# Semiclassical states for fractional logarithmic Schrödinger equations

Xiaoming An  School of Mathematics and Statistics & Guizhou University of
Finance and Economics, Guiyang, 550025, P. R. China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider the following fractional logarithmic Schrödinger
equation

$\varepsilon^{2s}(-\Delta)^{s}u+V(x)u=u\log|u|^{2}\ \ \text{in}\
\mathbb{R}^{N},$

where $\varepsilon>0$, $N\geq 1$, $V(x)\in C(\mathbb{R}^{N},[-1,+\infty))$. By
introducing an interesting penalized function, we show that the problem has a
positive solution $u_{\varepsilon}$ concentrating at a local minimum of $V$ as
$\varepsilon\to 0$. There is no restriction on decay rates of $V$, especially
it can be compactly supported.

Key words: Fractional Logarithmic Schrödinger; penalized; concentration;
compactly supported.

AMS Subject Classifications: 35J05, 35J20, 35J10.

##  1\. Introduction

We study the following fractional Schrödinger equation with logarithmic
nonlinear term:

$\varepsilon^{2s}(-\Delta)^{s}u+V(x)u=u\log|u|^{2},\ \ x\in\ \mathbb{R}^{N},$
(1.1)

where $\varepsilon>0$, $N\in\mathbb{N}$, $V(x)\in
C(\mathbb{R}^{N},\mathbb{R})$ is a continuous potential. This type of problem
comes from the study of standing waves $\psi(x,t)=e^{iEt/\varepsilon}u(x)$ of
the following fractional nonlinear Schrödinger equation:

$i\frac{\partial\psi}{\partial
t}=\varepsilon^{2s}(-\Delta)^{s}\psi+(V(x)+E)\psi-f(\psi),$ (1.2)

where $f:\mathbb{C}\to\mathbb{R}$ is a function with $f(z)=g(|z|)z$ and
$g:\mathbb{R}^{+}\to\mathbb{R}$ is a real function(which is $\log|\cdot|^{2}$
in (1.1)). For power type nonlinearities, the fractional Schrödinger equation
was introduced by Laskin ([11], [12]) as an extension of the classical
nonlinear Schrödinger equations $s=1$ in which the Brownian motion of the
quantum paths is replaced by a Lèvy flight.

Equation (1.1) is a generalization of the classical Nonlinear Schrödinger
Equation with logarithmic nonlinearity:

$-\varepsilon^{2}\Delta u+V(x)u=u\log|u|^{2},\ \ x\in\ \mathbb{R}^{N}$ (1.3)

which admits applications related to quantum mechanics, quantum optics,
nuclear physics, transport and diffusion phenomena, open quantum systems,
effective quantum gravity, theory of superfluidity and Bose-Einstein
condensation(see [20, 18] and the references therein for more details). For
this problem, one can check easily that there always exists $u\in
H^{1}(\mathbb{R}^{N})$ such that

$\int_{\mathbb{R}^{N}}|u|^{2}\log|u|^{2}=-\infty,$

which makes the natural functional corresponding to (1.3) is not well defined
in $H^{1}(\mathbb{R}^{N})$. To overcome this difficulty, considering the case
that $\varepsilon>0$ is fixed, M. Squassin et al. in [15], decompose the
functional $I$ into the sum of a $C^{1}$ functional and a convex l.s.c(short
for lower semicontinuous hereafter) functional and then use the Mountain Pass
Theorem 3.2 in [16] to find a critical point. W. Shuai in [14] added some
growth conditions such as
$\liminf\limits_{|x|\to+\infty}V(x)|x|^{2\sigma}>0$(which makes
$\int_{\mathbb{R}^{N}}|u|^{2}\log|u|^{2}$ is $C^{1}$ in
$H^{1}_{V}(\mathbb{R}^{N})=\\{u\in H^{1}(\mathbb{R}^{N}):V(x)|u|^{2}\in
L^{1}(\mathbb{R}^{N})\\}$) and then proved that the problem has a least energy
sign-changing solution and a positive ground state. Considering that
$\varepsilon>0$ is fixed and $V(x)\equiv\lambda>0$, Wiliam C. Troy in [17]
developed a new comparison method to show that the positive solution of (1.3)
is unique up to translation when $1\leq N\leq 9$; Using the result of Serrin-
Tang [13], D’Avenia et al. pointed out in [5] that the positive solution of
(1.3) is also unique up to translation. Following, letting $u_{p}$ be the
unique positive solution of (1.3), i.e.,

$-\Delta u+\lambda u=|u|^{p-2}{u},$

where $p>2$, Wang et al. in [18] proved that $u_{p}$ will converge to the
unique solution of (1.3) in the sense of $C^{2,\alpha}(\mathbb{R}^{N})$ if
$p\to 2$. Considering the case that $\varepsilon\to 0$, by penalized idea, C.
Zhang et al. in [21] showed that (1.3) has solutions $u_{\varepsilon}$
concentrating at various types of topological critical points of $V$ as
$\varepsilon\to 0$ provided that
$\lim\limits_{|x|\to\infty}V(x)|x|^{-2}>-\infty$($V$ can be unbounded below).

To our best knowledge, there are few results about (1.1) in the nonlocal case
$0<s<1$. When $\varepsilon>0$ is fixed, D’Avenia et al. in [6] obtained
existence of infinitely weak solutions. In [3], it was proved by compactness
method that (1.1) has ground states which are stable.

In this paper, we are interesting in semiclassical analysis of (1.1). From a
mathematical point of view, the transition from quantum to classical mechanics
can be formally performed by letting $\varepsilon\to 0$. For small
$\varepsilon>0$, solutions $u_{\varepsilon}$ are usually referred to as
semiclassical bound states.

In order to state our main result, we need to give some notations and
assumptions. For $s\in(0,1)$, the fractional Sobolev space
$H^{s}(\mathbb{R}^{N})$ is defined as

$H^{s}(\mathbb{R}^{N})=\Big{\\{}u\in
L^{2}(\mathbb{R}^{N}):\frac{u(x)-u(y)}{|x-y|^{N/2+s}}\in
L^{2}(\mathbb{R}^{N}\times\mathbb{R}^{N})\Big{\\}},$

endowed with the norm

$\|u\|_{H^{s}(\mathbb{R}^{N})}=\Big{(}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}+u^{2}\,{\mathrm{d}}x\Big{)}^{\frac{1}{2}},$

where

$\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}{\mathrm{d}}x=\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}\,{\mathrm{d}}x\,{\mathrm{d}}y.$

Like the classical case, we define the space $\dot{H}^{s}(\mathbb{R}^{N})$ as
the completion of $C^{\infty}_{c}(\mathbb{R}^{N})$ under the norm

$\|u\|^{2}=\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}{\mathrm{d}}x=\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}\,{\mathrm{d}}x\,{\mathrm{d}}y.$

We will use the following local fractional Sobolev space

$W^{s,2}(\Omega)=\Big{\\{}u\in
L^{2}(\Omega):\frac{|u(x)-u(y)|}{|x-y|^{\frac{N}{2}+s}}\in
L^{2}(\Omega\times\Omega)\Big{\\}}.$

It is easy to check that $W^{s,2}(\Omega)$ is a Hilbert space under the
following inner product

$(u,v)=\int_{\Omega}\int_{\Omega}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+2s}}dxdy+\int_{\Omega}uvdx\
\ \forall u,v\in W^{s,2}(\Omega),$

see [8] for more details. Also from [8], the fractional Laplacian is defined
as

$\displaystyle(-\Delta)^{s}u(x)\,$
$\displaystyle=\,C(N,s)P.V.\int_{\mathbb{R}^{N}}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,{\mathrm{d}}y$
$\displaystyle=\,C(N,s)\lim_{\varepsilon\to 0}\int_{\mathbb{R}^{N}\backslash
B_{\varepsilon}(x)}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,{\mathrm{d}}y.$

For the sake of simplicity, we define for every
$u\in\dot{H}^{s}(\mathbb{R}^{N})$ the fractional $(-\Delta)^{s}$ as

$(-\Delta)^{s}u(x)=2\int_{\mathbb{R}^{N}}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,{\mathrm{d}}y.$

Note that for every $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N})$, it holds

$\int_{\mathbb{R}^{N}}(-\Delta)^{s}u(x)\varphi(x)=\frac{d}{dt}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}(u+t\varphi)|^{2}\large|_{t=0}.$

Our solutions will be found in the following weighted fractional Sobolev
space:

$\mathcal{D}^{s}_{V,\varepsilon}(\mathbb{R}^{N})=\left\\{u\in\dot{H}^{s}(\mathbb{R}^{N}):\,\,u\in
L^{2}\big{(}\mathbb{R}^{N},\big{(}V(x)+1\big{)}\,{\mathrm{d}}x\big{)}\right\\},$

endowed with the norm

$\|u\|_{\mathcal{D}^{s}_{V,\varepsilon}(\mathbb{R}^{N})}=\Big{(}\int_{\mathbb{R}^{N}}\varepsilon^{2s}|(-\Delta)^{s/2}u|^{2}+(V(x)+1)u^{2}\,{\mathrm{d}}x\Big{)}^{\frac{1}{2}}.$

For the potential term $V$, we assume that $V$ is continuous and

$(\mathcal{V}_{1})$ $V(x)+1\geq 0$;

$(\mathcal{V}_{2})$ there exist open bounded sets $\Lambda\subset\subset U$
with smooth boundaries $\partial\Lambda,\ \partial U$, such that

$0<\lambda=\inf_{\Lambda}(V+1)<\inf_{U\backslash\Lambda}(V+1).$ (1.4)

Without loss of generality, we assume that $0\in\Lambda$.

Now, with the notations and assumptions above at hand, we are in a position to
state our main result:

######  Theorem 1.1.

Let $V$ satisfy $(\mathcal{V}_{1})$ and $(\mathcal{V}_{2})$. Then there exists
an $\varepsilon_{0}>0$ such that (1.1) has a positive solution
$u_{\varepsilon}$ if $\varepsilon\in(0,\varepsilon_{0})$. Moreover,
$u_{\varepsilon}$ has a global maximum point $x_{\varepsilon}$ which satisfies

$\lim_{\varepsilon\to 0}V(x_{\varepsilon})=\inf_{x\in\Lambda}V(x)$

and

$u_{\varepsilon}(x)\leq\frac{C\varepsilon^{N+2s}}{\varepsilon^{N+2s}+|x-x_{\varepsilon}|^{N+2s}},\
x\in\mathbb{R}^{N},$

where $C$ is a positive constant.

The difficulties in the proof of Theorem 1.1 are stated as follows. Firstly,
there exists $u\in\mathcal{D}^{s}_{V,\varepsilon}(\mathbb{R}^{N})$ such that

$\int_{\mathbb{R}^{N}}|u|^{2}\log|u|^{2}=-\infty,$

which makes the natural Euler-Lagrange functional corresponding to (1.1),
i.e.,

$I_{\varepsilon}(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}\big{(}\varepsilon^{2s}|(-\Delta)^{s/2}u|^{2}+(V(x)+1)|u|^{2}\big{)}-\frac{1}{2}\int_{\mathbb{R}^{N}}u^{2}\log
u^{2}\mathrm{d}x,$

is not well defined in $\mathcal{D}^{s}_{V,\varepsilon}(\mathbb{R}^{N})$. Then
all the methods developed for (1.1) in the case that the logarithmic term
replaced by power type nonlinear term failed(see [1, 2]). Moreover, the
expectation that the concentration should occur at a local minimum of $V$ in
$\Lambda$ makes us have to truncate the nonlinear term outside $\Lambda$.
Noting that since the ground states of limiting equation of (1.1) decay like
$\frac{1}{|x|^{N+2s}}$, we can not use the penalized idea in [21] which deal
with the case $s=1$ to truncate the nonlinear term or the potential term.

Creatively, to overcome the two difficulties above, we use the characteristic
function $4\chi_{\mathbb{R}^{N}\Lambda}$ to truncate the nonlinear term(see
(2) below), i.e., we firstly study the following penalized problem

$\varepsilon^{2s}(-\Delta)^{s}u_{\varepsilon}+(V(x)+1)u_{\varepsilon}=\chi_{\Lambda}(1+\log|u|^{2})u_{\varepsilon}-\chi_{\mathbb{R}^{N}\backslash\Lambda}\max\\{2u,-u(1+\log|u|^{2})\\}.$
(1.5)

The corresponding functional $J_{\varepsilon}$, all through is not well
defined in $\mathcal{D}^{s}_{V,\varepsilon}$, is the sum of a $C^{1}$
functional and a convex l.s.c. functional(see (2) and Remarks 2.3 below),
whose critical points can be found by the Mountain Pass Theorem 3.2 in [16].
Then, a penalized solution $u$ of (1.5) is indeed a solution of the origin
problem (1.1) if

$-(1+\log|u|^{2})\geq 2\ \ \text{on}\ \mathbb{R}^{N}\backslash\Lambda.$

This, after the proof of concentration of $u$, can be checked easily by the
well-known fact that a sub-solution of $(-\Delta)^{s}u+u\leq 0$ decays like
$\frac{1}{|x|^{N+2s}}$(see the last part of Section 3 for more details).

We need to emphasize that the nonlocal term $(-\Delta)^{s}$ makes the
estimates more difficult than the classical case. In fact, for a smooth
function $f\in C^{\infty}_{c}(\mathbb{R}^{N})$, one can not compute
$(-\Delta)^{s}f$ as precisely as $-\Delta f$(see Appendix A below for
example).

The nonlocal effect makes us have to know the global $L^{2}$-norm information
of penalized solution, which in [2] was given by the assumption
$\liminf_{|x|\to\infty}(V(x)+1)|x|^{2s}>0$. But, in the present paper, the
global $L^{2}$-norm information of penalized solution will be obtained by the
fractional Hardy inequality: There exists a positive constant $C_{N,s}$ such
that

$\int_{\mathbb{R}^{N}}\frac{|u(x)|^{2}}{|x|^{2s}}\,{\mathrm{d}}x\leq
C_{N,s}|(-\Delta)^{s/2}|^{2}_{2}$ (1.6)

for all $u\in\dot{H}^{s}(\mathbb{R}^{N})$(see the proof of (A) in Appendix A
for example).

Plan of the paper. In Section 2, we obtain the penalized problem by truncating
the logarithmic term in (1.1) outside by $4\chi_{\mathbb{R}^{N}\Lambda}u$,
then we use the Mountain Pass Theorem 3.2 in [16] to obtain a penalized
solution $u_{\varepsilon}$. In Section 3, we study the concentration of
$u_{\varepsilon}$ and then linearize the penalized equation in Section 2. At
the last part of Section 3, we use the well-known decay estimates of
fractional Schröinger equations to prove the asymptotic behaviour of
$u_{\varepsilon}$, which implies that $u_{\varepsilon}$ solves the origin
problem.

##  2\. The penalized problem

The following inequality exposes the relationship between
$H^{s}(\mathbb{R}^{N})$ and the Banach space $L^{q}(\mathbb{R}^{N})$.

######  Proposition 2.1.

(Fractional version of the Gagliardo$-$Nirenberg inequality [8]) For every
$u\in H^{s}(\mathbb{R}^{N})$,

$\displaystyle\|u\|_{q}\leq
C\|(-\Delta)^{s/2}u\|^{\theta}_{2}\|u\|^{1-\theta}_{2},$ (2.1)

where $q\in[2,2^{*}_{s}]$ and $\theta$ satisfies
$\frac{\theta}{2^{*}_{s}}+\frac{(1-\theta)}{2}=\frac{1}{q}$.

By the proposition above, we know that $H^{s}(\mathbb{R}^{N})$ is continuously
embedded into $L^{q}(\mathbb{R}^{N})$ for $\ q\in[2,2^{*}_{s}]$. Moreover, on
bounded set, the embedding is compact (see [8]), i.e.,

$H^{s}(\mathbb{R}^{N})\subset\subset L^{q}_{loc}(\mathbb{R}^{N})\
\text{compactly, if}\ q\in[1,2^{*}_{s}).$

Our proof will rely on the following fractional logarithmic Sobolev
inequality(see [4]):

######  Proposition 2.2.

For any $u\in H^{s}(\mathbb{R}^{N})$, it holds

$\int_{\mathbb{R}^{N}}|u|^{2}\log\Big{(}\frac{|u|^{2}}{\|u\|^{2}_{2}}\Big{)}+\Big{(}N+\frac{N}{s}\log
a+\log\frac{s\Gamma(\frac{N}{2})}{\Gamma(\frac{N}{2s})}\Big{)}\|u\|^{2}_{2}\leq\frac{a^{2}}{\pi^{s}}\|(-\Delta)^{s}\|^{2}_{2},\
a>0.$

Now we are going to modify the origin problem (1.1). In order to use the
Mountain Pass Theorem 3.2 in [16], considering the vanishing of $V$ and the
concentration should occur in $\Lambda$, we modified the nonlinear term as
follows. Define

$\displaystyle G_{1}(x,s)$
$\displaystyle=\frac{1}{2}\chi_{\Lambda}(x)s^{2}_{+}\log s^{2}_{+},$
$\displaystyle G_{2}(x,s)$
$\displaystyle=\frac{1}{2}\chi_{\mathbb{R}^{N}\backslash\Lambda}(x)\int_{0}^{s}\max\\{4t_{+},-2t_{+}(1+\log
t^{2}_{+})\\}dt.$ (2.2)

######  Remark 2.3.

Importantly, since $G^{\prime}_{2}(x,s)$ is nondecreasing on $s$,
$G_{2}(x,s)\geq 0$ is convex on $s$ for all
$x\in\mathbb{R}^{N}\backslash\Lambda$ and the functional
$\mathcal{G}^{2}:\mathcal{D}^{s}_{V,\varepsilon}(\mathbb{R}^{N})\to\mathbb{R}$
given by

$\mathcal{G}^{2}(u)=\int_{\mathbb{R}^{N}}G_{2}(x,u)$

is convex and l.s.c by Fatou’s Lemma.

Proposition 2.1 and the boundedness of $\Lambda$ imply that the functional
$\mathcal{G}^{1}:\mathcal{D}^{s}_{V,\varepsilon}(\mathbb{R}^{N})\to\mathbb{R}$
given by

$\mathcal{G}^{1}(u)=\int_{\mathbb{R}^{N}}G_{1}(x,u)$

is $C^{1}$. So the functional

$J_{\varepsilon}(u)=\Phi_{\varepsilon}(u)+\Psi(u)$

with

$\Phi_{\varepsilon}(u)=\frac{1}{2}\|u\|^{2}_{V,\varepsilon}-\mathcal{G}^{1}(u)\
\ \text{and}\ \ \Psi=\mathcal{G}^{2}(u)$

has the form stated in [16].

By the remark above, although $J_{\varepsilon}$ is not $C^{1}$, we can still
use the Mountain Pass Theorem 3.2 in [16] to find a critical point for
$J_{\varepsilon}$. We first state some necessary definitions corresponding to
those functionals has the form of $J_{\varepsilon}$.

######  Definition 2.4.

Let $E$ be a Banach space, $E^{\prime}$ be the dual space of $E$ and
$\langle\cdot,\cdot\rangle$ be the duality paring between $E^{\prime}$ and
$E$. Let $J:E\to\mathbb{R}$ be a functional of the form
$J(u)=\Phi(u)+\Psi(u)$, where $\Phi\in C^{1}(E,\mathbb{R})$ and $\Psi$ is
convex and l.s.c.. We have the following definitions:

$(i)$ A critical point of $J$ is a point $u\in E$ such that $J(u)<+\infty$ and
$0\in\partial J(u)$, i.e.

$\langle\Phi^{\prime}(u),v-u\rangle+\Psi(v)-\Psi(u)\geq 0,\ \forall v\in E.$

$(ii)$ A Palais-Smale sequence at level $c$ for $J$ is a sequence
$(u_{n})\subset E$ such that $J(u_{n})\to c$ and there is a numerical sequence
$\sigma_{n}\to 0^{+}$ with

$\langle\Phi^{\prime}(u_{n}),v-u_{n}\rangle+\Psi(v)-\Psi(u_{n})\geq-\sigma_{n}\|v-u_{n}\|,\
\forall v\in E.$

$(iii)$ The functional $J$ satisfies the Palais-Smale condition at level $c$
$(PS)_{c}$ condition if all Palais-Smale sequence at level $c$ has a
convergent subsequence.

$(iv)$ The set $D(J):=\\{u\in E:J(u)<+\infty\\}$ is called the effective
domain of $J$.

According to the Corollary 2.6 in [15], we have

######  Proposition 2.5.

Let $\sup_{n}J_{\varepsilon}(u_{n})<+\infty$. Then $(u_{n})$ is a Palais-Smale
sequence if and only if $J^{\prime}(u_{n})\to 0$ in
$\big{(}\mathcal{D}^{s}_{V,\varepsilon}(\mathbb{R}^{N})\big{)}^{\prime}$.

To use Theorem 3.2 in [16], we need to prove that $J_{\varepsilon}$ satisfies
the $(PS)_{c}$ condition $(iii)$ above.

######  Proposition 2.6.

$J_{\varepsilon}$ satisfies $(PS)_{c}$ condition, i.e., each sequence
$(u_{n})\subset\mathcal{D}^{s}_{V,\varepsilon}(\mathbb{R}^{N})$ with
$\lim\limits_{n\to\infty}J_{\varepsilon}(u_{n})\to c$ has a convergent
subsequence in $\mathcal{D}^{s}_{V,\varepsilon}(\mathbb{R}^{N})$.

###### Proof.

We first show that $(u_{n})$ is bounded in
$\mathcal{D}^{s}_{V,\varepsilon}(\mathbb{R}^{N})$. Observing firstly that
$\Psi\geq 0\ \forall u\in\mathcal{D}^{s}_{V,\varepsilon}(\mathbb{R}^{N})$,
hence we have

$J_{\varepsilon}(u_{n})\geq\Phi_{\varepsilon}(u_{n})=\frac{1}{2}\|u_{n}\|^{2}_{V,\varepsilon}-\frac{1}{2}\int_{\Lambda}|u_{n}|^{2}\log|u_{n}|^{2}.$
(2.3)

Moreover, by (2), Proposition 2.5 and the fact that
$\int\max\\{f,g\\}\geq\max\\{\int f,\int g\\}$, it holds

$\int_{\mathbb{R}^{N}}|u_{n}|^{2}\mathrm{d}x\leq
2J_{\varepsilon}(u_{n})-\langle
J^{\prime}_{\varepsilon}(u_{n}),u_{n}\rangle\leq
C+o_{n}(1)\|u_{n}\|_{V,\varepsilon}.$ (2.4)

For a set $A\in\mathbb{R}^{N}$, we define
$A^{d}:=\\{x\in\mathbb{R}^{N}:dist(A,x)<d\\}$. Let $\eta\in
C^{\infty}_{c}(\Lambda^{\delta})$ be a function satisfying $0\leq\eta\leq 1$
and $\eta\equiv 1$ on $\overline{\Lambda}$, where the $\delta$ is a small
parameter such that $\Lambda^{2\delta}\subset\subset U$. Defining
$v_{n}(x)=u_{n}(\varepsilon x)\eta(\varepsilon x)$, we have $v_{n}(x)\in
H^{s}(\mathbb{R}^{N})\backslash\\{0\\}$. Then, by Proposition 2.2, we have

$\displaystyle\int_{\mathbb{R}^{N}}|v_{n}|^{2}\log|v_{n}|^{2}\leq\frac{a^{s}}{\pi}\|(-\Delta)^{s/2}v_{n}\|^{2}+(C_{s,N}+\frac{N}{s}\log
a)\|v_{n}\|^{2}_{2}+\|v_{n}\|^{2}_{2}\log\|v_{n}\|^{2}_{2}.$

Resclaing back, we find

$\displaystyle\begin{split}&\quad\int_{\mathbb{R}^{N}}|\eta
u_{n}|^{2}\log\|\eta u_{n}\|^{2}\\\
&\leq\frac{a\varepsilon^{2s}}{2}\|(-\Delta)^{s/2}(\eta
u_{n})\|^{2}_{2}+(C_{s,N}+\frac{N}{s}\log a)\|\eta u_{n}\|_{2}^{2}+\|\eta
u_{n}\|^{2}_{2}\log\Big{(}\frac{1}{\varepsilon^{N}}\|\eta
u_{n}\|^{2}_{2}\Big{)}.\end{split}$ (2.5)

By fractional Hardy inequality (1.6) and some delicate nonlocal estimates, we
will prove in Appendix A that

$T_{1}(\eta):=\varepsilon^{2s}\|(-\Delta)^{s/2}(\eta u_{n})\|^{2}_{2}\leq
C\|u_{n}\|^{2}_{V,\varepsilon},$ (2.6)

where $C$ is a positive constant. Hence, returning back to (2.5), letting
$a>0$ be small enough, by (2.4), we get

$\displaystyle\begin{split}&\quad\int_{\mathbb{R}^{N}}|\eta
u_{n}|^{2}\log\|\eta
u_{n}\|^{2}\leq\frac{1}{2}\|u_{n}\|^{2}_{V,\varepsilon}+C(1+\|\eta
u_{n}\|_{2}^{2(1+\delta)})-N\|u_{n}\|^{2}_{2}\log\varepsilon\\\
&\leq\frac{1}{2}\|u_{n}\|^{2}_{V,\varepsilon}+C(1+\|u_{n}\|^{1+\delta}_{V,\varepsilon})-N\big{(}C+\|u_{n}\|_{V,\varepsilon}\big{)}\log\varepsilon,\end{split}$
(2.7)

where $\delta\in(0,1)$ is a parameter. Note that for every
$f:\mathbb{R}^{N}\to\mathbb{R}$, it holds

$\int_{U\backslash\Lambda}|f|^{2}\log|f|^{2}\geq-\frac{1}{e}|U\backslash\Lambda|.$

Then by (2.7) and (2.3), we have

$\widetilde{C}\geq\frac{1}{2}\|u_{n}\|^{2}_{V,\varepsilon}-\|u_{n}\|^{1+\delta}_{V,\varepsilon}+N(C+\|u_{n}\|_{V,\varepsilon})\log\varepsilon,$

where $\widetilde{C}$ is a positive constant. Consequently, since
$0<\delta<1$, we can conclude that $(u_{n})$ is bounded in
$\mathcal{D}^{s}_{V,\varepsilon}(\mathbb{R}^{N})$. Going if necessary to a
subsequence, we assume that

$u_{n}\rightharpoonup u\ \text{weakly in}\
\mathcal{D}^{s}_{V,\varepsilon}(\mathbb{R}^{N}).$

Next we show that $(u_{n})$ has a convergent subsequence. Since
$u_{n}\rightharpoonup u\in\mathcal{D}^{s}_{V,\varepsilon}(\mathbb{R}^{N})$, by
the boundedness of $\Lambda$ and the fact that $|G_{1}(x,s)|\leq
C_{1}|t|^{\frac{3}{2}}+C_{2}|t|^{\frac{5}{2}}$, where $C_{1},C_{2}>0$ are two
positive constants, we have

${G}_{1}^{\prime}(x,u_{n})u_{n}\to{G}_{1}^{\prime}(x,u)u.$

Noting that since
$J^{\prime}_{\varepsilon}(u_{n})\varphi=o_{n}(1)\|\varphi\|_{V,\varepsilon}$
for all $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N})$, we deduce that
$J^{\prime}_{\varepsilon}(u)\varphi=0$ for all $\varphi\in
C^{\infty}_{c}(\mathbb{R}^{N})$, and so, $J^{\prime}_{\varepsilon}(u)u=0$.
Combing with
$|J^{\prime}_{\varepsilon}(u_{n})u_{n}|\leq\|J^{\prime}_{\varepsilon}(u_{n})\|_{\mathcal{D}^{\prime}_{V,\varepsilon}}\|u_{n}\|_{V,\varepsilon}=o_{n}(1)\|u_{n}\|$,
we have

$\|u_{n}\|^{2}_{V,\varepsilon}+\int_{\mathbb{R}^{N}}\chi_{\mathbb{R}^{N}\backslash\Lambda}G_{2}^{\prime}(x,u_{n})u_{n}=\|u\|^{2}_{V,\varepsilon}+\int_{\mathbb{R}^{N}}\chi_{\mathbb{R}^{N}\backslash\Lambda}G_{2}^{\prime}(x,u)u+o_{n}(1).$
(2.8)

Finally, by
$\|u\|\leq\liminf\limits_{n\to\infty}\|u_{n}\|^{2}_{V,\varepsilon}$ and the
l.s.c property(by Fatou’s Lemma), we get that $u_{n}\to u$ strongly in
$\mathcal{D}^{s}_{V,\varepsilon}(\mathbb{R}^{N})$. This completes the proof. ∎

######  Remark 2.7.

The proof of (2.6) is trivial if $s=1$, but delicate if $0<s<1$. It will need
the global $L^{2}$ information of $u_{n}$, which is given by the nonlocal
operator $(-\Delta)^{s}$ and the fractional Hardy inequality (1.6)(see (A.2)
for example).

Obviously, there exists $\rho>0$ such that

$J_{\varepsilon}(u)\geq\Phi_{\varepsilon}(u)\geq\frac{1}{2}\|u\|^{2}_{V,\varepsilon}-C\|u\|^{p}_{V,\varepsilon}>0\
\text{for all}\ u\ \text{with}\ \|u\|^{2}_{V,\varepsilon}=\rho$

and for each $u\in C^{\infty}_{c}(\Lambda)\backslash\\{0\\}$, it holds

$J_{\varepsilon}(su)\to-\infty\ \text{as}\ s\to+\infty,$

i.e., $J_{\varepsilon}$ owns mountain pass geometry. Thus by Proposition 2.6
and Theorem 3.2 in [16], we immediately have:

######  Lemma 2.8.

The mountain pass value

$c_{\varepsilon}=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}J_{\varepsilon}(\gamma(t))$

is positive and can be achieved by a positive function $u_{\varepsilon}$ which
is a critical point of $J_{\varepsilon}$ and solves the following penalized
problem

$\varepsilon^{2s}(-\Delta)^{s}u_{\varepsilon}+(V(x)+1)u_{\varepsilon}=G_{1}^{\prime}(x,(u_{\varepsilon})_{+})-G_{2}^{\prime}(x,(u_{\varepsilon})_{+}).$
(2.9)

###### Proof.

By Theorem 3.2 in [16], $c_{\varepsilon}$ is a critical value, i.e., there
exists $u_{\varepsilon}\in
D(J_{\varepsilon})=\\{v\in\mathcal{D}^{s}_{V,\varepsilon}:J_{\varepsilon}(v)<+\infty\\}$
with $J_{\varepsilon}(u_{\varepsilon})=c_{\varepsilon}$ such that

$\langle\Phi^{\prime}_{\varepsilon}(u_{\varepsilon}),v-u_{\varepsilon}\rangle+\Psi(v)-\Psi(u_{\varepsilon})\geq
0\ \ \forall v\in\mathcal{D}^{s}_{V,\varepsilon}.$

In particular, letting $t>0$ and $v=u_{\varepsilon}+t\varphi$ with $\varphi\in
C^{\infty}_{c}(\mathbb{R}^{N})$, we have

$\langle\Phi^{\prime}_{\varepsilon}(u_{\varepsilon}),\varphi\rangle+\frac{\Psi(u_{\varepsilon}+t\varphi)-\Psi(u_{\varepsilon})}{t}\geq
0\ \ \forall t>0$

and then

$\langle
J^{\prime}_{\varepsilon}(u_{\varepsilon}),\varphi\rangle=\langle\Phi^{\prime}_{\varepsilon}(u_{\varepsilon}),\varphi\rangle+\int_{\mathbb{R}^{N}}G_{2}^{\prime}(x,u_{\varepsilon})\varphi\geq
0.$

Rearranging $\varphi=-\psi$, we eventually have

$\langle J^{\prime}_{\varepsilon}(u_{\varepsilon}),\varphi\rangle=0\
\forall\varphi\in C^{\infty}_{c}(\mathbb{R}^{N}),$

which implies (2.9).

Finally, letting $(u_{\varepsilon})_{-}$ be a test function to (2.9), we find
$u_{\varepsilon}\geq 0$. By the standard regularity assertion and maximum
principle in [10, Appendix D], we conclude that $u_{\varepsilon}$ is positive.
∎

##  3\. Concentration and the origin problem

In this section we will prove the concentration phenomenon of
$u_{\varepsilon}$ via energy comparison, by which we will prove at the last of
this section that

$-(1+\log|u_{\varepsilon}|^{2})\geq 2\ \ \forall
x\in\mathbb{R}^{N}\backslash\Lambda,$ (3.1)

which and (2.9) indicate that $u_{\varepsilon}$ solves the origin problem
(1.1).

###  3.1. Concentration

In the first subsection, we prove the concentration of $u_{\varepsilon}$ via
comparing the energy $c_{\varepsilon}$ with the least energy of the limiting
problem of (1.1).

The limiting problem corresponding to (1.1) is

$(-\Delta)^{s}u+\lambda u=u\log|u|^{2},$ (3.2)

where $\lambda>-1$. Its Euler-Lagrange functional is

$\mathcal{L}_{\lambda}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}+(\lambda+1)|u|^{2}-\frac{1}{2}\int_{\mathbb{R}^{N}}u^{2}\log
u^{2}.$

In [3], it was proved that the limiting problem (3.2) has a least energy
solution $U_{\lambda}$ with

$\mathcal{L}_{\lambda}(U_{\lambda})=\mathcal{C}_{\lambda}:=\inf_{\varphi\in
H^{s}(\mathbb{R}^{N})\backslash\\{0\\}}\max_{t>0}\mathcal{L}_{\lambda}(t\varphi)=\inf_{\varphi\in
C^{\infty}_{c}(\mathbb{R}^{N})\backslash\\{0\\},\varphi\geq
0}\max_{t>0}\mathcal{L}_{\lambda}(t\varphi).$

For $\mathcal{C}_{\lambda}$, we have

######  Proposition 3.1.

the function $\mathcal{C}_{\cdot}:(-1,+\infty)\to(0,+\infty)$ is continuous
and increasing.

###### Proof.

Let $-1<\lambda<\lambda^{\prime}<+\infty$ and $U_{\lambda^{\prime}}$ be the
ground solution of equation (3.2) with $\lambda=\lambda^{\prime}$. An easy
analysis shows that the following function $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$

$f(t)=\mathcal{L}_{\lambda}(tU_{\lambda^{\prime}})=\mathcal{L}_{\lambda^{\prime}}(tU_{\lambda^{\prime}})+\frac{t^{2}}{2}(\lambda-\lambda^{\prime})\int_{\mathbb{R}^{N}}|U_{\lambda^{\prime}}|^{2}$

has a unique maximum point $t^{\prime}\in(0,+\infty)$, from which we have

$\displaystyle\mathcal{C}_{\lambda}\leq\max_{t>0}f(t)$
$\displaystyle=\mathcal{L}_{\lambda^{\prime}}(t^{\prime}U_{\lambda^{\prime}})+\frac{(t^{\prime})^{2}}{2}(\lambda-\lambda^{\prime})\int_{\mathbb{R}^{N}}|U_{\lambda^{\prime}}|^{2}$
$\displaystyle\leq\mathcal{C}_{\lambda^{\prime}}+\frac{(t^{\prime})^{2}}{2}(\lambda-\lambda^{\prime})\int_{\mathbb{R}^{N}}|U_{\lambda^{\prime}}|^{2}$
$\displaystyle<\mathcal{C}_{\lambda^{\prime}}.$

Similarly, it holds

$\mathcal{C}_{\lambda^{\prime}}\leq\mathcal{C_{\lambda}}+\frac{\tilde{t}^{2}}{2}(\lambda^{\prime}-\lambda)\int_{\mathbb{R}^{N}}|U_{\lambda}|^{2}$

for some unique $\tilde{t}\in(0,+\infty)$. Then $\mathcal{C}_{\lambda}$ is
increasing and continuous. ∎

By the analysis above, we have the following upper bound of $c_{\varepsilon}$.

######  Proposition 3.2.

It holds

$\limsup_{\varepsilon\to
0}\frac{c_{\varepsilon}}{\varepsilon^{N}}\leq\min_{x\in\Lambda}\mathcal{C}_{V(x)}.$

###### Proof.

Let $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N})\backslash\\{0\\},\ \varphi\geq
0$ and define for each $x_{0}\in\Lambda$

$\varphi_{\varepsilon}(x)=\varphi\Big{(}\frac{x-x_{0}}{\varepsilon}\Big{)}.$

Obviously, supp$\varphi_{\varepsilon}\subset\Lambda$ for small $\varepsilon$
and
$\gamma_{\varepsilon}(t)=tT_{0}\varphi_{\varepsilon}\in\Gamma_{\varepsilon}$
for some $T_{0}$ large enough. Then we have

$\displaystyle\frac{c_{\varepsilon}}{\varepsilon^{N}}\leq\frac{\max_{t\in[0,1]}J_{\varepsilon}(\gamma_{\varepsilon}(t))}{\varepsilon^{N}}\leq\mathcal{L}_{V(x_{0})}(t\varphi)+o_{\varepsilon}(1)$

and

$\limsup_{\varepsilon\to
0}\frac{c_{\varepsilon}}{\varepsilon^{N}}\leq\inf_{{\varphi\in
C^{\infty}_{c}(\mathbb{R}^{N})\backslash\\{0\\}}\atop{\varphi\geq
0}}\mathcal{L}_{V(x_{0})}(t\varphi)=\mathcal{C}_{V(x_{0})},$

which completes the proof. ∎

Next, we give the lower bounds of solutions of (2.9).

######  Proposition 3.3.

Let $(u_{\varepsilon_{n}})$ with $\varepsilon_{n}>0,\ \varepsilon_{n}\to 0\
\text{as}\ n\to\infty$ be a family of solutions of $\eqref{eq2.1}$. If for
each $k\in\mathbb{N}$, there exists $k$ families of points
$\\{(x^{i}_{\varepsilon_{n}}):1\leq i\leq k\\}$ with
$\lim\limits_{n\to\infty}x^{i}_{\varepsilon_{n}}=x^{i}_{*}$ such that

$\liminf_{n\to\infty}\|u_{\varepsilon_{n}}\|_{L^{\infty}(B_{\varepsilon_{n}\rho})(x^{i}_{\varepsilon_{n}})}>0,\
V(x^{i}_{*})+1>0,\ \ 1\leq i\leq k,$
$\liminf_{n\to\infty}\frac{|x^{i}_{\varepsilon_{n}}-x^{j}_{\varepsilon_{n}}|}{\varepsilon_{n}}=+\infty,\
\ 1\leq i\neq j\leq k$

and

$\limsup_{n\to\infty}\frac{J_{\varepsilon_{n}}(u_{\varepsilon_{n}})}{\varepsilon^{N}_{n}}<+\infty,$

then

$\liminf_{n\to\infty}\frac{J_{\varepsilon_{n}}(u_{\varepsilon_{n}})}{\varepsilon^{N}}\geq\sum_{i=1}^{k}\mathcal{C}_{V(x^{i}_{*})}.$

###### Proof.

Fixing a $1\leq i\leq k$ and rescaling the function $u_{\varepsilon_{n}}$ as
$v^{i}_{n}(x)=u(\varepsilon_{n}x+x^{i}_{\varepsilon_{n}}),\
x\in\mathbb{R}^{N}$, we have by the estimate in Proposition 2.6 that

$\sup_{n}\int_{\mathbb{R}^{N}}\big{(}|(-\Delta)^{s/2}v^{i}_{n}|^{2}+\big{(}V^{i}_{n}(x)+1\big{)}|v^{i}_{n}|^{2}\big{)}<+\infty,$

where $V^{i}_{n}(\cdot)=V(\varepsilon_{n}\cdot+x^{i}_{\varepsilon_{n}})$.
Obviously, $v^{i}_{n}$ satisfies

$(-\Delta)^{s}v^{i}_{n}+\big{(}V^{i}_{n}(x)+1\big{)}v^{i}_{n}={G}_{1}^{\prime}(\varepsilon_{n}x+x^{i}_{\varepsilon_{n}},v^{i}_{n})-{G}_{2}^{\prime}(\varepsilon_{n}x+x^{i}_{n},v^{i}_{n})\
\ \text{in}\ \mathbb{R}^{N}.$ (3.3)

Fixing $R>0$, we have by continuity that

$\displaystyle\quad\limsup_{n\to\infty}\|v^{i}_{n}\|^{2}_{H^{s}(B_{R})}$
$\displaystyle\leq\limsup_{n\to\infty}(\inf_{\Lambda}(V(x)+1))^{-1}\int_{\mathbb{R}^{N}}\big{(}|(-\Delta)^{s/2}v^{i}_{n}|^{2}+\big{(}V^{i}_{n}(x)+1\big{)}|v^{i}_{n}|^{2}\big{)}<+\infty,$

which says that $(v^{i}_{n})$ is bounded in $H^{s}_{loc}(\mathbb{R}^{N})$ and
then by diagonal argument, we can assume without loss of generality that
$v^{i}_{n}\rightharpoonup v^{i}_{*}$ weakly in $H^{s}_{loc}$ as $n\to\infty$.
By

$\|v^{i}_{*}\|^{2}_{H^{s}(B_{R})}\leq\liminf_{n\to\infty}\|v^{i}_{n}\|^{2}_{H^{s}(B_{R})}<+\infty,$

we have $v^{i}_{*}\in H^{s}(\mathbb{R}^{N})$.

The smoothness of $\Lambda$ implies that the set
$\Lambda^{i}_{n}=\\{x:\varepsilon_{n}x+x^{i}_{\varepsilon_{n}}\in\Lambda\\}$
converges to a set $\Lambda^{i}_{*}\in\\{\emptyset,H,\mathbb{R}^{N}\\}$ as
$n\to\infty$, where $H$ is a half plane, by which we have

$\displaystyle\quad\int_{\mathbb{R}^{N}}\Big{(}{G}_{1}^{\prime}(\varepsilon
x+x^{i}_{\varepsilon_{n}},v^{i}_{n})\varphi-{G}_{2}^{\prime}(\varepsilon
x+x^{i}_{\varepsilon_{n}},v^{i}_{n})\varphi\Big{)}$
$\displaystyle\to\int_{\mathbb{R}^{N}}\chi_{\Lambda^{i}_{*}}(1+\log|v^{i}_{*}|^{2})v^{i}_{*}\varphi-\int_{\mathbb{R}^{N}}\chi_{\mathbb{R}^{N}\backslash\Lambda^{i}_{*}}\max\\{2v^{i}_{*},-(1+\log|v^{i}_{*}|^{2})v^{i}_{*}\\}\varphi$

for all $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N})$ as $n\to\infty$. Then we
conclude that $v^{i}_{*}$ satisfies the following equation:

$\displaystyle\ \quad(-\Delta)^{s}v^{i}_{*}+(V(x^{i}_{*})+1)v^{i}_{*}$
$\displaystyle=\chi_{\Lambda^{i}_{*}}(x)v^{i}_{*}(1+\log|v^{i}_{*}|^{2})-\chi_{\mathbb{R}^{N}\backslash\Lambda^{i}_{*}}\max\\{2v^{i}_{*},-(1+\log|v^{i}_{*}|^{2})v^{i}_{*}\\}\
\ \text{in}\ R^{N}.$ (3.4)

By the similar regularity argument in [10, Appendix D], we have

$\|v^{i}_{*}\|_{L^{\infty}(B_{\rho}(x^{i}_{*}))}=\lim_{n\to\infty}|v^{i}_{n}\|_{L^{\infty}(B_{\rho}(x^{i}_{n}))}=\lim_{n\to\infty}\|u^{i}_{n}\|_{L^{\infty}(B_{\varepsilon_{n}\rho}(x^{i}_{\varepsilon_{n}}))}>0,$

which implies that $v^{i}_{*}$ is nontrivial.

The Euler-Lagrange functional corresponding to (3.1) is

$\displaystyle J^{i}_{*}(u)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{N}}(|(-\Delta)^{s/2}u|^{2}+(V(x^{i}_{*})+1)|u|^{2})-\frac{1}{2}\int_{\mathbb{R}^{N}}\chi_{\Lambda^{i}_{*}}|u|^{2}\log|u|^{2}$
$\displaystyle\quad+\frac{1}{2}\int_{\mathbb{R}^{N}\backslash\Lambda^{i}_{*}}\mathrm{d}x\int_{0}^{|u(x)|}\max\\{4s,-2s(1+\log
s^{2})\\}ds,$

which implies

$\displaystyle
J^{i}_{*}(v^{i}_{*})=\max_{t>0}J^{i}_{*}(tv^{i}_{*})\geq\max_{t>0}\mathcal{L}_{V(x^{i}_{*})}(tv^{i}_{*})\geq\mathcal{C}_{V(x^{i}_{*})}.$

Then, after rescaling, we have

$\displaystyle\quad\liminf_{n\to\infty}\frac{1}{2\varepsilon^{N}_{n}}\int_{B_{\varepsilon_{n}R}(x^{i}_{\varepsilon_{n}})}\Big{(}((\varepsilon^{2s}_{n}|(-\Delta)^{s/2}u_{n}|^{2}+(V(x)+1)|u_{n}|^{2})$
$\displaystyle\qquad-\chi_{\Lambda}|u_{n}|^{2}\log|u_{n}|^{2}+\chi_{\mathbb{R}^{N}\backslash\Lambda}{G}_{2}(x,u_{n})\Big{)}$
$\displaystyle\geq
J^{i}_{*}(v^{i}_{*})+o_{R}(1)\geq\mathcal{C}_{V(x^{i}_{*})}+o_{R}(1).$

Now let us estimate the energy outside
$\bigcup_{i=1}^{k}B_{\varepsilon_{n}}(x^{i}_{\varepsilon_{n}})$. Choose $\eta$
as another cut-off function with $\eta_{R}\equiv 0$ in $B_{R}$ and
$\eta_{R}\equiv 1$ on $B^{c}_{2R}$ and define

$\eta_{n,R}(\cdot)=\prod_{i=1}^{k}\eta_{R}\Big{(}\frac{\cdot-x^{i}_{\varepsilon_{n}}}{\varepsilon_{n}}\Big{)}.$

Testing (2.9) against with $\eta_{n,R}u_{\varepsilon_{n}}$, by the definition
of penalized function in (2), we find

$\displaystyle\quad\frac{1}{2}\int_{\Big{(}\bigcup_{i=1}^{k}B_{\varepsilon_{n}R}(x^{i}_{\varepsilon_{n}})\Big{)}^{c}}(\varepsilon^{2s}_{n}|(-\Delta)^{s/2}u_{n}|^{2}+(V(x)+1)|u_{n}|^{2})$
$\displaystyle-\frac{1}{2}\int_{\Big{(}\bigcup_{i=1}^{k}B_{\varepsilon_{n}R}(x^{i}_{\varepsilon_{n}})\Big{)}^{c}\cap\Lambda}{G}_{1}(x,u_{n})+\frac{1}{2}\int_{\Big{(}\bigcup_{i=1}^{k}B_{\varepsilon_{n}R}(x^{i}_{\varepsilon_{n}})\Big{)}^{c}\cap\mathbb{R}^{N}\backslash\Lambda}{G}_{2}(x,u_{n})$
$\displaystyle:=T^{2}_{n,R}+\frac{1}{2}\int_{{\bigcup_{i=1}^{k}\Big{(}B_{2\varepsilon_{n}R}(x^{i}_{\varepsilon_{n}})\backslash
B_{\varepsilon_{n}R}(x^{i}_{\varepsilon_{n}})\Big{)}}}(1-\eta_{n,R})(\varepsilon^{2s}_{n}|(-\Delta)^{s/2}u_{n}|^{2}+(V(x)+1)|u_{n}|^{2})$
$\displaystyle\quad+\frac{1}{2}\int_{\Lambda}\eta_{n,R}|u_{n}|^{2}(1+\log|u_{n}|^{2})-\frac{1}{2}{\int_{\mathbb{R}^{N}}}\chi_{\Big{(}\bigcup_{i=1}^{k}B_{\varepsilon_{n}R}(x^{i}_{\varepsilon_{n}})\Big{)}^{c}\cap\Lambda}|u_{n}|^{2}\log|u_{n}|^{2}$
$\displaystyle\quad-\frac{1}{2}\int_{\mathbb{R}^{N}\backslash\Lambda}\eta_{n,R}(x)\max\\{2|u_{n}|^{2},-|u_{n}|^{2}(1+\log|u_{n}|^{2})\\}$
$\displaystyle\quad+\frac{1}{2}\int_{\mathbb{R}^{N}\backslash\Lambda}\chi_{\Big{(}\bigcup_{i=1}^{k}B_{\varepsilon_{n}R}(x^{i}_{\varepsilon_{n}})\Big{)}^{c}}\mathrm{d}x\int_{0}^{|u_{n}|}\max\\{4t_{+},-2t_{+}(1+\log|t_{+}|^{2})\\}$
(3.5) $\displaystyle\geq
T^{2}_{n,R}+\frac{1}{2}\int_{\Lambda}\eta_{n,R}|u_{n}|^{2}(1+\log|u_{n}|^{2})-\frac{1}{2}{\int_{\Lambda}}\chi_{\Big{(}\bigcup_{i=1}^{k}B_{\varepsilon_{n}R}(x^{i}_{\varepsilon_{n}})\Big{)}^{c}}|u_{n}|^{2}\log|u_{n}|^{2}$
$\displaystyle\quad-\frac{1}{2}\int_{\mathbb{R}^{N}\backslash\Lambda}\eta_{n,R}(x)\max\\{2|u_{n}|^{2},-|u_{n}|^{2}(1+\log|u_{n}|^{2})\\}$
$\displaystyle\quad+\frac{1}{2}\int_{\mathbb{R}^{N}\backslash\Lambda}\chi_{\Big{(}\bigcup_{i=1}^{k}B_{\varepsilon_{n}R}(x^{i}_{\varepsilon_{n}})\Big{)}^{c}}\max\\{2|u_{n}|^{2},-|u_{n}|^{2}(1+\log|u_{n}|^{2})\\}\mathrm{d}x$
$\displaystyle\geq
T^{2}_{n,R}+\frac{1}{2}\int_{\bigcup_{i=1}^{k}\Big{(}B_{2\varepsilon_{n}R}(x^{i}_{\varepsilon_{n}})\backslash
B_{\varepsilon_{n}R}(x^{i}_{\varepsilon_{n}})\Big{)}\cap\Lambda}(\eta_{n,R}-1)|u_{n}|^{2}\log|u_{n}|^{2}$
$\displaystyle\quad+\frac{1}{2}\int_{{\bigcup_{i=1}^{k}\Big{(}B_{2\varepsilon_{n}R}(x^{i}_{\varepsilon_{n}})\backslash
B_{\varepsilon_{n}R}(x^{i}_{\varepsilon_{n}})\Big{)}}\cap\mathbb{R}^{N}\backslash\Lambda}(\eta_{n,R}-1)\max\\{2|u_{n}|^{2},-|u_{n}|^{2}(1+\log|u_{n}|^{2})\\},$

where

$\displaystyle
T^{2}_{n,R}=\frac{\varepsilon^{2s}_{n}}{2}\int_{\mathbb{R}^{N}}\mathrm{d}x\int_{\mathbb{R}^{N}}\frac{(\eta_{n,R}(y)-\eta_{n,R}(x))u_{n}(y)(u_{n}(x)-u_{n}(y))}{|x-y|^{N+2s}}\mathrm{d}y.$

In Appendix A, we will prove by fractional Hardy inequality (1.6) that

$\limsup_{n\to\infty}\frac{T^{2}_{n,R}}{\varepsilon^{N}_{n}}\geq o_{R}(1).$
(3.6)

Hence, by the fact that $v^{i}_{n}\to v^{i}_{*}$ strongly in
$L^{q}_{loc}(\mathbb{R}^{N})$ with $1<q<2^{*}_{s}$, we conclude that

$\displaystyle\lim_{n\to\infty}\Big{(}\quad\frac{1}{2\varepsilon^{N}_{n}}\int_{\Big{(}\bigcup_{i=1}^{k}B_{\varepsilon_{n}R}(x^{i}_{\varepsilon_{n}})\Big{)}^{c}}(\varepsilon^{2s}_{n}|(-\Delta)^{s/2}u_{n}|^{2}+(V(x)+1)|u_{n}|^{2})$
$\displaystyle-\frac{1}{2\varepsilon^{N}_{n}}\int_{\Big{(}\bigcup_{i=1}^{k}B_{\varepsilon_{n}R}(x^{i}_{\varepsilon_{n}})\Big{)}^{c}}{G}_{1}(x,u_{n})+\frac{1}{2\varepsilon^{N}_{n}}\int_{\Big{(}\bigcup_{i=1}^{k}B_{\varepsilon_{n}R}(x^{i}_{\varepsilon_{n}})\Big{)}^{c}}{G}_{2}(x,u_{n})\Big{)}$
$\displaystyle\geq o_{R}(1)-C\int_{B_{2R}\backslash
B_{R}}(|v^{i}_{*}|+|v^{i}_{*}|^{q})$ $\displaystyle=o_{R}(1).$

Finally, by the analysis above, we have

$\liminf_{n\to\infty}\frac{J_{\varepsilon_{n}}(u_{\varepsilon_{n}})}{\varepsilon^{N}_{n}}\geq\sum_{i=1}^{k}\mathcal{C}_{V(x^{i}_{*})}+o_{R}(1),$

the conclusion then follows by letting $R\to\infty$.

∎

######  Remark 3.4.

It is easy to check that

$\int\max\\{f,g\\}\geq\max\\{\int f,\int g\\},$

which and the skillful choice of the truncated function in (2) play a key role
in the proof of (3.1). The estimates (3.6) is trivial in the classical case,
but delicate under the nonlocal effect of $(-\Delta)^{s}(0<s<1)$, some
skillful global estimates will be involved(e.g., the $L^{2}$ information of
$u_{\varepsilon_{n}}$ outside $\Lambda$, see (A) in Appendix for example).

Now we prove the concentration of $u_{\varepsilon}$.

######  Lemma 3.5.

Let $\rho>0$ and $u_{\varepsilon}$ be the penalized solution given by Lemma
2.8. There exists a family of points $(x_{\varepsilon})\subset\Lambda$ such
that

$(i)$ $\liminf\limits_{\varepsilon\to
0}\|u_{\varepsilon}\|_{L^{\infty}(B_{\varepsilon\rho}(x_{\varepsilon}))}>0$.

$(ii)$ $\lim\limits_{\varepsilon\to 0}V(x_{\varepsilon})=\inf_{\Lambda}V$.

$(iii)$ $\lim\limits_{{R\to\infty}\atop{\varepsilon\to
0}}\|u_{\varepsilon}\|_{L^{\infty}(U\backslash B_{\varepsilon
R}(x_{\varepsilon}))}=0$.

###### Proof.

Easily, we have

$\displaystyle 0$
$\displaystyle<\int_{\mathbb{R}^{N}}(\varepsilon^{2s}|(-\Delta)^{s/2}u_{\varepsilon}|^{2}+(V(x)+1)|u_{\varepsilon}|^{2})\leq\int_{\Lambda\cap\\{x:u_{\varepsilon}(x)>e^{-1/2}\\}}|u_{\varepsilon}|^{2}(1+\log|u_{\varepsilon}|^{2})$
$\displaystyle\qquad\leq\|1+\log|u_{\varepsilon}|^{2}\|_{L^{\infty}(\Lambda\cap\\{x:u_{\varepsilon}(x)>e^{-1/2}\\})}\int_{\Lambda}|u_{\varepsilon}|^{2},$

which and the similar regularity assertion in [10] imply that there exists
$x_{\varepsilon}\in\overline{\Lambda}$ such that

$u_{\varepsilon}(x_{\varepsilon})=\sup_{x\in\Lambda}u_{\varepsilon}(x)\ \
\text{and}\ \liminf_{\varepsilon\to
0}\|u_{\varepsilon}\|_{L^{\infty}(B_{\varepsilon\rho}(x_{\varepsilon}))}>0.$

This proves $(i)$.

For $(ii)$, assuming without loss of generality that
$\lim\limits_{\varepsilon\to 0}x_{\varepsilon}=x_{*}$, by the lower and upper
bounds of $u_{\varepsilon}$ in Propositions 3.2 and 3.3, we have

$\min_{x\in\Lambda}\mathcal{C}_{V(x)}\geq\liminf_{\varepsilon\to
0}\frac{J_{\varepsilon}(u_{\varepsilon})}{\varepsilon^{N}}\geq\mathcal{C}_{V(x_{*})},$

which implies $V(x_{*})=\min_{x\in\Lambda}V(x)$.

For $(iii)$, if it is not true, then one will get a contradiction like

$\min_{x\in\Lambda}\mathcal{C}_{V(x)}\geq\liminf_{\varepsilon\to
0}\frac{J_{\varepsilon}(u_{\varepsilon})}{\varepsilon^{N}}\geq\mathcal{C}_{V(x_{*})}+\mathcal{C}_{V(y_{*})}$

for some $y_{*}\in\overline{U}$ by Proposition 3.3. ∎

###  3.2. Back to the origin problem

In this subsection, we use Lemma 3.5 to linearize equation (2.9) and then use
the well-known decay estimates of positive solutions of fractional Schrödinger
equations to show that (3.1) is true.

Noting that by the regular assertion in Appendix D of [10], we can assume that

$\sup_{\Lambda}u_{\varepsilon}(x)\leq C<\infty,$ (3.7)

where $C$ is a positive constant. Hence by Lemma 3.5, we can linearize the
penalized equation (2.9) as follows.

######  Proposition 3.6.

Let $\varepsilon>0$ be small enough, $x_{\varepsilon}$ be the point given by
Lemma 3.5. Then there exists $R>0$ such that

$\left\\{\begin{array}[]{ll}\varepsilon^{2s}(-\Delta)^{s}u_{\varepsilon}+\min\\{\lambda,2\\}u_{\varepsilon}\leq
0,&\text{in}\ \mathbb{R}^{N}\backslash B_{\varepsilon R}(x_{\varepsilon})\\\
u_{\varepsilon}\leq C,&\text{in}\ B_{\varepsilon
R}(x_{\varepsilon})\end{array}\right.$

###### Proof.

For $\varepsilon>0$ small enough, by Lemma 3.5, there exists $R>0$ such that
$(1+\log|u_{\varepsilon}|^{2})\leq 0$ for all $x\in U\backslash B_{\varepsilon
R}(x_{\varepsilon})$, the conclusion then follows by the penalized function in
(2) and inserting (3.7) into (2.9). ∎

At last, we prove Theorem 1.1.

Completes the proof of Theorem 1.1.

By Proposition 3.6, the rescaling function
$v_{\varepsilon}=u_{\varepsilon}(\varepsilon x+x_{\varepsilon}$) satisfies

$\left\\{\begin{array}[]{ll}(-\Delta)^{s}v_{\varepsilon}+\lambda
v_{\varepsilon}\leq 0,&\text{in}\ \mathbb{R}^{N}\backslash B_{R}\\\
v_{\varepsilon}\leq C,&\text{in}\ B_{R}.\end{array}\right.$

Then by the well-known decay estimates of fractional Schrödinger equation(see
[10, Appendix D] for example), we have

$v_{\varepsilon}(x)\leq\frac{C}{1+|x|^{N+2s}},\ x\in\mathbb{R}^{N}.$ (3.8)

Consequently, it holds

$u_{\varepsilon}(x)\leq\frac{C\varepsilon^{N+2s}}{\varepsilon^{N+2s}+|x-x_{\varepsilon}|^{N+2s}},$

from which, we can conclude that

$\max\\{2u_{\varepsilon},-\big{(}1+\log(u_{\varepsilon})^{2}\big{)}\\}=-\big{(}1+\log(u_{\varepsilon})^{2}\big{)}\
\ \text{for all}\ x\in\mathbb{R}^{N}\backslash\Lambda$

if $\varepsilon>0$ is small. This proves that $u_{\varepsilon}$ solves the
origin problem (1.1) and complete the proof of this paper.

## Appendix A A

In this section, we are going to verify (2.6) and (3.6). The fractional Hardy
inequality (1.6) will be involved in the proof. We first give the proof of
(2.6). By Cauchy inequality, we have

$\displaystyle T_{1}(\eta)$ $\displaystyle\leq
2{\varepsilon^{2s}}\int_{\mathbb{R}^{N}}\mathrm{d}x\int_{\mathbb{R}^{N}}\frac{|u_{n}(x)|^{2}(\eta(x)-\eta(y))^{2}}{|x-y|^{N+2s}}\mathrm{d}y+C\|u_{n}\|^{2}_{V,\varepsilon}$
$\displaystyle:=T_{11}(\eta)+C\|u_{n}\|^{2}_{V,\varepsilon}.$

By decomposition, denoting
$f(x,y)=\frac{|u_{n}(x)|^{2}(\eta(x)-\eta(y))^{2}}{|x-y|^{N+2s}}$, we have

$\displaystyle\frac{1}{2}T_{11}(\eta)$
$\displaystyle=\varepsilon^{2s}\int_{\Lambda^{\delta}}\mathrm{d}x\int_{\mathbb{R}^{N}}f(x,y)\mathrm{d}y+\varepsilon^{2s}\int_{\mathbb{R}^{N}\backslash\Lambda^{\delta}}\mathrm{d}x\int_{\Lambda^{\delta}}f(x,y)\mathrm{d}y$
$\displaystyle:=\sum_{i=1}^{2}T_{11i}(\eta).$

Since

$\displaystyle\int_{\mathbb{R}^{N}}\frac{(\eta(x)-\eta(y))^{2}}{|x-y|^{N+2s}}\mathrm{d}y$
$\displaystyle=\int_{B_{1}(x)}\frac{(\eta(x)-\eta(y))^{2}}{|x-y|^{N+2s}}\mathrm{d}y+\int_{\mathbb{R}^{N}\backslash
B_{1}(x)}\frac{(\eta(x)-\eta(y))^{2}}{|x-y|^{N+2s}}\mathrm{d}y$
$\displaystyle\leq
C\Big{(}\int_{B_{1}(0)}\frac{1}{|z|^{N+2s-2}}\mathrm{d}y+\int_{\mathbb{R}^{N}\backslash
B_{1}(0)}\frac{1}{|z|^{N+2s}}\mathrm{d}y\Big{)}$ $\displaystyle\leq C,$

we have

$\displaystyle\begin{split}T_{111}(\eta)\leq
C\varepsilon^{2s}\int_{\Lambda^{\delta}}|u_{n}(x)|^{2}\leq
C\|u_{n}\|^{2}_{V,\varepsilon}.\end{split}$ (A.1)

For $T_{112}(\eta)$, similar to the estimates above, by fractional Hardy
inequality (1.6), we have

$\displaystyle\begin{split}T_{112}(\eta)&\leq
C\varepsilon^{2s}\int_{\mathbb{R}^{N}\backslash\Lambda^{2\delta}}|u_{n}(x)|^{2}\mathrm{d}x\int_{\Lambda^{\delta}}\frac{1}{|x-y|^{N+2s}}\mathrm{d}y\\\
&\leq
C\varepsilon^{2s}\int_{\mathbb{R}^{N}\backslash\Lambda^{2\delta}}\frac{|u_{n}(x)|^{2}}{|x|^{2s}}\mathrm{d}x\int_{\Lambda^{\delta}}\frac{|x|^{2s}}{|x-y|^{N+2s}}\mathrm{d}y\\\
&\leq
C\varepsilon^{2s}\int_{\mathbb{R}^{N}\backslash\Lambda^{2\delta}}\frac{|u_{n}(x)|^{2}}{|x|^{2s}}\mathrm{d}x\int_{\Lambda^{\delta}}\frac{|x|^{2s}}{\big{(}dist(x,\partial\Lambda^{2\delta})+\delta+dist(y,\partial\Lambda^{\delta})\big{)}^{N+2s}}\mathrm{d}y\\\
&\leq C\|u_{n}\|^{2}_{V,\varepsilon}.\end{split}$ (A.2)

By (A.1) and (A.2), we conclude that‘

$T_{1}(\eta)\leq C\|u_{n}\|^{2}_{V,\varepsilon}.$

Next, we give the proof of (3.6). A change of variable tells us

$\displaystyle\begin{split}T^{2}_{n,R}&=\int_{\mathbb{R}^{N}}u_{n}(y)\,{\mathrm{d}}y\int_{\mathbb{R}^{N}}\frac{(u_{n}(x)-u_{n}(y))(\psi_{n,R}(x)-\psi_{n,R}(y))}{|x-y|^{N+2s}}\,{\mathrm{d}}x\\\
&=\varepsilon^{N-2s}_{n}\sum_{l=1}^{k}\int_{\mathbb{R}^{N}}v^{l}_{n}(y)\beta^{l}_{n}(y)\,{\mathrm{d}}y\int_{\mathbb{R}^{N}}\frac{\alpha^{l}_{n}(x)(v^{l}_{n}(x)-v^{l}_{n}(y))(\eta_{R}(x)-\eta_{R}(y))}{|x-y|^{N+2s}}\,{\mathrm{d}}x,\end{split}$

where the functions $\beta^{l}_{n}$ and $\alpha^{l}_{n}$ are defined
skillfully as

$\beta^{l}_{n}(y)=\prod^{l-1}_{s=0}\eta\Big{(}\frac{y}{R}+\frac{x^{l}_{n}-x^{s}_{n}}{\varepsilon_{n}R}\Big{)},\,\,\,\beta^{0}_{n}(y)\equiv
1$

and

$\alpha^{l}_{n}(x)=\prod^{k}_{s=l+1}\eta\Big{(}\frac{x}{R}+\frac{x^{l}_{n}-x^{s}_{n}}{\varepsilon_{n}R}\Big{)},\,\,\,\alpha^{k}_{n}(x)\equiv
1.$

Following, we have

$\displaystyle\varepsilon^{2s-N}_{n}T^{2}_{n,R}$
$\displaystyle=\sum_{l=1}^{k}\int_{B_{2R}}v^{l}_{n}(y)\beta^{l}_{n}(y)\,{\mathrm{d}}y\int_{B^{c}_{2R}}\frac{\alpha^{l}_{n}(x)(v^{l}_{n}(x)-v^{l}_{n}(y))(\eta_{R}(x)-\eta_{R}(y))}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle\quad+\sum_{l=1}^{k}\int_{B^{c}_{2R}}v^{l}_{n}(y)\beta^{l}_{n}(y)\,{\mathrm{d}}y\int_{B_{2R}}\frac{\alpha^{l}_{n}(x)(v^{l}_{n}(x)-v^{l}_{n}(y))(\eta_{R}(x)-\eta_{R}(y))}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle\quad+\sum_{l=1}^{k}\int_{B_{2R}}v^{l}_{n}(y)\beta^{l}_{n}(y)\,{\mathrm{d}}y\int_{B_{2R}}\frac{\alpha^{l}_{n}(x)(v^{l}_{n}(x)-v^{l}_{n}(y))(\eta_{R}(x)-\eta_{R}(y))}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle:=T^{21}_{n,R}+T^{22}_{n,R}+T^{23}_{n,R}.$

By the choice of $\eta_{R}$ and
$\lim\limits_{n\to\infty}\frac{|x^{l}_{n}-x^{s}_{n}|}{\varepsilon_{n}}=\infty$
if $l\neq s$, for $n$ large, we have

$\displaystyle T^{21}_{n,R}$
$\displaystyle=\sum_{l=1}^{k}\int_{B_{2R}\backslash
B_{R}}v^{l}_{n}(y)\beta^{l}_{n}(y)\,{\mathrm{d}}y\int_{B^{c}_{2R}}\frac{\alpha^{l}_{n}(x)(v^{l}_{n}(x)-v^{l}_{n}(y))(1-\eta_{R}(y))}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle\quad+\sum_{l=1}^{k}\int_{{B_{R}}}v^{l}_{n}(y)\beta^{l}_{n}(y)\,{\mathrm{d}}y\int_{B^{c}_{2R}}\frac{\alpha^{l}_{n}(x)(v^{l}_{n}(x)-v^{l}_{n}(y))}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle=\sum_{l=1}^{k}\int_{B_{2R}\backslash
B_{R}}v^{l}_{n}(y)\,{\mathrm{d}}y\int_{B^{c}_{2R}}\frac{\alpha^{l}_{n}(x)(v^{l}_{n}(x)-v^{l}_{n}(y))(1-\eta_{R}(y))}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle\quad+\sum_{l=1}^{k}\int_{{B_{R}}}v^{l}_{n}(y)\,{\mathrm{d}}y\int_{B^{c}_{2R}}\frac{\alpha^{l}_{n}(x)(v^{l}_{n}(x)-v^{l}_{n}(y))}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle:$ $\displaystyle=T^{211}_{n,R}+T^{212}_{n,R}$

and

$\displaystyle T^{22}_{n,R}=$
$\displaystyle\sum_{l=1}^{k}\int_{B^{c}_{2R}}v^{l}_{n}(y)\beta^{l}_{n}(y)\,{\mathrm{d}}y\int_{B_{2R}\backslash
B_{R}}\frac{\alpha^{l}_{n}(x)(v^{l}_{n}(x)-v^{l}_{n}(y))(\eta_{R}(x)-1)}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle-\sum_{l=1}^{k}\int_{B^{c}_{2R}}v^{l}_{n}(y)\beta^{l}_{n}(y)\,{\mathrm{d}}y\int_{B_{R}}\frac{\alpha^{l}_{n}(x)(v^{l}_{n}(x)-v^{l}_{n}(y))}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle=$
$\displaystyle\sum_{l=1}^{k}\int_{B^{c}_{2R}}v^{l}_{n}(y)\beta^{l}_{n}(y)\,{\mathrm{d}}y\int_{B_{2R}\backslash
B_{R}}\frac{(v^{l}_{n}(x)-v^{l}_{n}(y))(\eta_{R}(x)-1)}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle-\sum_{l=1}^{k}\int_{B^{c}_{2R}}v^{l}_{n}(y)\beta^{l}_{n}(y)\,{\mathrm{d}}y\int_{B_{R}}\frac{v^{l}_{n}(x)-v^{l}_{n}(y)}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle:=$ $\displaystyle T^{221}_{n,R}+T^{222}_{n,R}.$

Also, for large $n$,

$\displaystyle T^{23}_{n,R}=$
$\displaystyle\sum_{l=1}^{k}\int_{B_{2R}\backslash{B_{R}}}v^{l}_{n}(y)\,{\mathrm{d}}y\int_{B_{2R}}\frac{(v^{l}_{n}(x)-v^{l}_{n}(y))(\eta_{R}(x)-\eta_{R}(y))}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle+\sum_{l=1}^{k}\int_{B_{2R}}v^{l}_{n}(y)\,{\mathrm{d}}y\int_{B_{2R}\backslash{B_{R}}}\frac{(v^{l}_{n}(x)-v^{l}_{n}(y))(\eta_{R}(x)-\eta_{R}(y))}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle+\sum_{l=1}^{k}\int_{B_{2R}\backslash{B_{R}}}v^{l}_{n}(y)\,{\mathrm{d}}y\int_{B_{2R}\backslash
B_{R}}\frac{(v^{l}_{n}(x)-v^{l}_{n}(y))(\eta_{R}(x)-\eta_{R}(y))}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle:=$ $\displaystyle T^{231}_{n,R}+T^{232}_{n,R}+T^{233}_{n,R}.$

For $|T^{2i2}_{n,R}|,\ i=1,2$, it holds

$\displaystyle\limsup_{n\to\infty}|T^{2i2}_{n,R}|\leq
CR^{-2s}+\limsup_{n\to\infty}2\sum_{l=1}^{k}\int_{B^{c}_{2R}}\,{\mathrm{d}}y\int_{B_{R}}\frac{(v^{l}_{n}(y))^{2}}{|x-y|^{N+2s}}\,{\mathrm{d}}x.$

By the fractional Hardy inequality (1.6) and letting
$\widetilde{R}=R^{\frac{N+1}{N}}$, we find

$\displaystyle\quad\limsup_{n\to\infty}\int_{B^{c}_{2R}}\big{(}v^{l}_{n}(y))^{2}\,{\mathrm{d}}y\int_{B_{R}}\frac{1}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle\leq
C\limsup_{n\to\infty}\int_{B^{c}_{2R}}\big{(}v^{l}_{n}(y))^{2}\frac{R^{N}}{|y|^{N+2s}}\,{\mathrm{d}}y$
$\displaystyle\leq C\limsup_{n\to\infty}\int_{B_{\widetilde{R}}\backslash
B_{2R}}\big{(}v^{l}_{n}(y))^{2}\frac{R^{N}}{|y|^{N+2s}}\,{\mathrm{d}}y+C\limsup_{n\to\infty}\int_{B^{c}_{\widetilde{R}}}\big{(}v^{l}_{n}(y))^{2}\frac{R^{N}}{|y|^{N+2s}}\,{\mathrm{d}}y$
$\displaystyle\leq C\limsup_{n\to\infty}\int_{B_{\widetilde{R}}\backslash
B_{2R}}\big{(}v^{l}_{n}(y))^{2}\,{\mathrm{d}}y+C\limsup_{n\to\infty}\int_{B^{c}_{\widetilde{R}}}\frac{(v^{l}_{n}(y))^{2}}{|y|^{2s}}\frac{R^{N}}{|y|^{N}}\,{\mathrm{d}}y$
(A.3) $\displaystyle\leq C\int_{B_{\widetilde{R}}\backslash
B_{2R}}\big{(}v^{l}_{*}(y))^{2}\,{\mathrm{d}}y+\frac{C}{R}$
$\displaystyle=o_{R}(1).$

Noting that for each $\Omega\subset\subset\mathbb{R}^{N}$ is smooth, it holds

$v^{l}_{n}\rightharpoonup v^{l}_{*}\ \text{weakly in}\ W^{s,2}(\Omega).$

Then, for $T^{211}_{n,R}$, by the estimates of $T^{2i2}_{n,R}$($i=1,2$), we
have

$\displaystyle\limsup_{n\to\infty}|T^{211}_{n,R}|$
$\displaystyle\leq\limsup_{n\to\infty}\sum_{l=1}^{k}\int_{B_{2R}\backslash
B_{R}}\,{\mathrm{d}}y\int_{B^{c}_{2R}}\frac{|v^{l}_{n}(x)-v^{l}_{n}(y)|^{2}}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle+\limsup_{n\to\infty}\sum_{l=1}^{k}\int_{B_{2R}\backslash
B_{R}}(v^{l}_{n}(y))^{2}\,{\mathrm{d}}y\int_{B^{c}_{2R}}\frac{(1-\eta_{R}(y))^{2}}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle\leq\limsup_{n\to\infty}\sum_{l=1}^{k}\int_{B_{2R}\backslash
B_{R}}\,{\mathrm{d}}y\int_{B^{c}_{2R}\cap
B_{4R}}\frac{|v^{l}_{n}(x)-v^{l}_{n}(y)|^{2}}{|x-y|^{N+2s}}\,{\mathrm{d}}x$
$\displaystyle+\limsup_{n\to\infty}\sum_{l=1}^{k}\int_{B_{2R}\backslash
B_{R}}\,{\mathrm{d}}y\int_{B^{c}_{4R}}\frac{|v^{l}_{n}(x)-v^{l}_{n}(y)|^{2}}{|x-y|^{N+2s}}\,{\mathrm{d}}x+C\limsup_{n\to\infty}\sum_{l=1}^{k}\int_{B_{2R}\backslash
B_{R}}(v^{l}_{n}(y))^{2}\,{\mathrm{d}}y$
$\displaystyle\leq\limsup_{n\to\infty}\sum_{l=1}^{k}\int_{B_{4R}\backslash
B_{R}}\,{\mathrm{d}}y\int_{B_{4R}\backslash
B_{R}}\frac{|v^{l}_{n}(x)-v^{l}_{n}(y)|^{2}}{|x-y|^{N+2s}}\,{\mathrm{d}}x+C\sum_{l=1}^{k}\int_{\mathbb{R}^{N}}(v^{*}_{l}(y))^{2}\mathrm{d}y$
$\displaystyle\leq C\int_{B_{4R}\backslash
B_{R}}\,{\mathrm{d}}y\int_{R^{N}}\frac{|v^{l}_{*}(x)-v^{l}_{*}(y)|^{2}}{|x-y|^{N+2s}}\,{\mathrm{d}}x+o_{R}(1)$
$\displaystyle=o_{R}(1).$

Similarly, we get

$\displaystyle\limsup_{n\to\infty}|T^{221}_{n,R}|$ $\displaystyle\leq
o_{R}(1)$

and

$\displaystyle\limsup_{n\to\infty}|T^{231}_{n,R}+T^{232}_{n,R}+T^{233}_{n,R}|\leq
o_{R}(1).$

Therefore

$\varepsilon^{2s-N}T^{2}_{n,R}\leq o_{R}(1).$

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Table 6. Lower bounds on $\operatorname{sys}(M)$ sufficient for $N_{\mathrm{small}}(M)/(2g-2)$ to be strictly smaller than given values. lower bound on $\operatorname{sys}(M)$ | strict upper bound on $N_{\mathrm{small}}(M)/(2g-2)$
---|---
2.317 | 1
3.234 | 1/2
3.919 | 1/3
4.486 | 1/4
4.978 | 1/5
5.409 | 1/6
5.818 | 1/7
6.180 | 1/8
6.505 | 1/9
6.894 | 1/10

###  10.4. Ramanujan surfaces

Borrowing terminology from graph theory, we say that a hyperbolic surface $M$
of finite area is _Ramanujan_ if $\lambda_{1}(M)\geq 1/4$. We will also say
that $M$ is _strictly Ramanujan_ if $\lambda_{1}(M)>1/4$. Selberg’s eigenvalue
conjecture [Sel65] states that all congruence covers of the modular curve are
Ramanujan. A related question is whether there exist closed Ramanujan surfaces
in every genus (see [Mon15, Question 1.1] and [Wri20, Problem 10.4]). Thanks
to the work of Hide and Magee [HM21], we now know that in large genus, there
exist closed surfaces that are _nearly Ramanujan_ in the sense that their
first eigenvalue is arbitrarily close to $1/4$.

Observe that

$\lambda_{1}(M)>L\ \Longleftrightarrow\ N_{[0,L]}(M)=1\ \Longleftrightarrow\
N_{[0,L]}(M)<2.$

This means that one can prove lower bounds on $\lambda_{1}$ by bounding
$N_{[0,L]}$ from above. In particular,

$\lambda_{1}(M)>\frac{1}{4}\ \Longleftrightarrow\ N_{\mathrm{small}}(M)<2\
\Longleftrightarrow\ \frac{N_{\mathrm{small}}(M)}{2g-2}<\frac{1}{g-1}$

so the values in Table 6 give lower bounds on the systole that are sufficient
for surfaces to be strictly Ramanujan in genus $2$ to $11$. However, in
obtaining these values we discarded the term $1-\widehat{f}(i/2)$ appearing in
Theorem 10.1. The lower bounds on the systole (sufficient to be strictly
Ramanujan) that we have obtained by taking this term into account are listed
in Table 7 for $g$ between $2$ and $20$ and plotted in Figure 2(b). The
corresponding ancillary file is verify_ramanujan.ipynb. According to Table 1,
there exist hyperbolic surfaces with systole larger than these bounds, hence
strictly Ramanujan, in genus $2$ to $7$, $14$, and $17$. For these specific
surfaces, we can increase $L$ further as long as $N_{[0,L]}<2$ to obtain
improved lower bounds on $\lambda_{1}$ still based only on the systole. The
resulting bounds are listed in Table 3 except in genus $2$ to $4$ where better
data was already available. The corresponding ancillary file is
verify_ramanujan_examples.ipynb. These bounds are rigorous modulo proving that
the lower bounds on the systole in Table 1 are correct (as pointed out
earlier, some are rigorous but not all).

Table 7. Lower bounds on $\operatorname{sys}(M)$ sufficient for $\lambda_{1}(M)>1/4$. genus | lower bound on $\operatorname{sys}(M)$ sufficient for $\lambda_{1}(M)>1/4$ | record systole
---|---|---
2 | 2.315 | 3.057141
3 | 3.218 | 3.983304
4 | 3.867 | 4.624499
5 | 4.380 | 4.91456
6 | 4.803 | 5.109
7 | 5.168 | 5.796298
8 | 5.482 |
9 | 5.760 |
10 | 6.010 |
11 | 6.236 |
12 | 6.443 |
13 | 6.632 |
14 | 6.808 | 6.887905
15 | 6.971 |
16 | 7.124 |
17 | 7.268 | 7.609407
18 | 7.403 |
19 | 7.531 |
20 | 7.651 |

It seems very likely that surfaces with systole larger than the values listed
in Table 7 exist in the remaining genera up to $20$ as well. However, our
numerical experiments suggest that this method cannot prove that the next
Hurwitz surface (of genus $118$) is Ramanujan using only its systole. Based on
Figure 1(a), Figure 2(b), and Theorem 6.4, it seems reasonable to make the
following conjecture.

######  Conjecture 10.4.

There exist constants $c_{1}>c_{2}$ such that the smallest upper bound on the
systole that can be obtained from Theorem 6.1 and the smallest lower bound on
the systole sufficient to prove $N_{\mathrm{small}}<2$ (equivalently
$\lambda_{1}>1/4$) using Theorem 10.1 are of the form

$2\log(g)+c_{j}+o(1)\quad\text{as}\quad g\to\infty.$

It is unknown if there exist closed hyperbolic surfaces with systole
asymptotic to $r\log(g)$ for any $r>4/3$, so even if the conjecture is true it
is unlikely to be good enough to prove the existence of Ramanujan surfaces in
large genus.

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