February  2013, 33(2): 837-859. doi: 10.3934/dcds.2013.33.837

Positive solutions for non local elliptic problems

1. 

Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 1680, Valparaíso

Received  June 2011 Revised  March 2012 Published  September 2012

We establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type for nonlinear problems involving the fractional power of the Dirichlet Laplacian.
Citation: Jinggang Tan. Positive solutions for non local elliptic problems. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 837-859. doi: 10.3934/dcds.2013.33.837
References:
[1]

D. Applebaum, Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.

[2]

H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Math., 22 (1991), 1-37.

[3]

K. Bogdan and B. Dyda, The best constant in a fractional Hardy inequalityarXiv:0807.1825.

[4]

C. Brändle, E. Colorado and A. de Pablo, A concave-convex elliptic problem involving the fractional laplacianarXiv:1006.4510.

[5]

X. Cabre and E. Cinti, Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian, Disc. Cont. Dyna. Syst., 28 (2010), 1179-1206. doi: 10.3934/dcds.2010.28.1179.

[6]

X. Cabre and J. Solà-Morales, Layer solutions in a halfspace for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093.

[7]

X. Cabre and Y. Sire, Nonlinear equations for fractional laplacians I: regularity, maximum principles, and hamiltonian estimates, preprint, arXiv:1012.0867.

[8]

X. Cabre and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Advances in Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.

[9]

L. Caffarelli, J. M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.

[10]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Part. Diff. Equa., 32 (2007), 1245-1260.

[11]

A. Capella, J. Davila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.

[12]

A. Chang, M. Gonzalez, Fractional Laplacian in conformal geometry, Advances in Mathematics, 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016.

[13]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 0.1002/cpa.20116.

[14]

M. Chipot, M. Chlebík, M. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $\mathbbR_{+}^n$ with a nonlinear boundary condition, J. Math. Anal. Appl., 223 (1998), 429-471. doi: 10.1006/jmaa.1998.5958.

[15]

P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, to appear in Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[16]

S. Filippas, L. Moschini and A. Tertikas, Sharp trace Hardy-Sobolev-Maz'ya inequalities and the fractional laplacian, preprint.

[17]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.

[18]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. in Part. Diff. Equa., 6 (1981), 883-901.

[19]

Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. doi: 10.4171/JEMS/6.

[20]

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417. doi: 10.1215/S0012-7094-95-08016-8.

[21]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.

[22]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2006), 67-112. doi: 10.1002/cpa.20153.

[23]

M. Struwe, "Variational Methods," Ergebnisse der Mathematik und ihrer Grenzgebiete 34, Springer-Verlag, 1996.

[24]

S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., 12 (1975), 45-51.

[25]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Vari. and Part. Diff. Equa., 42 (2011), 21-41.

[26]

J. Xiao, A sharp Sobolev trace inequality for the fractional-order derivatives, Bull. Sci. Math., 130 (2006), 87-96. doi: 10.1016/j.bulsci.2005.07.002.

show all references

References:
[1]

D. Applebaum, Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.

[2]

H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Math., 22 (1991), 1-37.

[3]

K. Bogdan and B. Dyda, The best constant in a fractional Hardy inequalityarXiv:0807.1825.

[4]

C. Brändle, E. Colorado and A. de Pablo, A concave-convex elliptic problem involving the fractional laplacianarXiv:1006.4510.

[5]

X. Cabre and E. Cinti, Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian, Disc. Cont. Dyna. Syst., 28 (2010), 1179-1206. doi: 10.3934/dcds.2010.28.1179.

[6]

X. Cabre and J. Solà-Morales, Layer solutions in a halfspace for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093.

[7]

X. Cabre and Y. Sire, Nonlinear equations for fractional laplacians I: regularity, maximum principles, and hamiltonian estimates, preprint, arXiv:1012.0867.

[8]

X. Cabre and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Advances in Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.

[9]

L. Caffarelli, J. M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.

[10]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Part. Diff. Equa., 32 (2007), 1245-1260.

[11]

A. Capella, J. Davila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.

[12]

A. Chang, M. Gonzalez, Fractional Laplacian in conformal geometry, Advances in Mathematics, 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016.

[13]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 0.1002/cpa.20116.

[14]

M. Chipot, M. Chlebík, M. Fila and I. Shafrir, Existence of positive solutions of a semilinear elliptic equation in $\mathbbR_{+}^n$ with a nonlinear boundary condition, J. Math. Anal. Appl., 223 (1998), 429-471. doi: 10.1006/jmaa.1998.5958.

[15]

P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, to appear in Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[16]

S. Filippas, L. Moschini and A. Tertikas, Sharp trace Hardy-Sobolev-Maz'ya inequalities and the fractional laplacian, preprint.

[17]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.

[18]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. in Part. Diff. Equa., 6 (1981), 883-901.

[19]

Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. doi: 10.4171/JEMS/6.

[20]

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417. doi: 10.1215/S0012-7094-95-08016-8.

[21]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.

[22]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2006), 67-112. doi: 10.1002/cpa.20153.

[23]

M. Struwe, "Variational Methods," Ergebnisse der Mathematik und ihrer Grenzgebiete 34, Springer-Verlag, 1996.

[24]

S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., 12 (1975), 45-51.

[25]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Vari. and Part. Diff. Equa., 42 (2011), 21-41.

[26]

J. Xiao, A sharp Sobolev trace inequality for the fractional-order derivatives, Bull. Sci. Math., 130 (2006), 87-96. doi: 10.1016/j.bulsci.2005.07.002.

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