-
Previous Article
Weak-strong uniqueness for the general Ericksen—Leslie system in three dimensions
- DCDS Home
- This Issue
-
Next Article
Existence of torsion-low maximal isotopies for area preserving surface homeomorphisms
Moving planes for nonlinear fractional Laplacian equation with negative powers
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 |
$u∈ C^{1, 1}_{loc}(\mathbb{R}^{n})\cap L_{α}$ |
$(-Δ)^{α/2}u(x)+u^{-β}(x) = 0, \ \ \ x∈ \mathbb{R}^n, $ |
$u(x) = a|x|^{m}+o(1), \ \ as \ \ |x| \to ∞, $ |
$\frac{α}{β+1}<m<1$ |
$a>0$ |
$u(x)$ |
$\mathbb{R}^{n}$ |
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. |
[1] |
Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015 |
[2] |
Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235 |
[3] |
Pengyan Wang, Pengcheng Niu. A direct method of moving planes for a fully nonlinear nonlocal system. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1707-1718. doi: 10.3934/cpaa.2017082 |
[4] |
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 |
[5] |
Xiaoxue Ji, Pengcheng Niu, Pengyan Wang. Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1111-1128. doi: 10.3934/cpaa.2020051 |
[6] |
Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393 |
[7] |
Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154 |
[8] |
Zhigang Wu, Hao Xu. Symmetry properties in systems of fractional Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1559-1571. doi: 10.3934/dcds.2019068 |
[9] |
Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125 |
[10] |
CÉSAR E. TORRES LEDESMA. Existence and symmetry result for fractional p-Laplacian in $\mathbb{R}^{n}$. Communications on Pure and Applied Analysis, 2017, 16 (1) : 99-114. doi: 10.3934/cpaa.2017004 |
[11] |
Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069 |
[12] |
Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071 |
[13] |
Wenxiong Chen, Shijie Qi. Direct methods on fractional equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1269-1310. doi: 10.3934/dcds.2019055 |
[14] |
Manli Song, Jinggang Tan. Hardy inequalities for the fractional powers of the Grushin operator. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4699-4726. doi: 10.3934/cpaa.2020192 |
[15] |
Maykel Belluzi, Flank D. M. Bezerra, Marcelo J. D. Nascimento. On spectral and fractional powers of damped wave equations. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2739-2773. doi: 10.3934/cpaa.2022071 |
[16] |
Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional $ p $-Laplacian. Communications on Pure and Applied Analysis, 2020, 19 (1) : 527-539. doi: 10.3934/cpaa.2020026 |
[17] |
Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3851-3863. doi: 10.3934/dcdss.2020445 |
[18] |
Mohammad Hadi Noori Skandari, Marzieh Habibli, Alireza Nazemi. A direct method based on the Clenshaw-Curtis formula for fractional optimal control problems. Mathematical Control and Related Fields, 2020, 10 (1) : 171-187. doi: 10.3934/mcrf.2019035 |
[19] |
Sanjay Dharmavaram, Timothy J. Healey. Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1669-1684. doi: 10.3934/dcdss.2019112 |
[20] |
Nicola Abatangelo, Sven Jarohs, Alberto Saldaña. Positive powers of the Laplacian: From hypersingular integrals to boundary value problems. Communications on Pure and Applied Analysis, 2018, 17 (3) : 899-922. doi: 10.3934/cpaa.2018045 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]