Skip to main content

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

• Published:

## 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.

This is a preview of subscription content, log in via an institution to check access.

## Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

## References

1. Alves, C.O., Miyagaki, O.H.: Existence and concentration of solution for a class of fractional elliptic equation in $${\mathbb{R} }^N$$ via penalization method. Calc. Var. Partial Differ. Equ. 55(3), 1–19 (2016)

2. Ambrosio, V.: Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method. Ann. Mat. Pura Appl. 196(6), 2043–2062 (2017)

3. An, X., Duan, L., Peng, Y.: Semi-classical analysis for fractional Schrödinger equations with fast decaying potentials. Appl. Anal. 101(14), 5138–5155 (2021)

4. An, X., Peng, S., Xie, C.: Semi-classical solutions for fractional Schrödinger equations with potential vanishing at infinity. J. Math. Phys. 60(2), 021501 (2019)

5. Chen, W., Li, Y., Ma, P.: The Fractional Laplacian. World Scientific Publishing, Singapore (2019)

6. Chen, W., Li, C., Ou, B.: Classification of solutions for an integral equation. Commun. Pure Appl. Math. 59(3), 330–343 (2006)

7. Dávila, J., del Pino, M., Wei, J.: Concentrating standing waves for the fractional nonlinear Schrödinger equation. J. Differ. Equ. 256(2), 858–892 (2014)

8. del Pino, M., Felmer, P.L.: Local Mountain pass for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4(2), 121–137 (1996)

9. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)

10. Dipierro, S., Medina, M., Valdinoci, E.: Fractional elliptic problems with critical growth in the whole of $${\mathbb{R} }^n$$. Lecture Notes, p. 162. Edizioni della Normale (2017)

11. Felmer, P., Quaas, A., Tan, J.: Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinb. Sect. A 142(6), 1237–1262 (2012)

12. Frank, L., Lenzmann, E., Silvestre, L.: Uniqueness of radial solutions for the fractional Laplacians. Commun. Pure. Appl. Math. 69, 1671–1726 (2016)

13. Frank, L., Seiringer, R.: Nonlinear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255, 3407–3430 (2008)

14. Laskin, N.: Fractional quantum mechanics and Levy path integrals. Phys. Lett. A 268, 298–305 (2000)

15. Palatucci, G., Pisante, A.: Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc. Var. Partial Differ. Equ. 50, 799–829 (2014)

16. Secchi, S.: Ground state solutions for nonlinear fractional Schrödinger equations in $${\mathbb{R} }^N$$. J. Math. Phys. 54(3), 031501 (2013)

17. Servadei, R., Valdinoci, E.: Weak and viscosity solutions of the fractional Laplace equation. Publ. Mat. 58(1), 133–154 (2014)

18. Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. PhD Thesis, University of Texas at Austin (2005)

Download references

## Author information

Authors

### Corresponding author

Correspondence to Shuangjie Peng.

## Additional information

Communicated by P. H. Rabinowitz.

### Publisher's Note

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. $$\square$$

## Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

## About this article

### Cite this article

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

Download citation

• Received:

• Accepted:

• Published:

• DOI: https://doi.org/10.1007/s00526-024-02728-2