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Elliptic equations with cylindrical potential and multiple critical exponents
1. | Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China |
2. | Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, PO Box 71010, Wuhan 430071E01103 |
References:
[1] |
J. Bellazzini and C. Bonanno, Nonlinear Schrödinger equations with strongly singular potentials, Proceedings of the Royal Society of Edinburgh, 140A (2010), 707-721.
doi: 10.1017/S0308210509001401. |
[2] |
M. Badiale, V. Bergio and S. Rolando, A nonlinear elliptic equation with singular potential and applications to nonlinear field equations, J. Eur. Math. Soc., 9 (2007), 355-381. |
[3] |
M. Badiale, M. Guida and S. Rolando, Elliptic equations with decaying cylindrical potentials and power-type nonlinearities, Adv. Diff. Equ., 12 (2007), 1321-1362.
doi: 10.1007/s00009-005-0055-5. |
[4] |
M. Badiale and S. Rolando, Nonlinear elliptic equations with subhomogeneous potentials, Nonlinear Analysis, 72 (2010), 602-617.
doi: 10.1016/j.na.2009.06.111. |
[5] |
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), 252-293.
doi: 10.1007/s002050200201. |
[6] |
M. Bhakta and K. Sandeep, Hardy-Sobolev-Maz'ya type equations in bounded domains, J. Differential Equations, 247 (2009), 119-139.
doi: 10.1016/j.jde.2008.12.011. |
[7] |
H. Brezis and E. Lieb, A relation between pointwise convergence of functionals and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[8] |
D. Castorina, I. Fabbri, G. Mancini and K. Sandeep, Hardy-Sobolev extremals, hyperbolic symmetry and scalar curvature equations, J. Differential Equations, 246 (2009), 1187-1206.
doi: 10.1016/j.jde.2008.09.006. |
[9] |
D. M. Cao and P. G. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials, J. Differential Equations, 224 (2006), 332-372.
doi: 10.1016/j.jde.2005.07.010. |
[10] |
D. M. Cao and Y. Y. Li, Results on positive solutions of elliptic equations with a critical Hardy-Sobolev operator, Methods and Applications of Analysis, 15 (2008), 081-096.
doi: 10.1.1.140.417. |
[11] |
L. D'Ambrosio, Hardy type inequalities related to degenerate elliptic differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci., 4 (2005), 451-486. |
[12] |
R. Filippucci, P. Pucci and F. Robert, On a p-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl., 91 (2009), 156-177.
doi: 10.1016/j.matpur.2008.09.008. |
[13] |
F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc., 356 (2004), 2149-2168.
doi: 10.1090/S0002-9947-03-03395-6. |
[14] |
N. Ghoussoub and X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire, 21 (2004), 767-793.
doi: 10.1016/j.anihpc.2003.07.002. |
[15] |
Y. Y. Li and C. S. Lin, A nonlinear Elliptic PDE with two Sobolev-Hardy critical exponents, Arch. Rational Mech. Anal., 203 (2012), 943-968.
doi: 10.1007/s00205-011-0467-2. |
[16] |
G. Mancini, I. Fabbri and K. Sandeep, Classification of solutions of a critical Hardy-Sobolev operator, J. Differential Equations, 224 (2006), 258-276.
doi: 10.1016/j.jde.2005.07.001. |
[17] |
G. Mancini and K. Sandeep, On a semilinear elliptic equation in $\Bbb H^n$, Ann. Sc. Norm. Super. Pisa Cl. Sci., 5 (2008), 635-671. |
[18] |
R. Musina, Ground state solutions of a critical problem involving cylindrical weights, Nonlinear Anal., 68 (2008), 3972-3986.
doi: 10.1016/j.na.2007.04.034. |
[19] |
R. S. Palais, The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30.
doi: 10.1007/BF01941322. |
[20] |
J. B. Su and Z. Q. Wang, Sobolev type embedding and quasilinear elliptic equations with radial potentials, J. Differential Equa., 250 (2011), 223-242.
doi: 10.1016/j.jde.2010.08.025. |
[21] |
J. B. Su, Z. Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equa., 238 (2007), 201-219.
doi: 10.1016/j.jde.2007.03.018. |
[22] |
G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.
doi: 10.1007/BF02418013. |
[23] |
S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 2 (1996), 241-264. |
[24] |
A. Tertikas and K. Tintarev, On existence of minimizers for the Hardy-Sobolev-Maz'ya inequality, Ann. Mat. Pura e Appl., 186 (2007), 645-662.
doi: 10.1007/s10231-006-0024-z. |
[25] |
S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space, Ann. Inst. Henry Poincaré-Analyse Nonlinéaire, 12 (1995), 319-337. |
[26] |
M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Third edition, Springer-Verlag, Berlin, 2000. |
show all references
References:
[1] |
J. Bellazzini and C. Bonanno, Nonlinear Schrödinger equations with strongly singular potentials, Proceedings of the Royal Society of Edinburgh, 140A (2010), 707-721.
doi: 10.1017/S0308210509001401. |
[2] |
M. Badiale, V. Bergio and S. Rolando, A nonlinear elliptic equation with singular potential and applications to nonlinear field equations, J. Eur. Math. Soc., 9 (2007), 355-381. |
[3] |
M. Badiale, M. Guida and S. Rolando, Elliptic equations with decaying cylindrical potentials and power-type nonlinearities, Adv. Diff. Equ., 12 (2007), 1321-1362.
doi: 10.1007/s00009-005-0055-5. |
[4] |
M. Badiale and S. Rolando, Nonlinear elliptic equations with subhomogeneous potentials, Nonlinear Analysis, 72 (2010), 602-617.
doi: 10.1016/j.na.2009.06.111. |
[5] |
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), 252-293.
doi: 10.1007/s002050200201. |
[6] |
M. Bhakta and K. Sandeep, Hardy-Sobolev-Maz'ya type equations in bounded domains, J. Differential Equations, 247 (2009), 119-139.
doi: 10.1016/j.jde.2008.12.011. |
[7] |
H. Brezis and E. Lieb, A relation between pointwise convergence of functionals and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[8] |
D. Castorina, I. Fabbri, G. Mancini and K. Sandeep, Hardy-Sobolev extremals, hyperbolic symmetry and scalar curvature equations, J. Differential Equations, 246 (2009), 1187-1206.
doi: 10.1016/j.jde.2008.09.006. |
[9] |
D. M. Cao and P. G. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials, J. Differential Equations, 224 (2006), 332-372.
doi: 10.1016/j.jde.2005.07.010. |
[10] |
D. M. Cao and Y. Y. Li, Results on positive solutions of elliptic equations with a critical Hardy-Sobolev operator, Methods and Applications of Analysis, 15 (2008), 081-096.
doi: 10.1.1.140.417. |
[11] |
L. D'Ambrosio, Hardy type inequalities related to degenerate elliptic differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci., 4 (2005), 451-486. |
[12] |
R. Filippucci, P. Pucci and F. Robert, On a p-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl., 91 (2009), 156-177.
doi: 10.1016/j.matpur.2008.09.008. |
[13] |
F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc., 356 (2004), 2149-2168.
doi: 10.1090/S0002-9947-03-03395-6. |
[14] |
N. Ghoussoub and X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire, 21 (2004), 767-793.
doi: 10.1016/j.anihpc.2003.07.002. |
[15] |
Y. Y. Li and C. S. Lin, A nonlinear Elliptic PDE with two Sobolev-Hardy critical exponents, Arch. Rational Mech. Anal., 203 (2012), 943-968.
doi: 10.1007/s00205-011-0467-2. |
[16] |
G. Mancini, I. Fabbri and K. Sandeep, Classification of solutions of a critical Hardy-Sobolev operator, J. Differential Equations, 224 (2006), 258-276.
doi: 10.1016/j.jde.2005.07.001. |
[17] |
G. Mancini and K. Sandeep, On a semilinear elliptic equation in $\Bbb H^n$, Ann. Sc. Norm. Super. Pisa Cl. Sci., 5 (2008), 635-671. |
[18] |
R. Musina, Ground state solutions of a critical problem involving cylindrical weights, Nonlinear Anal., 68 (2008), 3972-3986.
doi: 10.1016/j.na.2007.04.034. |
[19] |
R. S. Palais, The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30.
doi: 10.1007/BF01941322. |
[20] |
J. B. Su and Z. Q. Wang, Sobolev type embedding and quasilinear elliptic equations with radial potentials, J. Differential Equa., 250 (2011), 223-242.
doi: 10.1016/j.jde.2010.08.025. |
[21] |
J. B. Su, Z. Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equa., 238 (2007), 201-219.
doi: 10.1016/j.jde.2007.03.018. |
[22] |
G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.
doi: 10.1007/BF02418013. |
[23] |
S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 2 (1996), 241-264. |
[24] |
A. Tertikas and K. Tintarev, On existence of minimizers for the Hardy-Sobolev-Maz'ya inequality, Ann. Mat. Pura e Appl., 186 (2007), 645-662.
doi: 10.1007/s10231-006-0024-z. |
[25] |
S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space, Ann. Inst. Henry Poincaré-Analyse Nonlinéaire, 12 (1995), 319-337. |
[26] |
M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Third edition, Springer-Verlag, Berlin, 2000. |
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