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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 & 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

M. Montenegro and E. Valdinoci, Pointwise estimates and monotonicity formulas without maximum principle, J. Convex Anal., 20 (2013), 199-220.   Google Scholar

[27]

N. Soave and E. Valdinoci, Overdetermined problems for the fractional Laplacian in exterior and annular sets, Preprint arXiv: 1412.5074. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

M. Montenegro and E. Valdinoci, Pointwise estimates and monotonicity formulas without maximum principle, J. Convex Anal., 20 (2013), 199-220.   Google Scholar

[27]

N. Soave and E. Valdinoci, Overdetermined problems for the fractional Laplacian in exterior and annular sets, Preprint arXiv: 1412.5074. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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