# American Institute of Mathematical Sciences

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

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##### 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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