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April  2021, 41(4): 1605-1626. doi: 10.3934/dcds.2020333

Classification of nonnegative solutions to an equation involving the Laplacian of arbitrary order

Department of Economic Mathematics, Banking University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

Received  April 2020 Revised  August 2020 Published  April 2021 Early access  October 2020

Fund Project: This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2020.22

We classify all nonnegative nontrivial classical solutions to the equation
$ (-\Delta)^{\frac{\alpha}{2}} u = c_1 \left(\frac{1}{|x|^{n-\beta}} * f(u)\right) g(u) + c_2 h(u) \quad\text{ in } \mathbb{R}^n, $
where
$ 0<\alpha,\beta<n $
,
$ c_1,c_2\ge0 $
,
$ c_1+c_2>0 $
and
$ f,g,h \in C([0, +\infty),[0, +\infty)) $
are increasing functions such that
$ {f(t)}/{t^{\frac{n+\beta}{n-\alpha}}} $
,
$ {g(t)}/{t^{\frac{\alpha+\beta}{n-\alpha}}} $
,
$ {h(t)}/{t^{\frac{n+\alpha}{n-\alpha}}} $
are nonincreasing in
$ (0, +\infty) $
. We also derive a Liouville type theorem for the equation in the case
$ \alpha\ge n $
. The main tool we use is the method of moving spheres in integral forms.
Citation: Phuong Le. Classification of nonnegative solutions to an equation involving the Laplacian of arbitrary order. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1605-1626. doi: 10.3934/dcds.2020333
References:
[1]

G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on ${\bf R}^n$ or ${\bf R}^n_{+}$ through the method of moving planes, Comm. Partial Differential Equations, 22 (1997), 1671-1690.  doi: 10.1080/03605309708821315.

[2]

L. A. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.

[3]

D. Cao and W. Dai, Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 979-994.  doi: 10.1017/prm.2018.67.

[4]

D. Cao, W. Dai and G. Qin, Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving higher-order fractional Laplacians, Preprint, arXiv: 1905.04300.

[5]

W. Chen, W. Dai and G. Qin, Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy-Hénon equations in $\mathbb{R}^{n}$, Preprint, arXiv: 1808.06609.

[6]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.

[7]

W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12 (2013), 2497-2514.  doi: 10.3934/cpaa.2013.12.2497.

[8]

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.

[9]

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.

[10]

W. ChenY. Li and R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157.  doi: 10.1016/j.jfa.2017.02.022.

[11]

W. DaiY. Fang and G. Qin, Classification of positive solutions to fractional order {H}artree equations via a direct method of moving planes, J. Differential Equations, 265 (2018), 2044-2063.  doi: 10.1016/j.jde.2018.04.026.

[12]

W. DaiJ. HuangY. QinB. Wang and Y. Fang, Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete Contin. Dyn. Syst., 39 (2019), 1389-1403.  doi: 10.3934/dcds.2018117.

[13]

W. Dai and Z. Liu, Classification of nonnegative solutions to static Schrödinger–Hartree and Schrödinger–Maxwell equations with combined nonlinearities, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 156, 24pp. doi: 10.1007/s00526-019-1595-z.

[14]

W. Dai, Z. Liu and G. Qin, Classification of nonnegative solutions to static Schrödinger-Hartree-Maxwell type equations, Preprint, arXiv: 1909.00492.

[15]

W. Dai and G. Qin, Classification of nonnegative classical solutions to third-order equations, Adv. Math., 328 (2018), 822-857.  doi: 10.1016/j.aim.2018.02.016.

[16]

L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. Mat. Iberoamericana, 20 (2004), 67-86.  doi: 10.4171/RMI/380.

[17]

P. d'AveniaG. Siciliano and M. Squassina, On fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447-1476.  doi: 10.1142/S0218202515500384.

[18]

M. Ghimenti and J. Van Schaftingen, Nodal solutions for the Choquard equation, J. Funct. Anal., 271 (2016), 107-135.  doi: 10.1016/j.jfa.2016.04.019.

[19]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.

[21]

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.

[22]

L. Guo, T. Hu, S. Peng and W. Shuai, Existence and uniqueness of solutions for Choquard equation involving Hardy–Littlewood–Sobolev critical exponent, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 128, 34pp. doi: 10.1007/s00526-019-1585-1.

[23]

T. Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist., 17 (1997), 339-364. 

[24]

P. Le, Liouville theorem and classification of positive solutions for a fractional Choquard type equation, Nonlinear Anal., 185 (2019), 123-141.  doi: 10.1016/j.na.2019.03.006.

[25]

P. Le, Symmetry and classification of solutions to an integral equation of the Choquard type, C. R. Math. Acad. Sci. Paris, 357 (2019), 878-888.  doi: 10.1016/j.crma.2019.11.005.

[26]

P. Le, Liouville theorems for an integral equation of Choquard type, Commun. Pure Appl. Anal., 19 (2020), 771-783.  doi: 10.3934/cpaa.2020036.

[27]

P. Le, On classical solutions to the Hartree equation, J. Math. Anal. Appl., 485 (2020), 123859, 10pp. doi: 10.1016/j.jmaa.2020.123859.

[28]

Y. Lei, Liouville theorems and classification results for a nonlocal Schrödinger equation, Discrete Contin. Dyn. Syst., 38 (2018), 5351-5377.  doi: 10.3934/dcds.2018236.

[29]

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.

[30]

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.

[31]

E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.  doi: 10.1007/BF01609845.

[32]

C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in Rn, Comment. Math. Helv., 73 (1998), 206-231.  doi: 10.1007/s000140050052.

[33]

S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 1796-1806.  doi: 10.1016/j.na.2009.01.014.

[34]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.

[35]

I. M. MorozR. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019.

[36]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.

[37]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12pp. doi: 10.1142/S0219199715500054.

[38]

V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1.

[39]

S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.

[40]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[41]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.  doi: 10.1007/s002080050258.

[42]

D. Xu and Y. Lei, Classification of positive solutions for a static Schrödinger-Maxwell equation with fractional Laplacian, Appl. Math. Lett., 43 (2015), 85-89.  doi: 10.1016/j.aml.2014.12.007.

[43]

L. Zhang and Y. Wang, Symmetry of solutions to semilinear equations involving the fractional laplacian on ${\mathbb R}^n$ and ${\mathbb R}^n_+$, Preprint, arXiv: 1610.00122.

[44]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.

show all references

References:
[1]

G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on ${\bf R}^n$ or ${\bf R}^n_{+}$ through the method of moving planes, Comm. Partial Differential Equations, 22 (1997), 1671-1690.  doi: 10.1080/03605309708821315.

[2]

L. A. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.

[3]

D. Cao and W. Dai, Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 979-994.  doi: 10.1017/prm.2018.67.

[4]

D. Cao, W. Dai and G. Qin, Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving higher-order fractional Laplacians, Preprint, arXiv: 1905.04300.

[5]

W. Chen, W. Dai and G. Qin, Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy-Hénon equations in $\mathbb{R}^{n}$, Preprint, arXiv: 1808.06609.

[6]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.

[7]

W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12 (2013), 2497-2514.  doi: 10.3934/cpaa.2013.12.2497.

[8]

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.

[9]

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.

[10]

W. ChenY. Li and R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157.  doi: 10.1016/j.jfa.2017.02.022.

[11]

W. DaiY. Fang and G. Qin, Classification of positive solutions to fractional order {H}artree equations via a direct method of moving planes, J. Differential Equations, 265 (2018), 2044-2063.  doi: 10.1016/j.jde.2018.04.026.

[12]

W. DaiJ. HuangY. QinB. Wang and Y. Fang, Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete Contin. Dyn. Syst., 39 (2019), 1389-1403.  doi: 10.3934/dcds.2018117.

[13]

W. Dai and Z. Liu, Classification of nonnegative solutions to static Schrödinger–Hartree and Schrödinger–Maxwell equations with combined nonlinearities, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 156, 24pp. doi: 10.1007/s00526-019-1595-z.

[14]

W. Dai, Z. Liu and G. Qin, Classification of nonnegative solutions to static Schrödinger-Hartree-Maxwell type equations, Preprint, arXiv: 1909.00492.

[15]

W. Dai and G. Qin, Classification of nonnegative classical solutions to third-order equations, Adv. Math., 328 (2018), 822-857.  doi: 10.1016/j.aim.2018.02.016.

[16]

L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. Mat. Iberoamericana, 20 (2004), 67-86.  doi: 10.4171/RMI/380.

[17]

P. d'AveniaG. Siciliano and M. Squassina, On fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447-1476.  doi: 10.1142/S0218202515500384.

[18]

M. Ghimenti and J. Van Schaftingen, Nodal solutions for the Choquard equation, J. Funct. Anal., 271 (2016), 107-135.  doi: 10.1016/j.jfa.2016.04.019.

[19]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.

[21]

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.

[22]

L. Guo, T. Hu, S. Peng and W. Shuai, Existence and uniqueness of solutions for Choquard equation involving Hardy–Littlewood–Sobolev critical exponent, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 128, 34pp. doi: 10.1007/s00526-019-1585-1.

[23]

T. Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist., 17 (1997), 339-364. 

[24]

P. Le, Liouville theorem and classification of positive solutions for a fractional Choquard type equation, Nonlinear Anal., 185 (2019), 123-141.  doi: 10.1016/j.na.2019.03.006.

[25]

P. Le, Symmetry and classification of solutions to an integral equation of the Choquard type, C. R. Math. Acad. Sci. Paris, 357 (2019), 878-888.  doi: 10.1016/j.crma.2019.11.005.

[26]

P. Le, Liouville theorems for an integral equation of Choquard type, Commun. Pure Appl. Anal., 19 (2020), 771-783.  doi: 10.3934/cpaa.2020036.

[27]

P. Le, On classical solutions to the Hartree equation, J. Math. Anal. Appl., 485 (2020), 123859, 10pp. doi: 10.1016/j.jmaa.2020.123859.

[28]

Y. Lei, Liouville theorems and classification results for a nonlocal Schrödinger equation, Discrete Contin. Dyn. Syst., 38 (2018), 5351-5377.  doi: 10.3934/dcds.2018236.

[29]

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.

[30]

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.

[31]

E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.  doi: 10.1007/BF01609845.

[32]

C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in Rn, Comment. Math. Helv., 73 (1998), 206-231.  doi: 10.1007/s000140050052.

[33]

S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 1796-1806.  doi: 10.1016/j.na.2009.01.014.

[34]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.

[35]

I. M. MorozR. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019.

[36]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.

[37]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12pp. doi: 10.1142/S0219199715500054.

[38]

V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1.

[39]

S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.

[40]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[41]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.  doi: 10.1007/s002080050258.

[42]

D. Xu and Y. Lei, Classification of positive solutions for a static Schrödinger-Maxwell equation with fractional Laplacian, Appl. Math. Lett., 43 (2015), 85-89.  doi: 10.1016/j.aml.2014.12.007.

[43]

L. Zhang and Y. Wang, Symmetry of solutions to semilinear equations involving the fractional laplacian on ${\mathbb R}^n$ and ${\mathbb R}^n_+$, Preprint, arXiv: 1610.00122.

[44]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.

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