Multi-peak semiclassical bound states for Fractional Schrödinger Equations with fast decaying potentials

1.   Introduction and main results

In this paper, we consider the fractional Schrödinger equation

where $N > 2s$, $s\in (0,1)$, $V$ is a continuous function, $\varepsilon > 0$ is a small parameter, $f: \mathbb{R}^N\to \mathbb{R}$ is a nonlinear function. Problem (1.1) is derived from the study of time-independent waves $\psi(x,t) = e^{-iEt}u(x)$ of the following nonlinear fractional Schrödinger equation

For example, letting $f(t) = |t|^{p - 2}t$, $V(x) = U(x) - E$ and inserting $\psi(x,t) = e^{-iEt}u(x)$ into $(NLFS)$, one can show that $(NLFS)$ is

In physics, Eq (1.1) can be used to describe some properties of Einstein's theory of relativity and also has been derived as models of many physical phenomena, such as phase transition, conservation laws, especially in fractional quantum mechanics, etc., ^[1]. $(NLFS)$ was introduced by Laskin ^[2,3] 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. To see more physical backgrounds, we refer to ^[4].

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 the local case $s = 1$, the study of the nonlinear Schrödinger equation

has been extensively investigated in the semiclassical regime and a considerable amount of work has been done, showing that existence and concentration phenomena of single- and multi-bump solutions occur at critical points of the electric potential $V$ when $\varepsilon\to 0$, see ^{[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]} and the references therein for example.

To our best knowledge, there are few results on the semiclassical bound states to problem (1.1) in the nonlocal case $s\in (0,1)$. Basing on the well-known non-degenerate results in ^[20,21] and the mathematical reduction method, it was proved in ^[22,23,24] that problem (1.2) has solutions concentrating at the prescribed non-degenerate critical points of $V$ when $\varepsilon\to 0$. When $\inf_{x\in \mathbb{R}^N}V(x) > 0$ and $V$ has local minimum which may be degenerate, Alves et al. in ^[25] used the penalized method developed by del Pino et al. in ^[10] and the extension method developed by Caffarelli et al. in ^[26] to construct solutions concentrating at a local minimum of $V$ when $\varepsilon\to 0$. Successively, assuming more weakly that $\liminf_{|x|\to\infty}V(x)|x|^{2s} \ge 0$, in ^[27,28], solutions concentrating at a local minimum of $V$ were also obtained. We point out here that the solutions found in ^[25] and ^[27] have exactly one local maximum and hence are single-peaked.

However, concerning (1.1), up to now there are no research on the multi-bump solutions in the case that the potentials $V(x)$ vanish at infinity and critical points of $V(x)$ are degenerate. The main difficulty lies in that for a suitable function $u:\ \mathbb{R}^N\to \mathbb{R}$, under the nonlocal effects of $(-\Delta)^s$, one can not compute $(-\Delta )^su$ as precisely as $-\Delta u$. Moreover, the nonlocal operator $(-\Delta)^s$ makes the peaks of the candidate solutions affect mutually, which causes more difficulties in finding solutions with multiple bumps (see the estimates of (2.23), (2.26) and (2.29) in Lemma A.2 for example).

This paper devotes to finding solutions with multiple bumps for more general potentials including fast decaying potentials, i.e.,

in which, a typical case is that $V$ is compactly supported.

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

endowed with the norm

where

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

Define the following fractional Sobolev space

It is easy to check that with the inner product

$W^{s,2}(\Omega)$ is a Hilbert space (see ^[4] for details). According to ^[4], the fractional Laplacian is defined as

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

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

endowed with the norm

For the nonlinear term $f(u)$, we assume

A typical case of $f(t)$ is: $f(t) = |t|^{p-2}t$ with $2+\frac{2s}{N-2s}<p<2^*_s$.

For the potential term $V$, we assume that $V\in C\big(\mathbb{R}^N,[0,\infty)\big)$ and

$(\textbf{V})$ There exist open bounded sets $\Lambda_i\subset\subset S_i\subset\subset U_i$ with smooth boundaries, such that

Denote $\Lambda = \bigcup_{i = 1}^k\Lambda_i$, $S = \bigcup_{i = 1}^kS_i$ and $U = \bigcup_{i = 1}^kU_i$. Without loss of generality, we assume that $0\in \Lambda$.

Theorem 1.1. Let $N > 2s$, $s\in(0,1)$, $V$ satisfy (V) and $f$ satisfy the assumptions $(f_1)-(f_4)$. Then problem (1.1) has a positive solution $u_{\varepsilon}\in \mathcal{D}^s_{V,\varepsilon}(\mathbb{R}^N)$ if $\varepsilon>0$ is small enough. Moreover, there exists $k$ families of points $\{\{x^i_{\varepsilon}\}:1\le i\le k\}$ and an $\alpha$ close to $N - 2s$, such that

where $C$ and $\rho$ are positive constants.

Now we introduce the main idea of the proof. For the local case $s = 1$, certain penalized functional like

was usually employed to prove that the penalized solution $u_{\varepsilon}$ has exactly one peak in each $\Lambda_i$, see ^[17,18,29] for example. But, the key step of this argument is to eliminate the effect of $K_{\varepsilon}(u)$ to the equation, which needs a type of isolated property of the least energy of $-\Delta u + u = g(u)$. However, for our case $0<s<1$, this type of isolated property is still unknown. To overcome this difficulty, we use the method developed by Byeon and Jeanjean in ^[30], which proves the existence of multi-peak solutions of following equation

by using only the compactness of the set consisting of the radial positive least energy solutions of the following limiting problem of (1.6):

where $a>0$ is a constant and $g$ is a nonlinear term satisfying some subcritical conditions. For more application of this methods, see ^[31]. Roughly speaking, by the compact property, we use the deformation ideas of Lemma 2.2 in ^[32] to construct a $(PS)_c$ sequence near the least energy solutions of the following $k$ problems:

It is worth mentioning that the compact property can be obtained by the decay estimates of positive radial least energy solutions(see Proposition 2.4 below). However, the vanishing of $V$ and the nonlocal effect of $(-\Delta)^s$ makes the construction of multi-peak solutions more difficult than the classical case $s = 1$, the non-vanishing case ^[30] and the single peak case ^[28]. Firstly, an elementary(but tedious) calculations show that when $V(x)$ vanishes faster than $|x|^{-2s}$, the natural functional $I_{\varepsilon}:\mathcal{D}^s_{V,\varepsilon}(\mathbb{R}^N)\to \mathbb{R}$ corresponding to (1.1) defined as

whose critical points are solutions of Eq (1.1), is not well-defined in $\mathcal{D}^s_{V,\varepsilon}(\mathbb{R}^N)$, where $F(t) = \int_{0}^tf(s)ds$. Moreover, the fact that $V(x)$ may be compactly supported makes it impossible that $V$ can dominated the nonlinear term $|u|^{p-2}u$ like ^[27]. Hence we have to introduce a different penalized idea from ^[30] to cut-off the nonlinear term. More precisely, we will first use the nonlocal part $(-\Delta)^s$ to modify the problem by the following fractional Hardy inequality

for all $u\in \dot{H}^s( \mathbb{R}^N)$(see ^[35]), and then construct a sup-solution and estimate the energy of multi-peak solutions.

The celebrated paper ^[26] provides an easy way to understand the nonlocal problem (see ^[25] for example), by which, one can convert the nonlocal problem (1.1) into a local problem. But we do not use this method in our paper. Indeed, if problem (1.1) becomes a local problem, the vanishing of $V$ and the added variable "$t>0$" (which comes from extending the problem into $\mathbb{R}^{N + 1}_+$, see ^[25] for instance) will make it difficult to construct precise penalized functions.

The paper is organized as follows: in Section 2, we establish the penalized scheme. By using the compact property of the set consisting of positive radial least energy solutions and the deformation idea in Lemma 2.2 of ^[32], we construct a $(PS)_c$ sequence with $k$-peaks in $\Lambda$, and then get a penalized multi-peak solution. In Section 3, we construct a penalized function to prove that the penalized solution is indeed a solution of the original equation (1.1). In the Appendix we will give some tedious energy estimates caused by the nonlocal operator.

2.   The penalized problem

In this section, we first establish a penalized problem by using the fractional Hardy inequality (1.7) to cut off the nonlinear term $f$. A well-defined smooth penalized functional in $\mathcal{D}^s_{V,\varepsilon}( \mathbb{R}^N)$ will be obtained. Secondly, we use the compact property of set consisting of least energy solutions and the deformation lemma [32, Lemma 2.2] to construct a $(PS)_c$ sequence near the least energy solutions. A penalized solution with $k$ peaks for the penalized problem will be obtained by passing limit on the $(PS)_c$ sequence.

2.1.   The Penalized Functional

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) ^[37]For every $u\in H^s(\mathbb{R}^N)$,

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

The above inequality implies 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 ^[4]), i.e.,

2.2.   The penalized functional

Now we are going to modify the original problem (1.1). According to the fractional Hardy inequality (1.7), we choose a family of penalized potentials $\mathcal{P}_{\varepsilon}\in L^{\infty}(\mathbb{R}^N,[0,\infty))$ for $\varepsilon > 0$ small in such a way that

where $\kappa>0$ is the same parameter in $(f_1)$. Noting that by (1.7), when $\varepsilon>0$ is small enough, it holds that for any $A\subset \mathbb{R}^N$,

where $C_{N,s}$ is the constant in (1.7). This type of estimate plays a key role in the paper(see (2.10) below for example).

Now we give the penalized problem according to the choice of $\mathcal{P}_{\varepsilon}$:

It is easy to check that if a solution $u_{\varepsilon}$ of (2.3) satisfies

then $u_{\varepsilon}$ is a solution of (1.1).

Given a penalized potential $\mathcal{P}_{\varepsilon}$ that satisfies (2.1), we define the penalized nonlinearity $g_{\varepsilon}:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$ as

We denote $G_{\varepsilon}(x,t) = \int_{0}^{t}g_{\varepsilon}(x,s)ds$.

Accordingly, the penalized superposition operators $\mathfrak{g}_{\varepsilon}$ and $\mathfrak{G}_{\varepsilon}$ are given by

Following, we define the penalized functional $J_{\varepsilon}:\mathcal{D}^s_{V,\varepsilon}(\mathbb{R}^N)\to \mathbb{R}$ as

The strong assumption (2.1) can help to check that $J_{\varepsilon}$ is $C^1$ and satisfies (P.S.) condition.

Lemma 2.2. (1) If $2 < p< 2^*_s$ and (2.1) hold, then $J_{\varepsilon}\in C^1(\mathcal{D}^s_{V,\varepsilon}(\mathbb{R}^N), \mathbb{R})$ and for $u\in \mathcal{D}^s_{V,\varepsilon}(\mathbb{R}^N)$, $\varphi\in \mathcal{D}^s_{V,\varepsilon}(\mathbb{R}^N)$,

Here $\langle\cdot,\cdot\rangle$ denotes the duality product between the dual space $\mathcal{D}^s_{V,\varepsilon}(\mathbb{R}^N)'$ and the space $\mathcal{D}^s_{V,\varepsilon}(\mathbb{R}^N)$. In particular, $u\in \mathcal{D}^s_{V,\varepsilon}(\mathbb{R}^N)$ is a critical point of $J_{\varepsilon}$ if and only if $u$ is a weak solution of the penalized equation

(2) ((P.S.) condition) If $2 < p <2^*_s$ and (2.1) holds, then $J_{\varepsilon}$ owns the mountain pass geometry and satisfies the Palais-Smale condition.

Proof. We omit the proof since it is quite similar to that in [28, Lemma 2.4].

2.3.   Construction of solutions with $k$ peaks

Definition 2.3. For $a>0$, we define the value $c_a$ as

where $L_{a}:H^s(\mathbb{R}^N)\to \mathbb{R}$ and $\Gamma_a$ are given by

and

where $F(t) = \int_{0}^tf(s)ds$. From ^[1,36], we know that $c_{a}$ is continuous, increasing on $a$ and can be achieved by a positive radial solution $U_a$ which satisfies the following limiting problem

Moreover, there exist two positive constants $\tilde{c}_a,\ \widetilde{C}_a$ such that

Then, letting $S_a = \{U_a:U_a\ {\rm{is\;positive\;radial\;and\;achieves}}\ c_a\}$, by the decay estimate (2.5), we have

Proposition 2.4. The set $S_a$ is compact in $H^s( \mathbb{R}^N)$.

Proof. If $S_a$ contains finitely many elements, then it is compact. Otherwise, taking a sequence $\{U_n\}\subset S_a$, since $\{U_n\}$ is bounded in $H^s( \mathbb{R}^N)$, there exists a $\overline{U}\in H^s( \mathbb{R}^N)$ such that

Then, by (2.5), we have $U_n\to\ \overline{U}\ {\rm{strongly in}}\ L^p( \mathbb{R}^N)$. Obviously, $\overline{U}$ is nonnegative and satisfies

Furthermore, by standard regularity argument(see Appendix D in ^[21] for example), we have $\overline{U}>0$. Then, by Definition 2.3, we have $\liminf_{n\to\infty}L_{a}(U_n)\ge L_{a}(\overline{U})\ge c_a$. Then $L_{a}(\overline{U}) = c_a$, $\overline{U}\in S_a$ and

as $n\to\infty$. This completes the proof.

From now on we define

Let $\eta(x) = \eta(|x|)\in C^{\infty}_c( \mathbb{R}^N)$ satisfy $0\le \eta\le 1$, $\eta \equiv 1$ on $\overline{B}_{\beta}(0)$ and $\eta\equiv 0$ on $\mathbb{R}^N\backslash B_{2\beta}(0)$, where $\beta>0$ is a small parameter satisfying $\mathcal{M}^{2\beta}\subset \Lambda$. For each $p_i\in\mathcal{M}_i$ and $U_{\lambda_i}\in S_{\lambda_i}$ given by Definition 2.3, we define

We will find a solution to (2.4), for sufficiently small $\varepsilon>0$, near the set

For each $1\le i\le k$, we also define

We have:

Proposition 2.5. For each $i\in \{1,\ldots,k\}$, it holds

if $t_i>T$ for some $T\in(0,+\infty)$.

Proof. By the choice of $W^i_{\varepsilon}$, there exists a positive constant $C$ such that

By decomposition, we have

But, arguing as done in the proof of the following (2.23), (2.26) and (2.29) in Lemma A.2, we know that

Hence

Then, by the assumption on $f$ and $\max\limits_{{t>0}\atop{1\le i\le k}}\big(Ct^2_i\|U_{\lambda_i}(x)\|^2 - \int_{B_1(0)}F(t_iU_{\lambda_i}(x))\big)<+\infty$, we get the conclusion.

As a result of Proposition 2.5, we know that the following definition is reasonable: for $\tau = (t_1,\ldots,t_k)\in[0,T]^k$, let $\gamma_{\varepsilon}(\tau) = \sum_{i = 1}^kt_iW^i_{\varepsilon}$ and define

We have the following estimate for $\mathcal{D}_{\varepsilon}$.

Proposition 2.6. (i) $\lim\limits_{\varepsilon\to 0}\frac{\mathcal{D}_{\varepsilon}}{\varepsilon^N} = \sum_{i = 1}^kc_{\lambda_i}$.

(ii) $\limsup\limits_{\varepsilon\to 0}\frac{\max_{\tau\in\partial[0,T]^k}J_{\varepsilon}\big(\gamma_{\varepsilon}(\tau)\big)}{\varepsilon^N}\le\sum_{i = 1}^kc_{\lambda_i}-\min\limits_{1\le i\le k}c_{\lambda_i}$.

(iii) For each $\delta>0$, there exists $\alpha>0$ such that for sufficiently small $\varepsilon>0$,

implies that $\gamma_{\varepsilon}(\tau)\in \mathcal{X}^{\frac{\delta\varepsilon^{N/2}}{2}}_{\varepsilon}$.

Proof. By the decay rates of $U_{\lambda_i}$ and the analysis of (2.6), we have

where $\eta_{\varepsilon}(x) = \eta(\varepsilon x)$. Choosing $\varepsilon>0$ be small enough such that $supp \eta_{\varepsilon}\cap supp\eta_{\varepsilon}\big(\cdot+ \frac{p_i-p_j}{\varepsilon}\big) = \emptyset$, we have

Then by the fact that $p_i\in\mathcal{M}_i$ and $t_i\le T$, $1\le i\le k$, we have

Hence we get (i) and obviously (ii) is true.

Finally, (2.7) implies that if $\tau_{\varepsilon}\in[0,T]^k$ satisfies $\lim\limits_{\varepsilon\to 0}\Big(\frac{J_{\varepsilon}\big(\gamma_{\varepsilon}(\tau_{\varepsilon})\big)}{\varepsilon^N} - \frac{\mathcal{D}_{\varepsilon}}{\varepsilon^N}\Big) = 0$, then it must hold

which implies (iii).

Consequently, we complete the proof.

Next, we define

where

where $\nu>0$ is large positive constant. Obviously, $\Psi_{\varepsilon}$ is nonempty since $\gamma_{\varepsilon}\in \Psi_{\varepsilon}$. We now prove the following property of $\mathcal{C}_{\varepsilon}$.

Lemma 2.7.

The proof will rely on the following lemma, whose proof, for the sake of continuity, is postponed to the appendix. We define for every $i\in\{1,\ldots,k\}$, the functional $J^i_{\varepsilon}:W^{s,2}(S_i)\to \mathbb{R}$ as

We have

Lemma 2.8. The mountain pass value

can be achieved, where

Moreover,

Now we prove Lemma 2.7:

Proof of Lemma 2.7. By Proposition (2.6), we have the upper bounds

It remains to prove the lower estimate, i.e.,

We first observe that given any $\psi_{\varepsilon}\in \Psi_{\varepsilon}$ and any continuous curve $c:[0,1]\to [0,T]^k$ with $c(0)\in \{0\}\times[0,T]^{k - 1}$ and $c(1)\in \{T\}\times [0,T]^{k - 1}$, we have $\gamma^1_{\varepsilon} = \psi_{\varepsilon} \circ c|_{S_1}\in \Gamma^1_{\varepsilon}$. In fact, by the definition of $\Psi_{\varepsilon}$, we have

Lemma 2.8 implies that

Similarly, for every $\gamma^j_{\varepsilon} = \gamma \circ c|_{S_j}$ belongs to $\Gamma^j_{\varepsilon}$, where $c$ is arbitrary continuous path which joint $[0,T]^{j - 1}\times\{0\}\times[0,T]^{k - j}$ with $[0,T]^{j - 1}\times\{T\}\times[0,T]^{k - j}$, it holds

Thus we can repeat the argument of Coti-Zetali and Rabinowitz in ^[34] to prove, for every path $\psi_{\varepsilon}\in \Gamma$, the existence of a point $\hat{\tau}\in [0,1]^k$ satisfying

Consequently, by (2.1), (2.2) and the fact that $\psi_{\varepsilon}(\tau)\in \mathcal{X}^{\nu\varepsilon^{N/2}}_{\varepsilon}$, we get

which is exactly the required lower estimate.

Next, we are going to construct a penalized solution for the penalized problem (2.3). We first prove that the limit of a $(PS)_c$ sequence near the set $\mathcal{X}_{\varepsilon}$ must own $k$-peaks.

Proposition 2.9. Let $\{\varepsilon_j\}_j$ with $\lim\limits_{j\to\infty}\varepsilon_j = 0$ and $\{u_{\varepsilon_j}\}\subset \mathcal{X}^{d\varepsilon^{N/2}_j}_{\varepsilon_j}$ satisfy

Then for sufficiently small $d>0$, there exist, up to subsequence, $\{x^i_j\}_j\subset \mathbb{R}^3$, $i = 1,\ldots,k$, $x_i\in \mathcal{M}_i$, $\overline{U}_{\lambda_i}\in S_{\lambda_i}$ such that

and

Proof. For the sake of convenience, we write $\varepsilon$ for $\varepsilon_j$. Since $S_{\lambda_i}$, $i = 1,\ldots,k$ are compact in $H^s( \mathbb{R}^N)$, there exist ${U}_{\lambda_i}\in S_{\lambda_i}$ and $p^i_{\varepsilon}\in \mathcal{M}_i$ such that

Letting $R_0\ge 1$ be a fixed positive constant and $\varepsilon R_0\le \beta$, for each $i = 1,\ldots,k$, we have

As a result, we can let $d>0$ be small enough so that

for all $1\le i\le k$.

Denote $u^{1,i}_{\varepsilon}(x) = \eta(x-p^i_{\varepsilon})u_{\varepsilon}(x)$, $u^1_{\varepsilon}(x) = \sum_{i = 1}^ku^{1,i}_{\varepsilon}(x)$ and $u^2_{\varepsilon}(x) = u_{\varepsilon}(x)-u^1_{\varepsilon}(x)$. Denote $v^{1,i}_{\varepsilon}(x) = u^{1,i}_{\varepsilon}(\varepsilon x + p^i_{\varepsilon})$ and $v^{2,i}_{\varepsilon}(x) = v^i_{\varepsilon}(x) - v^{1,i}_{\varepsilon}(x)$, where $v^i_{\varepsilon}(x) = u_{\varepsilon}(\varepsilon x + p^i_{\varepsilon})$. Fix arbitrarily an $i\in\{1,\ldots,k\}$. Obviously, by assumption, for each $\varphi\in C^{\infty}_c( \mathbb{R}^N)$ and $\varepsilon$ small enough, testing $J'_{\varepsilon}(u_{\varepsilon})$ with $\varphi\Big(\frac{x-p^i_{\varepsilon}}{\varepsilon}\Big)$, we find

Since $\{u_{\varepsilon}\}\subset \mathcal{X}^{d\varepsilon^{N/2}}_{\varepsilon}$, by fractional Hardy inequality (1.7), we have

Then, since $\{v^{1,i}_{\varepsilon}\}$ is bounded in $H^s( \mathbb{R}^N)$, by the Liouville type Theorem 3.3 of ^[27], we have

where $v^{1,i}_*$ is the weak limit of some subsequence of $v^{1,i}_{\varepsilon}$ in $H^s( \mathbb{R}^N)$ and $p^i_*\in\mathcal{M}_i$ is limit of $p^i_{\varepsilon}$. Consequently, according to the argument of Proposition 3.4 in ^[28], we have for every $R>0$ that

Consequently, by Lemma 2.8, we have $\lambda_i = V(p^i_*)$, $p^i_*\in\mathcal{M}_i$ and $v^{1,i}_*(\cdot+z_i)\in S_{\lambda_i}$ for some $z_i\in \mathbb{R}^N$. Denote

In the following we show that

By the same argument of Lemma 3.4 in ^[28], we can conclude that

and for any $r>0$, $y_{\varepsilon}\in \mathbb{R}^N$ with $\lim_{\varepsilon\to 0}\frac{|p^i_{\varepsilon}-y_{\varepsilon}|}{\varepsilon} = +\infty$, it holds

Then according to Proposition 2.1 and the Concentration-Compactness Lemma 1.21 of ^[32], we have

By decomposition, one find

But, with (2.19) at hand, we can use the same method in the proof of (2.24)(which needs only (2.25)) to show that

which and (2.2) imply that

From (2.19), we have

Hence, by the analysis above, we have

Decomposing again, we find

where

But, it has been proved in Appendix that

Hence, it holds

So

which combining with the analysis of (2.17) yields

Consequently, by (2.21), we have

which gives (2.18).

Now from (2.18), we have

It remains to show that

By the same blow-up analysis of lemmas 3.3 and 3.4 in ^[28], it holds

Consequently, denoting $\tilde{\eta}_{\varepsilon} = 1 - \sum_{i = 1}^k\eta\big(2({x-p^i_{\varepsilon}-\varepsilon z_i})\big)$ and testing $J'_{\varepsilon}(u_{\varepsilon})$ against with $\bar{\eta}_{\varepsilon}u_{\varepsilon}$, we have, for $\varepsilon>0$ small enough,

which implies

However, we have proved in the Appendix that

and

Hence, by the choice of $\mathcal{P}_{\varepsilon}$ and fractional Hardy inequality (1.7), it holds

Noting the following estimate proved in the Appendix

where $\breve{\eta}_{\varepsilon}(x) = 1-\sum_{i = 1}^k\eta(x-p^i_{\varepsilon}-\varepsilon z_i)$, we find

which is exactly (2.24). Letting $x^i_{\varepsilon} = p^i_{\varepsilon} + \varepsilon z_i$, we get

Hence we complete the proof.

Proposition 2.10. For $d>0$ sufficiently small, there exist constants $\sigma>0$ and $\varepsilon_0>0$, such that

where $J^{\mathcal{D}_{\varepsilon}}_{\varepsilon} = \{u\in \mathcal{D}^s_{V,\varepsilon}( \mathbb{R}^N):J_{\varepsilon}(u)\le \mathcal{D}_{\varepsilon}\}$.

Proof. To the contrary, suppose that for small $d_1>d_2>0$, there exist $\{\varepsilon_j\}^{\infty}_{j = 1}$ with $\lim\limits_{j\to\infty}\varepsilon_j = 0$ and $u_{\varepsilon_j}\in \mathcal{X}^{d_1\varepsilon^{N/2}_j}_{\varepsilon_j}\backslash \mathcal{X}^{d_2\varepsilon^{N/2}_j}_{\varepsilon_j}$ satisfying $\lim\limits_{j\to\infty}J_{\varepsilon_j}(u_{\varepsilon_j})/\varepsilon^N_j\le \sum_{i = 1}^kc_{\lambda_i}$ and $\lim\limits_{j\to\infty}\frac{J'_{\varepsilon_j}(u_{\varepsilon_j})}{\varepsilon^{N/2}_j} = 0$. By Proposition 2.9, there exists $\{x^i_j\}^{\infty}_{j = 1}\subset \mathbb{R}^N$, $i = 1,\ldots,k$, $x_i\in \mathcal{M}_i$, such that

Hence, by the definition of $\mathcal{X}_{\varepsilon}$, we see that $\lim\limits_{j\to\infty}{\rm{dist}}(u_{\varepsilon_j},\mathcal{X}_{\varepsilon_j})/\varepsilon^{N/2}_j = 0$. This is a contradiction to $u_{\varepsilon_j}\not\in \mathcal{X}^{d_2 \varepsilon^{N/2}_j/2}_{\varepsilon_j}$.

Now, we use Proposition 2.10 and the Deformation Lemma 2.2 in ^[32] to construct a $(PS)_c$ sequence near the set $\mathcal{X}_{\varepsilon}$.

Define

Fix $d_0\in(0,\frac{\mu}{2})$ such that Propositions 2.9 and 2.10 hold for $d\in(0,d_0]$.

Proposition 2.11. For sufficiently small fixed $\varepsilon>0$, there exists a sequence $\{u_n\}^{\infty}_{n = 1}\subset J^{\mathcal{D}_{\varepsilon}}_{\varepsilon}\cap\mathcal{X}^{d\varepsilon^{N/2}}_{\varepsilon}$ such that $J'_{\varepsilon}(u_n)\to 0$ as $n\to\infty$.

Proof. By Proposition 2.10, there exists a constant $\sigma\in(0,1)$, such that

From Proposition 2.6(iii), there exist constants $\alpha>0$, $\varepsilon_1(\alpha)>0$ such that for $\varepsilon\in(0,\varepsilon_1]$ and $d\in(0,d_0]$, that

Now, set

where $\rho = \min\limits_{1\le i\le k}c_{\lambda_i}$. We choose $0<\bar{\varepsilon}<\min\{\varepsilon_0,\varepsilon_1\}$ such that for $\varepsilon\in(0,\bar{\varepsilon}]$

We assume to the contrary that for some $\varepsilon\in(0,\bar{\varepsilon}]$, $d\in(0,d_0)$, there exist $\beta = \beta(\varepsilon)\in(0,1)$ such that

By Lemma 2.2 in ^[32], we can choose $g_{\varepsilon}$ be a pseudo-gradient vector field for $J'_{\varepsilon}$ on a neighbourhood $N_{\varepsilon}$ of $J^{\mathcal{D}_{\varepsilon}}_{\varepsilon}\cap \mathcal{X}^{d\varepsilon^{N/2}}_{\varepsilon}$, which satisfies

Let $\zeta_{\varepsilon}$ be a Lipschitz continuous function on $\mathcal{D}^s_{V,\varepsilon}( \mathbb{R}^N)$ such that $0\le \zeta_{\varepsilon}\le 1,\ \zeta_{\varepsilon}\equiv 1$ on $\mathcal{X}^{d\varepsilon^{N/2}}_{\varepsilon}\cap J^{\mathcal{D}_{\varepsilon}}_{\varepsilon}$ and $\zeta_{\varepsilon}\equiv 0$ on $\mathcal{D}^s_{V,\varepsilon}( \mathbb{R}^N)\backslash N_{\varepsilon}$. Let $\xi_{\varepsilon}$ be a Lipschitz continuous function on $\mathbb{R}$ such that $0\le \xi_{\varepsilon}\le 1$, $\xi_{\varepsilon}(l)\equiv 1$ if $|l-\mathcal{D}_{\varepsilon}\varepsilon^{-N}|\le \frac{\alpha}{2}$ and $\xi_{\varepsilon}(l)\equiv 0$ if $|l-\mathcal{D}_{\varepsilon}\varepsilon^{-N}|\ge \alpha$. Set

Then there exists a unique solution $\Phi_{\varepsilon}:\mathcal{D}^s_{V,\varepsilon}\times[0,+\infty)\to \mathcal{D}^s_{V,\varepsilon}$ to the following initial value problem

(See the proof of Lemma 2.3 in ^[32]). It can be easily check that $\Phi_{\varepsilon}$ has the following properties:

Claim 1 For any $\tau\in[0,T]^k$, there exists $\theta_{\tau}\in[0,+\infty)$ such that

Proof of Claim 1. Assume by contradiction that there exists $\tau_0\in[0,T]^k$ such that

for all $\theta>0$. Then, by the property (3) in (2.33), we have

which and the choice of $\alpha_0$ imply that $\xi_{\varepsilon}(\varepsilon^{-N}J_{\varepsilon}(\Phi_{\varepsilon}(\gamma_{\varepsilon}(\tau_0),\theta)))\equiv 1$.

If $\Phi_{\varepsilon}(\gamma_{\varepsilon}(\tau_0),\theta)\in \mathcal{X}^{d\varepsilon^{N/2}}_{\varepsilon}$ for all $\theta\ge 0$, then by (2.35), we have $\Phi_{\varepsilon}(\gamma_{\varepsilon}(\tau_0),\theta)\in \mathcal{X}^{d\varepsilon^{N/2}}_{\varepsilon}\cap J^{\mathcal{D}_{\varepsilon}}_{\varepsilon}$ for all $\theta \ge 0$. Then $\zeta_{\varepsilon}(\Phi_{\varepsilon}(\gamma_{\varepsilon}(\tau_0),\theta))\equiv 1$ and $|\frac{d}{d\theta}J_{\varepsilon}(\Phi_{\varepsilon}(\gamma_{\varepsilon}(\tau_0),\theta))|\ge\beta^2\varepsilon^N$ for all $\theta\ge 0$. Hence

a contradiction to (2.35).

Assume that $\Phi_{\varepsilon}(\gamma_{\varepsilon}(\tau_0),\theta_0)\not\in \mathcal{X}^{d\varepsilon^{N/2}}_{\varepsilon}$ for some $\theta_0>0$. Note that (2.34), (2.35) and (2.30) imply that $\gamma_{\varepsilon}(\tau_0) \in \mathcal{X}^{\frac{d}{2}\varepsilon^{N/2}}_{\varepsilon}$. Then there exist $0<\theta^1_0<\theta^2_0$ such that $\Phi_{\varepsilon}(\gamma_{\varepsilon}(\tau_0),\theta^1_0)\in \partial \mathcal{X}^{\frac{d}{2}\varepsilon^{N/2}}_{\varepsilon}$, $\Phi_{\varepsilon}(\gamma_{\varepsilon}(\tau_0),\theta^2_0)\in \partial \mathcal{X}^{d\varepsilon^{N/2}}_{\varepsilon}$ and $\Phi_{\varepsilon}(\gamma_{\varepsilon}(\tau_0),\theta)\in \mathcal{X}^{d\varepsilon^{N/2}}_{\varepsilon}\backslash\mathcal{X}^{\frac{d}{2}\varepsilon^{N/2}}_{\varepsilon}$ for all $\theta\in(\theta^1_0,\theta^2_0)$. Then by Proposition 2.10, we have $|\frac{d}{d\theta}J_{\varepsilon}(\Phi_{\varepsilon}(\gamma_{\varepsilon}(\tau_0),\theta)|\ge \sigma^2\varepsilon^{N}$ for all $\theta\in(\theta^1_0,\theta^2_0)$. By property (2) of (2.33) and mean value theorem, we have

which implies

Hence

which is a contradiction to (2.35). This completes the proof of Claim 1.

By Claim 1, we can define $\theta(\tau): = \inf\{\theta\ge 0:J_{\varepsilon}\big(\Phi_{\varepsilon}(\gamma_{\varepsilon}(\tau),\theta)\big)\le \mathcal{D}_{\varepsilon} - \alpha_0\varepsilon^N\}$ and let $\bar{\gamma}_{\varepsilon}(\tau): = \Phi_{\varepsilon}(\gamma_{\varepsilon}(\tau),\theta(\tau))$. We have

Claim 2 $\bar{\gamma}_{\varepsilon}(\tau)\in\Psi_{\varepsilon}$.

Proof of Claim 2. Firstly, for any $\tau\in\partial[0,T]^k$, by Proposition 2.6, we have $\gamma_{\varepsilon}(\tau)\in J^{\mathcal{D}_{\varepsilon}-\alpha_0\varepsilon^N}_{\varepsilon}$. Hence $\theta(\tau) = 0$ and $\bar{\gamma}_{\varepsilon}(\tau) = \gamma_{\varepsilon}(\tau)$ if $\tau\in\partial[0,T]^k$. If $J_{\varepsilon}(\gamma_{\varepsilon}(\gamma_{\varepsilon}(\tau)) \le\mathcal{D}_{\varepsilon}-\alpha_0\varepsilon^N$, then $\vartheta(\tau) = 0$ and so $\bar{\gamma}_{\varepsilon}(\tau) = \gamma_{\varepsilon}(\tau)\in\mathcal{X}^{\nu\varepsilon^{N/2}}_{\varepsilon}$ for large $\nu>0$. If $J_{\varepsilon}(\gamma_{\varepsilon}(\tau)) >\mathcal{D}_{\varepsilon}-\alpha_0\varepsilon^N$, then by (2.30), $\gamma_{\varepsilon}(\tau)\in \mathcal{X}^{d\varepsilon^{N/2}/2}$ and by property (3) in (2.33)

This implies $\xi_{\varepsilon}(\varepsilon^{-N}J_{\varepsilon}(\Phi_{\varepsilon}(\gamma_{\varepsilon}(\tau_0),\theta)))\equiv 1$ for all $\theta\in[0,\theta(\tau))$. Consequently, if $\bar{\gamma}_{\varepsilon}(\tau) = \Phi_{\varepsilon}(\gamma_{\varepsilon}(\tau),\vartheta(\tau)) \not\in \mathcal{X}^{d\varepsilon^N}_{\varepsilon}$, then by the same argument of (2.36), there exists a $\theta\in(0,\theta(\tau))$ such that

This contradicts the definition of $\theta(\tau)$. Hence $\bar{\gamma}_{\varepsilon}(\tau)\in \mathcal{X}^{d\varepsilon^{N/2}}_{\varepsilon}\subset\mathcal{X}^{\nu\varepsilon^{N/2}}_{\varepsilon}$.

Secondly, we prove that $\bar{\gamma}_{\varepsilon}(\tau)$ is continuous. We fix any $\bar{\tau}\in[0,1]^k$. If $J_{\varepsilon}(\gamma_{\varepsilon}(\bar{\tau}))<\mathcal{D}_{\varepsilon} - \alpha_0\varepsilon^N$, then $\theta(\bar{\tau}) = 0$. Then by the continuity of $\gamma_{\varepsilon}$, we conclude that $\bar{\gamma}_{\varepsilon}(\tau)$ is continuous at $\bar{\tau}$. If $J_{\varepsilon}(\gamma_{\varepsilon}(\bar{\tau})) = \mathcal{D}_{\varepsilon} - \alpha_0\varepsilon^N$, then from the proof of (2.36), we know that $\gamma_{\varepsilon}(\bar{\tau})\in \mathcal{X}^{d\varepsilon^{N/2}}_{\varepsilon}$, and so

Thus, from the property (3) in (2.33), we have $J_{\varepsilon}\big(\Phi_{\varepsilon}(\gamma_{\varepsilon}(\bar{\tau}),\theta(\bar{\tau})+\omega\big) <\mathcal{D}_{\varepsilon}-\alpha_0\varepsilon^N$. By the continuity of $\gamma_{\varepsilon}$, we choose $r>0$ as the constants such that $J_{\varepsilon}(\Phi_{\varepsilon}(\gamma_{\varepsilon}(\tau),\theta(\bar{\tau}))\big) <\mathcal{D}_{\varepsilon}-\alpha_0\varepsilon^N$ for all $\tau\in B_r(\bar{\tau})$. Then by the definition of $\theta(\tau)$, we have $\theta({\tau})<\theta(\bar{\tau})$ for all $\tau\in B_r(\bar{\tau})\cap[0,T]^k$, and then

If $\theta({\bar{\tau}}) = 0$, we immediately have

If $\theta({\tau})>0$, then for any $0<\omega<\theta(\bar{\tau})$, similarly we have $J_{\varepsilon}(\Phi_{\varepsilon}(\gamma_{\varepsilon}(\tau),\theta(\bar{\tau})-\omega)\big) >\mathcal{D}_{\varepsilon}-\alpha_0\varepsilon^N$. By the continuity of $\gamma_{\varepsilon}$ again, we see that

So $\theta({\cdot})$ is continuous at $\bar{\tau}$. This completes the proof of Claim 2.

Now we have proved that $\bar{\gamma}_{\varepsilon}(\tau)\in \Psi_{\varepsilon}$ and $\max_{\tau\in[0,T]^k}\le \mathcal{D}_{\varepsilon}-\alpha_0\varepsilon^N$, which contradicts the definition of $\mathcal{C}_{\varepsilon}$. This completes the proof.

Lemma 2.12. Let $\{u_n\}^{\infty}_{n = 1}$ be the sequence given by Proposition 2.11. Then $\{u_n\}$ has a subsequence which converges to $u_{\varepsilon}$ in $\mathcal{D}^s_{V,\varepsilon}( \mathbb{R}^N)$. Moreover, there hold $u_{\varepsilon}>0$, $u_{\varepsilon} \in \mathcal{D}^s_{V,\varepsilon}(\mathbb{R}^N)\cap C^{1,\beta}(\mathbb{R}^N)$ for some $\beta\in (0,1)$ and $u_{\varepsilon}$ is a solution to the penalized problem (2.3)(or (2.4)).

Proof. The convergence is from Lemma 2.2. The regularity result follows from Appendix D in ^[21]. Testing the penalized equation (2.4) with $(u_{\varepsilon})_{-}$ and integrating, we can see that $u_{\varepsilon}\geq 0$. Suppose to the contrary that there exists $x_0\in \mathbb{R}^N$ such that $u_{\varepsilon}(x_0) = 0$, then we have

which is a contradiction. Therefore, $u_{\varepsilon}>0$.

To end this section, we prove that $u_{\varepsilon}$ owns $k$-peaks.

Lemma 2.13. Let $\rho > 0$ and $u_{\varepsilon}$ be the solution of (2.3) given by Lemma 2.12. Then there exists $k$ families of points $\{x^i_{\varepsilon}\}$, $i = 1,\ldots,k$, such that

Proof. The proof is trivial by the fact that the $(PS)$ sequence given by Proposition 2.11 satisfies the assumptions of Proposition 2.9.

3.   Back to the original problem

In this section we show that $u_{\varepsilon}$ solves the original problem (1.1). For this purpose, basing on the penalized equation (2.4), all we need to do is to prove that

We use comparison principle to prove (3.1), for which we should first linearize the penalized equation (2.4) outside small balls.

Proposition 3.1. Let $\{x^i_{\varepsilon}\}, i = 1,\ldots,k$ be the $k$ families of points given by Lemma 2.13. Then for $\varepsilon > 0$ small enough and $\delta\in (0,1)$, there exist $C_{\infty}>0$ and $R > 0$ such that

Proof. That $u_{\varepsilon}\leq C_{\infty}$ in $\Lambda$ is from Lemma 2.13 and the $L^{\infty}$ estimate in [21, Appendix D]. By the assumption on $f$, $\inf_{U} V(x) > 0$ and Lemma 2.13, there exists $R > 0$ such that

Obviously

Hence we conclude our result by inserting the previous pointwise bounds into the penalized equation (2.4).

Next, we construct a suitable sup-solution to Eq (2.31). Some of the the details are similar to that in Proposition 4.2 of ^[28]. Let $\tilde\eta_{\beta}(s), s\ge 0$ be a smooth non-increasing function with $\tilde{\eta}_{\beta}\equiv 1$ on $[0,1]$ and $\tilde{\eta}_{\beta}\equiv 0$ on $(1+\beta,+\infty)$, where $\beta$ is a small parameter. Define $\eta_{\beta,R}(|x|) = \tilde{\eta}_{\beta}(|x|/R)$. Setting $0<\alpha <N-2s$ and denoting

We have

Proposition 3.2. Let $\varepsilon>0$ be small enough. Then for every $x\in \mathbb{R}^N\backslash \bigcup_{i = 1}^kB_{R\varepsilon}(x^i_{\varepsilon})$, it holds

Proof. Fixing any $i\in\{1,\ldots,k\}$, a computation shows that

where $V^i_{\varepsilon}(\cdot) = V(\varepsilon x\cdot + x^i_{\varepsilon})$ and $\widehat{\mathcal{P}}^i_{\varepsilon}(\cdot) = \mathcal{P}_{\varepsilon}(\varepsilon \cdot + x^i_{\varepsilon})$. But, using the non-increasing property of $\eta_{\beta}$ and the computation of Proposition 4.2 of ^[28], for any $y\in \mathbb{R}^N\backslash B_R(0)$, when $\varepsilon>0$ is small enough, we can conclude that

Then for all $x\in \mathbb{R}^N\backslash B_{R\varepsilon}(x^i_{\varepsilon})$, it holds

As a result, we have

for all $x\in \mathbb{R}^N\backslash \bigcup_{i = 1}^kB_{R\varepsilon}(x^i_{\varepsilon})$. This completes the proof.

At last, we give the proof of Theorem 1.1.

Proof of Theorem 1.1. Let

It is easy to check that $\mathcal{P}_{\varepsilon}$ satisfies the assumption (2.1).

By Proposition 3.2, choosing the constant $C>0$ large enough and letting $v_{\varepsilon}(x) = u_{\varepsilon}(x)-\overline{U}_{\varepsilon}(x)$, we have

Since ${v}^+_{\varepsilon}\in \mathcal{D}^s_{V,\varepsilon}$(when $\alpha$ is closed to $N - 2s$), testing the equation above against with ${v}^+_{\varepsilon}(x)$, by the fractional Hardy inequality in (1.7), we find ${v}^+_{\varepsilon}(x) = 0,\ x\in \mathbb{R}^N$. Hence ${v}_{\varepsilon}(x)\le 0,\ x\in \mathbb{R}^N$. Especially, we have

Moreover, letting $\alpha$ be closed to $N-2s$, for all $x\in \mathbb{R}^N\backslash\Lambda$, it holds

This gives (3.1). As a result, $u_{\varepsilon}$ solves the original problem.

Remark 3.3. In the local case $s = 1$, we can prove the same result more easily by introducing the same penalized function $\mathcal{P}_{\varepsilon}$ in this paper. We point out here that we also answer positively to the conjecture proposed by Ambrosetti and Malchiodi in ^[33] in the nonlocal case.

A.   Appendix

In this section we are going to verify Lemma 2.8, (2.23), (2.26), (2.27) and (2.29).

Proposition A.1. For every $i = 1,\ldots,k$, it holds

Proof. The achievement of $c^i_{\varepsilon}$ is easily from the fact that the embedding

is compact for $1 \leq p < 2^*_s$(see ^[4] for more details). Thus we only need to prove (2.9).

For every nonnegative $v\in C^{\infty}_c(\mathbb{R}^N)\backslash\{0\}$ and $x_0\in\Lambda_i$, let $v_{\varepsilon}(x) = v\big(\frac{x - x_0}{\varepsilon}\big)$. Obviously, $supp\ v_{\varepsilon}\subset\Lambda_i$ and $\gamma(t) = tTv_{\varepsilon}\in \Gamma^i_{\varepsilon}$ for $\varepsilon$ small enough and $T$ large enough. Therefore,

and then

Hence, by the arbitrariness of $v$ and $x_0$, we have

On the other hand, let $w_{\varepsilon}$ be a critical point corresponding to $c^i_{\varepsilon}$, i.e., $J^i_{\varepsilon}(w_{\varepsilon}) = c^i_{\varepsilon}$ and

It follows that

Then by (2.2), it holds

from which we conclude that there exists $x^i_{\varepsilon}\in\overline{\Lambda_i}$ such that for $\rho > 0$,

Going if necessary to a subsequence, we assume that

Now, let $\tilde{w}_{\varepsilon}(x) = w_{\varepsilon}(x^i_{\varepsilon} + \varepsilon x)$, then $\tilde{w}_{\varepsilon}$ satisfies

where $V_{\varepsilon}(x) = V(x^i_{\varepsilon} + \varepsilon x)$, $S^i_{\varepsilon} = \{x\in\mathbb{R}^N:\varepsilon x + x^i_{\varepsilon}\in S\}$ and $\tilde{\mathfrak{g}}_{\varepsilon}(\tilde{w}_{\varepsilon}) = g(\varepsilon x + x^i_{\varepsilon},\tilde{w}_{\varepsilon})$. Moreover, by (A.1), we have

for every $R\in(0,+\infty)$. Thus, by diagonal argument, we conclude that $\tilde{w}_{\varepsilon}\rightharpoonup \tilde{w}$ weakly in $W^{s,2}(B_R)$ for every $R>0$. Moreover, it is easy to check by Fatou's Lemma that $\tilde{w}\in H^s(\mathbb{R}^N)$. Then, by (A.4), using Corollary 7.2 in ^[4] and taking limit in (A.5), we conclude that

where $\Lambda^*_i$ is the limit of the set $\Lambda^i_{\varepsilon} = \{x\in\mathbb{R}^N:\varepsilon x + x^i_{\varepsilon}\in\Lambda_i\}$. But by (A.3) and using the standard bootstrap argument in Appendix D in ^[21], we have

which combined with the Liouville-type results (see Lemma 3.3 in ^[27]) implies that $\Lambda^i_* = \mathbb{R}^N$. Hence we have

Proceeding as one proves Lemma 3.3 of ^[28], we have

Therefore,

which and (A.1) complete the proof.

Lemma A.2. The estimates (2.23), (2.26), (2.27) and (2.29) hold.

Proof. Hereafter, we define $\hat{\eta}_{\varepsilon}(x) = \eta(2\varepsilon x) = \eta_{\varepsilon}(2x)$ for all $x\in \mathbb{R}^N$. We first give the proof of (2.26). By the definition of $\bar{\eta}_{\varepsilon}$, we have

For each $i = 1,\ldots,k$, dividing $\mathbb{R}^N$ into several regions, we have

For $T^{2,i,1}_{\varepsilon}(\eta)$, by Cauchy inequality, we have

For $T^{2,i,2}_{\varepsilon}(\eta)$, by the definition of $\eta$, we have

But, using the similar estimate of $T^{2,i,1}_{\varepsilon}(\eta)$ and fractional Hardy inequality (1.7), we have

Hence, it holds

Similarly, one has

So

and

Secondly, we prove (2.23). By the definition of $\eta$, we have

Then we have

which gives (2.23).

Thirdly, we give the proof of (2.29). Denoting $A_{\varepsilon} = \mathbb{R}^N\backslash\bigcup_{i = 1}^kB_{2\beta}(p^i_{\varepsilon}+\varepsilon z_i)$, one can check that

As a result, we get (2.29).

The proof of (2.27) is similar and we omit it.

Acknowledgments

The authors are grateful to the referee for carefully reading the manuscript and for many valuable comments which largely improved the article. This work was partially supported by NSFC grants (No.12101150; No.11831009) and the Science and Technology Foundation of Guizhou Province ([2021]ZK008).

Conflict of interest

The authors declare there is no conflicts of interest.