leq r_{1}<R_{0},\ \text{for\ any\ }v\in\Omega_{i}.$

Let $a_{1}>a$ be such that
$\bar{g}_{a_{1}}(R_{0})=\frac{\beta_{\epsilon}^{i}}{2}$. There exists
$\delta>0$ such that if

$u_{0}\in H^{1}(\mathbb{R}^{N})\ \text{and}\
\text{dist}_{H^{1}(\mathbb{R}^{N})}(u_{0},\Omega_{i})<\delta,$

then

$\|u_{0}\|_{2}\leq a_{1},\ \|\nabla u_{0}\|_{2}\leq
r_{1}+\frac{R_{0}-r_{1}}{2},\
E_{\epsilon,T}(u_{0})\leq\frac{2}{3}\beta_{\epsilon}^{i}.$

Denoting by $\psi(t,\cdot)$ the solution to (5.1) with initial data $u_{0}$
and denoting by $[0,T_{\text{max}})$ the maximal existence interval for
$\psi(t,\cdot)$, we have classically that either $\psi(t,\cdot)$ is globally
defined for positive times, or $\|\nabla\psi(t,\cdot)\|_{2}=+\infty$ as $t\to
T_{\text{max}}^{-}$, see ([32], Section 3). Set $\tilde{a}=\|u_{0}\|_{2}$.
Note that $\|\psi(t,\cdot)\|_{2}=\|u_{0}\|_{2}$ for all
$t\in(0,T_{\text{max}})$ by the conservation of the mass. If there exists
$\tilde{t}\in(0,T_{\text{max}})$ such that
$\|\nabla\psi(\tilde{t},\cdot)\|_{2}=R_{0}$, then

$E_{\epsilon}(\psi(\tilde{t},\cdot))=E_{\epsilon,T}(\psi(\tilde{t},\cdot))\geq\bar{g}_{\tilde{a}}(R_{0})\geq\bar{g}_{a_{1}}(R_{0})=\frac{\beta_{\epsilon}^{i}}{2},$

which contradicts the conservation of the energy

$E_{\epsilon}(\psi(t,\cdot))=E_{\epsilon}(u_{0})\leq\frac{2}{3}\beta_{\epsilon}^{i},\
\text{for\ all\ }t\in(0,T_{\text{max}}).$

Thus,

(5.2) $\|\nabla\psi(t,\cdot)\|_{2}<R_{0}\ \text{for\ all\
}t\in[0,T_{\text{max}}),$

which implies that $\psi(t,\cdot)$ is globally defined for positive times.

Next we prove that $\Omega_{i}$ is orbitally stable. The validity of Lemma 4.5
for complex valued function can be proved exactly as in Theorem 3.1 in [11].
Thus, the orbital stability of $\Omega_{i}$ can be proved by modifying the
classical Cazenave-Lions argument [9] (see also [21]). For the completeness,
we give the proof here. Suppose by contradiction that there exist sequences
$\\{u_{0,n}\\}\subset H^{1}(\mathbb{R}^{N})$ and
$\\{t_{n}\\}\subset\mathbb{R}^{+}$ and a constant $\theta_{0}>0$ such that for
all $n\geq 1$,

(5.3) $\inf_{v\in\Omega_{i}}\|u_{0,n}-v\|<\frac{1}{n}$

and

(5.4) $\inf_{v\in\Omega_{i}}\|\psi_{n}(t_{n},\cdot)-v\|\geq\theta_{0},$

where $\psi_{n}(t,\cdot)$ is the solution to (5.1) with initial data
$u_{0,n}$. By (5.2), there exists $n_{0}$ such that for $n>n_{0}$ it holds
that $\|\nabla\psi_{n}(t,\cdot)\|_{2}<R_{0}$ for all $t\geq 0$.

By (5.3), there exists $\\{v_{n}\\}\subset\Omega_{i}$ such that

(5.5) $\|u_{0,n}-v_{n}\|<\frac{2}{n}.$

That $\\{v_{n}\\}\subset\Omega_{i}$ implies that
$\\{v_{n}\\}\subset\theta_{\epsilon}^{i}$ is a $(PS)_{\beta_{\epsilon}^{i}}$
sequence of $E_{\epsilon,T}$ restricted to $S(a)$. From the proof of Theorem
1.1, there exists $v\in\Omega_{i}$ such that

$\lim_{n\to+\infty}\|v_{n}-v\|=0,$

which combined with (5.5) gives that

(5.6) $\lim_{n\to+\infty}\|u_{0,n}-v\|=0.$

Hence,

$\lim_{n\to+\infty}\|u_{0,n}\|_{2}=\|v\|_{2}=a,\ \
\lim_{n\to+\infty}E_{\epsilon}(u_{0,n})=E_{\epsilon}(v)={\beta}_{\epsilon}^{i}<\tilde{\beta}_{\epsilon}^{i}.$

Then by the conservation of mass and energy, we obtain that

(5.7) $\lim_{n\to+\infty}\|\psi_{n}(t,\cdot)\|_{2}=a,\ \
\lim_{n\to+\infty}E_{\epsilon}(\psi_{n}(t,\cdot))={\beta}_{\epsilon}^{i},\
\text{for\ any\ }t\geq 0.$

Define

$\varphi_{n}(t,\cdot)=\frac{a\psi_{n}(t,\cdot)}{\|\psi_{n}(t,\cdot)\|_{2}},\
t\geq 0.$

Then $\varphi_{n}(t,\cdot)\in S(a)$ and

(5.8) $\|\varphi_{n}(t,\cdot)-\psi_{n}(t,\cdot)\|\to 0\ \text{as}\
n\to+\infty\ \text{uniformly\ in\ }t\geq 0,$

which combined with (5.6) gives that

$\lim_{n\to+\infty}\|\varphi_{n}(0,\cdot)-v\|=\lim_{n\to+\infty}\left\|\frac{au_{0,n}}{\|u_{0,n}\|_{2}}-v\right\|=0.$

Hence,
$\varphi_{n}(0,\cdot)\in\theta_{\epsilon}^{i}\backslash\partial\theta_{\epsilon}^{i}$
for $n$ large enough because
$v\in\theta_{\epsilon}^{i}\backslash\partial\theta_{\epsilon}^{i}$. Using the
method of continuity,
$\lim_{n\to+\infty}E_{\epsilon}(\varphi_{n}(t,\cdot))={\beta}_{\epsilon}^{i}$
for all $t\geq 0$, and ${\beta}_{\epsilon}^{i}<\tilde{\beta}_{\epsilon}^{i}$,
we obtain that

(5.9) $\text{for}\ n\ \text{large\ enough},\
\varphi_{n}(t,\cdot)\in\theta_{\epsilon}^{i}\backslash\partial\theta_{\epsilon}^{i}\
\text{for\ all\ }t\geq 0.$

From (5.7)-(5.9), $\\{\varphi_{n}(t_{n},\cdot)\\}\subset\theta_{\epsilon}^{i}$
is a minimizing sequence of $E_{\epsilon,T}$ at level $\beta_{\epsilon}^{i}$,
and from the proof of Theorem 1.1, there exists
$\tilde{v}\in\theta_{\epsilon}^{i}$ such that

(5.10) $\lim_{n\to+\infty}\|\varphi_{n}(t_{n},\cdot)-\tilde{v}\|=0,$

which combined with (5.8) gives that

$\lim_{n\to+\infty}\|\psi_{n}(t_{n},\cdot)-\tilde{v}\|=0.$

That contradicts (5.4). Hence $\Omega_{i}$ is orbitally stable for any
$i=1,2,\cdots,l$. ∎

######  Theorem 5.3.

Let $N,a,\eta,q,h,\epsilon_{0}$ be as in Theorem 1.1, $p=2^{*}$,
$\epsilon\in(0,\epsilon_{0})$, (2.3) and (2.6) hold. Further assume that
$h(x)\in C^{1}(\mathbb{R}^{N})$ and $h^{\prime}(x)\in
L^{\infty}(\mathbb{R}^{N})$. Then $\Omega_{i}\,(i=1,\cdots,l)$ is orbitally
stable under the flow associated with the problem (5.1).

###### Proof.

The proof can be done by modifying the arguments of Sections 3 and 4 in [16]
and using the arguments of the proof of Theorem 5.2, so we omit it. ∎

Acknowledgements. This work is supported by the National Natural Science
Foundation of China (No. 12001403).

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