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Multiplicity results for classes of singular problems on an exterior domain
On the nonexistence of positive solutions to doubly nonlinear equations for Baouendi-Grushin operators
1. | Department of Mathematics, Faculty of Science and Letters, Istanbul Commerce University, Uskudar, Istanbul, Turkey |
References:
[1] |
J. A. Aguilar Crespo and I. Peral Alonso, Global behaviour of the Cauchy problem for some critical nonlinear parabolic equations,, SIAM J. Math. Anal., 31 (2000), 1270.
doi: 10.1137/S0036141098341137. |
[2] |
B. Abdellaoui, Eduardo Colorado and I. Peral, Existence and nonexistence results for a class of linear and semilinear parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities,, Journal of the European Mathematical Society, 6 (2004), 119.
|
[3] |
P. Baras and J. A. Goldstein, The heat equation with a singular potential,, Trans. AMS, 284 (1984), 121.
doi: 10.1090/S0002-9947-1984-0742415-3. |
[4] |
M. S. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés,, Bull. Soc. Math. France, 95 (1967), 45.
|
[5] |
W. Beckner, On the Grushin operator and hyperbolic symmetry,, Proc. Amer. Math. Soc., 129 (2001), 1233.
doi: 10.1090/S0002-9939-00-05630-6. |
[6] |
A. Bellaiche and J. J. Risler, "Sub-Riemannian Geometry,", Birkhauser, (1996).
doi: 10.1007/978-3-0348-9210-0. |
[7] |
T. Bieske, Viscosity solutions on Grushin-planes,, Illinois J. Math, 46 (2002), 893.
|
[8] |
X. Cabré and Y. Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier,, C. R. Acad. Sci. Paris, 329 (1999), 973.
doi: 10.1016/S0764-4442(00)88588-2. |
[9] |
Lorenzo D'Ambrosio, Hardy inequalities related to Grushin type operators,, Proc. Amer. Math. Soc., 132 (2004), 725.
doi: 10.1090/S0002-9939-03-07232-0. |
[10] |
A. Dall' Aglio, D. Giachetti and I. Peral, Results on Parabolic Equations Related to some Caffarelli-Kohn-Nirenberg inequalities,, SIAM J. Math. Anal., 36 (2004), 691.
doi: 10.1137/S0036141003432353. |
[11] |
J. Garcia Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems,, J. Diff. Equations, 144 (1998), 441.
doi: 10.1006/jdeq.1997.3375. |
[12] |
C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems,, in, (1981), 590.
|
[13] |
F. Ferrari and B. Franchi, Geometry of the boundary and doubling property of the harmonic measure for Grushin type operators,, Rend. Sem. Univ. e Politec. Torino, 58 (2000), 281.
|
[14] |
B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of linear nonlinear uniformly elliptic operators with measurable coefficients,, Ann. Scuola Norm. Sup. Pisa (4), 10 (1983), 523.
|
[15] |
B. Franchi, C. E. Gutiérrez and R. L. Wheeden, Weighted Sobolev-Poincare inequalities for Grushin type operators,, Comm. Partial Differential Equations, 19 (1994), 523.
doi: 10.1080/03605309408821025. |
[16] |
N. Garofalo, Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension,, J. Diff. Equations, 104 (1993), 117.
doi: 10.1006/jdeq.1993.1065. |
[17] |
N. Garofalo and Z. Shen, Absence of positive eigenvalues for a class of subelliptic operators,, Math. Ann., 304 (1996), 701. Google Scholar |
[18] |
J. A. Goldstein, G. Ruiz Goldstein and I. Kombe, Nonlinear parabolic equations with singular coefficient and critical exponent,, Applicable Analysis, 84 (2005), 571.
doi: 10.1080/00036810500047709. |
[19] |
J. A. Goldstein and I. Kombe, Instantaneous blow up,, Contemp. Math, 327 (2003), 141.
doi: 10.1090/conm/327/05810. |
[20] |
J. A. Goldstein and I. Kombe, Nonlinear parabolic differentail equations with singular lower order term,, Adv. Differential Equations, 10 (2003), 1153.
|
[21] |
J. A. Goldstein and I. Kombe, Nonlinear degenerate differential equations with singular lower order term on the Heisenberg group,, International Journal of Evolution Equations, 1 (2005), 1.
|
[22] |
J. A. Goldstein and Q. S. Zhang, On a degenerate heat equation with a singular potential,, J. Functional Analysis, 186 (2001), 342.
doi: 10.1006/jfan.2001.3792. |
[23] |
J. A. Goldstein and Q. S. Zhang, Linear parabolic equations with strong singular potentials,, Trans. AMS, 355 (2003), 197.
doi: 10.1090/S0002-9947-02-03057-X. |
[24] |
V. Grushin, A certain class of hypoelliptic operators,, Mat. Sb. (N.S), 83 (1970), 456.
|
[25] |
V. Grushin, A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold,, Mat. Sb., 84 (1971), 163.
|
[26] |
A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations,, Russian Math. Surveys, 42 (1987), 169.
|
[27] |
J. L. Lions, "Quelque Methodes de Resolution des Problemes aux Limites Nonlineaire,", Springer, (). Google Scholar |
[28] |
J. Juan Manfredi and V. Vespri, Large time behavior of solutions to a class of doubly nonlinear parabolic equations,, Electron. J. Differential Equations, (1994), 1.
|
[29] |
G. Savar and V. Vespri, The asymptotic profile of solutions of a class of doubly nonlinear equations,, Nonlinear Anal., 22 (1994), 1553.
doi: 10.1016/0362-546X(94)90188-0. |
[30] |
I. Kombe, The linear heat equation with a highly oscillating potential,, Proc. Amer. Math. Soc., 132 (2004), 2683.
doi: 10.1090/S0002-9939-04-07392-7. |
[31] |
I. Kombe, Doubly nonlinear parabolic equations with singular lower order term,, Nonlinear Analysis, 56 (2004), 185.
doi: 10.1016/j.na.2003.09.006. |
[32] |
I. Kombe, Nonlinear degenerate parabolic equations for Baouendi-Grushin operators,, Mathematische Nachrichten, 279 (2006), 756.
doi: 10.1002/mana.200310391. |
[33] |
I. Kombe, Cauchy-Dirichlet problem for the nonlinear degenerate parabolic equations,, Abstract and Applied Analysis, 6 (2005), 607.
doi: 10.1155/AAA.2005.607. |
[34] |
I. Kombe, Hardy and Rellich type inequalities with remainders for Baouendi-Grushin vector fields,, to appear in Houston Journal of Mathematics., (). Google Scholar |
[35] |
F. Lascialfari and D. Pardo, Compact embedding of a degenerate Sobolev space and existence of entire solutions to semilinear equation for a Grushin-type operator,, Rend. Sem. Mat. Univ. Padova, 107 (2002), 139.
|
[36] |
R. Monti and D. Morbidelli, Isoperimetric inequality in the Grushin plane,, J. Geom. Anal., 14 (2004), 355.
doi: 10.1007/BF02922077. |
[37] |
S. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields, I: Basic properties,, Acta Math., 155 (1985), 103.
doi: 10.1007/BF02392539. |
show all references
References:
[1] |
J. A. Aguilar Crespo and I. Peral Alonso, Global behaviour of the Cauchy problem for some critical nonlinear parabolic equations,, SIAM J. Math. Anal., 31 (2000), 1270.
doi: 10.1137/S0036141098341137. |
[2] |
B. Abdellaoui, Eduardo Colorado and I. Peral, Existence and nonexistence results for a class of linear and semilinear parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities,, Journal of the European Mathematical Society, 6 (2004), 119.
|
[3] |
P. Baras and J. A. Goldstein, The heat equation with a singular potential,, Trans. AMS, 284 (1984), 121.
doi: 10.1090/S0002-9947-1984-0742415-3. |
[4] |
M. S. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés,, Bull. Soc. Math. France, 95 (1967), 45.
|
[5] |
W. Beckner, On the Grushin operator and hyperbolic symmetry,, Proc. Amer. Math. Soc., 129 (2001), 1233.
doi: 10.1090/S0002-9939-00-05630-6. |
[6] |
A. Bellaiche and J. J. Risler, "Sub-Riemannian Geometry,", Birkhauser, (1996).
doi: 10.1007/978-3-0348-9210-0. |
[7] |
T. Bieske, Viscosity solutions on Grushin-planes,, Illinois J. Math, 46 (2002), 893.
|
[8] |
X. Cabré and Y. Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier,, C. R. Acad. Sci. Paris, 329 (1999), 973.
doi: 10.1016/S0764-4442(00)88588-2. |
[9] |
Lorenzo D'Ambrosio, Hardy inequalities related to Grushin type operators,, Proc. Amer. Math. Soc., 132 (2004), 725.
doi: 10.1090/S0002-9939-03-07232-0. |
[10] |
A. Dall' Aglio, D. Giachetti and I. Peral, Results on Parabolic Equations Related to some Caffarelli-Kohn-Nirenberg inequalities,, SIAM J. Math. Anal., 36 (2004), 691.
doi: 10.1137/S0036141003432353. |
[11] |
J. Garcia Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems,, J. Diff. Equations, 144 (1998), 441.
doi: 10.1006/jdeq.1997.3375. |
[12] |
C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems,, in, (1981), 590.
|
[13] |
F. Ferrari and B. Franchi, Geometry of the boundary and doubling property of the harmonic measure for Grushin type operators,, Rend. Sem. Univ. e Politec. Torino, 58 (2000), 281.
|
[14] |
B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of linear nonlinear uniformly elliptic operators with measurable coefficients,, Ann. Scuola Norm. Sup. Pisa (4), 10 (1983), 523.
|
[15] |
B. Franchi, C. E. Gutiérrez and R. L. Wheeden, Weighted Sobolev-Poincare inequalities for Grushin type operators,, Comm. Partial Differential Equations, 19 (1994), 523.
doi: 10.1080/03605309408821025. |
[16] |
N. Garofalo, Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension,, J. Diff. Equations, 104 (1993), 117.
doi: 10.1006/jdeq.1993.1065. |
[17] |
N. Garofalo and Z. Shen, Absence of positive eigenvalues for a class of subelliptic operators,, Math. Ann., 304 (1996), 701. Google Scholar |
[18] |
J. A. Goldstein, G. Ruiz Goldstein and I. Kombe, Nonlinear parabolic equations with singular coefficient and critical exponent,, Applicable Analysis, 84 (2005), 571.
doi: 10.1080/00036810500047709. |
[19] |
J. A. Goldstein and I. Kombe, Instantaneous blow up,, Contemp. Math, 327 (2003), 141.
doi: 10.1090/conm/327/05810. |
[20] |
J. A. Goldstein and I. Kombe, Nonlinear parabolic differentail equations with singular lower order term,, Adv. Differential Equations, 10 (2003), 1153.
|
[21] |
J. A. Goldstein and I. Kombe, Nonlinear degenerate differential equations with singular lower order term on the Heisenberg group,, International Journal of Evolution Equations, 1 (2005), 1.
|
[22] |
J. A. Goldstein and Q. S. Zhang, On a degenerate heat equation with a singular potential,, J. Functional Analysis, 186 (2001), 342.
doi: 10.1006/jfan.2001.3792. |
[23] |
J. A. Goldstein and Q. S. Zhang, Linear parabolic equations with strong singular potentials,, Trans. AMS, 355 (2003), 197.
doi: 10.1090/S0002-9947-02-03057-X. |
[24] |
V. Grushin, A certain class of hypoelliptic operators,, Mat. Sb. (N.S), 83 (1970), 456.
|
[25] |
V. Grushin, A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold,, Mat. Sb., 84 (1971), 163.
|
[26] |
A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations,, Russian Math. Surveys, 42 (1987), 169.
|
[27] |
J. L. Lions, "Quelque Methodes de Resolution des Problemes aux Limites Nonlineaire,", Springer, (). Google Scholar |
[28] |
J. Juan Manfredi and V. Vespri, Large time behavior of solutions to a class of doubly nonlinear parabolic equations,, Electron. J. Differential Equations, (1994), 1.
|
[29] |
G. Savar and V. Vespri, The asymptotic profile of solutions of a class of doubly nonlinear equations,, Nonlinear Anal., 22 (1994), 1553.
doi: 10.1016/0362-546X(94)90188-0. |
[30] |
I. Kombe, The linear heat equation with a highly oscillating potential,, Proc. Amer. Math. Soc., 132 (2004), 2683.
doi: 10.1090/S0002-9939-04-07392-7. |
[31] |
I. Kombe, Doubly nonlinear parabolic equations with singular lower order term,, Nonlinear Analysis, 56 (2004), 185.
doi: 10.1016/j.na.2003.09.006. |
[32] |
I. Kombe, Nonlinear degenerate parabolic equations for Baouendi-Grushin operators,, Mathematische Nachrichten, 279 (2006), 756.
doi: 10.1002/mana.200310391. |
[33] |
I. Kombe, Cauchy-Dirichlet problem for the nonlinear degenerate parabolic equations,, Abstract and Applied Analysis, 6 (2005), 607.
doi: 10.1155/AAA.2005.607. |
[34] |
I. Kombe, Hardy and Rellich type inequalities with remainders for Baouendi-Grushin vector fields,, to appear in Houston Journal of Mathematics., (). Google Scholar |
[35] |
F. Lascialfari and D. Pardo, Compact embedding of a degenerate Sobolev space and existence of entire solutions to semilinear equation for a Grushin-type operator,, Rend. Sem. Mat. Univ. Padova, 107 (2002), 139.
|
[36] |
R. Monti and D. Morbidelli, Isoperimetric inequality in the Grushin plane,, J. Geom. Anal., 14 (2004), 355.
doi: 10.1007/BF02922077. |
[37] |
S. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields, I: Basic properties,, Acta Math., 155 (1985), 103.
doi: 10.1007/BF02392539. |
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