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

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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 and Continuous Dynamical Systems - S, 2021, 14 (6) : 1871-1897. doi: 10.3934/dcdss.2020462

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.
[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.
[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.
[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.
[5]	J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co. Pte. Ltd., Singapore, 2020. doi: 10.1142/10550.
[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.
[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.
[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.
[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.
[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.
[20]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDEs, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[38]	E. De Giorgi, Convergence problems for functionals and operators, in Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, Rome, (1978), Pitagora, (1979).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[5]	J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co. Pte. Ltd., Singapore, 2020. doi: 10.1142/10550.
[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.
[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.
[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.
[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.
[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.
[20]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDEs, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[38]	E. De Giorgi, Convergence problems for functionals and operators, in Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, Rome, (1978), Pitagora, (1979).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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