Moving planes for nonlinear fractional Laplacian equation with negative powers

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September 2018, 38(9): 4603-4615. doi: 10.3934/dcds.2018201

Moving planes for nonlinear fractional Laplacian equation with negative powers

Miaomiao Cai ^1, and Li Ma ^2,,

1.	School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
2.	School of Mathematics and Physics, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian District, Beijing 100083, China

* Corresponding author: Li Ma

Received November 2017 Revised April 2018 Published June 2018

Fund Project: The research of L.Ma is partially supported by the National Natural Science Foundation of China (No. 11771124, No.11271111).

In this paper, we study symmetry properties of positive solutions to the fractional Laplace equation with negative powers on the whole space. We can use the direct method of moving planes introduced by Jarohs-Weth-Chen-Li-Li to prove one particular result below. If

$u∈ C^{1, 1}_{loc}(\mathbb{R}^{n})\cap L_{α}$

satisfies

$(-Δ)^{α/2}u(x)+u^{-β}(x) = 0, \ \ \ x∈ \mathbb{R}^n, $

with the growth/decay property

$u(x) = a|x|^{m}+o(1), \ \ as \ \ |x| \to ∞, $

where

$\frac{α}{β+1}<m<1$

,

$a>0$

is a constant, then the positive solution

$u(x)$

must be radially symmetric about some point in

$\mathbb{R}^{n}$

. Similar result is also true for Hénon type nonlinear fractional Laplace equation with negative powers.

Keywords: Fractional Laplacian, negative powers, the direct method of moving planes, symmetry.

Mathematics Subject Classification: Primary: 35R11, 35J60, 53C21; Secondary: 58J05.

Citation: Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201

References:

[1]	A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51 (1998), 625-661. doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9.
[2]	A. L. Bertozzi and M. C. Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J., 49 (2000), 1323-1366. doi: 10.1512/iumj.2000.49.1887.
[3]	C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc Royal Soc. of Edinburgh, 143 (2013), 39-71. doi: 10.1017/S0308210511000175.
[4]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[5]	W. X. Chen, C. M. 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.
[6]	W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.
[7]	W. X. Chen, Y. Li and R. B. 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.
[8]	R. Dal Passo, L. Giacomelli and A. Shishkov, The thin film equation with nonlinear diffusion, Comm. Partial Differential Equations, 26 (2001), 1509-1557. doi: 10.1081/PDE-100107451.
[9]	J. Davila, K. Wang and J. C. Wei, Qualitative analysis of rupture solutions for a MEMS problem, Ann. Inst. H. Poincare Anal. Non Lineaire, 33 (2016), 221-242. doi: 10.1016/j.anihpc.2014.09.009.
[10]	S. Dipierro, L. Montoro, I. Peral and B. Sciunzi, Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential, Calc. Var. Partial Differential Equations, 55 (2016), Art. 99, 29 pp. doi: 10.1007/s00526-016-1032-5.
[11]	Y. H. Du and Z. M. Guo, Positive solutions of an elliptic equation with negative exponent: stability and critical power, J. Differential Equations, 246 (2009), 2387-2414. doi: 10.1016/j.jde.2008.08.008.
[12]	P. Felmer and Y. Wang, Radial symmetry of positive solutions to equa- tions involving the fractional Laplacian, Commun. Contemp. Math., 16 (2014), 1350023, 24pp. doi: 10.1142/S0219199713500235.
[13]	N. Ghoussoub and Y. J. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM J. Math. Anal., 38 (2006), 1423-1449. doi: 10.1137/050647803.
[14]	Z. M. Guo and J. C. Wei, Symmetry of non-negative solutions of a semilinear elliptic equation with singular nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 963-994. doi: 10.1017/S0308210505001083.
[15]	S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Annali di Math. Pura ed Appl., 195 (2016), 273-291. doi: 10.1007/s10231-014-0462-y.
[16]	H. Q. Jiang and W. M. Ni, On steady states of van der Waals force driven thin film equations, European J. Appl. Math., 18 (2007), 153-180. doi: 10.1017/S0956792507006936.
[17]	R. S. Laugesen and M. C. Pugh, Linear stability of steady states for thin film and Cahn-Hilliard type equations, Arch. Ration. Mech. Anal., 154 (2000), 3-51. doi: 10.1007/PL00004234.
[18]	Y. T. Lei, On the integral systems with negative exponents, Discrete Contin. Dyn. Syst., 35 (2015), 1039-1057. doi: 10.3934/dcds.2015.35.1039.
[19]	B. Y. 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.
[20]	L. Ma, Liouville type theorem and uniform bound for the Lichnerowicz equation and the Ginzburg-Landau equation, C. R. Math. Acad. Sci. Paris., 348 (2010), 993-996. doi: 10.1016/j.crma.2010.07.031.
[21]	L. Ma and J. C. Wei, Properties of positive solutions to an elliptic equation with negative exponent, J. Funct. Anal., 254 (2008), 1058-1087. doi: 10.1016/j.jfa.2007.09.017.
[22]	L. Ma and J. C. Wei, Stability and multiple solutions to Einstein-scalar field Lichnerowicz equation on manifolds, J. Math. Pures Appl., 99 (2013), 174-186. doi: 10.1016/j.matpur.2012.06.009.
[23]	L. Ma and X. W. Xu, Uniform bound and a non-existence result for Lichnerowicz equation in the whole n-space, C. R. Math. Acad. Sci. Paris, 347 (2009), 805-808. doi: 10.1016/j.crma.2009.04.017.
[24]	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.
[25]	A. Meadows, Stable and singular solutions of the equation $Δ u = \frac{1}{u}$, Indiana Univ. Math. J., 53 (2004), 1681-1703. doi: 10.1512/iumj.2004.53.2560.
[26]	M. Montenegro and E. Valdinoci, Pointwise estimates and monotonicity formulas without maximum principle, J. Convex Anal., 20 (2013), 199-220.
[27]	N. Soave and E. Valdinoci, Overdetermined problems for the fractional Laplacian in exterior and annular sets, Preprint arXiv: 1412.5074.
[28]	X. F. Song and L. Zhao, Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds, Z. Angew. Math. Phys., 61 (2010), 655-662. doi: 10.1007/s00033-009-0047-6.
[29]	X. Xu, Uniqueness theorem for integral equations and its application, J. Funct. Anal., 247 (2007), 95-109. doi: 10.1016/j.jfa.2007.03.005.

show all references

References:

[1]	A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51 (1998), 625-661. doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9.
[2]	A. L. Bertozzi and M. C. Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J., 49 (2000), 1323-1366. doi: 10.1512/iumj.2000.49.1887.
[3]	C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc Royal Soc. of Edinburgh, 143 (2013), 39-71. doi: 10.1017/S0308210511000175.
[4]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[5]	W. X. Chen, C. M. 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.
[6]	W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.
[7]	W. X. Chen, Y. Li and R. B. 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.
[8]	R. Dal Passo, L. Giacomelli and A. Shishkov, The thin film equation with nonlinear diffusion, Comm. Partial Differential Equations, 26 (2001), 1509-1557. doi: 10.1081/PDE-100107451.
[9]	J. Davila, K. Wang and J. C. Wei, Qualitative analysis of rupture solutions for a MEMS problem, Ann. Inst. H. Poincare Anal. Non Lineaire, 33 (2016), 221-242. doi: 10.1016/j.anihpc.2014.09.009.
[10]	S. Dipierro, L. Montoro, I. Peral and B. Sciunzi, Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential, Calc. Var. Partial Differential Equations, 55 (2016), Art. 99, 29 pp. doi: 10.1007/s00526-016-1032-5.
[11]	Y. H. Du and Z. M. Guo, Positive solutions of an elliptic equation with negative exponent: stability and critical power, J. Differential Equations, 246 (2009), 2387-2414. doi: 10.1016/j.jde.2008.08.008.
[12]	P. Felmer and Y. Wang, Radial symmetry of positive solutions to equa- tions involving the fractional Laplacian, Commun. Contemp. Math., 16 (2014), 1350023, 24pp. doi: 10.1142/S0219199713500235.
[13]	N. Ghoussoub and Y. J. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM J. Math. Anal., 38 (2006), 1423-1449. doi: 10.1137/050647803.
[14]	Z. M. Guo and J. C. Wei, Symmetry of non-negative solutions of a semilinear elliptic equation with singular nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 963-994. doi: 10.1017/S0308210505001083.
[15]	S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Annali di Math. Pura ed Appl., 195 (2016), 273-291. doi: 10.1007/s10231-014-0462-y.
[16]	H. Q. Jiang and W. M. Ni, On steady states of van der Waals force driven thin film equations, European J. Appl. Math., 18 (2007), 153-180. doi: 10.1017/S0956792507006936.
[17]	R. S. Laugesen and M. C. Pugh, Linear stability of steady states for thin film and Cahn-Hilliard type equations, Arch. Ration. Mech. Anal., 154 (2000), 3-51. doi: 10.1007/PL00004234.
[18]	Y. T. Lei, On the integral systems with negative exponents, Discrete Contin. Dyn. Syst., 35 (2015), 1039-1057. doi: 10.3934/dcds.2015.35.1039.
[19]	B. Y. 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.
[20]	L. Ma, Liouville type theorem and uniform bound for the Lichnerowicz equation and the Ginzburg-Landau equation, C. R. Math. Acad. Sci. Paris., 348 (2010), 993-996. doi: 10.1016/j.crma.2010.07.031.
[21]	L. Ma and J. C. Wei, Properties of positive solutions to an elliptic equation with negative exponent, J. Funct. Anal., 254 (2008), 1058-1087. doi: 10.1016/j.jfa.2007.09.017.
[22]	L. Ma and J. C. Wei, Stability and multiple solutions to Einstein-scalar field Lichnerowicz equation on manifolds, J. Math. Pures Appl., 99 (2013), 174-186. doi: 10.1016/j.matpur.2012.06.009.
[23]	L. Ma and X. W. Xu, Uniform bound and a non-existence result for Lichnerowicz equation in the whole n-space, C. R. Math. Acad. Sci. Paris, 347 (2009), 805-808. doi: 10.1016/j.crma.2009.04.017.
[24]	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.
[25]	A. Meadows, Stable and singular solutions of the equation $Δ u = \frac{1}{u}$, Indiana Univ. Math. J., 53 (2004), 1681-1703. doi: 10.1512/iumj.2004.53.2560.
[26]	M. Montenegro and E. Valdinoci, Pointwise estimates and monotonicity formulas without maximum principle, J. Convex Anal., 20 (2013), 199-220.
[27]	N. Soave and E. Valdinoci, Overdetermined problems for the fractional Laplacian in exterior and annular sets, Preprint arXiv: 1412.5074.
[28]	X. F. Song and L. Zhao, Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds, Z. Angew. Math. Phys., 61 (2010), 655-662. doi: 10.1007/s00033-009-0047-6.
[29]	X. Xu, Uniqueness theorem for integral equations and its application, J. Funct. Anal., 247 (2007), 95-109. doi: 10.1016/j.jfa.2007.03.005.

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