June  2022, 21(6): 2079-2100. doi: 10.3934/cpaa.2022050

Liouville type theorem for Hartree-Fock Equation on half space

School of Mathematics and Statistics, Shenzhen University, Shenzhen, Guangdong, 518060, China

* Corresponding author

Received  October 2021 Revised  January 2022 Published  June 2022 Early access  March 2022

Fund Project: The second author is supported by NSFC grant 12171212

In this paper, we study the Liouville type theorem for the following Hartree-Fock equation in half space
$ \begin{align*} \begin{cases} - \Delta {u_i}(y) = \sum\limits_{j = 1}^n {{\int _{\partial \mathbb{R}_ + ^N}}} \frac{{{u_j}(\bar x, 0){F_1}({u_j}(\bar x, 0))}} {{|(\bar x, 0) - y{|^{N - \alpha }}}}d\bar x{f_2}({u_i}(y)) \\ \qquad \qquad \qquad + \sum\limits_{j = 1}^n {{\int _{\partial \mathbb{R}_ + ^N}}} \frac{{{u_j}(\bar x, 0){F_2}({u_i}(\bar x, 0))}} {{|(\bar x, 0) - y{|^{N - \alpha }}}}d\bar x{f_1}({u_j}(y)), \ y \in \mathbb{R}_ + ^N, \hfill \\ \frac{{\partial {u_i}}} {{\partial \nu }}(\bar x, 0) = \sum\limits_{j = 1}^n {{\int _{ \mathbb{R}_ + ^N}}} \frac{{{u_j}(y){G_1}({u_j}(y))}} {{|(\bar x, 0) - y{|^{N - \alpha }}}}dy{g_2}({u_i}(\bar x, 0)) \\ \qquad \qquad \qquad + \sum\limits_{j = 1}^n {{\int _{ \mathbb{R}_ + ^N}}} \frac{{{u_j}(y){G_2}({u_i}(y))}} {{|(\bar x, 0) - y{|^{N - \alpha }}}}dy{g_1}({u_j}(\bar x, 0)), \quad \quad(\bar x, 0) \in \partial \mathbb{R}_ + ^N, \end{cases} \end{align*} $
where
$ \mathbb{R}_+^N = \{x\in{\mathbb{R}^N}: x_N > 0\}, f_1, f_2, g_1, g_2, F_1, F_2, G_1, G_2 $
are some nonlinear functions. Under some assumptions on the nonlinear functions
$ F, G, f, g $
, we will prove the above equation only possesses trivial positive solution. We use the moving plane method in an integral form to prove our result.
Citation: Xiaomei Chen, Xiaohui Yu. Liouville type theorem for Hartree-Fock Equation on half space. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2079-2100. doi: 10.3934/cpaa.2022050
References:
[1]

W. ChenY. Fang and C. Li, Super poly-harmonic property of solutions for navier boundary problems on a half space, J. Funct. Anal., 265 (2013), 1522-1555.  doi: 10.1016/j.jfa.2013.06.010.

[2]

W. ChenY. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013.

[3]

W. ChenC. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038.

[4]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[5]

J. Dou and M. Zhu, Sharp Hardy-Littlewood-Sobolev inequality on the upper half space, Int. Math. Res. Not. IMRN, 3 (2015), 651-687.  doi: 10.1093/imrn/rnt213.

[6]

V. Fock, Naherungsmethode zur Losung des quantenmechanischen Mehrkorperproblems, Z. Phys., 61 (1930), 126-148.  doi: 10.1007/BF01340294.

[7]

D. Hartree, The wave mechanics of an atom with a non-coulomb central field, Part Ⅰ. Theory and methods, Proc. Camb. Phil. Soc., 24 (1928), 89-312.  doi: 10.1017/S0305004100011919.

[8]

H. Li, Liouville Type Theorem for Hartree Equations on Half Spaces, Acta Math. Sci.(in Chinese), 41A (2021), 388–401.

[9]

J. LiuY. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbb R^N$, J. Differ. Equ., 225 (2006), 685-709.  doi: 10.1016/j.jde.2005.10.016.

[10]

S. Luo and W. Zou, Liouville theorems for integral systems related to fractional lane-emden systems in $\mathbb R_+^N$, Differ. Integral Equ., 29 (2016), 1107-1138. 

[11]

J. C. Slater, Note on Hartree's method, Phys. Rev., 35 (1930), 210-211.  doi: 10.1103/physrev.35.210.2.

[12]

J. Yang and X. Yu, Liouville type theorems for Hartree and Hartree-Fock equations, Nonlinear Anal., 183 (2019), 191-213.  doi: 10.1016/j.na.2019.01.012.

[13]

X. Yu, Liouville type theorems for singular integral equations and integral systems, Commun. Pure Appl. Anal., 15 (2016), 1825-1840.  doi: 10.3934/cpaa.2016017.

[14]

X. Yu, Liouville type theorem for some nonlocal elliptic equations, J. Differ. Equ., 263 (2017), 6805-6820.  doi: 10.1016/j.jde.2017.07.028.

show all references

References:
[1]

W. ChenY. Fang and C. Li, Super poly-harmonic property of solutions for navier boundary problems on a half space, J. Funct. Anal., 265 (2013), 1522-1555.  doi: 10.1016/j.jfa.2013.06.010.

[2]

W. ChenY. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013.

[3]

W. ChenC. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038.

[4]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[5]

J. Dou and M. Zhu, Sharp Hardy-Littlewood-Sobolev inequality on the upper half space, Int. Math. Res. Not. IMRN, 3 (2015), 651-687.  doi: 10.1093/imrn/rnt213.

[6]

V. Fock, Naherungsmethode zur Losung des quantenmechanischen Mehrkorperproblems, Z. Phys., 61 (1930), 126-148.  doi: 10.1007/BF01340294.

[7]

D. Hartree, The wave mechanics of an atom with a non-coulomb central field, Part Ⅰ. Theory and methods, Proc. Camb. Phil. Soc., 24 (1928), 89-312.  doi: 10.1017/S0305004100011919.

[8]

H. Li, Liouville Type Theorem for Hartree Equations on Half Spaces, Acta Math. Sci.(in Chinese), 41A (2021), 388–401.

[9]

J. LiuY. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbb R^N$, J. Differ. Equ., 225 (2006), 685-709.  doi: 10.1016/j.jde.2005.10.016.

[10]

S. Luo and W. Zou, Liouville theorems for integral systems related to fractional lane-emden systems in $\mathbb R_+^N$, Differ. Integral Equ., 29 (2016), 1107-1138. 

[11]

J. C. Slater, Note on Hartree's method, Phys. Rev., 35 (1930), 210-211.  doi: 10.1103/physrev.35.210.2.

[12]

J. Yang and X. Yu, Liouville type theorems for Hartree and Hartree-Fock equations, Nonlinear Anal., 183 (2019), 191-213.  doi: 10.1016/j.na.2019.01.012.

[13]

X. Yu, Liouville type theorems for singular integral equations and integral systems, Commun. Pure Appl. Anal., 15 (2016), 1825-1840.  doi: 10.3934/cpaa.2016017.

[14]

X. Yu, Liouville type theorem for some nonlocal elliptic equations, J. Differ. Equ., 263 (2017), 6805-6820.  doi: 10.1016/j.jde.2017.07.028.

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