.

# On concentration of real solutions for fractional Helmholtz equation

Zifei Shen  and  Shuijin Zhang  Zifei Shen, Shuijin Zhang
Department of Mathematics, Zhejiang Normal University,
Jinhua, Zhejiang, 321004, People’s Republic of China  Z. Shen:
<EMAIL_ADDRESS>S. Zhang<EMAIL_ADDRESS>
###### Abstract.

This paper studies the nonlinear fractional Helmholtz equation

(0.1) $(-\Delta)^{s}u-k^{2}u=Q(x)|u|^{p-2}u,\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{in}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{R}^{N},\leavevmode\nobreak\ \leavevmode\nobreak\ N\geq 3,$

where $\frac{N}{N+1}<s<\frac{N}{2}$, $\frac{2(N+1)}{N-1}<p<\frac{2N}{N-2s}$
are two real exponents, and the coefficient $Q$ is bounded continuous,
nonnegative and satisfies the condition

(0.2) $\mathop{\mathrm{lim\leavevmode\nobreak\
sup}}\limits_{|x|\longrightarrow\infty}Q(x)<\mathop{\mathrm{sup}}\limits_{x\in\mathbb{R}^{N}}Q(x).$

For $k>0$ large, the existence of real-valued solutions for (0.1) are proved,
and in the limit $k\longrightarrow\infty$, sequence of solutions associated
with ground states of a dual equation are shown to concentrate, after
rescaling, at global maximum points of the function $Q$.

###### Key words and phrases:

Fractional Helmholtz equation; Concentration of solutions; Dual variational
method; Lusternik-Schnirelman category

###### 2010 Mathematics Subject Classification:

35J15, 45E10, 45G05

##  1\. Introduction and Main Results

In this paper, we are concerned with the nonlinear fractional Helmholtz
equation

(1.1) $(-\Delta)^{s}u-k^{2}u=Q(x)|u|^{p-2}u,\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{in}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbb{R}^{N},\leavevmode\nobreak\ \leavevmode\nobreak\ N\geq 3,$

where $\frac{N}{N+1}<s<\frac{N}{2}$, $\frac{2(N+1)}{N-1}<p<\frac{2N}{N-2s}$
are two real exponents, $Q$ is a bounded continuous function.

When $s=1$, the Helmholtz equation

(1.2) $-\Delta u-k^{2}u=f(x,u)\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{in}\leavevmode\nobreak\ \mathbb{R}^{N},$

has attracted immense attention in the recent decades due to the importance in
Scattering and Optics. The main feature of this problem is that the parameter
$k^{2}>0$ is contained in the essential spectrum of negative Laplace
$-\Delta$. A general method to detect the existence of weak solutions is the
Linking argument, that is find the critical point of the corresponding
functional. However, one can not find a appropriate space in where the
associated functional of (1.2) can be well defined, this is due to that the
oscillating solutions with slow decay which, in general, are not elements of
$H^{1}(\mathbb{R}^{N})$. Therefore, the direct variational approach is
invalid.

To overcome this difficult, a dual invariant method has been proposed, which
is based on the “Limiting Absorption Principle”. By constructing the auxiliary
problems

(1.3) $-\Delta u-(\lambda+i\varepsilon)u=f(x,u)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{in}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathbb{R}^{N},$

one can obtain the boundedness estimate for the resolvent operator

$\mathcal{R}_{\lambda,\varepsilon}=(-\Delta-(\lambda+i\varepsilon))^{-1},$

and as $\varepsilon\longrightarrow 0^{+}$, one can also obtain the boundedness
estimate for the resolvent $\mathcal{R}_{\lambda}=(-\Delta-\lambda)^{-1}$, see
[27, Theorem 6] or see [32, 35, 39, 14, 33]. Based on the boundedness
estimate, Evéquoz and Weth [20] (see [22]) set up a dual variational framework
for (1.2). Correspondingly, the nontrivial real-valued solutions of equation
(1.2) with $f(x,u)=Q(x)|u|^{p-2}u$ are detected via the mountain pass
argument, where $Q(x)$ is a positive weight function, see also [24, 21, 34,
25, 31] for the other cases. By a similar way, Shen and the second author [10]
also obtained the real valued solutions for the fractional Helmholtz equation
(1.1) in the case that $0<k^{2}<+\infty$ and $Q(x)$ is assumed to be a
periodic or decay function.

Recently, Evéquoz [23] considered (1.2) in a limit case and obtain some
surprising results on the solutions. If $Q$ is assumed to be a bounded
continuous function, equation (1.2) still possess a real valued solution for
$k$ large enough. Furthermore, the solutions concentrate at the global maximum
points of the function $Q(x)$ as the frequency $\lambda$ tends infinity.
Actually, the concentrating solutions is also a big and time honored topic for
the nonlinear Schrödinger equation

(1.4) $-\varepsilon^{2}\Delta u+V(x)u=Q(x)|u|^{p-2}u\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{in}\leavevmode\nobreak\
\mathbb{R}^{N},$

where $V(x)\geq 0$ is a potential function. Basically, there are two main
routes have been pursued to investigate the concentrating solutions. One is
the Lyapunov-Schmidt reduction scheme proposed by Floer and Weinstein [26],
which has been further extended and combined with variational arguments by
Ambrosetti et al. [2, 3, 4, 5], see also for example [30, 36] for multibump
solutions. Another one is the purely variational approach initiated by
Rabinowitz [38], which is mainly relayed by del Pino and Felmer [16, 17, 18,
19]. More precisely, under the global condition

(1.5) $\mathop{\mathrm{lim\leavevmode\nobreak\
inf}}\limits_{|x|\longrightarrow\infty}\leavevmode\nobreak\
V(x)>\mathop{\mathrm{inf}}\limits_{x\in\mathbb{R}^{N}}\leavevmode\nobreak\
V(x),$

it was proved in [37] that a ground state (i.e., positive least-energy
solution) of (1.4) exists for small $\varepsilon>0$. In the limit
$\varepsilon\longrightarrow 0$, Wang [41] showed that sequences of ground
states concentrate at a global minimum point $x_{0}$ of $V$ and converge,
after rescaling, toward the ground state of the limit problem

(1.6) $-\Delta u+V(x_{0})u=|u|^{p-2}u\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{in}\leavevmode\nobreak\ \mathbb{R}^{N}.$

These results are also extended to the fractional Shrödinger equation, that is

(1.7) $\varepsilon^{2s}(-\Delta)^{s}u+V(x)u=f(x,u),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{in}\leavevmode\nobreak\
\mathbb{R}^{N}.$

We would refer the papers [15, 11] and [1, 28] to readers.

Motivated by these works, we also consider in this paper that the existence
and concentrating phenomenon of the solutions of (1.1) as
$k\longrightarrow\infty$. However, as we introduced before, the structure of
the Helmholtz equation is vary complex. There is no uniqueness and
nondegeneracy result for the real valued solutions, therefore, the classical
methods of reduction may be not available to construct the concentrating
solutions. This impels us to consider the another method what proposed by
Rabinowitz. Actually, the variational method also can not be directly adapt in
our case, since there is no natural concept of ground state associated the
direct variational formulation.

Follow the idea in [23], we define the dual ground state for (1.1) as follow.
Setting $\varepsilon=k^{-1}$,
$u_{\varepsilon}(x)=\varepsilon^{\frac{2s}{p-2}}u(\varepsilon x)$ and
$Q_{\varepsilon}(x)=Q(\varepsilon x)$, $x\in\mathbb{R}^{N}$, (1.1) can be
rewritten as

(1.8)
$(-\Delta)^{s}u_{\varepsilon}-u_{\varepsilon}=Q_{\varepsilon}(x)|u_{\varepsilon}|^{p-2}u_{\varepsilon}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{in}\leavevmode\nobreak\
\mathbb{R}^{N}.$

Furthermore, setting
$v=Q^{\frac{1}{p^{\prime}}}|u_{\varepsilon}|^{p-2}u_{\varepsilon}$, we are led
to consider the integral equation

(1.9)
$|v|^{p-2}v=Q_{\varepsilon}^{\frac{1}{p}}[\mathbf{R}^{s}\ast(Q_{\varepsilon}^{\frac{1}{p}}v)]$

where $p^{\prime}=\frac{p}{p-1}$ and $\mathbf{R}^{s}$ denotes the real part of
the fractional Helmholtz resolvent operator, see [40]. The solutions of this
integral equation are critical points of the so-called dual energy functional
$J_{\varepsilon}:L^{p^{\prime}}(\mathbb{R}^{N})\longrightarrow\mathbb{R}$
given by

(1.10)
$J_{\varepsilon}(v)=\frac{1}{p^{\prime}}\int_{\mathbb{R}^{N}}|v|^{p^{\prime}}\mathrm{d}x-\frac{1}{2}\int_{\mathbb{R}^{N}}Q_{\varepsilon}^{\frac{1}{p}}v\mathbf{R}^{s}\big{(}Q_{\varepsilon}^{\frac{1}{p}}v\big{)}\mathrm{d}x.$

Furthermore, every critical point $v$ of $J_{\varepsilon}$ gives rise to a
strong solution $u$ of (1.1) with $k=\frac{1}{\varepsilon}$, by setting

(1.11)
$u(x)=k^{\frac{2s}{p-2}}\mathbf{R}^{s}\big{(}Q_{\varepsilon}^{\frac{1}{p}}v\big{)}(kx),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ x\in\mathbb{R}^{N}.$

This correspondence allows us to define a notion of ground state for (1.1) as
follows. If $\varepsilon=\frac{1}{k}$, and $v$ is a nontrivial critical point
for $J_{\varepsilon}$ at the mountain pass level, the function $u$ given by
(1.11) will be called a $dual\leavevmode\nobreak\ \leavevmode\nobreak\
ground\leavevmode\nobreak\ \leavevmode\nobreak\ state$ of (1.1).

Apparently, once we obtain the existence and concentration of $v$, we then
obtain the existence and concentration of $u$, up to rescaling, of sequences
of dual ground states. There we present our first main result.

######  Theorem 1.1.

Let $N\geq 3$, $\frac{N}{N+1}<s<\frac{N}{2}$,
$\frac{2(N+1)}{N-1}<p<\frac{2N}{N-2s}$ and consider a function $Q$ satisfying

(Q0) Q is continuous, bounded and $Q\geq 0$ on $\mathbb{R}^{N}$;

(Q1) $Q_{\infty}:=\mathop{\mathrm{lim\leavevmode\nobreak\
sup}}\limits_{|x|\longrightarrow\infty}\leavevmode\nobreak\
Q(x)<Q_{0}:=\mathop{\mathrm{sup}}\limits_{x\in\mathbb{R}^{N}}\leavevmode\nobreak\
Q(x)$.

(i) There is $k_{0}>0$ such that for all $k>k_{0}$, the problem (1.1) admits a
dual ground state.

(ii) Let $(k_{n})_{n}\subset(k_{0},\infty)$ satisfy
$\mathop{\mathrm{lim}}\limits_{n\longrightarrow\infty}k_{n}=\infty$ and
consider for each $n$, a dual ground state $u_{n}$ of

(1.12) $(-\Delta)^{s}u-k_{n}^{2}u=Q(x)|u|^{p-2}u\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{in}\leavevmode\nobreak\
\mathbb{R}^{N}.$

Then there is a maximum point $x_{0}$ of $Q$, a dual ground state $u_{0}$ of

(1.13) $(-\Delta)^{s}u-u=Q_{0}|u|^{p-2}u\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{in}\leavevmode\nobreak\
\mathbb{R}^{N}$

and a sequence $(x_{n})_{n}\subset\mathbb{R}^{N}$ such that (up to a
subsequence)
$\mathop{\mathrm{lim}}\limits_{n\longrightarrow\infty}x_{n}=x_{0}$ and

(1.14)
$k_{n}^{-\frac{2}{p-2}}u_{n}\big{(}\frac{\cdot}{k_{n}}+x_{n}\big{)}\longrightarrow
u_{0}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{in}\leavevmode\nobreak\ L^{p}(\mathbb{R}^{N}),\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ n\longrightarrow\infty.$

Due to the assumption on $Q$, we show that, for some $\varepsilon$ small
enough, the dual energy functional is strictly below the least among all
possible energy levels for the problem at infinity, see Section 2.
Correspondingly, we prove that all the dual energy functional satisfies the
Palais-Smale condition, and therefore prove the first assertion of the above
theorem, see Section 3. The proof of the second assertion depends on a
representation lemma and we finish it in Section 4.

Follow the same idea in [23, Theorem 2], we can also obtain the multiplicity
result for (1.1). Let $M=\\{x\in\mathbb{R}^{N}:Q(x)=Q_{0}\\}$ denotes the set
of maximum points of $Q$, and for $\delta>0$ we let
$M_{\delta}=\\{x\in\mathbb{R}^{N}:\mathrm{dist}(x,M)\leq\delta\\}$. Also, for
a closed subset $Y$ of a metric space $X$ we denote by $\mathrm{cat_{X}}(Y)$
the Lusternik-Schnirelmann category of $Y$ with respect to $X$, i.e., the
least number of closed contractible sets in $X$ which cover $Y$.

######  Theorem 1.2.

Let $N\geq 3$, $\frac{N}{N+1}<s<\frac{N}{2}$,
$\frac{2(N+1)}{N-1}<p<\frac{2N}{N-2s}$ and consider a function $Q$ satisfying
(Q0) and (Q1). For every $\delta>0$, there exists $k(\delta)>0$ such that
$(\ref{main1})$ has at least $\mathrm{cat}_{M_{\delta}}(M)$ nontrivial
solutions for all $k>k_{\delta}$.

The proof of the above result depends on a topological arguments close to the
ones developed by Cingolani and Lazzo [12] for (1.4) (see also [13]) and based
on ideas of Benci, Cerami and Passaseo [7, 8] for problems on bounded domains.
The main point lies in the construction of two maps whose composition is
homotopic to the inclusion $M\hookrightarrow M_{\delta}$, we omit it here and
refer readers to [23, Theorem 2].

We close the introduction by some notations. For $1\leq q\leq\infty$, we write
$||\cdot||_{q}$ instead $||\cdot||_{L^{q}(\mathbb{R}^{N})}$ for the standard
norm of the Lebesgue space $L^{q}(\mathbb{R}^{N})$. In addition, for $r>0$ and
$x\in\mathbb{R}^{N}$, we denote by $B_{r}(x)$ the open ball in
$\mathbb{R}^{N}$ of radius $r$ centered at $x$, and let $B_{r}=B_{r}(0)$.

##  2\. The variational framework

###  2.1. Dual Functional

Before we compare the energy functional, we recall some properties of the dual
functional (1.10). Since $p^{\prime}<2$ and since the kernel of the operator
$\mathbf{R}^{s}$ is positive close to the origin, the geometry of the
functional $J_{\varepsilon}$ is of mountain pass type:

(2.1) $\exists\alpha>0\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{and}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\rho>0\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{such\leavevmode\nobreak\ \leavevmode\nobreak\
that}\leavevmode\nobreak\ \leavevmode\nobreak\
J_{\varepsilon}(v)\geq\alpha>0,\leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ v\in
L^{p^{\prime}}(\mathbb{R}^{N})\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{with}\leavevmode\nobreak\ \leavevmode\nobreak\
||v||_{p^{\prime}}=\rho.$

(2.2) $\exists v_{0}\in L^{p^{\prime}}(\mathbb{R}^{N})\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{such\leavevmode\nobreak\ \leavevmode\nobreak\
that}\leavevmode\nobreak\ \leavevmode\nobreak\
||v_{0}||_{p^{\prime}}>\rho\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{and}\leavevmode\nobreak\ \leavevmode\nobreak\
J_{\varepsilon}(v_{0})<0.$

As a consequence, the Nehari set associated to $J_{\varepsilon}$:

(2.3) $\mathcal{N}_{\varepsilon}:=\\{v\in
L^{p^{\prime}}(\mathbb{R}^{N})\setminus\\{0\\}:J^{\prime}_{\varepsilon}(v)v=0\\},$

is not empty. More precisely, by (2.1), the set

(2.4) $U^{+}_{\varepsilon}:=\\{v\in
L^{p^{\prime}}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}Q_{\varepsilon}^{\frac{1}{p}}v\mathbf{R}^{s}\big{(}Q_{\varepsilon}^{\frac{1}{p}}v\big{)}\mathrm{d}x>0\\}$

is not empty and for each $v\in U^{+}_{\varepsilon}$ there is a unique
$t_{v}>0$ such that $t_{v}v\in\mathcal{N}_{\varepsilon}$ holds. It is given by

(2.5)
$t_{v}^{2-p^{\prime}}=\frac{\int_{\mathbb{R}^{N}}|v|^{p^{\prime}}\mathrm{d}x}{\int_{\mathbb{R}^{N}}Q_{\varepsilon}^{\frac{1}{p}}v\mathbf{R}^{s}\big{(}Q_{\varepsilon}^{\frac{1}{p}}v\big{)}\mathrm{d}x}.$

In addition, $t_{v}$ is the unique maximum point of $t\mapsto
J_{\varepsilon}(tv)$, $t\geq 0$. Using (2.1), we obtain in particular

(2.6)
$c_{\varepsilon}:=\mathop{\mathrm{inf}}\limits_{\mathcal{N}_{\varepsilon}}\leavevmode\nobreak\
J_{\varepsilon}=\mathop{\mathrm{inf}}\limits_{v\in
U^{+}_{\varepsilon}}\leavevmode\nobreak\ J_{\varepsilon}(t_{v}v)>0.$

Moreover, for every $v\in\mathcal{N}_{\varepsilon}$ we have
$c_{\varepsilon}\leq
J_{\varepsilon}(v)=(\frac{1}{p^{\prime}}-\frac{1}{2})||v||_{p^{\prime}}^{p^{\prime}}$.
Hence, $0$ is isolated in the set $\\{v\in
L^{p^{\prime}}(\mathbb{R}^{N}):\leavevmode\nobreak\
J^{\prime}_{\varepsilon}(v_{n})\longrightarrow 0\\}$ and as a consequence, the
$C^{1}$-submanifold $\mathcal{N}_{\varepsilon}$ of
$L^{p^{\prime}}(\mathbb{R}^{N})$ is complete.

We recall that $(v_{n})_{n}\subset L^{p^{\prime}}(\mathbb{R}^{N})$ is termed a
Palais-Smale sequence, or a (PS)-sequence, for $J_{\varepsilon}$ if
$(J_{\varepsilon}(v_{n}))_{n}$ is bounded and
$J^{\prime}_{\varepsilon}(v_{n})\longrightarrow 0$ as
$n\longrightarrow\infty$. Also, for $d>0$, we say that $(v_{n})_{n}$ is a
$(\mathrm{PS})_{d}$-sequence for $J_{\varepsilon}$ if it is a
$(\mathrm{PS})$-sequence and if $J_{\varepsilon}\longrightarrow d$ as
$n\longrightarrow\infty$. The following properties hold (see [40, Sect.2]).

######  Lemma 2.1.

Let $(v_{n})_{n}\subset L^{p^{\prime}}(\mathbb{R}^{N})$ be a Palais-Smale
sequence for $J_{\varepsilon}$. Then $(v_{n})_{n}$ is bounded and there exists
$v\in L^{p^{\prime}}(\mathbb{R}^{N})$ such that
$J^{\prime}_{\varepsilon}(v)=0$ and, up to a subsequence,
$v_{n}\rightharpoonup v$ weakly in $L^{p^{\prime}}(\mathbb{R}^{N})$ and
$J_{\varepsilon}(v)\leq\mathop{\mathrm{lim\leavevmode\nobreak\
inf}}\limits_{n\longrightarrow\infty}\leavevmode\nobreak\
J_{\varepsilon}(v_{n})$.

Moreover, for every bounded and measurable set $B\subset\mathbb{R}^{N}$,
$1_{B}v_{n}\longrightarrow 1_{B}v$ strongly in
$L^{p^{\prime}}(\mathbb{R}^{N})$.

As a consequence, we obtain the following characterization of the infimum
$c_{\varepsilon}$ of $J_{\varepsilon}$ over the Nehari manifold
$\mathcal{N}_{\varepsilon}$ (see [40, Sect.4]).

######  Lemma 2.2.

(i) $c_{\varepsilon}$ coincides with the mountain pass level, i.e.,

$c_{\varepsilon}=\mathop{\mathrm{inf}}\limits_{\gamma\in\Gamma}\mathop{\mathrm{max}}\limits_{t\in[0,1]}J_{\varepsilon}(\gamma(t)),\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{where}$ $\Gamma=\\{\gamma\in
C([0,1],L^{p^{\prime}}(\mathbb{R}^{N})):\gamma(0)=0,\leavevmode\nobreak\
\mathrm{and}\leavevmode\nobreak\ J(\gamma(1))<0\\}.$

(ii) If $c_{\varepsilon}$ is attained, then
$c_{\varepsilon}=\mathrm{min}\\{J_{\varepsilon}(v):v\in
L^{p^{\prime}}(\mathbb{R}^{N})\setminus\\{0\\},\leavevmode\nobreak\
J^{\prime}_{\varepsilon}(v)=0\\}$.

(iii) If $Q_{\varepsilon}$ is constant or $\mathbb{Z}^{N}-$periodic, then
$c_{\varepsilon}$ is attained.

###  2.2. Energy Compare

Consider the functional

(2.7)
$J_{0}(v):=\frac{1}{p^{\prime}}\int_{\mathbb{R}^{N}}|v|^{p^{\prime}}dx-\frac{1}{2}\int_{\mathbb{R}^{N}}Q_{0}^{\frac{1}{p}}v\mathbf{R}^{s}(Q_{0}^{\frac{1}{p}}v)\mathrm{d}x,\leavevmode\nobreak\
\leavevmode\nobreak\ v\in L^{p^{\prime}}(\mathbb{R}^{N})$

and the corresponding Nehari manifold

(2.8) $\mathcal{N}_{0}:=\\{v\in
L^{p^{\prime}}(\mathbb{R}^{N})\setminus\\{0\\}:J^{\prime}_{0}(v)v=0\\},$

associated to the limit problem

(2.9) $(-\Delta)^{s}u-u=Q_{0}|u|^{p-2}u,\leavevmode\nobreak\
\leavevmode\nobreak\ x\in\mathbb{R}^{N}.$

Lemma 2.2 implies that the level
$c_{0}:=\mathop{\mathrm{inf}}\limits_{\mathcal{N}_{0}}J_{0}$ is attained and
coincides with the least-energy level, i.e.,

(2.10) $c_{0}=\mathrm{inf}\\{J_{0}(v):v\in
L^{p^{\prime}}(\mathbb{R}^{N}),v\neq 0\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{and}\leavevmode\nobreak\ J^{\prime}_{0}(v)=0\\}.$

Denote the set of maximum points of $Q$ by

(2.11) $M:=\\{x\in\mathbb{R}^{N}:Q(x)=Q_{0}\\}.$

It then follows from (Q0) and (Q1) that $M\neq\emptyset$. We start by studying
the projection on the Nehari manifold of truncation of of translated and
rescaled ground states of $J_{0}$. Take a cutoff function $\eta\in
C_{c}^{\infty}(\mathbb{R}^{N})$, $0\leq\eta\leq 1$, such that $\eta\equiv 1$
in $B_{1}(0)$ and $\eta\equiv 0$ in $\mathbb{R}^{N}\setminus B_{2}(0)$. For
$y\in M$, $\varepsilon>0$ we let

(2.12) $\varphi_{\varepsilon,y}(x):=\eta(\varepsilon
x-y)w(x-\varepsilon^{-1}y),$

where $w\in L^{p^{\prime}}(\mathbb{R}^{N})$ is some fixed least-energy
critical point of $J_{0}$.

######  Lemma 2.3.

There is $\varepsilon^{\ast}>0$ such that for all
$0<\varepsilon\leq\varepsilon^{\ast}$, $y\in M$, a unique
$t_{\varepsilon,y}>0$ satisfying
$t_{\varepsilon,y}\varphi_{\varepsilon,y}\in\mathcal{N}_{\varepsilon}$ exists.
Moreover,

(2.13) $\mathop{\mathrm{lim}}\limits_{\varepsilon\longrightarrow
0^{+}}J_{\varepsilon}(t_{\varepsilon,y}\varphi_{\varepsilon,y})=c_{0},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{uniformly\leavevmode\nobreak\ for\leavevmode\nobreak\ }y\in M.$

###### Proof.

Since $M$ is compact and $Q$ is continuous by assumption, we have
$Q(y+\varepsilon\cdot)\eta(\varepsilon\cdot)w\longrightarrow Q_{0}w$ in
$L^{p^{\prime}}(\mathbb{R}^{N})$ as $\varepsilon\longrightarrow 0^{+}$,
uniformly with respect to $y\in M$, Consequently, as
$\varepsilon\longrightarrow 0^{+}$,

(2.14)
$\displaystyle\int_{\mathbb{R}^{N}}Q_{\varepsilon}^{\frac{1}{p}}\varphi_{\varepsilon,y}\mathbf{R}^{s}(Q_{\varepsilon}^{\frac{1}{p}}\varphi_{\varepsilon,y})\mathrm{d}x$
$\displaystyle=\int_{\mathbb{R}^{N}}Q^{\frac{1}{p}}(y+\varepsilon
z)\eta(\varepsilon
z)w(z)\mathbf{R}^{s}(Q^{\frac{1}{p}}(y+\varepsilon\cdot)\eta(\varepsilon\cdot)w)(x)\mathrm{d}z$
$\displaystyle\longrightarrow\int_{\mathbb{R}^{N}}Q_{0}^{\frac{1}{p}}w\mathbf{R}^{s}(Q_{0}^{\frac{1}{p}}w)\mathrm{d}z=(\frac{1}{p^{\prime}}-\frac{1}{2})^{-1}c_{0}>0,$

uniformly for $y\in M$. Therefore, for all $y\in M$ and $\varepsilon>0$ small
enough, we deduce that $\varphi_{\varepsilon,y}\in U_{\varepsilon}^{+}$, this
implies the first assertion with $t_{\varepsilon,y}$ given by (2.5). In
addition, for all $y\in M$,

(2.15)
$\int_{\mathbb{R}^{N}}|\varphi_{\varepsilon,y}|^{p^{\prime}}\mathrm{d}x=\int_{\mathbb{R}^{N}}|\eta(\varepsilon
z)w(z)|^{p^{\prime}}\mathrm{d}z\longrightarrow\int_{\mathbb{R}^{N}}|w|^{p^{\prime}}\mathrm{d}z=(\frac{1}{p^{\prime}}-\frac{1}{2})^{-1}c_{0},\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\
\varepsilon\longrightarrow 0^{+}.$

As a consequence, $t_{\varepsilon,y}\longrightarrow 1$ as
$\varepsilon\longrightarrow 0^{+}$, uniformly for $y\in M$, and we obtain
$J_{\varepsilon}(t_{\varepsilon,y}\varphi_{\varepsilon,y})\longrightarrow
c_{0}$ as $\varepsilon\longrightarrow 0^{+}$, uniformly for $y\in M$. The
second assertion follows. ∎

######  Lemma 2.4.

For all $\varepsilon>0$ there holds $c_{0}\leq c_{\varepsilon}$. Moreover,
$\mathop{\mathrm{lim}}\limits_{\varepsilon\longrightarrow
0^{+}}c_{\varepsilon}=c_{0}$.

###### Proof.

Consider $v_{\varepsilon}\in\mathcal{N}_{\varepsilon}$ and set
$v_{0}:=(\frac{Q_{\varepsilon}}{Q_{0}})^{\frac{1}{p}}v_{\varepsilon}$. Notice
that $|v_{0}|\leq|v_{\varepsilon}|$ a.e. on $\mathbb{R}^{N}$. Since
$v_{\varepsilon}\in U_{\varepsilon}^{+}$, we find

(2.16)
$\int_{\mathbb{R}^{N}}Q_{0}^{\frac{1}{p}}v_{0}\mathbf{R}^{s}(Q_{0}^{\frac{1}{p}}v_{0})=\int_{\mathbb{R}^{N}}Q_{\varepsilon}^{\frac{1}{p}}v_{\varepsilon}\mathbf{R}^{s}(Q_{\varepsilon}^{\frac{1}{p}}v_{\varepsilon})>0,$

i.e., $v_{0}\in U_{0}^{+}$. By (2.5) we deduce

(2.17)
$t_{\varepsilon}^{2-p^{\prime}}=\frac{\int_{\mathbb{R}^{N}}|v_{0}|^{p^{\prime}}\mathrm{d}x}{\int_{\mathbb{R}^{N}}Q_{0}^{\frac{1}{p}}v_{0}\mathbf{R}^{s}(Q_{0}^{\frac{1}{p}}v_{0})\mathrm{d}x}\leq\frac{\int_{\mathbb{R}^{N}}|v_{\varepsilon}|^{p^{\prime}}\mathrm{d}x}{\int_{\mathbb{R}^{N}}Q_{0}^{\frac{1}{p}}v_{\varepsilon}\mathbf{R}^{s}(Q_{\varepsilon}^{\frac{1}{p}}v_{\varepsilon})\mathrm{d}x}=1.$

This implies that $t_{\varepsilon}v_{0}\in\mathcal{N}_{0}$. Follow the
definition of the dual functional, we yield that

(2.18) $c_{0}\leq
J_{0}(t_{\varepsilon}v_{0})=(\frac{1}{p^{\prime}}-\frac{1}{2})t_{\varepsilon}^{p^{\prime}}\int_{\mathbb{R}^{N}}|v_{0}|^{p^{\prime}}\mathrm{d}x\leq(\frac{1}{p^{\prime}}-\frac{1}{2})\int_{\mathbb{R}^{N}}|v_{\varepsilon}|^{p^{\prime}}\mathrm{d}x=J_{\varepsilon}(v_{\varepsilon}).$

Since $v_{\varepsilon}\in\mathcal{N}_{\varepsilon}$ was arbitrarily chosen, we
conclude that
$c_{0}\leq\mathop{\mathrm{inf}}\limits_{\mathcal{N}_{\varepsilon}}=c_{\varepsilon}$.
On the other hand, Lemma 2.3 gives for $y\in M$, $c_{\varepsilon}\leq
J_{\varepsilon}(t_{\varepsilon,y}\varphi_{\varepsilon,y})\longrightarrow
c_{0}$ as $\varepsilon\longrightarrow 0^{+}$. Hence,
$\mathop{\mathrm{lim}}\limits_{\varepsilon\longrightarrow
0^{+}}c_{\varepsilon}=c_{0}$, as claimed. ∎

Now, consider the energy functional
$J_{\infty}:L^{p^{\prime}}(\mathbb{R}^{N})\longrightarrow\mathbb{R}$ given by

(2.19)
$J_{\infty}(v)=\frac{1}{p^{\prime}}\int_{\mathbb{R}^{N}}|v|^{p^{\prime}}\mathrm{d}x-\frac{1}{2}\int_{\mathbb{R}^{N}}Q_{\infty}^{\frac{1}{p}}v\mathbf{R}^{s}\big{(}Q_{\infty}^{\frac{1}{p}}v\big{)}\mathrm{d}x,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ v\in
L^{p^{\prime}}(\mathbb{R}^{N}).$

The corresponding Nehari manifold

(2.20) $\mathcal{N}_{\infty}:=\\{v\in
L^{p^{\prime}}(\mathbb{R}^{N})\setminus\\{0\\}:J^{\prime}_{\infty}(v)v=0\\},$

has the same structure as $\mathcal{N}_{\varepsilon}$ and, since $Q_{\infty}$
is constant, Lemma 2.2 implies that
$c_{\infty}:=\mathop{\mathrm{inf}}\limits_{\mathcal{N}_{\infty}}J_{\infty}$ is
attained and coincides with the least energy level for nontrivial critical
points of $J_{\infty}$.

######  Proposition 2.5.

There is $\varepsilon_{0}>0$ such that for all $\varepsilon<\varepsilon_{0}$,
$c_{\varepsilon}<c_{\infty}$.

###### Proof.

By Lemma 2.4 and Condition (Q1), there is $\varepsilon_{0}>0$ such that
$c_{\varepsilon}<c_{\infty}$ for all $0<\varepsilon<\varepsilon_{0}$. ∎

##  3\. Exsitence of dual ground ststes

In this section, we proof the $(PS)_{c_{\varepsilon}}$ condition for the
energy functional $J_{\varepsilon}$. Since the resolvent Helmholtz operator is
not positive definite, we need to handle the nonlocal interaction between
functions with disjoint support.

######  Lemma 3.1.

there exists a constant $C=C(N,p)>0$ such that for any $R>0$, $r\geq 1$ and
$u,v\in L^{p^{\prime}}(\mathbb{R}^{N})$ with $\mathrm{supp}(u)\subset B_{R}$
and $\mathrm{supp}(v)\subset\mathbb{R}^{N}\setminus B_{R+r}$,

(3.1) $\big{|}\int_{\mathbb{R}^{N}}u\mathbf{R}^{s}v\leavevmode\nobreak\
\mathrm{d}x\big{|}\leq
Cr^{-\lambda_{p}}||u||_{p^{\prime}}||v||_{p^{\prime}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{where}\leavevmode\nobreak\
\lambda_{p}=\frac{N-1}{2}-\frac{N+1}{p}.$

###### Proof.

Let $\mathcal{R}^{s}$ denote the resolvent of the Fractional Helmholtz
eqaution, which is given by the convolution with the kernel $K(x)$ (see [29]
and [40] for more details). Since $\mathbf{R}^{s}$ is the real part of
$\mathcal{R}^{s}$ and since $u,v$ are real-valued, we prove the lemma for the
nonlocal term $\int_{\mathbb{R}^{N}}v\mathcal{R}^{s}u\mathrm{d}x$. By density,
it suffices to prove the estimate for Schwartz function.

Let $M_{R+r}:=\mathbb{R}^{N}\setminus B_{R+r}$ and let
$u,v\in\mathcal{S}(\mathbb{R}^{N})$ be such that $\mathrm{supp}(u)\subset
B_{R}$ and $\mathrm{supp}(v)\subset M_{R+r}$. The symmetry of the operator
$\mathcal{R}^{s}$ and Hölder’s inequality gives

(3.2)
$\big{|}\int_{\mathbb{R}^{N}}u\mathcal{R}^{s}v\mathrm{d}x\big{|}=\big{|}\int_{\mathbb{R}^{N}}v\mathcal{R}^{s}u\mathrm{d}x\big{|}\leq||v||_{p^{\prime}}||K\ast
u||_{L^{p}(M_{R+r})}.$

This reduces us to estimating the second factor on the right-hand side. For
this, we decomposition $K$ as follow. Fix $\psi\in\mathcal{S}(\mathbb{R}^{N})$
such that $\widehat{\psi}\in\mathcal{C}_{c}^{\infty}(\mathbb{R}^{N})$ is
radial, $0\leq\widehat{\psi}\leq 1$, $\widehat{\psi}(\xi)=1$ for
$||\xi|-1|\leq\frac{1}{6}$ and $\widehat{\psi}(\xi)=0$ for
$||\xi|-1|\geq\frac{1}{4}$. Writing $K=K_{1}+K_{2}$ with

(3.3) $K_{1}:=(2\pi)^{-\frac{N}{2}}(\psi\ast K),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
K_{2}:=K-K_{1}.$

It follows from the estimate in [40] that

(3.4) $|K_{1}(x)|\leq C_{0}(1+|x|)^{\frac{1-N}{2}}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\ x\in\mathbb{R}^{N},$
(3.5) $\mathrm{and}\leavevmode\nobreak\ \leavevmode\nobreak\ |K_{2}(x)|\leq
C_{0}|x|^{-N}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{for}\leavevmode\nobreak\ x\neq 0.$

Since the support of $u$ is contained in $B_{R}$, we find

(3.6) $\displaystyle||K_{2}\ast u||_{L^{p}(M_{R+r})}$
$\displaystyle\leq\big{[}\int_{|x|\geq R+r}\big{(}\int_{|y|\leq
R}|K_{2}(x-y)||u(y)|\mathrm{d}y\big{)}^{p}\mathrm{d}x\big{]}^{\frac{1}{p}}$
$\displaystyle\leq\big{[}\int_{\mathbb{R}^{N}}\big{(}\int_{|x-y|\geq
r}|K_{2}(x-y)||u(y)|\mathrm{d}y\big{)}^{p}\mathrm{d}x\big{]}^{\frac{1}{p}}$
$\displaystyle=||(1_{M_{r}}|K_{2}|\ast|u|)||_{p}\leq||1_{M_{r}}K_{2}||_{\frac{p}{2}}||u||_{p^{\prime}}.$

Moreover, (3.5) gives

(3.7) $||1_{M_{r}}K_{2}||_{\frac{p}{2}}\leq
C_{0}\big{(}\omega_{N}\int_{r}^{\infty}s^{N-1-\frac{Np}{2}}ds\big{)}^{\frac{2}{p}}\leq
Cr^{-\frac{N(p-2)}{p}}\leq Cr^{-\lambda_{p}},$

since $r\geq 1$, and therefore

(3.8) $||K_{2}\ast u||_{L^{p}(M_{R+r})}\leq
Cr^{-\lambda_{p}}||u||_{p^{\prime}}.$

It remains to prove the estimate for $K_{1}$. Fix a radial function
$K\in\mathcal{S}(\mathbb{R}^{N})$ such that
$\widehat{K}\in\mathcal{C}_{c}^{\infty}(\mathbb{R}^{N})$ is radial,
$0\leq\widehat{K}\leq 1$, $\widehat{K}(\xi)=1$ for $||\xi|-1|\leq\frac{1}{4}$
and $\widehat{K}(\xi)=0$ for $||\xi|-1|\geq\frac{1}{2}$, let $\tilde{u}:=K\ast
u\in\mathcal{S}(\mathbb{R}^{N})$, we then have $K_{1}\ast
u=(2\pi)^{-\frac{N}{2}}(K_{1}\ast\tilde{u})$, since
$\widehat{K_{1}}\widehat{K}=\widehat{K_{1}}$ by construction. Now write

(3.9)
$K_{1}\ast\tilde{u}=\big{[}1_{B_{\frac{r}{2}}}K_{1}\big{]}\ast\tilde{u}+\big{[}1_{M_{\frac{r}{2}}}K_{1}\big{]}\ast\tilde{u}$

and let $g_{r}:=\big{[}1_{B_{\frac{r}{2}}}K_{1}\big{]}\ast K$. Since
$\mathrm{supp}(u)\subset B_{R}$, we find as above

(3.10) $\displaystyle||1_{M_{r}}g_{r}||_{\frac{p}{2}}^{\frac{p}{2}}$
$\displaystyle\leq C_{0}^{\frac{p}{2}}\int_{|x|\geq
r}\big{(}\int_{|y|\leq\frac{r}{2}}|K(x-y)|\mathrm{d}y\big{)}^{\frac{p}{2}}\mathrm{d}x$
$\displaystyle\leq C\int_{|x|\geq
r}\big{(}\int_{|y|\leq\frac{r}{2}}|x-y|^{-m}\mathrm{d}y\big{)}^{\frac{p}{2}}\mathrm{d}x\leq
C|B_{\frac{r}{2}}|^{\frac{p}{2}}\int_{|x|\geq
r}(|x|-\frac{r}{2})^{-\frac{mp}{2}}dx$
$\displaystyle=Cr^{\frac{(N-m)p}{2}+N}\int_{|z|\geq
1}(|z|-\frac{1}{2})^{-\frac{mp}{2}}\mathrm{d}z=Cr^{\frac{(N-m)p}{2}+N},$

where $C$ is independent of $r$ and where $m$ may be fixed so large that
$\frac{(m-N)p}{2}-N\geq\lambda_{p}$. As a consequence of [20, Proposition
3.3], we have moreover

(3.11)
$||\big{[}1_{M_{\frac{r}{2}}}K_{1}\big{]}\ast\tilde{u}||_{L^{p}(M_{R+r})}\leq||\big{[}1_{M_{\frac{r}{2}}}K_{1}\ast\tilde{u}\big{]}||_{p}\leq
Cr^{-\lambda_{p}}||\tilde{u}||_{p^{\prime}}\leq
Cr^{-\lambda_{p}}||u||_{p^{\prime}}$

and we conclude that

(3.12) $||K_{1}\ast u||_{L^{p}(M_{R+r})}\leq
Cr^{-\lambda_{p}}||u||_{p^{\prime}}.$

Combining (3.2), (3.12) and (3.8) yields the claim. ∎

######  Lemma 3.2.

Let $\varepsilon>0$ and assume $Q_{\infty}>0$ and
$c_{\varepsilon}<c_{\infty}$. Then $J_{\varepsilon}$ satisfies the Palais-
Smale condition on $\mathcal{N}_{\varepsilon}$ at level below $c_{\infty}$,
i.e., every sequence $(v_{n})_{n}\subset\mathcal{N}_{\varepsilon}$ such that
$J_{\varepsilon}(v_{n})\longrightarrow d<c_{\infty}$ and
$(J_{\varepsilon}|{\mathcal{N}_{\varepsilon}})^{\prime}(v_{n})\longrightarrow
0$ as $n\longrightarrow\infty$ has a convergent subsequence.

###### Proof.

Since $c_{\varepsilon}<c_{\infty}$, the set
$\\{v\in\mathcal{N}_{\varepsilon}:J_{\varepsilon}(v)<c_{\infty}\\}$ is not
empty. If $d<c_{\varepsilon}$, all is done. It remains to consider the case
$c_{\varepsilon}\leq d<c_{\infty}$. Let $(v_{n})_{n}$ be a
$(\mathrm{PS})_{d}$-sequence for
$J_{\varepsilon}|{\mathcal{N}_{\varepsilon}}$. Since
$\mathcal{N}_{\varepsilon}$ is a natural constraint and a $C^{1}-$manifold, we
find that $(v_{n})_{n}$ is a $(PS)_{d}$-sequence for the unconstrained
functional $J_{\varepsilon}$. Using Lemma 2.1, we obtain that (up to a
subsequence) $v_{n}\rightharpoonup v$ and $1_{B_{R}}v_{n}\longrightarrow
1_{B_{R}}v$ in $L^{p^{\prime}}(\mathbb{R}^{N})$ for all $R>0$, where $v\in
L^{p^{\prime}}(\mathbb{R}^{N})$ is a critical point of $J_{\varepsilon}$ with
$J_{\varepsilon}(v)\leq d$. In order to conclude that $v_{n}\longrightarrow v$
strongly in $L^{p^{\prime}}(\mathbb{R}^{N})$, it suffices to show that

(3.13) $\forall\leavevmode\nobreak\ \zeta>0,\leavevmode\nobreak\
\leavevmode\nobreak\ \exists\leavevmode\nobreak\ R>0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{such\leavevmode\nobreak\
that}\leavevmode\nobreak\ \leavevmode\nobreak\
\int_{|x|>R}|v_{n}|^{p^{\prime}}dx<\zeta,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \forall\leavevmode\nobreak\
\leavevmode\nobreak\ n.$

We prove (3.13) by contradiction. Assuming that there exists a subsequence
$(n_{n_{k}})_{k}$ and $\zeta_{0}>0$ such that

(3.14) $\int_{|x|>k}|v_{n_{k}}|^{p^{\prime}}\geq\zeta_{0},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \forall\leavevmode\nobreak\ k.$

Firstly, for a annular region, we claim that

(3.15) $\forall\leavevmode\nobreak\ \eta>0\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ R>0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \exists\leavevmode\nobreak\ r>R\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{such\leavevmode\nobreak\
that}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathop{\mathrm{lim\leavevmode\nobreak\
inf}}\limits_{n\longrightarrow\infty}\int_{r<|x|<2r}|v_{n}|^{p^{\prime}}\mathrm{d}x<\eta.$

Otherwise, for every $m>R_{0}$, $n_{0}=n_{0}(m)$, we can find $\eta_{0},R_{0}$
such that $\int_{m<|x|<2m}|v_{n}|^{p^{\prime}}\mathrm{d}x\geq\eta_{0}$ for all
$n\geq n_{0}$. Without loss of generality, we assume that $n_{0}(m+1)\geq
n_{0}(m)$ for all $m$. Hence, for every $l\in\mathbb{N}$ there is
$N_{0}=N_{0}(l)$ such that

(3.16)
$\int_{\mathbb{R}^{N}}|v_{n}|^{p^{\prime}}\mathrm{d}x\geq\mathop{\sum}\limits_{k=0}^{l-1}\int_{2^{k}([R_{0}]+1)<|x|<2^{k+1}([R_{0}]+1)}|v_{n}|^{p^{\prime}}\mathrm{d}x\geq
l\eta_{0},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ n\geq N_{0}.$

Letting $l\longrightarrow\infty$, we obtain a contradiction to the fact that
$(v_{n})_{n}$ is bounded and this gives (3.15).

Now fix $0<\eta<\mathrm{min}\\{1,(\frac{\zeta_{0}}{3C_{1}})^{p^{\prime}}\\}$,
where
$C_{1}=2C(N,p)||Q||_{\infty}^{\frac{2}{p}}\mathrm{max}\\{1,\mathop{\mathrm{sup}}\limits_{k\in\mathbb{N}}||v_{n_{k}}||_{p^{\prime}}^{2}\\}$,
the constant $C(N,p)$ being chosen such that Lemma 3.1 holds and
$||\mathbf{R}^{s}v||_{p}\leq C(N,p)||v||_{p^{\prime}}$ for all $u\in
L^{p^{\prime}}(\mathbb{R}^{N})$. By definition of $Q_{\infty}$ and since
$\varepsilon>0$ is fixed, there exists $R(\eta)>0$ such that

(3.17) $Q_{\varepsilon}\leq Q_{\infty}+\eta\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\
all}\leavevmode\nobreak\ |x|\geq R(\eta).$

Also, from (3.15), we can find
$r>\mathrm{max}\\{R(\eta),\eta^{-\frac{1}{\lambda_{p}}}\\}$ and a subsequence,
still denoted by $(v_{n_{k}})_{k}$, such that

(3.18)
$\int_{r<|x|<2r}|v_{n_{k}}|^{p^{\prime}}\mathrm{d}x<\eta\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{for\leavevmode\nobreak\ all}\leavevmode\nobreak\ k.$

Setting $w_{n_{k}}:=1_{\\{|x|\geq 2r\\}}v_{n_{k}}$ we can write for all $k$

(3.19)
$\displaystyle\big{|}J^{\prime}_{\varepsilon}(v_{n_{k}})w_{n_{k}}-J^{\prime}_{\varepsilon}(w_{n_{k}})w_{n_{k}}\big{|}$
$\displaystyle=\big{|}\int_{|x|<r}Q_{\varepsilon}^{\frac{1}{p}}v_{n_{k}}\mathbf{R}^{s}\big{(}Q_{\varepsilon}^{\frac{1}{p}}w_{n_{k}}\big{)}\mathrm{d}x+\int_{r<|x|<2r}Q_{\varepsilon}^{\frac{1}{p}}v_{n_{k}}\mathbf{R}^{s}\big{(}Q_{\varepsilon}^{\frac{1}{p}}w_{n_{k}}\big{)}\mathrm{d}x\big{|}$
$\displaystyle\leq
C(N,p)r^{-\lambda_{p}}||Q||_{\infty}^{\frac{2}{p}}||v_{n_{k}}||_{p^{\prime}}^{2}+C(N,p)||Q||_{\infty}^{\frac{2}{p}}||v_{n_{k}}||_{p^{\prime}}\big{(}\int_{r<|x|<2r}|v_{n_{k}}|^{p^{\prime}}\mathrm{d}x\big{)}^{\frac{1}{p^{\prime}}}$
$\displaystyle\leq C_{1}\eta^{\frac{1}{p^{\prime}}},$

using Lemma 3.1. In addition, by (3.14) and the definition of $w_{n_{k}}$,
there holds

(3.20)
$\int_{\mathbb{R}^{N}}|w_{n_{k}}|^{p^{\prime}}\mathrm{d}x\geq\zeta_{0}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\
all}\leavevmode\nobreak\ k\geq 2r.$

Recalling our choice of $\eta$, we know that
$C_{1}\eta^{\frac{1}{p^{\prime}}}<\frac{\zeta_{0}}{3}$, and we find some
$k_{0}=k_{0}(r,\eta,\zeta_{0})\geq 2r$ such that

(3.21)
$\displaystyle\int_{\mathbb{R}^{N}}Q_{\varepsilon}^{\frac{1}{p}}w_{n_{k}}\mathbf{R}^{s}\big{(}Q_{\varepsilon}^{\frac{1}{p}}w_{n_{k}}\big{)}\mathrm{d}x$
$\displaystyle=\int_{\mathbb{R}^{N}}|w_{n_{k}}|^{p^{\prime}}\mathrm{d}x-J^{\prime}_{\varepsilon}(v_{n_{k}})w_{n_{k}}+[J^{\prime}_{\varepsilon}(v_{n_{k}})w_{n_{k}}-J^{\prime}_{\varepsilon}(w_{n_{k}})w_{n_{k}}]$
$\displaystyle\geq\int_{\mathbb{R}^{N}}|w_{n_{k}}|^{p^{\prime}}\mathrm{d}x-|J^{\prime}_{\varepsilon}(v_{n_{k}})w_{n_{k}}|-C_{1}\eta^{\frac{1}{p^{\prime}}}\geq\frac{\zeta_{0}}{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\
\leavevmode\nobreak\ all}\leavevmode\nobreak\ k\geq k_{0},$

since $J^{\prime}_{\varepsilon}(v_{n_{k}})w_{n_{k}}\longrightarrow 0$ as
$k\longrightarrow\infty$.

For $k\geq k_{0}$, let now
$\tilde{w}_{k}:=(\frac{Q_{\varepsilon}}{Q_{\infty}})^{\frac{1}{p}}w_{n_{k}}$
and notice that
$|\tilde{w}_{k}|\leq(1+\frac{\eta}{Q_{\infty}})^{\frac{1}{p}}|w_{n_{k}}|$. In
view of (3.21), there is $t_{k}^{\infty}>0$, for which
$t^{\infty}_{k}\tilde{w}_{k}\in\mathcal{N}_{\infty}$ and there holds

(3.22) $\displaystyle(t_{k}^{\infty})^{2-p^{\prime}}$
$\displaystyle\leq\frac{(1+\frac{\eta}{Q_{\infty}})^{p^{\prime}-1}\int_{\mathbb{R}^{N}}|w_{n_{k}}|^{p^{\prime}}\mathrm{d}x}{\int_{\mathbb{R}^{N}}Q_{\varepsilon}^{\frac{1}{p}}w_{n_{k}}\mathbf{R}^{s}(Q_{\varepsilon}^{\frac{1}{p}}w_{n_{k}})\mathrm{d}x}$
$\displaystyle\leq(1+\frac{\eta}{Q_{\infty}})^{p^{\prime}-1}\big{(}1+\frac{|J^{\prime}_{\varepsilon}(w_{n_{k}})w_{n_{k}}|+C_{1}\eta^{\frac{1}{p^{\prime}}}}{\int_{\mathbb{R}^{N}}Q_{\varepsilon}^{\frac{1}{p}}w_{n_{k}}\mathbf{R}^{s}(Q_{\varepsilon}^{\frac{1}{p}}w_{n_{k}})\mathrm{d}x}\big{)}$
$\displaystyle\leq(1+\frac{\eta}{Q_{\infty}})^{p^{\prime}-1}\big{(}1+\frac{2|J^{\prime}_{\varepsilon}(w_{n_{k}})w_{n_{k}}+2C_{1}\eta^{\frac{1}{p^{\prime}}}|}{\zeta_{0}}\big{)}.$

Since $v_{n_{k}}\in\mathcal{N}_{\varepsilon}$, there holds

(3.23)
$\int_{\mathbb{R}^{N}}|w_{n_{k}}|^{p^{\prime}}\mathrm{d}x\leq\int_{\mathbb{R}^{N}}|v_{n_{k}}|^{p^{\prime}}\mathrm{d}x=(\frac{1}{p^{\prime}}-\frac{1}{2})^{-1}J_{\varepsilon}(v_{n_{k}}).$

Consequently, for all $k\geq k_{0}$,

(3.24) $\displaystyle c_{\infty}$ $\displaystyle\leq
J_{\infty}(t_{k}^{\infty}\tilde{w}_{k})$
$\displaystyle\leq(\frac{1}{p^{\prime}}-\frac{1}{2})(t_{k}^{\infty})^{p^{\prime}}(1+\frac{\eta}{Q_{\infty}})^{p^{\prime}-1}\int_{\mathbb{R}^{N}}|w_{n_{k}}|^{p^{\prime}}\mathrm{d}x$
$\displaystyle\leq(1+\frac{\eta}{Q_{\infty}})^{\frac{2(p^{\prime}-1)}{2-p^{\prime}}}\big{(}1+\frac{2|J^{\prime}_{\varepsilon}(w_{n_{k}})w_{n_{k}}+2C_{1}\eta^{\frac{1}{p^{\prime}}}|}{\zeta_{0}}\big{)}^{\frac{p^{\prime}}{2-p^{\prime}}}J_{\varepsilon}(v_{n_{k}}).$

Letting $k\longrightarrow\infty$, we find

(3.25)
$c_{\infty}\leq(1+\frac{\eta}{Q_{\infty}})^{\frac{2(p^{\prime}-1)}{2-p^{\prime}}}\big{(}1+\frac{2C_{1}\eta^{\frac{1}{p^{\prime}}}}{\zeta_{0}}\big{)}^{\frac{p^{\prime}}{2-p^{\prime}}}d,$

and letting $\eta\longrightarrow 0$ we obtain

(3.26) $c_{\infty}\leq d,$

which contradicts the assumption $d<\infty$ and prove (3.13). From this, we
conclude the strong convergence $v_{n}\longrightarrow v$ in
$L^{p^{\prime}}(\mathbb{R}^{N})$ and the assertion follows. ∎

###### Proof of Theorem 1.1 (i).

Fix $\varepsilon_{0}$ in Proposition (2.5). For any
$\varepsilon\leq\varepsilon_{0}$, using the fact that
$\mathcal{N}_{\varepsilon}$ is a $C^{1}-$submanifold of
$L^{p^{\prime}}(\mathbb{R}^{N})$, we obtain from Ekeland’s variational
principle the existence of Palais-Smale sequence for $J_{\varepsilon}$ on
$\mathcal{N}_{\varepsilon}$, at level $c_{\varepsilon}$, and by Lemma 3.2,
$c_{\varepsilon}$ is attained. ∎

##  4\. Concentration of dual ground states

To show the concentration behaviour of the solutions of (1.1), we first prove
a representation lemma for the Palais-Smale sequences of the functional
$J_{\varepsilon}$, which is in the spirit of and Benci and Cerami [6]. A
crucial ingredient related to the nonlocal quadratic part of the energy
functional is the nonvanishing theory proved in [40, Sect.4]. For simplicity,
we drop the subscript $\varepsilon$.

######  Lemma 4.1.

Suppose $Q\equiv Q(0)>0$ on $\mathbb{R}^{N}$. Consider for some $d>0$ a
$(PS)_{d}$-sequence $(v_{n})_{n}\subset L^{p^{\prime}}(\mathbb{R}^{N})$ for
$J$. Then there is an integer $m\geq 1$, critical points $w^{(1)},...,w^{(m)}$
of $J$ and sequence
$(x_{n}^{(1)})_{n},....,(x_{n}^{(m)})_{n}\subset\mathbb{R}^{N}$ such that (up
to a subsequence)

(4.1) $\left\\{\begin{aligned}
&||v_{n}-\mathop{\sum}\limits_{j=1}^{m}w^{j}(\cdot-
x_{n}^{j})||_{p^{\prime}}\longrightarrow 0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\
n\longrightarrow\infty,\\\
&|x_{n}^{(i)}-x_{n}^{(j)}|\longrightarrow\infty\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\
n\longrightarrow\infty,\leavevmode\nobreak\ \mathrm{if}\leavevmode\nobreak\
i\neq j,\\\ &\mathop{\sum}\limits_{j=1}^{m}J(w^{(j)})=d.\\\
\end{aligned}\right.$

###### Proof.

For any bounded $(\mathrm{PS})_{d}$-sequenc $(v_{n})_{n}$, we have

(4.2)
$\mathop{\mathrm{lim}}\limits_{n\longrightarrow\infty}\int_{\mathbb{R}^{N}}Q^{\frac{1}{p}}v_{n}\mathbf{R}^{s}\big{(}Q^{\frac{1}{p}}v_{n}\big{)}\mathrm{d}x=\frac{2p^{\prime}}{2-p^{\prime}}\mathop{\mathrm{lim}}\limits_{n\longrightarrow\infty}\big{[}J(v_{n})-\frac{1}{p^{\prime}}J^{\prime}(v_{n})v_{n}\big{]}=\frac{2p^{\prime}d}{2-p^{\prime}}>0.$

It then follows from the nonvanishing theorem [40, Theorem 4.1] that there are
$R,\zeta>0$ and a sequence $(x_{n}^{(1)})_{n}$ such that, up to a subsequence,

(4.3)
$\int_{B_{r}(x_{n}^{(1)})}|v_{n}|^{p^{\prime}}\mathrm{d}x\geq\zeta>0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\
all}\leavevmode\nobreak\ n.$

Setting $v_{n}^{(1)}=v_{n}(\cdot+x_{n}^{(1)})$, then by the invariance of the
energy functional, $(v_{n}^{1})_{n}$ is also a $(\mathrm{PS})_{d}$-sequence
for $J$. By Lemma 2.1, going to a further subsequence, we may assume
$v_{n}^{(1)}\rightharpoonup w^{(1)}$ weakly,
$1_{B_{R}}v_{n}^{(1)}\longrightarrow 1_{B_{R}}w^{(1)}$ strongly in
$L^{p^{\prime}}(\mathbb{R}^{N})$, and
$J(w^{(1)})\leq\mathop{\mathrm{lim}}\limits_{n\longrightarrow\infty}J(v_{n}^{(1)})=d$.
These properties and the definition of $v_{n}^{(1)}$ imply that $w^{(1)}$ is a
nontrivial critical point of $J$.

If $J(w^{(1)})=d$, we obtain

(4.4)
$\displaystyle\big{(}\frac{1}{p^{\prime}}-\frac{1}{2}\big{)}||w^{(1)}||_{p^{\prime}}^{p^{\prime}}$
$\displaystyle=J(w^{(1)})-\frac{1}{2}J^{\prime}(w^{(1)})w^{(1)}$
$\displaystyle=d=\mathop{\mathrm{lim}}\limits_{n\longrightarrow\infty}\big{[}J(v_{n})-\frac{1}{2}J^{\prime}(v_{n})v_{n}\big{]}=\big{(}\frac{1}{p^{\prime}}-\frac{1}{2}\big{)}\mathop{\mathrm{lim}}\limits_{n\longrightarrow\infty}||v_{n}||_{p^{\prime}}^{p^{\prime}},$

i.e., $v_{n}^{(1)}\longrightarrow w^{(1)}$ strongly in
$L^{p^{\prime}}(\mathbb{R}^{N})$, then the lemma is proved.

Otherwise, $J(w^{(1)})<d$ and we set $v_{n}^{(2)}=v_{n}^{(1)}-w^{(1)}$. The
weak convergence $v_{n}^{(1)}\rightharpoonup w^{(1)}$ then implies

(4.5)
$\displaystyle\int_{\mathbb{R}^{N}}Q^{\frac{1}{p}}v_{n}^{(2)}\mathbf{R}^{s}\big{(}Q^{\frac{1}{p}}v_{n}^{(2)}\big{)}\mathrm{d}x$
$\displaystyle=\int_{\mathbb{R}^{N}}Q^{\frac{1}{p}}v_{n}^{(1)}\mathbf{R}^{s}\big{(}Q^{\frac{1}{p}}v_{n}^{(1)}\big{)}\mathrm{d}x$
$\displaystyle-\int_{\mathbb{R}^{N}}Q^{\frac{1}{p}}w_{n}^{(1)}\mathbf{R}^{s}\big{(}Q^{\frac{1}{p}}w_{n}^{(1)}\big{)}\mathrm{d}x+o(1),$

as $n\longrightarrow\infty$. Moreover, by the Brézis-Lieb Lemma [9],

(4.6)
$\int_{\mathbb{R}^{N}}|v_{n}^{(2)}|^{p^{\prime}}\mathrm{d}x=\int_{\mathbb{R}^{N}}|v_{n}^{(1)}|^{p^{\prime}}\mathrm{d}x-\int_{\mathbb{R}^{N}}|w^{(1)}|^{p^{\prime}}\mathrm{d}x+o(1),\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ n\longrightarrow\infty.$

These properties and the translation invariance of $J$ together give

(4.7)
$J(v_{n}^{(2)})=J(v_{n}^{(1)})-J(w^{(1)})+o(1)=d-J(w^{(1)})+o(1),\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ n\longrightarrow\infty.$

Since by Lemma 2.1, $1_{B_{r}}v_{n}^{(1)}\longrightarrow 1_{B_{r}}w^{(1)}$
strongly in $L^{p^{\prime}}(\mathbb{R}^{N})$ for all $r>0$, we find

(4.8)
$1_{B_{r}}|v_{n}^{(2)}|^{p^{\prime}-2}v_{n}^{(2)}-1_{B_{r}}|v_{n}^{(1)}|^{p^{\prime}-2}v_{n}^{(1)}+1_{B_{r}}|w^{(1)}|^{p^{\prime}-2}w^{(1)}\longrightarrow
0\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{in}\leavevmode\nobreak\
L^{p}(\mathbb{R}^{N}),\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{as}\leavevmode\nobreak\ n\longrightarrow\infty.$

On the other hand, since $||a|^{q-1}a-|b|^{q-1}b|\leq 2^{1-q}|a-b|^{q}$ for
all $a,b\in\mathbb{R}$ and $0<q<1$, it follows that

(4.9) $\int_{\mathbb{R}^{N}\setminus
B_{r}}\big{|}|v_{n}^{(2)}|^{p^{\prime}-2}v_{n}^{(2)}-|v_{n}^{(1)}|^{p^{\prime}-2}v_{n}^{(1)}\big{|}^{p}\mathrm{d}x\leq
2^{(2-p^{\prime})p}\int_{\mathbb{R}^{N}\setminus
B_{r}}|w^{(1)}|^{p^{\prime}}\mathrm{d}x\longrightarrow 0,$

as $n\longrightarrow\infty$, uniformly in $n$. The both estimates then give
the strong convergence

(4.10)
$|v_{n}^{(2)}|^{p^{\prime}-2}v_{n}^{(2)}-|v_{n}^{(1)}|^{p^{\prime}-2}v_{n}^{(1)}+|w^{(1)}|^{p^{\prime}-2}w^{(1)}\longrightarrow
0\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{in}\leavevmode\nobreak\
L^{p}(\mathbb{R}^{N}),\leavevmode\nobreak\ \leavevmode\nobreak\
\mathrm{as}\leavevmode\nobreak\ n\longrightarrow\infty,$

and therefore,

(4.11)
$J^{\prime}(v_{n}^{(2)})=J^{\prime}(v_{n}^{(1)})-J^{\prime}(w^{(1)})+o(1),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\
n\longrightarrow\infty.$

This implies that $(v_{n}^{(2)})_{n}$ is a (PS)-sequence for $J$ at level
$d-J(w^{(1)})>0$. Thus, the nonvanishing theorem again gives the existence of
$R_{1}$, $\zeta_{1}>0$ and of a sequence $(y_{n})_{n}\subset\mathbb{R}^{N}$
such that, going to a subsequence

(4.12)
$\int_{B_{R_{1}}(y_{n})}|v_{n}^{(2)}|^{p^{\prime}}\mathrm{d}x\geq\zeta_{1}>0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{for\leavevmode\nobreak\
all}\leavevmode\nobreak\ n.$

By Lemma 2.1, there is a critical point $w^{(2)}$ of $J$ such that (taking a
further subsequence) $v_{n}^{(2)}(\cdot+y_{n})\rightharpoonup w^{(2)}$ weakly
and $1_{B}v_{n}^{(2)}(\cdot+y_{n})\longrightarrow 1_{B}w^{(2)}$ strongly in
$L^{p^{\prime}}(\mathbb{R}^{N})$, for all bounded and measurable set
$B\subset\mathbb{R}^{N}$. In particular, $w^{(2)}\neq 0$ and since
$v_{n}^{(2)}\rightharpoonup 0$, we see that $|y_{n}|\longrightarrow\infty$ as
$n\longrightarrow\infty$.

Setting $x_{n}^{(2)}=x_{n}^{(1)}=y_{n}$, we obtain
$|x_{n}^{(2)}-x_{n}^{(1)}|\longrightarrow\infty$ as $n\longrightarrow\infty$,
and

(4.13)
$v_{n}-\big{(}w^{(1)}(\cdot+x_{n}^{(1)})+w_{n}^{(2)}(\cdot+x_{n}^{(2)})\big{)}=v_{n}^{(2)}(\cdot+y_{n}-x_{n}^{(2)})-w^{(2)}(\cdot-
x_{n}^{(2)})\rightharpoonup 0,$

weakly in $L^{p^{\prime}}(\mathbb{R}^{N})$. In addition, the same argument as
before show that

(4.14) $J(w^{(2)})\leq\mathop{\mathrm{lim\leavevmode\nobreak\
inf}}\limits_{n\longrightarrow\infty}J(v_{n}^{(2)})=d-J(w^{(1)})$

with equality if and only $v_{n}^{(2)}(\cdot+y_{n})\longrightarrow w^{(2)}$
strongly in $L^{p^{\prime}}(\mathbb{R}^{N})$. If the inequality is strict, we
can iterate the procedure. Since for every nontrivial critical point $w$ of
$J$ we have $J(w)\geq c=\mathop{\mathrm{inf}}\limits_{\mathcal{N}}J>0$, the
iterate has to stop after finitely many steps, and we obtain the desired
result. ∎

######  Proposition 4.2.

Let $(\varepsilon_{n})_{n}\subset(0,\infty)$ satisfy
$\varepsilon_{n}\longrightarrow 0$ as $n\longrightarrow\infty$. Consider for
each $n$ some $v_{n}\in\mathcal{N}_{\varepsilon_{n}}$ and assume that
$J_{\varepsilon_{n}}(v_{n})\longrightarrow c_{0}$ as $n\longrightarrow\infty$.
Then, there is $x_{0}\in M$, a critical point $w_{0}$ of $J_{0}$ at level
$c_{0}$ and a sequence $(y_{n})_{n}\subset\mathbb{R}^{N}$ such that (up to a
subsequence)

(4.15) $\varepsilon_{n}y_{n}\longrightarrow x_{0}\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{and}\leavevmode\nobreak\ \leavevmode\nobreak\
||v_{n}(\cdot+y_{n})-w_{0}||_{p^{\prime}}\longrightarrow 0\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ n\longrightarrow\infty.$

###### Proof.

For each $n\in\mathbb{N}$, set
$v_{0,n}:=(\frac{Q_{\varepsilon_{n}}}{Q_{0}})^{\frac{1}{p}}v_{n}$. It follows
that $|v_{0,n}|\leq|v_{n}|$ a.e. on $\mathbb{R}^{N}$ and that

(4.16)
$\int_{\mathbb{R}^{N}}Q_{0}^{\frac{1}{p}}v_{0,n}\mathbf{R}^{s}\big{(}Q_{0}^{\frac{1}{p}}v_{0,n}\big{)}\mathrm{d}=\int_{\mathbb{R}^{N}}Q_{\varepsilon_{n}}^{\frac{1}{p}}v_{n}\mathbf{R}^{s}(Q_{\varepsilon_{n}}^{\frac{1}{p}}v_{n})\mathrm{d}x>0.$

Therefore, setting

(4.17)
$t_{0,n}^{2-p^{\prime}}=\frac{\int_{\mathbb{R}^{N}}|v_{0,n}|^{p^{\prime}}\mathrm{d}x}{\int_{\mathbb{R}^{N}}Q_{0}^{\frac{1}{p}}v_{0,n}\mathbf{R}^{s}\big{(}Q_{0}^{\frac{1}{p}}v_{0,n}\big{)}\mathrm{d}x}$

we find that $t_{0,n}v_{0,n}\in\mathcal{N}_{0}$ and $0<t_{0,n}\leq 1$. As a
consequence, we can write

(4.18) $\displaystyle c_{0}$ $\displaystyle\leq
J_{0}(t_{0,n}v_{0,n})=(\frac{1}{p^{\prime}}-\frac{1}{2})t_{0,n}^{2}\int_{\mathbb{R}^{N}}Q_{0}^{\frac{1}{p}}v_{0,n}\mathbf{R}^{s}\big{(}Q_{0}^{\frac{1}{p}}v_{0,n}\big{)}\mathrm{d}x$
$\displaystyle=(\frac{1}{p^{\prime}}-\frac{1}{2})t_{0,n}^{2}\int_{\mathbb{R}^{N}}Q_{\varepsilon_{n}}^{\frac{1}{p}}v_{n}\mathbf{R}^{s}\big{(}Q_{\varepsilon_{n}}^{\frac{1}{p}}v_{n}\big{)}\mathrm{d}x$
$\displaystyle=t_{0,n}^{2}J_{\varepsilon_{n}}(v_{n})\leq
J_{\varepsilon_{n}}(v_{n})\longrightarrow c_{0}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\
n\longrightarrow\infty.$

In particular, we find

(4.19)
$\mathop{\mathrm{lim}}\limits_{n\longrightarrow\infty}\leavevmode\nobreak\
t_{0,n}=1,$

and $(t_{0,n}v_{0,})_{n}\subset\mathcal{N}_{0}$ is thus a minimizing sequence
for $J_{0}$ on $\mathcal{N}_{0}$. Using Ekeland’s variational principle and
the fact that $\mathcal{N}_{0}$ is a natural constraint, we obtain the
existence of a $(\mathrm{PS})_{c_{0}}$-sequence $(w_{n})_{n}\subset
L^{p^{\prime}}(\mathbb{R}^{N})$ for $J_{0}$ with the property that
$||v_{0,n}-w_{n}||_{p^{\prime}}\longrightarrow 0$, as
$n\longrightarrow\infty$.

By Lemma 4.1, there exists a critical point $w_{0}$ for $J_{0}$ at level
$c_{0}$ and a sequence $(y_{n})_{n}\subset\mathbb{R}^{N}$ such that (up to a
subsequence) $||w_{n}(\cdot+y_{n})-w_{0}||_{p^{\prime}}\longrightarrow 0$, as
$n\longrightarrow\infty$. Therefore,

(4.20) $v_{0,n}(\cdot+y_{n})\longrightarrow w_{0}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{strongly\leavevmode\nobreak\
in\leavevmode\nobreak\ }L^{p^{\prime}}(\mathbb{R}^{N}),\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ n\longrightarrow\infty.$

We are going to show that $(\varepsilon_{n}y_{n})_{n}$ is bounded. Suppose
not, there exist some subsequence (which we still call
$(\varepsilon_{n}y_{n})_{n})$ such that
$\mathop{\mathrm{lim}}\limits_{n\longrightarrow\infty}|\varepsilon_{n}y_{n}|=\infty$.
We consider the following two cases.

(1) $Q_{\infty}=0$. In this case, by the assumption on $Q$, we have
$Q(\varepsilon_{n}\cdot+\varepsilon_{n}y_{n})\longrightarrow 0$, as
$n\longrightarrow\infty$, holds uniformly on bounded sets of $\mathbb{R}^{N}$.
From the definition of $v_{0,n}$, we infer that
$v_{0,n}(\cdot+y_{n})\rightharpoonup 0$ and therefore $w_{0}=0$, in
contradiction to $J_{0}(w_{0})=c_{0}>0$. Hence, $(\varepsilon_{n}y_{n})_{n}$
is bounded in this case.

(2) $Q_{\infty}>0$. By the Fatou’s lemma and the strong convergence
$v_{0,n}(\cdot+y_{n})\longrightarrow w_{0}$, we deduce that

(4.21) $\displaystyle c_{0}$
$\displaystyle=\mathop{\mathrm{lim}}\limits_{n\longrightarrow\infty}J_{\varepsilon_{n}}(v_{n})=\mathop{\mathrm{lim}}\limits_{n\longrightarrow\infty}(\frac{1}{p^{\prime}}-\frac{1}{2})\int_{\mathbb{R}^{N}}|v_{n}|^{p^{\prime}}\mathrm{d}x$
$\displaystyle=\mathop{\mathrm{lim}}\limits_{n\longrightarrow\infty}(\frac{1}{p^{\prime}}-\frac{1}{2})\int_{\mathbb{R}^{N}}|v_{n}(x+y_{n})|^{p^{\prime}}\mathrm{d}x$
$\displaystyle=\mathop{\mathrm{lim\leavevmode\nobreak\
inf}}\limits_{n\longrightarrow\infty}(\frac{1}{p^{\prime}}-\frac{1}{2})\int_{\mathbb{R}^{N}}\big{(}\frac{Q_{0}}{Q(\varepsilon_{n}x+\varepsilon_{n}y_{n})}\big{)}^{p^{\prime}-1}|v_{0,n}(x+y_{n})|^{p^{\prime}}\mathrm{d}x$
$\displaystyle\geq(\frac{1}{p^{\prime}}-\frac{1}{2})\int_{\mathbb{R}^{N}}\big{(}\frac{Q_{0}}{Q_{\infty}}\big{)}^{p^{\prime}-1}|w_{0}|^{p^{\prime}}\mathrm{d}x$
$\displaystyle=\big{(}\frac{Q_{0}}{Q_{\infty}}\big{)}^{p^{\prime}-1}c_{0},$

and this contradicts (Q1). Therefore, $(\varepsilon_{n}y_{n})_{n}$ is a
bounded sequence, and we may assume (going to a subsequence) that
$\varepsilon_{n}y_{n}\longrightarrow x_{0}\in\mathbb{R}^{N}$.

Since $Q(\varepsilon_{n}x+\varepsilon_{n}y_{n})\longrightarrow Q_{x_{0}}$, as
$n\longrightarrow\infty$, uniformly on bounded set, the argument of case (1)
above gives $Q(x_{0})>0$ and, using the Dominated Convergence Theorem, we see
that $Q(x_{0})=Q_{0}$, since the following holds.

(4.22) $\displaystyle c_{0}$
$\displaystyle=\mathop{\mathrm{lim}}\limits_{n\longrightarrow\infty}J_{\varepsilon_{n}}(v_{n})=\mathop{\mathrm{lim}}\limits_{n\longrightarrow\infty}(\frac{1}{p^{\prime}}-\frac{1}{2})\int_{\mathbb{R}^{N}}|v_{n}|^{p^{\prime}}\mathrm{d}x$
$\displaystyle=\mathop{\mathrm{lim}}\limits_{n\longrightarrow\infty}(\frac{1}{p^{\prime}}-\frac{1}{2})\int_{\mathbb{R}^{N}}\big{(}\frac{Q_{0}}{Q(\varepsilon_{n}x+\varepsilon_{n}y_{n})}\big{)}^{p^{\prime}-1}|v_{0,n}(x+y_{n})|^{p^{\prime}}\mathrm{d}x$
$\displaystyle=(\frac{1}{p^{\prime}}-\frac{1}{2})\int_{\mathbb{R}^{N}}\big{(}\frac{Q_{0}}{Q(x_{0})}\big{)}^{p^{\prime}-1}|w_{0}|^{p^{\prime}}\mathrm{d}x$
$\displaystyle=\big{(}\frac{Q_{0}}{Q_{x_{0}}}\big{)}^{p^{\prime}-1}c_{0}.$

Going back to the original sequence we obtain

(4.23)
$v_{n}(\cdot+y_{n})=\big{(}\frac{Q_{0}}{Q(\varepsilon_{n}+\varepsilon_{n}y_{n})}\big{)}^{\frac{1}{p}}v_{0,n}(\cdot+y_{n})\longrightarrow\big{(}\frac{Q_{0}}{Q(x_{0})}\big{)}^{\frac{1}{p}}w_{0}=w_{0},\leavevmode\nobreak\
\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ n\longrightarrow\infty,$

strongly in $L^{p^{\prime}}(\mathbb{R}^{N})$, using again the Dominated
Convergence Theorem. The proof is complete. ∎

###### Proof of Theorem 1.1 (ii).

By (1.11), the dual ground state $u_{n}$ can be represented as

(4.24)
$u_{n}(x)=k_{n}^{\frac{2s}{p-2}}\mathbf{R}^{s}\big{(}Q_{\varepsilon_{n}}^{\frac{1}{p}}v_{n}\big{)}(k_{n}x),\leavevmode\nobreak\
\leavevmode\nobreak\ x\in\mathbb{R}^{N},$

where $\varepsilon_{n}=k_{n}^{-1}$ and $v_{n}\in
L^{p^{\prime}}(\mathbb{R}^{N})$ is a least-energy critical point of
$J_{\varepsilon}$, i.e., $J^{\prime}_{\varepsilon}(v_{n})=0$ and
$J_{\varepsilon_{n}}(v_{n})=c_{\varepsilon_{n}}$. By Lemma 2.4 and Proposition
4.2, there is $x_{0}\in M$ and a sequence $(y_{n})_{n}\subset\mathbb{R}^{N}$
such that, as $n\longrightarrow\infty$,
$x_{n}:=\varepsilon_{n}y_{n}\longrightarrow x_{0}$ and, going to a
subsequence, $v(\cdot+y_{n})\longrightarrow w_{0}$ in
$L^{p^{\prime}}(\mathbb{R}^{N})$ for some least-energy critical point $w_{0}$
of $J_{0}$. Therefore, for $x\in\mathbb{R}^{N}$,

(4.25)
$k_{n}^{-\frac{2s}{p-2}}u_{n}\big{(}\frac{x}{k_{n}}+x_{n}\big{)}=\mathbf{R}^{s}\big{(}Q_{\varepsilon_{n}}^{\frac{1}{p}}v_{n}\big{)}(x+y_{n})=\mathbf{R}^{s}\big{(}Q_{\varepsilon_{n}}^{\frac{1}{p}}(\cdot+y_{n})v_{n}(\cdot+y_{n})\big{)}(x).$

On the other hand, by the continuity of $\mathbf{R}^{s}$ and the pointwise
convergence $Q_{\varepsilon_{n}}(x+y_{n})\longrightarrow Q(x_{0})=Q_{0}$ as
$n\longrightarrow\infty$ for all $x\in\mathbb{R}^{N}$, we have the following
strong convergence

(4.26)
$k_{n}^{-\frac{2s}{p-2}}u_{n}\big{(}\frac{x}{k_{n}}+x_{n}\big{)}\longrightarrow\mathbf{R}^{s}\big{(}Q_{0}^{\frac{1}{p}}w_{0}\big{)}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{in}\leavevmode\nobreak\
L^{p}(\mathbb{R}^{N}).$

Setting $u_{0}=\mathbf{R}^{s}(Q_{0}^{\frac{1}{p}}w_{0})$, the properties
$J_{0}(w_{0})=c_{0}$ and $J^{\prime}_{0}(w_{0})=0$ imply that $u_{0}$ is a
dual ground state solution of (2.9) and this conclude the proof. ∎

## Acknowledgements

The research of Minbo Yang was partially supported by NSFC (11971436,
12011530199) and ZJNSF (LD19A010001).

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