American Institute of Mathematical Sciences

Advanced
June  2021, 14(6): 1871-1897. doi: 10.3934/dcdss.2020462

A direct method of moving planes for fully nonlinear nonlocal operators and applications

Yuxia Guo 1,, and  Shaolong Peng 1,

Department of Mathematics, Tsinghua University, Beijing, 100084, China

* Corresponding author: Yuxia Guo

Received  July 2020 Published  June 2021 Early access  November 2020

Fund Project: The first author is supported by NSFC (No. 11771235) and the second author supported by NSFC (No. 11971049)

In this paper, we are concerned with the following generalized fully nonlinear nonlocal operators:
$ F_{s,m}(u(x)) = c_{N,s} m^{ \frac{N}{2}+s} P.V. \int_{\mathbb{R}^{N}} \frac{G(u(x)-u(y))}{|x-y|^{ \frac{N}{2}+s}} K_{ \frac{N}{2}+s}(m|x-y|)dy+m^{2s}u(x), $
where
$ s\in (0,1) $
 and mass
$ m>0 $
. By establishing various maximal principle and using the direct method of moving plane, we prove the monotonicity, symmetry and uniqueness for solutions to fully nonlinear nonlocal equation in unit ball,
$ \mathbb{R}^{N} $
,
$ \mathbb{R}^{N}_{+} $
 and a coercive epigraph domain
$ \Omega $
 in
$ \mathbb{R}^N $
 respectively.
Keywords: Fully nonlinear nonlocal operators, Direct methods of moving planes, Maximal principle, Liouville theorem, Symmetry and Monotonicity.
Mathematics Subject Classification: Primary:35R11;Secondary:35B06, 35B53.
Citation: Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1871-1897. doi: 10.3934/dcdss.2020462
References:
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V. Ambrosio, Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator, J. Math. Phys., 57 (2016), 051502, 18 pp. doi: 10.1063/1.4949352.  Google Scholar

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H. Berestycki, L. A. Caffarelli and L. Nirenberg, Inequalitites for second-order elliptic equations with applications to unbounded domains I, Duke Math. J., 81 (1996), 467-494.  doi: 10.1215/S0012-7094-96-08117-X.  Google Scholar

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H. Berestycki, L. A. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math., 50 (1997), 1089-1111.  doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6.  Google Scholar

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C. Brandle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. of Edinburgh-A: Math., 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.  Google Scholar

[5] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996.   Google Scholar
[6]

H. Berestycki, F. Hamel and R. Monneau, One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., 103 (2000), 375-396.  doi: 10.1215/S0012-7094-00-10331-6.  Google Scholar

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H. Berestycki and L. Nirenberg, Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. Phys., 5 (1988), 237-275.  doi: 10.1016/0393-0440(88)90006-X.  Google Scholar

[8] H. Berestycki and L. Nirenberg, Some Qualitative Properties of Solutions of Semilinear Elliptic Equations in Cylindrical Domains, Analysis, et Cetera, Academic Press, Boston, MA, 1990.   Google Scholar
[9]

H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), 1-37.  doi: 10.1007/BF01244896.  Google Scholar

[10]

S.-Y. A. Chang and M. del Mar Gonzàlez, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432.  doi: 10.1016/j.aim.2010.07.016.  Google Scholar

[11]

Y. Chen and B. Liu, Symmetry and non-existence of positive solutions for fractional p-Laplacian systems, Nonlinear Anal., 183 (2019), 303-322.  doi: 10.1016/j.na.2019.02.023.  Google Scholar

[12]

W. Chen and C. Li, Maximum principles for the fractional $p$-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.  doi: 10.1016/j.aim.2018.07.016.  Google Scholar

[13]

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

[14]

W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co. Pte. Ltd., Singapore, 2020. doi: 10.1142/10550.  Google Scholar

[15]

W. Chen and C. Li, Moving planes, moving spheres, and a priori estimates, J. Differential Equations, 195 (2003), 1-13.  doi: 10.1016/j.jde.2003.06.004.  Google Scholar

[16]

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

[17]

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

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W. Chen and S. Qi, Direct methods on fractional equations, Disc. Cont. Dyn. Syst.-A, 39 (2019), 1269-1310.  doi: 10.3934/dcds.2019055.  Google Scholar

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L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDEs, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[21]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025.  Google Scholar

[22]

L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Math., 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903.  Google Scholar

[23]

W. Chen and L. Wu, The sliding methods for the fractional $p$-Laplacian, Adv. Math., 361 (2020), 106933, 26 pp. doi: 10.1016/j.aim.2019.106933.  Google Scholar

[24]

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

[25]

R. Carmona, W. C. Masters and B. Simon, Relativistic Schrödinger operators: Asymptotic behavior of the eigenfunctions, J. Funct. Anal., 91 (1990), 117-142.  doi: 10.1016/0022-1236(90)90049-Q.  Google Scholar

[26]

W. Dai, Y. Fang, J. Huang, Y. Qin and B. Wang, Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete and Continuous Dynamical Systems-A, 39 (2019), 1389-1403.  doi: 10.3934/dcds.2018117.  Google Scholar

[27]

W. Dai and Z. Liu, Classification of nonnegative solutions to static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities, Calc. Var. & PDEs, 58 (2019), Art. 156, 24pp. doi: 10.1007/s00526-019-1595-z.  Google Scholar

[28]

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

[29]

W. Dai and G. Qin, Liouville type theorems for fractional and higher order Hénon-Hardy equations via the method of scaling spheres, preprint, submitted for publication, arXiv: 1810.02752. Google Scholar

[30]

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

[31]

W. Dai and G. Qin, Liouville type theorems for elliptic equations with Dirichlet conditions in exterior domains, J. Differential Equations, 269 (2020), 7231-7252.  doi: 10.1016/j.jde.2020.05.026.  Google Scholar

[32]

W. Dai, G. Qin and D. Wu, Direct methods for pseudo-relativistic Schrödinger operators, Journal of Geometric Analysis, (2020). doi: 10.1007/s12220-020-00492-1.  Google Scholar

[33]

S. Dipierro, N. Soave and E. Valdinoci, On fractional elliptic equations in Lipschitz sets and epigraphs: Regularity, monotonicity and rigidity results, Math. Ann., 369 (2017), 1283-1326.  doi: 10.1007/s00208-016-1487-x.  Google Scholar

[34]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions: Vol. Ⅱ, Journal of the Franklin Institute, McGraw-Hill, New York, 257 (1954), 150. doi: 10.1016/0016-0032(54)90080-0.  Google Scholar

[35]

M. M. Fall and V. Felli, Sharp essential self-adjointness of relativistic Schrödinger operators with a singular potential, J. Funct. Anal., 267 (2014), 1851-1877.  doi: 10.1016/j.jfa.2014.06.010.  Google Scholar

[36]

M. M. Fall and V. Felli, Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst.-A, 35 (2015), 5827-5867.  doi: 10.3934/dcds.2015.35.5827.  Google Scholar

[37]

R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional laplacian, Comm. Pure Appl. Math., 69 (2013), 1671-1726.  doi: 10.1002/cpa.21591.  Google Scholar

[38]

E. De Giorgi, Convergence problems for functionals and operators, in Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, Rome, (1978), Pitagora, (1979).  Google Scholar

[39]

Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $R^{N}$, Commun. Partial Differ. Equ., 33 (2008), 263-284.  doi: 10.1080/03605300701257476.  Google Scholar

[40]

I. W. Herbst, Spectral theory of the operator $(p^{2}+m^{2})^{1/2}-Ze^{2}/r$, Comm. Math. Phys., 53 (1977), 285-294.   Google Scholar

[41]

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

[42]

B. Liu and L. Ma, Radial symmetry results for fractional Laplacian systems,, Nonlinear Anal., 146 (2016), 120-135.  doi: 10.1016/j.na.2016.08.022.  Google Scholar

[43]

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

[44]

S. Peng, Liouville theorems for fractional and higher order Hénon-Hardy systems on $\mathbb{R}^{n}$, Complex Variables and Elliptic Equations, 11 (2020), 1-25.  doi: 10.1080/17476933.2020.1783661.  Google Scholar

[45]

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

[46]

M. Qu and L. Yang, Solutions to the nonlinear Schrödinger systems involving the fractional Laplacian, J. Inequal. Appl., 297 (2018), 16pp. doi: 10.1186/s13660-018-1874-9.  Google Scholar

show all references

References:
[1]

V. Ambrosio, Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator, J. Math. Phys., 57 (2016), 051502, 18 pp. doi: 10.1063/1.4949352.  Google Scholar

[2]

H. Berestycki, L. A. Caffarelli and L. Nirenberg, Inequalitites for second-order elliptic equations with applications to unbounded domains I, Duke Math. J., 81 (1996), 467-494.  doi: 10.1215/S0012-7094-96-08117-X.  Google Scholar

[3]

H. Berestycki, L. A. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math., 50 (1997), 1089-1111.  doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6.  Google Scholar

[4]

C. Brandle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. of Edinburgh-A: Math., 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.  Google Scholar

[5] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996.   Google Scholar
[6]

H. Berestycki, F. Hamel and R. Monneau, One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., 103 (2000), 375-396.  doi: 10.1215/S0012-7094-00-10331-6.  Google Scholar

[7]

H. Berestycki and L. Nirenberg, Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. Phys., 5 (1988), 237-275.  doi: 10.1016/0393-0440(88)90006-X.  Google Scholar

[8] H. Berestycki and L. Nirenberg, Some Qualitative Properties of Solutions of Semilinear Elliptic Equations in Cylindrical Domains, Analysis, et Cetera, Academic Press, Boston, MA, 1990.   Google Scholar
[9]

H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), 1-37.  doi: 10.1007/BF01244896.  Google Scholar

[10]

S.-Y. A. Chang and M. del Mar Gonzàlez, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432.  doi: 10.1016/j.aim.2010.07.016.  Google Scholar

[11]

Y. Chen and B. Liu, Symmetry and non-existence of positive solutions for fractional p-Laplacian systems, Nonlinear Anal., 183 (2019), 303-322.  doi: 10.1016/j.na.2019.02.023.  Google Scholar

[12]

W. Chen and C. Li, Maximum principles for the fractional $p$-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.  doi: 10.1016/j.aim.2018.07.016.  Google Scholar

[13]

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

[14]

W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co. Pte. Ltd., Singapore, 2020. doi: 10.1142/10550.  Google Scholar

[15]

W. Chen and C. Li, Moving planes, moving spheres, and a priori estimates, J. Differential Equations, 195 (2003), 1-13.  doi: 10.1016/j.jde.2003.06.004.  Google Scholar

[16]

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

[17]

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

[18]

P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in mathematical foundation of turbulent viscous flows, Lecture Notes in Math., Springer, Berlin, 1871 (2006), 1-43. doi: 10.1007/11545989_1.  Google Scholar

[19]

W. Chen and S. Qi, Direct methods on fractional equations, Disc. Cont. Dyn. Syst.-A, 39 (2019), 1269-1310.  doi: 10.3934/dcds.2019055.  Google Scholar

[20]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDEs, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[21]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025.  Google Scholar

[22]

L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Math., 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903.  Google Scholar

[23]

W. Chen and L. Wu, The sliding methods for the fractional $p$-Laplacian, Adv. Math., 361 (2020), 106933, 26 pp. doi: 10.1016/j.aim.2019.106933.  Google Scholar

[24]

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

[25]

R. Carmona, W. C. Masters and B. Simon, Relativistic Schrödinger operators: Asymptotic behavior of the eigenfunctions, J. Funct. Anal., 91 (1990), 117-142.  doi: 10.1016/0022-1236(90)90049-Q.  Google Scholar

[26]

W. Dai, Y. Fang, J. Huang, Y. Qin and B. Wang, Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete and Continuous Dynamical Systems-A, 39 (2019), 1389-1403.  doi: 10.3934/dcds.2018117.  Google Scholar

[27]

W. Dai and Z. Liu, Classification of nonnegative solutions to static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities, Calc. Var. & PDEs, 58 (2019), Art. 156, 24pp. doi: 10.1007/s00526-019-1595-z.  Google Scholar

[28]

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

[29]

W. Dai and G. Qin, Liouville type theorems for fractional and higher order Hénon-Hardy equations via the method of scaling spheres, preprint, submitted for publication, arXiv: 1810.02752. Google Scholar

[30]

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

[31]

W. Dai and G. Qin, Liouville type theorems for elliptic equations with Dirichlet conditions in exterior domains, J. Differential Equations, 269 (2020), 7231-7252.  doi: 10.1016/j.jde.2020.05.026.  Google Scholar

[32]

W. Dai, G. Qin and D. Wu, Direct methods for pseudo-relativistic Schrödinger operators, Journal of Geometric Analysis, (2020). doi: 10.1007/s12220-020-00492-1.  Google Scholar

[33]

S. Dipierro, N. Soave and E. Valdinoci, On fractional elliptic equations in Lipschitz sets and epigraphs: Regularity, monotonicity and rigidity results, Math. Ann., 369 (2017), 1283-1326.  doi: 10.1007/s00208-016-1487-x.  Google Scholar

[34]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions: Vol. Ⅱ, Journal of the Franklin Institute, McGraw-Hill, New York, 257 (1954), 150. doi: 10.1016/0016-0032(54)90080-0.  Google Scholar

[35]

M. M. Fall and V. Felli, Sharp essential self-adjointness of relativistic Schrödinger operators with a singular potential, J. Funct. Anal., 267 (2014), 1851-1877.  doi: 10.1016/j.jfa.2014.06.010.  Google Scholar

[36]

M. M. Fall and V. Felli, Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst.-A, 35 (2015), 5827-5867.  doi: 10.3934/dcds.2015.35.5827.  Google Scholar

[37]

R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional laplacian, Comm. Pure Appl. Math., 69 (2013), 1671-1726.  doi: 10.1002/cpa.21591.  Google Scholar

[38]

E. De Giorgi, Convergence problems for functionals and operators, in Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, Rome, (1978), Pitagora, (1979).  Google Scholar

[39]

Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $R^{N}$, Commun. Partial Differ. Equ., 33 (2008), 263-284.  doi: 10.1080/03605300701257476.  Google Scholar

[40]

I. W. Herbst, Spectral theory of the operator $(p^{2}+m^{2})^{1/2}-Ze^{2}/r$, Comm. Math. Phys., 53 (1977), 285-294.   Google Scholar

[41]

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

[42]

B. Liu and L. Ma, Radial symmetry results for fractional Laplacian systems,, Nonlinear Anal., 146 (2016), 120-135.  doi: 10.1016/j.na.2016.08.022.  Google Scholar

[43]

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

[44]

S. Peng, Liouville theorems for fractional and higher order Hénon-Hardy systems on $\mathbb{R}^{n}$, Complex Variables and Elliptic Equations, 11 (2020), 1-25.  doi: 10.1080/17476933.2020.1783661.  Google Scholar

[45]

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

[46]

M. Qu and L. Yang, Solutions to the nonlinear Schrödinger systems involving the fractional Laplacian, J. Inequal. Appl., 297 (2018), 16pp. doi: 10.1186/s13660-018-1874-9.  Google Scholar

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