doi: 10.3934/dcds.2022070
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Classifications of positive solutions to an integral system involving the multilinear fractional integral inequality

1. 

School of Statistics, Xi'an University of Finance and Economics, Xi'an, Shaanxi 710100, China

2. 

School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, Shaanxi 710119, China

*Corresponding author: Jingbo Dou

Received  November 2021 Revised  April 2022 Early access May 2022

In this paper, we classify the positive solutions to the following integral system
$ \begin{eqnarray*} \begin{cases} u_s(x_s) = \int_{\mathbb{R}^{N(k-1)}}\frac{\prod\limits_{j\neq s}u^{p_j}_j(x_j)}{\prod_{1\leq i<j\leq k}|x_i-x_j|^{N-h_{ij}}}dX_{\widehat{s}}, \\ u_s\geq0, \; \rm{in}\quad\mathbb{R}^N, \quad s = 1, 2, \cdots, k, \end{cases} \end{eqnarray*} $
where
$ N\ge1, p_j>1, 0<h_{ij}<N $
for all
$ i, j\in\{1, 2, \cdots, k\} $
. Up to a positive constant multiplier, this system is the Euler-Lagrangian equations associated to the multilinear fractional integral inequality established by Beckner. Employing the method of moving spheres, we give the explicit form of positive solutions to the above system with
$ p_j = \frac{\sum_{1\leq i<j\leq k} \ \ h_{ij}-(k-3)N}{(k-1)N-\sum_{1\leq i<j\leq k} \ \ h_{ij}} $
satisfying
$ \sum\limits^k_{j = 1}\frac{1}{p_j+1} = \sum\limits_{1\leq i<j\leq k}\frac{N-h_{ij}}{N}, $
and show the nonexistence of positive solutions for
$ p_j>1 $
with
$ \sum\limits^k_{j = 1}\frac{1}{p_j+1}>\sum\limits_{1\leq i<j\leq k}\frac{N-h_{ij}}{N}. $
Citation: Huaiyu Zhou, Jingbo Dou. Classifications of positive solutions to an integral system involving the multilinear fractional integral inequality. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022070
References:
[1]

W. Beckner, Geometric inequalities in Fourier analysis, Essays on Fourier Analysis in Honor of Elias M. Stein, Princeton Math. Ser., Princeton Univ. Press, Princeton, NJ, 42 (1995), 36-68. 

[2]

W. Beckner, Functionals for multilinear fractional embedding, Acta Math. Sin. (Engl. Ser.), 31 (2015), 1-28.  doi: 10.1007/s10114-015-4321-6.

[3]

W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math., 138 (1993), 213-242.  doi: 10.2307/2946638.

[4]

G. CaristiL. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systemsand related Liouville theorems, Milan J. Math., 76 (2008), 27-67.  doi: 10.1007/s00032-008-0090-3.

[5]

L. ChenZ. LiuG. Lu and C. Tao, Stein-Weiss inequalities with the fractional Poisson kernel, Rev. Mat. Iberoam., 36 (2020), 1289-1308.  doi: 10.4171/rmi/1167.

[6]

L. ChenZ. LiuG. Lu and C. Tao, Reverse Stein-Weiss inequalities and existence of their extremal functions, Trans. Amer. Math. Soc., 370 (2018), 8429-8450.  doi: 10.1090/tran/7273.

[7]

L. ChenG. Lu and C. Tao, Reverse Stein-Weiss inequalities on the upper half space and the existence of their extremals, Adv. Nonlinear Stud., 19 (2019), 475-494.  doi: 10.1515/ans-2018-2038.

[8]

L. ChenG. Lu and C. Tao, Hardy-Littlewood-Sobolev inequalities with the fractional Poisson kernel and their applications in PDEs, Acta Math. Sin. (Engl. Ser.), 35 (2019), 853-875.  doi: 10.1007/s10114-019-8417-2.

[9]

L. ChenG. Lu and C. Tao, Existence of extremal functions for the Stein-Weiss inequalities on the Heisenberg group, J. Funct. Anal., 277 (2019), 1112-1138.  doi: 10.1016/j.jfa.2019.01.002.

[10]

S. Chen, A new family of sharp conformally invariant integral inequalities, Int. Math. Res. Not. IMRN, (2014), 1205–1220. doi: 10.1093/imrn/rns248.

[11]

W. Chen and C. Li, An integral system and the Lane-Emdem conjecture, Disc. Cont. Dyn. Sys., 24 (2009), 1167-1184.  doi: 10.3934/dcds.2009.24.1167.

[12]

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

[13]

W. ChenC. Li and B. Ou, Classiffication of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65.  doi: 10.1081/PDE-200044445.

[14]

M. Christ, On the restriction of the Fourier transform to curves: Endpoint results and the degenerate case, Trans. Amer. Math. Soc., 287 (1985), 223-238.  doi: 10.1090/S0002-9947-1985-0766216-6.

[15]

M. Christ, Estimates for the $k$-plane transform, Indiana Univ. Math. J., 33 (1984), 891-910.  doi: 10.1512/iumj.1984.33.33048.

[16]

J. Dou, Weighted Hardy-Littlewood-Sobolev inequalities on the upper half space, Comm. Cont. Math., 18 (2016), 1550067, 20 pp. doi: 10.1142/S0219199715500674.

[17]

J. DouQ. Guo and M. Zhu, Subcritical approach to sharp Hardy-Littlewood-Sobolev type inequalities on the upper half space, Adv. Math., 312 (2017), 1-45.  doi: 10.1016/j.aim.2017.07.009.

[18]

J. DouQ. Guo and M. Zhu, Negative power nonlinear integral equations on bounded domains, J. Diff. Eqs., 269 (2020), 10527-10557.  doi: 10.1016/j.jde.2020.07.021.

[19]

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

[20]

J. Dou and M. Zhu, Reversed Hardy-Littewood-Sobolev inequality, Int. Math. Res. Not. IMRN, (2015), 9696–9726. doi: 10.1093/imrn/rnu241.

[21]

J. Dou and M. Zhu, Nonlinear integral equations on bounded domains, J. Funct. Anal., 277 (2019), 111-134.  doi: 10.1016/j.jfa.2018.05.020.

[22]

S. W. Drury, $L^p$ estimates for the $X$-ray transform, Illinois J. Math., 27 (1983), 125-129. 

[23]

R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39 (2010), 85-99.  doi: 10.1007/s00526-009-0302-x.

[24]

R. L. Frank and E. H. Lieb, Sharp constants in several inequalities on the Heisenberg group, Ann. of Math., 176 (2012), 349-381.  doi: 10.4007/annals.2012.176.1.6.

[25]

R. L. Frank and E. H. Lieb, A New, Rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality, Spectral Theory, Function Spaces and Inequalities, Oper. Theory Adv. Appl., Birkhäuser/Springer Basel AG, Basel, 219 (2011), 55-67.  doi: 10.1007/978-3-0348-0263-5_4.

[26]

M. Gluck, Subcritical approach to conformally invariant extension operators on the upper half space, J. Funct. Anal., 278 (2020), 108082, 46 pp. doi: 10.1016/j.jfa.2018.08.012.

[27]

M. Gluck and M. Zhu, An extension operator on bounded domains and applications, Calc. Var. PDE, 58 (2019), Paper No. 79, 27 pp. doi: 10.1007/s00526-019-1513-4.

[28]

L. Grafakos and C. Morpurgo, A Selberg integral formula and applications, Pacific J. Math., 191 (1999), 85-94.  doi: 10.2140/pjm.1999.191.85.

[29]

L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1976), 1061-1083.  doi: 10.2307/2373688.

[30]

Y. Han and M. Zhu, Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications, J. Diff. Eqs., 260 (2016), 1-25.  doi: 10.1016/j.jde.2015.06.032.

[31]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.  doi: 10.4310/MRL.2007.v14.n3.a2.

[32]

F. HangX. Wang and X. Yan, Sharp integral inequalities for harmonic functions, Comm. Pure Appl. Math., 61 (2008), 54-95.  doi: 10.1002/cpa.20193.

[33]

G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Zeitschr., 27 (1928), 565-606.  doi: 10.1007/BF01171116.

[34]

G. H. Hardy and J. E. Littlewood, On certain inequalities connected with the calculus of variations, J. London Math. Soc., 5 (1930), 34-39.  doi: 10.1112/jlms/s1-5.1.34.

[35]

Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc. (JEMS), 6 (2004), 153-180.  doi: 10.4171/jems/6.

[36]

Y. Y. Li and L. Zhang, Liouville type theorems and Harnack type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), 27-87.  doi: 10.1007/BF02786551.

[37]

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8.

[38]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.  doi: 10.2307/2007032.

[39]

Q. N. Ngô and V. H. Nguyen, Sharp reversed Hardy-Littlewood-Sobolev inequality on $\mathbb{R}^n$, Israel J. Math., 220 (2017), 189-223.  doi: 10.1007/s11856-017-1515-x.

[40]

Q. N. Ngô and V. H. Nguyen, Sharp reversed Hardy-Littlewood-Sobolev inequality on the half space $\mathbb{R}^n_+$, Int. Math. Res. Not. IMRN, (2017), 6187–6230. doi: 10.1093/imrn/rnw108.

[41]

F. Pezzolo, On some multilinear type integral systems, Nonl. Anal., 198 (2020), 111890, 23 pp. doi: 10.1016/j.na.2020.111890.

[42]

S. L. Sobolev, On a theorem of functional analysis, A. M. S. Transl. Ser. 2, 34 (1963), 39-68. 

[43]

M. Zhu, Prescribing integral curvature equation, Diff. Inte. Eqs., 29 (2016), 889-904. 

show all references

References:
[1]

W. Beckner, Geometric inequalities in Fourier analysis, Essays on Fourier Analysis in Honor of Elias M. Stein, Princeton Math. Ser., Princeton Univ. Press, Princeton, NJ, 42 (1995), 36-68. 

[2]

W. Beckner, Functionals for multilinear fractional embedding, Acta Math. Sin. (Engl. Ser.), 31 (2015), 1-28.  doi: 10.1007/s10114-015-4321-6.

[3]

W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math., 138 (1993), 213-242.  doi: 10.2307/2946638.

[4]

G. CaristiL. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systemsand related Liouville theorems, Milan J. Math., 76 (2008), 27-67.  doi: 10.1007/s00032-008-0090-3.

[5]

L. ChenZ. LiuG. Lu and C. Tao, Stein-Weiss inequalities with the fractional Poisson kernel, Rev. Mat. Iberoam., 36 (2020), 1289-1308.  doi: 10.4171/rmi/1167.

[6]

L. ChenZ. LiuG. Lu and C. Tao, Reverse Stein-Weiss inequalities and existence of their extremal functions, Trans. Amer. Math. Soc., 370 (2018), 8429-8450.  doi: 10.1090/tran/7273.

[7]

L. ChenG. Lu and C. Tao, Reverse Stein-Weiss inequalities on the upper half space and the existence of their extremals, Adv. Nonlinear Stud., 19 (2019), 475-494.  doi: 10.1515/ans-2018-2038.

[8]

L. ChenG. Lu and C. Tao, Hardy-Littlewood-Sobolev inequalities with the fractional Poisson kernel and their applications in PDEs, Acta Math. Sin. (Engl. Ser.), 35 (2019), 853-875.  doi: 10.1007/s10114-019-8417-2.

[9]

L. ChenG. Lu and C. Tao, Existence of extremal functions for the Stein-Weiss inequalities on the Heisenberg group, J. Funct. Anal., 277 (2019), 1112-1138.  doi: 10.1016/j.jfa.2019.01.002.

[10]

S. Chen, A new family of sharp conformally invariant integral inequalities, Int. Math. Res. Not. IMRN, (2014), 1205–1220. doi: 10.1093/imrn/rns248.

[11]

W. Chen and C. Li, An integral system and the Lane-Emdem conjecture, Disc. Cont. Dyn. Sys., 24 (2009), 1167-1184.  doi: 10.3934/dcds.2009.24.1167.

[12]

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

[13]

W. ChenC. Li and B. Ou, Classiffication of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65.  doi: 10.1081/PDE-200044445.

[14]

M. Christ, On the restriction of the Fourier transform to curves: Endpoint results and the degenerate case, Trans. Amer. Math. Soc., 287 (1985), 223-238.  doi: 10.1090/S0002-9947-1985-0766216-6.

[15]

M. Christ, Estimates for the $k$-plane transform, Indiana Univ. Math. J., 33 (1984), 891-910.  doi: 10.1512/iumj.1984.33.33048.

[16]

J. Dou, Weighted Hardy-Littlewood-Sobolev inequalities on the upper half space, Comm. Cont. Math., 18 (2016), 1550067, 20 pp. doi: 10.1142/S0219199715500674.

[17]

J. DouQ. Guo and M. Zhu, Subcritical approach to sharp Hardy-Littlewood-Sobolev type inequalities on the upper half space, Adv. Math., 312 (2017), 1-45.  doi: 10.1016/j.aim.2017.07.009.

[18]

J. DouQ. Guo and M. Zhu, Negative power nonlinear integral equations on bounded domains, J. Diff. Eqs., 269 (2020), 10527-10557.  doi: 10.1016/j.jde.2020.07.021.

[19]

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

[20]

J. Dou and M. Zhu, Reversed Hardy-Littewood-Sobolev inequality, Int. Math. Res. Not. IMRN, (2015), 9696–9726. doi: 10.1093/imrn/rnu241.

[21]

J. Dou and M. Zhu, Nonlinear integral equations on bounded domains, J. Funct. Anal., 277 (2019), 111-134.  doi: 10.1016/j.jfa.2018.05.020.

[22]

S. W. Drury, $L^p$ estimates for the $X$-ray transform, Illinois J. Math., 27 (1983), 125-129. 

[23]

R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39 (2010), 85-99.  doi: 10.1007/s00526-009-0302-x.

[24]

R. L. Frank and E. H. Lieb, Sharp constants in several inequalities on the Heisenberg group, Ann. of Math., 176 (2012), 349-381.  doi: 10.4007/annals.2012.176.1.6.

[25]

R. L. Frank and E. H. Lieb, A New, Rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality, Spectral Theory, Function Spaces and Inequalities, Oper. Theory Adv. Appl., Birkhäuser/Springer Basel AG, Basel, 219 (2011), 55-67.  doi: 10.1007/978-3-0348-0263-5_4.

[26]

M. Gluck, Subcritical approach to conformally invariant extension operators on the upper half space, J. Funct. Anal., 278 (2020), 108082, 46 pp. doi: 10.1016/j.jfa.2018.08.012.

[27]

M. Gluck and M. Zhu, An extension operator on bounded domains and applications, Calc. Var. PDE, 58 (2019), Paper No. 79, 27 pp. doi: 10.1007/s00526-019-1513-4.

[28]

L. Grafakos and C. Morpurgo, A Selberg integral formula and applications, Pacific J. Math., 191 (1999), 85-94.  doi: 10.2140/pjm.1999.191.85.

[29]

L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1976), 1061-1083.  doi: 10.2307/2373688.

[30]

Y. Han and M. Zhu, Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications, J. Diff. Eqs., 260 (2016), 1-25.  doi: 10.1016/j.jde.2015.06.032.

[31]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.  doi: 10.4310/MRL.2007.v14.n3.a2.

[32]

F. HangX. Wang and X. Yan, Sharp integral inequalities for harmonic functions, Comm. Pure Appl. Math., 61 (2008), 54-95.  doi: 10.1002/cpa.20193.

[33]

G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Zeitschr., 27 (1928), 565-606.  doi: 10.1007/BF01171116.

[34]

G. H. Hardy and J. E. Littlewood, On certain inequalities connected with the calculus of variations, J. London Math. Soc., 5 (1930), 34-39.  doi: 10.1112/jlms/s1-5.1.34.

[35]

Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc. (JEMS), 6 (2004), 153-180.  doi: 10.4171/jems/6.

[36]

Y. Y. Li and L. Zhang, Liouville type theorems and Harnack type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), 27-87.  doi: 10.1007/BF02786551.

[37]

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8.

[38]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.  doi: 10.2307/2007032.

[39]

Q. N. Ngô and V. H. Nguyen, Sharp reversed Hardy-Littlewood-Sobolev inequality on $\mathbb{R}^n$, Israel J. Math., 220 (2017), 189-223.  doi: 10.1007/s11856-017-1515-x.

[40]

Q. N. Ngô and V. H. Nguyen, Sharp reversed Hardy-Littlewood-Sobolev inequality on the half space $\mathbb{R}^n_+$, Int. Math. Res. Not. IMRN, (2017), 6187–6230. doi: 10.1093/imrn/rnw108.

[41]

F. Pezzolo, On some multilinear type integral systems, Nonl. Anal., 198 (2020), 111890, 23 pp. doi: 10.1016/j.na.2020.111890.

[42]

S. L. Sobolev, On a theorem of functional analysis, A. M. S. Transl. Ser. 2, 34 (1963), 39-68. 

[43]

M. Zhu, Prescribing integral curvature equation, Diff. Inte. Eqs., 29 (2016), 889-904. 

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