A direct method of moving planes for fractional Laplacian equations in the unit ball. (English) Zbl 1352.35218
Summary: In this paper, we employ a direct method of moving planes for the fractional Laplacian equation in the unit ball. Instead of using the conventional extension method introduced by L. Caffarelli and L. Silvestre [Commun. Partial Differ. Equations 32, No. 8, 1245–1260 (2007; Zbl 1143.26002)], W. Chen, C. Li and Y. Li [“A direct method of moving planes for the fractional Laplacian”, Preprint, arXiv:1411.1697] developed a direct method of moving planes for the fractional Laplacian. Inspired by this new method, in this paper we deal with the semilinear pseudo-differential equation in the unit ball directly. We first review key ingredients needed in the method of moving planes in a bounded domain, such as the narrow region principle for the fractional Laplacian. Then, by using this new method, we obtain the radial symmetry and monotonicity of positive solutions for some interesting semi-linear equations.
MSC:
| 35R11 | Fractional partial differential equations |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 58J05 | Elliptic equations on manifolds, general theory |
| 35B07 | Axially symmetric solutions to PDEs |
Keywords:
fractional Laplacian; narrow region principle; method of moving planes; radial symmetry; monotonicityCitations:
Zbl 1143.26002References:
| [1] | D. Applebaum, <em>Lévy Processes and Stochastic Calculus</em>,, \(2^{st}\) edition (2009) · Zbl 1200.60001 · doi:10.1017/CBO9780511809781 |
| [2] | J. Bertoin, <em>Lévy Processes, Cambridge Tracts in Mathematics</em>,, 121 (1996) · Zbl 0861.60003 |
| [3] | J. P. Bouchard, Anomalous diffusion in disordered media,, Statistical mechanics, 195, 127 (1990) · doi:10.1016/0370-1573(90)90099-N |
| [4] | \(C. Br \ddota\) ndle, A concave-convex elliptic problem involving the fractional Laplacian,, Proc Royal Soc Edinburgh, A143, 39 (2013) · Zbl 1290.35304 |
| [5] | X. Cabr \(\acutee \), Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv in Math, 224, 2052 (2010) · Zbl 1198.35286 · doi:10.1016/j.aim.2010.01.025 |
| [6] | L. Caffarelli, An extension problem related to the fractional Laplacian,, Comm Partial Differential Equations, 32, 1245 (2007) · Zbl 1143.26002 · doi:10.1080/03605300600987306 |
| [7] | L. Caffarelli, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation,, Ann Math, 171, 1903 (2010) · Zbl 1204.35063 · doi:10.4007/annals.2010.171.1903 |
| [8] | W. X. Chen, A direct method of moving planes for the fractional Laplacian,, preprint · Zbl 1362.35320 |
| [9] | W. X. Chen, Classification of solutions for an integral equation,, Comm Pure Appl Math, 59, 330 (2006) · Zbl 1093.45001 · doi:10.1002/cpa.20116 |
| [10] | W. X. Chen, Qualitative properities of solutions for an integral equation,, Disc Cont Dyn Sys, 12, 347 (2005) · Zbl 1081.45003 |
| [11] | W. X. Chen, Indefinite fractional elliptic problem and Liouville theorems,, preprint · Zbl 1336.35089 |
| [12] | P. Constantin, Euler equations, Navier-Stokes equations and turbulence,, Mathematical Foundation of Turbulent Viscous Flows Lecture Notes in Mathematics, 1871, 1 (2006) · Zbl 1190.76146 · doi:10.1007/11545989_1 |
| [13] | B. Gidas, Symmetry and related properties via the maximum principle,, Comm Math Phys, 68, 209 (1979) · Zbl 0425.35020 |
| [14] | B. Gidas, Symmetry of positive solutions of nonlinear elliptic equations in \(\mathbbR^n\),, Mathematical Analysis and Applications, 369 (1981) · Zbl 0469.35052 |
| [15] | C. C. Li, Uniqueness of positive bound states to Schrödinger systems with critical exponents,, SIAM Journal on Mathematical Analysis, 40, 1049 (2008) · Zbl 1167.35347 · doi:10.1137/080712301 |
| [16] | L. Ma, Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application,, Journal of Differential Equations, 245, 2551 (2008) · Zbl 1154.35083 · doi:10.1016/j.jde.2008.04.008 |
| [17] | E. Nezza, Hitchhikers guide to the fractional Sobolev spaces,, Bull Sci Math, 136, 521 (2012) · Zbl 1252.46023 · doi:10.1016/j.bulsci.2011.12.004 |
| [18] | V. E. Tarasov, Fractional dynamics of systems with long-range interaction,, Comm Non1 Sci Numer Simul, 11, 885 (2006) · Zbl 1106.35103 · doi:10.1016/j.cnsns.2006.03.005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
