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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[2]

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

[3]

K. Bogdan and B. Dyda, The best constant in a fractional Hardy inequality,, , ().   Google Scholar

[4]

C. Brändle, E. Colorado and A. de Pablo, A concave-convex elliptic problem involving the fractional laplacian,, , ().   Google Scholar

[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.  doi: 10.3934/dcds.2010.28.1179.  Google Scholar

[6]

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

[7]

X. Cabre and Y. Sire, Nonlinear equations for fractional laplacians I: regularity, maximum principles, and hamiltonian estimates,, preprint, ().   Google Scholar

[8]

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

[9]

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

[10]

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

[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.   Google Scholar

[12]

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

[13]

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

[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.  doi: 10.1006/jmaa.1998.5958.  Google Scholar

[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., ().   Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

M. Struwe, "Variational Methods,", Ergebnisse der Mathematik und ihrer Grenzgebiete 34, (1996).   Google Scholar

[24]

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

[25]

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

[26]

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

show all references

References:
[1]

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

[2]

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

[3]

K. Bogdan and B. Dyda, The best constant in a fractional Hardy inequality,, , ().   Google Scholar

[4]

C. Brändle, E. Colorado and A. de Pablo, A concave-convex elliptic problem involving the fractional laplacian,, , ().   Google Scholar

[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.  doi: 10.3934/dcds.2010.28.1179.  Google Scholar

[6]

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

[7]

X. Cabre and Y. Sire, Nonlinear equations for fractional laplacians I: regularity, maximum principles, and hamiltonian estimates,, preprint, ().   Google Scholar

[8]

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

[9]

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

[10]

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

[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.   Google Scholar

[12]

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

[13]

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

[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.  doi: 10.1006/jmaa.1998.5958.  Google Scholar

[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., ().   Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

M. Struwe, "Variational Methods,", Ergebnisse der Mathematik und ihrer Grenzgebiete 34, (1996).   Google Scholar

[24]

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

[25]

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

[26]

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

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