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Existence and decays of solutions for fractional Schrödinger equations with general potentials

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Abstract

We revisit the following fractional Schrödinger equation

$$\begin{aligned} \varepsilon ^{2s}(-\Delta )^su +Vu=u^{p-1},\,\,\,u>0,\ \ \ \textrm{in}\ {\mathbb {R}}^N, \end{aligned}$$
(0.1)

where \(\varepsilon >0\) is a small parameter, \((-\Delta )^s\) denotes the fractional Laplacian, \(s\in (0,1)\), \(p\in (2, 2_s^*)\), \(2_s^*=\frac{2N}{N-2s}\), \(N>2s\), \(V\in C\big ({\mathbb {R}}^N, [0, +\infty )\big )\) is a general potential. Under various assumptions on V(x) at infinity, including V(x) decaying with various rate at infinity, we introduce a unified penalization argument and give a complete result on the existence and nonexistence of positive solutions. More precisely, we combine a comparison principle with iteration process to detect an explicit threshold value \(p_*\), such that the above problem admits positive concentration solutions if \(p\in (p_*, \,2_s^*)\), while it has no positive weak solutions for \(p\in (2,\,p_*)\) if \(p_*>2\), where the threshold \(p_*\in [2, 2^*_s)\) can be characterized explicitly by

$$\begin{aligned} p_*=\left\{ \begin{array}{ll} 2+\frac{2s}{N-2s} &{}\quad \text{ if } \lim \limits _{|x| \rightarrow \infty } (1+|x|^{2s})V(x)=0,\\ 2+\frac{\omega }{N+2s-\omega } &{}\quad \text{ if } 0\!<\!\inf (1\!+\!|x|^\omega )V(x)\!\le \! \sup (1\!+\!|x|^\omega )V(x)\!<\! \infty \text{ for } \text{ some } \omega \!\in \! [0, 2s],\\ 2&{}\quad \text{ if } \inf V(x)\log (e+|x|^2)>0. \end{array}\right. \end{aligned}$$

Moreover, corresponding to the various decay assumptions of V(x), we obtain the decay properties of the solutions at infinity. Our results reveal some new phenomena on the existence and decays of the solutions to this type of problems.

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Correspondence to Shuangjie Peng.

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Communicated by P. H. Rabinowitz.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Y. Deng and S. Peng were supported by National R &D Program of China (Grant 2023YFA1010002). The research was also supported by the NSF of China (No.12271196, 11931012).

Appendix A

Appendix A

In this appendix, we give a complete proof of Proposition 3.3.

Proof of Proposition 3.3

The estimate of \((-\Delta )^sw_\mu \) base on the estimate of \((-\Delta )^sh_\mu \), where \(h_\mu =|x|^{-\mu }\ (x\ne 0)\). So we first estimate \((-\Delta )^sh_\mu \).

For any given \(x\in \mathbb {R}^N\) such that \(|x|>1\), by changes of variable we have

$$\begin{aligned} \frac{1}{2}(-\Delta )^sh_\mu =&\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_r(x)}\frac{|x|^{-\mu }-|y|^{-\mu }}{|x-y|^{N+2s}}\textrm{d}y\quad (\textrm{set}\ y=|x|y')\nonumber \\ =&\frac{1}{|x|^{\mu +2s}}\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_{r}(\vec {e}_1)}\frac{|y'|^{\mu }-1}{|y'|^{\mu }|y'-\vec {e}_1|^{N+2s}}\textrm{d}y'\nonumber \\ =&\frac{1}{|x|^{\mu +2s}}\lim _{r\rightarrow 0}\Bigg (\int _{B_1(0){\setminus } B_{r}(\vec {e}_1)}\frac{|y|^{\mu }-1}{|y|^{\mu }|y-\vec {e}_1|^{N+2s}}\textrm{d}y\nonumber \\&\quad +\int _{(\mathbb {R}^N{\setminus } B_1(0)){\setminus } B_{r}(\vec {e}_1)}\frac{|y|^{\mu }-1}{|y|^{\mu }|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg ) \end{aligned}$$
(A.1)

where \(\vec {e}_1:=\frac{x}{|x|}\) is a unit vector.

For the coordinate transformation \(y=\frac{y'}{|y'|^2}\) (a inversion of a sphere), we have

$$\begin{aligned} |y||y'|=1,\ dy=|y'|^{-2N}dy'\ (\mathrm{see \ [6, Section\ 2.2.2]}),\ |y-\vec {e}_1||y'|=|y'-\vec {e}_1|, \end{aligned}$$

where we have used that

$$\begin{aligned} |y-\vec {e}_1|^2|y'|^2=(|y|^2-2y\cdot \vec {e}_1+1)|y'|^2=1-2y'\cdot \vec {e}_1+|y'|^2=|y'-\vec {e}_1|^2. \end{aligned}$$

It follows that

$$\begin{aligned} \int _{B_1(0){\setminus } B_{r}(\vec {e}_1)}\frac{|y|^{\mu }-1}{|y|^{\mu }|y-\vec {e}_1|^{N+2s}}\textrm{d}y \nonumber \\ =&\int _{(\mathbb {R}^N\backslash B_1(0)){\setminus } (B_{r}(\vec {e}_1))^*}\frac{1-|y'|^{\mu }}{|y'|^{N-2s}|y'-\vec {e}_1|^{N+2s}}\textrm{d}y' \nonumber \\ =&\int _{(\mathbb {R}^N\backslash B_1(0)){\setminus } B_{r}(\vec {e}_1))}\frac{1-|y|^{\mu }}{|y|^{N-2s}|y-\vec {e}_1|^{N+2s}}\textrm{d}y \nonumber \\&+\int _{B_{r}(\vec {e}_1){\setminus } (B_{r}(\vec {e}_1))^*}\frac{1-|y|^{\mu }}{|y|^{N-2s}|y-\vec {e}_1|^{N+2s}}\textrm{d}y \nonumber \\&-\int _{(B_{r}(\vec {e}_1))^*{\setminus } B_{r}(\vec {e}_1)}\frac{1-|y|^{\mu }}{|y|^{N-2s}|y-\vec {e}_1|^{N+2s}}\textrm{d}y, \end{aligned}$$
(A.2)

where

$$\begin{aligned}(B_{r}(\vec {e}_1))^*:=\Bigg \{\frac{y}{|y|^2}\mid y\in B_r(\vec {e}_1)\Bigg \}. \end{aligned}$$

Since \(B_{r/2}(\vec {e}_1)\subset (B_{r}(\vec {e}_1))^*\subset B_{3r/2}(\vec {e}_1)\ \textrm{as}\ r\rightarrow 0,\) we have that

$$\begin{aligned}\frac{|1-|y|^{\mu }|}{|y|^{N-2s}|y-\vec {e}_1|^{N+2s}}\le O\Bigg (\frac{1}{r^{N+2s-1}}\Bigg ),\quad y\in \big (B_{r}(\vec {e}_1){\setminus } (B_{r}(\vec {e}_1))^*\big )\cup \big (B_{r}(\vec {e}_1))^*{\setminus } B_{r}(\vec {e}_1)\big ).\end{aligned}$$

On the other hand, we can verify that

$$\begin{aligned}|B_{r}(\vec {e}_1){\setminus } (B_{r}(\vec {e}_1))^*|+|B_{r}(\vec {e}_1))^*{\setminus } B_{r}(\vec {e}_1)|=O(r^{N+1})\ \textrm{as}\ r\rightarrow 0,\end{aligned}$$

it follows that

$$\begin{aligned}&\int _{B_{r}(\vec {e}_1){\setminus } (B_{r}(\vec {e}_1))^*}\frac{1-|y|^{\mu }}{|y|^{N-2s}|y-\vec {e}_1|^{N+2s}}\textrm{d}y=O(r^{2-2s})\rightarrow 0\ \textrm{as}\ r\rightarrow 0, \\&\int _{(B_{r}(\vec {e}_1))^*{\setminus } B_{r}(\vec {e}_1)}\frac{1-|y|^{\mu }}{|y|^{N-2s}|y-\vec {e}_1|^{N+2s}}\textrm{d}y=O(r^{2-2s})\rightarrow 0\ \textrm{as}\ r\rightarrow 0. \end{aligned}$$

Substituting the estimates above into (A.2), we obtain that

$$\begin{aligned}&\int _{B_1(0){\setminus } B_{r}(\vec {e}_1)}\frac{|y|^{\mu }-1}{|y|^{\mu }|y-\vec {e}_1|^{N+2s}}\textrm{d}y \nonumber \\ =&\int _{(\mathbb {R}^N\backslash B_1(0)){\setminus } B_{r}(\vec {e}_1))}\frac{1-|y|^{\mu }}{|y|^{N-2s}|y-\vec {e}_1|^{N+2s}}\textrm{d}y+O(r^{2-2s})\ \textrm{as}\ r\rightarrow 0. \end{aligned}$$
(A.3)

Putting (A.3) into (A.1) yields

$$\begin{aligned} \frac{1}{2}(-\Delta )^sh_\mu =&\frac{1}{|x|^{\mu +2s}}\lim _{r\rightarrow 0}\int _{(\mathbb {R}^N{\setminus } B_1(0)){\setminus } B_{r}(\vec {e}_1)}\frac{|y|^{\mu }-1}{|y-\vec {e}_1|^{N+2s}}\Bigg ( \frac{1}{|y|^{\mu }}-\frac{1}{|y|^{N-2s}}\Bigg )\textrm{d}y\nonumber \\ =&\frac{1}{|x|^{\mu +2s}}\int _{\mathbb {R}^N{\setminus } B_1(0)}\frac{|y|^{\mu }-1}{|y-\vec {e}_1|^{N+2s}}\Bigg ( \frac{1}{|y|^{\mu }}-\frac{1}{|y|^{N-2s}}\Bigg )\textrm{d}y :=A_\mu \frac{1}{|x|^{\mu +2s}}, \end{aligned}$$
(A.4)

where \(\vec {e}_1\) is not a singular point for the last integral in (A.4) since

$$\begin{aligned}\frac{|y|^{\mu }-1}{|y-\vec {e}_1|^{N+2s}}\Bigg ( \frac{1}{|y|^{\mu }}-\frac{1}{|y|^{N-2s}}\Bigg )=O\Bigg (\frac{1}{|x-\vec {e}_1|^{N+2s-2}}\Bigg )\ \ \textrm{as}\ y\rightarrow \vec {e}_1.\end{aligned}$$

Noting the asymptotic behavior of \(\frac{|y|^{\mu }-1}{|y-\vec {e}_1|^{N+2s}}( \frac{1}{|y|^{\mu }}-\frac{1}{|y|^{N-2s}})\) as \(|y|\rightarrow \infty \), it is easy to check that \(A_\mu \) satisfies

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} A_\mu \in (0,+\infty ),&{}\quad \textrm{if }\ 0<\mu<N-2s,\\ A_\mu =0, &{} \quad \textrm{if }\ \mu =N-2s;\\ A_\mu \in (-\infty ,0),&{} \quad \textrm{if }\ N-2s<\mu <N\\ A_\mu =-\infty ,&{}\quad \textrm{if}\ \mu \ge N. \end{array} \right. \end{aligned} \end{aligned}$$
(A.5)

Now we are ready to estimate \((-\Delta )^sw_{\mu }\) according to different cases stated in (A.5).

Case 1. \(\mu \in (0,N){\setminus }\{N-2s\}\).

By changing variable as in (A.1), we have

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\left| (-\Delta )^sw_\mu -(-\Delta )^sh_\mu \right| \\ \quad \le&\frac{1}{|x|^{\mu +2s}}\Bigg |\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_{r}(\vec {e}_1)}\frac{(|x|^{-2}+1)^{-\frac{\mu }{2}}-(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}-1+|y|^{-\mu }}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\\:=&\frac{1}{|x|^{\mu +2s}}\Bigg |\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_{r}(\vec {e}_1)}\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |. \end{aligned} \end{aligned}$$
(A.6)

For any \(M>2\) and \(\rho '\in (r,1/2)\), letting \(|x|>2M\), we have

$$\begin{aligned}&\Bigg |\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_{r}(\vec {e}_1)}\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\nonumber \\ \le&\int _{|y-\vec {e}_1|>M}\frac{|L(x,y)|}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y+\Bigg |\lim _{r\rightarrow 0}\int _{r<|y-\vec {e}_1|<\rho '}\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\nonumber \\&+\int _{\{\rho '\le |y-\vec {e}_1|\le M\}\cap \{|y|\ge \rho '\}}\frac{|L(x,y)|}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y +\int _{\{\rho '\le |y-\vec {e}_1|\le M\}\cap \{|y|<\rho '\}}\frac{|L(x,y)|}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y. \end{aligned}$$
(A.7)

Clearly, \(|L(x,y)|\le 4\) for \(|y-\vec {e}_1|>M\), then

$$\begin{aligned} \int _{|y-\vec {e}_1|>M}\frac{|L(x,y)|}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\le 4\int _{|y-\vec {e}_1|>M}\frac{1}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\le \frac{C}{M^{2s}}. \end{aligned}$$
(A.8)

For \(|y-\vec {e}_1|<\rho '\), we have \(1/2 \le |y|\le 3/2\), then by Taylor expansion, we have

$$\begin{aligned} |y|^{-\mu }-|\vec {e}_1|=-\mu \vec {e}_1\cdot (y-\vec {e}_1)+O(|y-\vec {e}_1|^2), \end{aligned}$$
(A.9)
$$\begin{aligned}(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}-(|x|^{-2}+|\vec {e}_1|^2)^{-\frac{\mu }{2}}= -\mu (|x|^{-2}+1)^{-\frac{\mu }{2}-1}\vec {e}_1\cdot (y-\vec {e}_1)+O(|y-\vec {e}_1|^2). \end{aligned}$$

By symmetry,

$$\begin{aligned} \int _{r<|y-\vec {e}_1|<\rho '}\frac{\vec {e}_1\cdot (y-\vec {e}_1)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y=0. \end{aligned}$$
(A.10)

Therefore,

$$\begin{aligned} \Bigg |\lim _{r\rightarrow 0}\int _{r<|y-\vec {e}_1|<\rho '}\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg | \le C\lim _{r\rightarrow 0}\int _{r<|y-\vec {e}_1|<\rho '}\frac{1}{|y-\vec {e}_1|^{N+2s-2}}\textrm{d}y\le C(\rho ')^{2-2s}. \end{aligned}$$
(A.11)

For \(y\in \{\rho '\le |y-\vec {e}_1|\le M\}\cap \{|y|\ge \rho '\}\), we have

$$\begin{aligned}{} & {} |(|x|^{-2}+1)^{-\mu }-1|\le C|x|^{-2},\\{} & {} \big |(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}-|y|^{-\mu }\big |\le \frac{C}{(\rho ')^{\mu +1}}|x|^{-2}, \end{aligned}$$

and thereby,

$$\begin{aligned} \int _{\{\rho '\le |y-\vec {e}_1|\le M\}\cap \{|y|\ge \rho '\}}\frac{|L(x,y)|}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y&\le C|x|^{-2}\Bigg (1+\frac{C}{(\rho ')^{\mu +1}}\Bigg )\int _{\{\rho '\le |y-\vec {e}_1|\}}\frac{1}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\nonumber \\&\le C|x|^{-2}\Bigg (1+\frac{C}{(\rho ')^{\mu +1}}\Bigg )\frac{1}{(\rho ')^{2s}}. \end{aligned}$$
(A.12)

For \(y\in \{\rho '\le |y-\vec {e}_1|\le M\}\cap \{|y|<\rho '\}\), then \(|y-\vec {e}_1|\ge 1-|y|\ge 1/2\) and

$$\begin{aligned}|L(x,y)|=1-(|x|^{-2}+1)^{-\frac{\mu }{2}}+|y|^{-\mu }-(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}\le 1+|y|^{-\mu },\end{aligned}$$

and consequently,

$$\begin{aligned} \int _{\{\rho '\le |y-\vec {e}_1|\le M\}\cap \{|y|<\rho '\}}\frac{|L(x,y)|}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\le&\frac{1}{(1/2)^{N+2s}}\int _{\{|y|<\rho '\}}(1+|y|^{-\mu })\textrm{d}y\nonumber \\ =&C(\rho ')^{N}+C(\rho ')^{N-\mu }. \end{aligned}$$
(A.13)

As a result, we conclude from (A.17)–(A.8) and (A.11)–(A.13) that

$$\begin{aligned}&\Bigg |\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_{r}(\vec {e}_1)}\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\nonumber \\&\quad \le C\Bigg (\frac{1}{M^{2s}}+(\rho ')^{2-2s}+(\rho ')^{N}+(\rho ')^{N-\mu }+|x|^{-2}\big (1+\frac{1}{(\rho ')^{\mu +1}}\big )\frac{1}{(\rho ')^{2s}}\Bigg ) \end{aligned}$$

for a constant \(C>0\) independent of \(M>2\) and \(\rho '\in (0,1/2)\). Letting \(M\rightarrow +\infty \) and \(\rho '\rightarrow 0_+\), we have

$$\begin{aligned}&\lim _{|x|\rightarrow \infty }\Bigg |\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_{r}(\vec {e}_1)}\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\nonumber \\&\quad \le \mathop {\lim }\limits _{\begin{array}{c} M\rightarrow +\infty \\ {\rho '\rightarrow 0_+} \end{array}} \lim _{|x|\rightarrow \infty }C\Bigg (\frac{1}{M^{2s}}+(\rho ')^{2-2s}+(\rho ')^{N}+(\rho ')^{N-\mu } +|x|^{-2}\big (1+\frac{1}{(\rho ')^{\mu +1}}\big )\frac{1}{(\rho ')^{2s}}\Bigg )\nonumber \\&\quad =0. \end{aligned}$$
(A.14)

Then there exists \(R_\mu >0\) such that

$$\begin{aligned} \begin{aligned} \Bigg |\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_{r}(\vec {e}_1)}\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\le \frac{1}{2}|A_\mu |,\ \ |x|>R_\mu . \end{aligned} \end{aligned}$$
(A.15)

Putting (A.1)–(A.6) and (A.15) together, we infer that

$$\begin{aligned}&0<\frac{A_\mu }{2}\frac{1}{|x|^{\mu +2s}}\le \frac{1}{2}(-\Delta )^sw_\mu \le \frac{3A_\mu }{2}\frac{1}{|x|^{\mu +2s}},\ \textrm{if }\ |x|>R_\mu \ \textrm{and}\ \mu \in (0,N-2s);\\&\frac{3A_\mu }{2}\frac{1}{|x|^{\mu +2s}}\le \frac{1}{2}(-\Delta )^sw_\mu \le \frac{A_\mu }{2}\frac{1}{|x|^{\mu +2s}}<0,\ \textrm{if }\ |x|>R_\mu \ \textrm{and}\ \mu \in (N-2s,N). \end{aligned}$$

Case 2. \(\mu \ge N\).

Also by changing variable as in (A.1), there holds

$$\begin{aligned} \frac{1}{2}(-\Delta )^sw_\mu =&\frac{1}{|x|^{\mu +2s}}\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_{r}(\vec {e}_1)}\frac{(|x|^{-2}+1)^{-\frac{\mu }{2}}-(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\nonumber \\ =&\frac{1}{|x|^{\mu +2s}}\Bigg (\lim _{r\rightarrow 0}\int _{(\mathbb {R}^N{\setminus } B_{1/2}(0))\backslash B_{r}(\vec {e}_1) }\frac{(|x|^{-2}+1)^{-\frac{\mu }{2}}-(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\nonumber \\&\ +\int _{B_{1/2}(0)}\frac{(|x|^{-2}+1)^{-\frac{\mu }{2}}}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y -\int _{B_{1/2}(0)}\frac{(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg ). \end{aligned}$$
(A.16)

Same as (A.6), we denote \(L(x,y):=(|x|^{-2}+1)^{-\frac{\mu }{2}}-(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}-1+|y|^{-\mu }\). For any \(M>2\) and \(\rho '\in (r,1/2)\), letting \(|x|>2M\), be the same arguments as A.8, (A.11) and (A.12), we have

$$\begin{aligned}&\Bigg |\lim _{r\rightarrow 0}\int _{(\mathbb {R}^N{\setminus } B_{1/2}(0))\backslash B_{r}(\vec {e}_1) }\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\nonumber \\&\quad \le \int _{|y-\vec {e}_1|>M}\frac{|L(x,y)|}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y+\Bigg |\lim _{r\rightarrow 0}\int _{r<|y-\vec {e}_1|<\rho '}\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\nonumber \\&\qquad +\int _{\{\rho '\le |y-\vec {e}_1|\le M\}\cap \{|y|\ge \frac{1}{2}\}}\frac{|L(x,y)|}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\nonumber \\&\quad \le \frac{C}{M^{2s}}+C(\rho ')^{2-2s}+C|x|^{-2}\frac{1}{(\rho ')^{2s}}, \end{aligned}$$
(A.17)

which implies that

$$\begin{aligned}&\lim _{|x|\rightarrow \infty }\Bigg |\lim _{r\rightarrow 0}\int _{(\mathbb {R}^N{\setminus } B_{1/2}(0))\backslash B_{r}(\vec {e}_1) }\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\nonumber \\&\quad \le \mathop {\lim }\limits _{\begin{array}{c} M\rightarrow +\infty \\ {\rho '\rightarrow 0_+} \end{array}}\lim _{|x|\rightarrow \infty }C\Bigg (\frac{1}{M^{2s}}+(\rho ')^{2-2s}+|x|^{-2}\frac{1}{(\rho ')^{2s}}\Bigg )=0. \end{aligned}$$
(A.18)

In view of (A.9) and (A.10), the following integral converges to a constant independent of x as \(r\rightarrow 0\),

$$\begin{aligned}\lim _{r\rightarrow 0}\int _{(\mathbb {R}^N{\setminus } B_{1/2}(0))\backslash B_{r}(\vec {e}_1)}\frac{1-|y|^{-\mu }}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y:=C^{*}\in {\mathbb {R}},\end{aligned}$$

and thereby from (A.18),

$$\begin{aligned}&\lim _{|x|\rightarrow \infty }\lim _{r\rightarrow 0}\int _{(\mathbb {R}^N{\setminus } B_{1/2}(0))\backslash B_{r}(\vec {e}_1) }\frac{(|x|^{-2}+1)^{-\frac{\mu }{2}}-(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y=C^*. \end{aligned}$$
(A.19)

Obviously,

$$\begin{aligned} \int _{B_{1/2}(0)}\frac{(|x|^{-2}+1)^{-\frac{\mu }{2}}}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\le 2^{2s}\omega _N(|x|^{-2}+1)^{-\frac{\mu }{2}}\le 2^{2s}\omega _N. \end{aligned}$$
(A.20)

Letting \(|x|>4\), we have

$$\begin{aligned}&\int _{B_{1/2}(0)}\frac{(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\nonumber \\&\quad \le 2^{N+2s}\int _{B_{1/2}(0)}(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}\textrm{d}y =2^{N+2s}|x|^{\mu -N}\int _{B_{|x|/2}(0)}\frac{1}{(1+|y|^2)^{\frac{\mu }{2}}}\textrm{d}y\nonumber \\&\quad \le 2^{N+2s}|x|^{\mu -N}\Bigg (\int _{B_1(0)}\frac{1}{(1+|y|^2)^{\frac{\mu }{2}}}+\omega _N\int ^{|x|/2}_1\frac{1}{r^{\mu -N+1}}\textrm{d}r\Bigg )\nonumber \\&\quad \le \left\{ \begin{array}{ll} 2^{N+2s}(\omega _N\ln |x|+C),&{} \quad \mu =N, \\ 2^{N+2s}(\frac{\omega _N}{\mu -N}+C)|x|^{\mu -N},&{} \quad \mu >N, \end{array} \right. \end{aligned}$$
(A.21)

where \(\omega _N:=\int _{\partial B_1(0)}dS\). On the other hand,

$$\begin{aligned}&\int _{B_{1/2}(0)}\frac{(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\nonumber \\&\quad \ge (1/2)^{N+2s}\int _{B_{1/2}(0)}(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}\textrm{d}y=(1/2)^{N+2s}|x|^{\mu -N}\int _{B_{|x|/2}(0)}\frac{1}{(1+|y|^2)^{\frac{\mu }{2}}}\textrm{d}y\nonumber \\&\quad \ge (1/2)^{N+2s}|x|^{\mu -N}\Bigg (\int _{B_1(0)}\frac{1}{(1+|y|^2)^{\frac{\mu }{2}}}+\frac{\omega _N}{2^\mu }\int ^{|x|/2}_1\frac{1}{r^{\mu -N+1}}\textrm{d}r\Bigg )\nonumber \\&\quad \ge \left\{ \begin{array}{ll} (1/2)^{N+2s}\left( \frac{\omega _N}{2^N}\ln |x|-C\right) ,&{} \quad \mu =N, \\ \frac{N\omega _N}{2^{2N+2s}}|x|^{\mu -N},&{} \quad \mu >N. \end{array} \right. \end{aligned}$$
(A.22)

Summing up the estimates (A.16)–(A.22) above, there exists \(R_\mu >0\) and \({\tilde{C}}_1,{\tilde{C}}_2, {\tilde{C}}_3, {\tilde{C}}_4>0\) such that

$$\begin{aligned} \begin{aligned}&-\frac{{\tilde{C}}_2\ln |x|}{|x|^{N+2s}}\le \frac{1}{2}(-\Delta )^sw_\mu \le -\frac{{\tilde{C}}_1\ln |x|}{|x|^{N+2s}}<0,\ \textrm{if }\ |x|>R_\mu \ \textrm{and}\ \mu =N;\\&-\frac{{\tilde{C}}_4}{|x|^{N+2s}}\le \frac{1}{2}(-\Delta )^sw_\mu \le -\frac{{\tilde{C}}_3}{|x|^{N+2s}}<0,\ \textrm{if }\ |x|>R_\mu \ \textrm{and}\ \mu >N, \end{aligned} \end{aligned}$$

where

$$\begin{aligned}{\tilde{C}}_1=\frac{\omega _N}{2^{2N+2s+1}},\, {\tilde{C}}_2=2^{N+2s+1}\omega _N,\, {\tilde{C}}_3=\frac{\omega _N}{2^{2N+2s+1}},\,{\tilde{C}}_4=\frac{2^{N+2s+1}}{\mu -N}\omega _N.\end{aligned}$$

Case 3. \(\mu =N-2s\).

In this case, \(w_\mu =(1+|x|^2)^{-\frac{N-2s}{2}}\) is the fundamental solution of the critical fractional equation

$$\begin{aligned}(-\Delta )^su=C_{N-2s}u^{2_s^*-1}\end{aligned}$$

for some constant \(C_{N-2s}>0\) (see [6]).

As a consequence, the proof is completed. \(\square \)

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Deng, Y., Peng, S. & Yang, X. Existence and decays of solutions for fractional Schrödinger equations with general potentials. Calc. Var. 63, 128 (2024). https://doi.org/10.1007/s00526-024-02728-2

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