On a semilinear elliptic equation in n
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 4, pp. 635-671.

We prove existence/nonexistence and uniqueness of positive entire solutions for some semilinear elliptic equations on the Hyperbolic space.

Classification : 35J60, 35B05, 35A15
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     title = {On a semilinear elliptic equation in $\mathbb {H}^n$},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {635--671},
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     volume = {Ser. 5, 7},
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Mancini, Gianni; Sandeep, Kunnath. On a semilinear elliptic equation in $\mathbb {H}^n$. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 4, pp. 635-671. http://www.numdam.org/item/ASNSP_2008_5_7_4_635_0/

[1] L. Almeida, L. Damascelli and Y. Ge, A few symmetry results for nonlinear elliptic PDE on noncompact manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 313-342. | EuDML | Numdam | MR | Zbl

[2] W. Beckner, On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math. Soc. 129 (2001), 1233-1246. | MR | Zbl

[3] R. D. Benguria, R. L. Frank and M. Loss, The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space, Math. Res. Lett. 15 (2008), 613-622. | MR | Zbl

[4] H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994), 47-92. | MR | Zbl

[5] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437-477. | MR | Zbl

[6] M. Badiale and G. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal. 163 (2002), 259-293. | MR | Zbl

[7] A. Bonfiglioli and F. Uguzzoni, Nonlinear Liouvile Theorems for some critical problems on H-type groups, J. Funct. Anal. 207 (2004), 161-215. | MR | Zbl

[8] D. Cao and Y.Y. Li, Results on positive solutions of elliptic equations with a critical Hardy-Sobolev operator, Methods Appl. Anal., in press. | MR | Zbl

[9] D. Castorina, I. Fabbri, G. Mancini and K. Sandeep, Hardy Sobolev inequalities and hyperbolic symmetry, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.19 (2008), 189-197. | MR | Zbl

[10] I. Chavel, “Eigenvalues in Riemannian Geometry", Pure and Applied Mathematics, 115, Academic Press, Inc., Orlando, FL, 1984. | MR | Zbl

[11] A. Coddington-Earl and N. Levinson, “Theory of Ordinary Differential Equations", TATA McGraw-Hill Publishing Co. LTD, New Delhi, 1985. | MR | Zbl

[12] L. D'Ambrosio, Hardy inequalities related to Grushin type operators, Proc. Amer. Math. Soc. 132 (2003), 725-734. | MR | Zbl

[13] I. Fabbri, G. Mancini and K. Sandeep, Classification of solutions of a critical Hardy-Sobolev operator, J. Differential Equations 224 (2006), 258-276. | MR | Zbl

[14] V. Felli and F. Uguzzoni, Some existence results for the Webster scalar curvature problem in presence of symmetry, Ann. Mat. Pura Appl. 183 (2004), 469-493. | MR | Zbl

[15] N. Garofalo and D. Vassilev, Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type, Duke Math. J. 106 (2001), 411-448. | MR | Zbl

[16] M. Gazzini and R. Musina, Hardy-Sobolev-Maz'ya inequalities: symmetry and breaking symmetry of extremal functions, to appear on Contemp. Math. | MR | Zbl

[17] E. Hebey, “Sobolev Spaces on Riemannian Manifolds", Lecture Notes in Mathematics, Vol. 635, Springer-Verlag, 1996. | MR | Zbl

[18] Kwong, M. Kam-Li and Yi, Uniqueness of radial solutions of semilinear elliptic equations, Trans. Amer. Math. Soc. 333 (1992), 339-363. | MR | Zbl

[19] A. Malchiodi and F. Uguzzoni, A perturbation result for the Webster scalar curvature problem on the CR sphere, J. Math. Pure Appl. 81 (2002), 983-997. | MR | Zbl

[20] Maz'ya and G. Vladimir, “Sobolev Spaces", Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. | MR

[21] R. Musina, Ground state solutions of a critical problem involving cylindrical weights, Nonlinear Anal. 68 (2008), 3972-3986. | MR | Zbl

[22] R. Monti, Sobolev inequalities for weighted gradients, Comm. Partial Differential Equations 31 (2006), 1479-1504. | MR | Zbl

[23] R. Monti and D. Morbidelli, Kelvin transform for Grushin operators and critical semilinear equations, Duke Math. J. 131 (2006), 167-202. | MR | Zbl

[24] G. Mancini and K. Sandeep, Cylindrical symmetry of extremals of a Hardy-Sobolev inequality, Ann. Mat. Pura Appl. (4) 183 (2004), 165-172. | MR | Zbl

[25] S. Secchi, D. Smets and M. Willem, Remarks on a Hardy-Sobolev inequality, C. R. Math. Acad. Sci. Paris 336 (2003), 811-815. | MR | Zbl

[26] S. Stapelkamp, The Brézis-Nirenberg problem on n . Existence and uniqueness of solutions. Elliptic and parabolic problems, (Rolduc/Gaeta, 2001), 283-290, World Sci. Publ., River Edge, NJ, 2002. | MR | Zbl

[27] S. Stapelkamp, “Das Brezis-Nirenberg Problem im Hn”, PhD thesis, Universität Basel, 2003.

[28] A. Tertikas and K. Tintarev, On existence of minimizers for the Hardy-Sobolev-Maz'ya inequality, Ann. Mat. Pura Appl. (4) 186 (2007), 645-662. | MR | Zbl