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A strongly singular parabolic problem on an unbounded domain
Radial and non radial ground states for a class of dilation invariant fourth order semilinear elliptic equations on $R^n$
1. | Dipartimento di Matematica, Università di Torino, via Carlo Alberto, 10-10123 Torino, Italy |
References:
[1] |
Adimurthi, M. Grossi and S. Santra, Optimal Hardy-Rellich inequalities, maximum principle and related eigenvalue problem, J. Funct. Anal., 240 (2006), 36-83.
doi: 10.1016/j.jfa.2006.07.011. |
[2] |
Adimurthi and S. Santra, Generalized Hardy-Rellich inequalities in critical dimensions and its applications, Commun. Contemp. Math., 11 (2009), 367-394.
doi: 10.1142/S0219199709003405. |
[3] |
C. O. Alves and J. M. do Ò, Positive solutions of a fourth-order semilinear problem involving critical growth, Adv. Nonlinear Stud., 2 (2002), 437-458. |
[4] |
M. Bhakta and R. Musina, Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials, Nonlinear Anal. T.M.A., 75 (2012), 3836-3848.
doi: 10.1016/j.na.2012.02.005. |
[5] |
P. Caldiroli and R. Musina, On Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic operator in cones, Milan J. Math., 79 (2011), 657-687.
doi: 10.1007/s00032-011-0167-2. |
[6] |
P. Caldiroli and R. Musina, A class of second order dilation invariant inequalities,, in, ().
|
[7] |
F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258.
doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I. |
[8] |
N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann., 349 (2011), 1-57.
doi: 10.1007/s00208-010-0510-x. |
[9] |
N. Ghoussoub and A. Moradifam, "Functional Inequalities: New Perspectives and New Applications," Mathematical Surveys and Monographs, vol. 187. American Mathematical Society, (2013). |
[10] |
C.-S. Lin, Interpolation inequalities with weights, Comm. Part. Diff. Eq., 11 (1986), 1515-1538.
doi: 10.1080/03605308608820473. |
[11] |
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The Limit Case, Part 1, Rev. Mat. Iberoam., 1 (1985), 145-201.
doi: 10.4171/RMI/6. |
[12] |
E. Mitidieri, A Rellich type identity and applications, Comm. Part. Diff. Eq., 18 (1993), 125-151.
doi: 10.1080/03605309308820923. |
[13] |
E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\R^N$, Diff. Int. Eq., 9 (1996), 465-479. |
[14] |
A. Moradifam, Optimal weighted Hardy-Rellich inequalities on $H^2 \cap H^1_0$, J. London. Math. Soc., 85 (2011), 22-40.
doi: 10.1112/jlms/jdr045. |
[15] |
R. Musina, Weighted Sobolev spaces of radially symmetric functions,, Ann. Mat. Pura Appl., ().
doi: 10.1007/s10231-013-0348-4. |
[16] |
E. S. Noussair, C. A. Swanson and J. Yang, Transcritical Biharmonic Equations in $R^N$, Funkcialaj Ekvacioj, 35 (1992), 533-543. |
[17] |
F. Rellich, Halbbeschränkte Differentialoperatoren höherer Ordnung, in "Proceedings of the International Congress of Mathematicians (1954)'' (Gerretsen, J.C.H., de Groot, J., eds.), vol. III, pp. 243-250. Noordhoff, Groningen (1956). |
[18] |
F. Rellich, "Perturbation Theory of Eigenvalue Problems," Gordon and Breach, New York (1969). |
[19] |
M. Struwe, "Variational Methods," fourth edition, Springer, 2008.
doi: PMCid:PMC2582268. |
[20] |
C. A. Swanson, The best Sobolev constant, Appl. Anal., 47 (1992), 227-239.
doi: 10.1080/00036819208840142. |
[21] |
S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Eq., 1 (1996), 241-264. |
[22] |
A. Tertikas and N. B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improvements, Adv. Math., 209 (2007), 407-459.
doi: 10.1016/j.aim.2006.05.011. |
show all references
References:
[1] |
Adimurthi, M. Grossi and S. Santra, Optimal Hardy-Rellich inequalities, maximum principle and related eigenvalue problem, J. Funct. Anal., 240 (2006), 36-83.
doi: 10.1016/j.jfa.2006.07.011. |
[2] |
Adimurthi and S. Santra, Generalized Hardy-Rellich inequalities in critical dimensions and its applications, Commun. Contemp. Math., 11 (2009), 367-394.
doi: 10.1142/S0219199709003405. |
[3] |
C. O. Alves and J. M. do Ò, Positive solutions of a fourth-order semilinear problem involving critical growth, Adv. Nonlinear Stud., 2 (2002), 437-458. |
[4] |
M. Bhakta and R. Musina, Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials, Nonlinear Anal. T.M.A., 75 (2012), 3836-3848.
doi: 10.1016/j.na.2012.02.005. |
[5] |
P. Caldiroli and R. Musina, On Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic operator in cones, Milan J. Math., 79 (2011), 657-687.
doi: 10.1007/s00032-011-0167-2. |
[6] |
P. Caldiroli and R. Musina, A class of second order dilation invariant inequalities,, in, ().
|
[7] |
F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258.
doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I. |
[8] |
N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann., 349 (2011), 1-57.
doi: 10.1007/s00208-010-0510-x. |
[9] |
N. Ghoussoub and A. Moradifam, "Functional Inequalities: New Perspectives and New Applications," Mathematical Surveys and Monographs, vol. 187. American Mathematical Society, (2013). |
[10] |
C.-S. Lin, Interpolation inequalities with weights, Comm. Part. Diff. Eq., 11 (1986), 1515-1538.
doi: 10.1080/03605308608820473. |
[11] |
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The Limit Case, Part 1, Rev. Mat. Iberoam., 1 (1985), 145-201.
doi: 10.4171/RMI/6. |
[12] |
E. Mitidieri, A Rellich type identity and applications, Comm. Part. Diff. Eq., 18 (1993), 125-151.
doi: 10.1080/03605309308820923. |
[13] |
E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\R^N$, Diff. Int. Eq., 9 (1996), 465-479. |
[14] |
A. Moradifam, Optimal weighted Hardy-Rellich inequalities on $H^2 \cap H^1_0$, J. London. Math. Soc., 85 (2011), 22-40.
doi: 10.1112/jlms/jdr045. |
[15] |
R. Musina, Weighted Sobolev spaces of radially symmetric functions,, Ann. Mat. Pura Appl., ().
doi: 10.1007/s10231-013-0348-4. |
[16] |
E. S. Noussair, C. A. Swanson and J. Yang, Transcritical Biharmonic Equations in $R^N$, Funkcialaj Ekvacioj, 35 (1992), 533-543. |
[17] |
F. Rellich, Halbbeschränkte Differentialoperatoren höherer Ordnung, in "Proceedings of the International Congress of Mathematicians (1954)'' (Gerretsen, J.C.H., de Groot, J., eds.), vol. III, pp. 243-250. Noordhoff, Groningen (1956). |
[18] |
F. Rellich, "Perturbation Theory of Eigenvalue Problems," Gordon and Breach, New York (1969). |
[19] |
M. Struwe, "Variational Methods," fourth edition, Springer, 2008.
doi: PMCid:PMC2582268. |
[20] |
C. A. Swanson, The best Sobolev constant, Appl. Anal., 47 (1992), 227-239.
doi: 10.1080/00036819208840142. |
[21] |
S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Eq., 1 (1996), 241-264. |
[22] |
A. Tertikas and N. B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improvements, Adv. Math., 209 (2007), 407-459.
doi: 10.1016/j.aim.2006.05.011. |
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