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doi: 10.3934/dcds.2020333

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

Phuong Le

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

Received  April 2020 Revised  August 2020 Published  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.
Keywords: fractional order equation, higher order equation, Choquard-Hartree type equation, classification of solutions, Liouville theorem.
Mathematics Subject Classification: 35R11, 35J30, 35B06, 35B53, 35A02.
Citation: Phuong Le. Classification of nonnegative solutions to an equation involving the Laplacian of arbitrary order. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[2]

L. A. Caffarelli, B. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. Chen, C. 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.  Google Scholar

[9]

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

[10]

W. Chen, Y. 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.  Google Scholar

[11]

W. Dai, Y. 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.  Google Scholar

[12]

W. Dai, J. Huang, Y. Qin, B. 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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

I. M. Moroz, R. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[44]

R. Zhuo, W. Chen, X. 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.  Google Scholar

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.  Google Scholar

[2]

L. A. Caffarelli, B. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. Chen, C. 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.  Google Scholar

[9]

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

[10]

W. Chen, Y. 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.  Google Scholar

[11]

W. Dai, Y. 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.  Google Scholar

[12]

W. Dai, J. Huang, Y. Qin, B. 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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

I. M. Moroz, R. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[44]

R. Zhuo, W. Chen, X. 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.  Google Scholar

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