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Isolated singularities for elliptic equations with hardy operator and source nonlinearity
1. | Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China |
2. | Center for PDEs and Department of Mathematics, East China Normal University, Shanghai 200241, China |
$- \Delta u + \frac{\mathit{\mu }}{{|x{|^2}}}u = {u^p}\;\;\;{\rm{in }}\;\;\;\Omega \setminus \{ 0\} ,\;\;\;u = 0\;\;\;{\rm{on}}\;\;\;\partial \Omega .\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right)$ |
References:
[1] |
O. Adimurthi, N. Chaudhuri and M. Ramaswamy,
An improved Hardy-Sobolev inequality and its application, Proc. Amer. Math. Soc., 130 (2002), 489-505.
doi: 10.1090/S0002-9939-01-06132-9. |
[2] |
P. Aviles,
Local behaviour of the solutions of some elliptic equations, Comm. Math. Phys., 108 (1987), 177-192.
doi: 10.1007/BF01210610. |
[3] |
L. Boccardo, L. Orsina and I. Peral,
A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential, Discrete Contin. Dyn. Syst. A, 16 (2006), 513-523.
doi: 10.3934/dcds.2006.16.513. |
[4] |
H. Brezis and P. Lions,
A note on isolated singularities for linear elliptic equations, in Mathematical Analysis and Applications, Acad. Press, 7 (1981), 263-266.
|
[5] |
H. Brezis and M. Marcus,
Hardy's inequalities revisited, Ann. Sc. Norm. Super. Pisa Cl. Sci., 25 (1997), 217-237.
|
[6] |
H. Brezis and L. Vázquez,
Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
|
[7] |
L. Caffarelli, B. Gidas and J. Spruck,
Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.
doi: 10.1002/cpa.3160420304. |
[8] |
D. Cao and Y. Li,
Results on positive solutions of elliptic equations with a critical Hardy-Sobolev operator, Methods Appl. Anal., 15 (2008), 81-95.
doi: 10.4310/MAA.2008.v15.n1.a8. |
[9] |
N. Chaudhuri and F. Cîrstea,
On trichotomy of positive singular solutions associated with the Hardy-Sobolev operator, C. R. Math. Acad. Sci. Paris, 347 (2009), 153-158.
doi: 10.1016/j.crma.2008.12.018. |
[10] |
H. Chen, A. Quaas and F. Zhou, On nonhomogeneous elliptic equations with the Hardy-Leray potentials, arXiv: 1705.08047. |
[11] |
H. Chen and F. Zhou,
Classification of isolated singularities of positive solutions for Choquard equations, J. Diff. Eq., 261 (2016), 6668-6698.
doi: 10.1016/j.jde.2016.08.047. |
[12] |
H. Chen and F. Zhou, Isolated singularities of positive solutions for Choquard equations in sublinear case, Comm. Cont. Math., (2017). |
[13] |
F. Cîrstea, A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials,
Mem. Amer. Math. Soc., 227 (2014), ⅵ+85 pp. |
[14] |
J. Davila and L. Dupaigne,
Hardy-type inequalities, J. Eur. Math. Soc., 6 (2004), 335-365.
|
[15] |
L. Dupaigne,
A nonlinear elliptic PDE with the inverse square potential, J. d'Analyse Mathématique, 86 (2002), 359-398.
doi: 10.1007/BF02786656. |
[16] |
M. Fall and R. Musina, Sharp nonexistence results for a linear elliptic inequality involving Hardy and Leray potentials,
J. Inequal. Appl., (2011), Art. ID 917201, 21 pp. |
[17] |
V. Felli and A. Ferrero,
On semilinear elliptic equations with borderline Hardy potentials, J. d'Analyse Mathématique, 123 (2014), 303-340.
doi: 10.1007/s11854-014-0022-9. |
[18] |
S. Filippas and A. Tertikas,
Optimizing improved Hardy inequalities, J. Func. Anal., 192 (2002), 186-233.
doi: 10.1006/jfan.2001.3900. |
[19] |
A. García and G. Peral,
Hardy inequalities and some critical elliptic and parabolic problems, J. Diff. Eq., 144 (1998), 441-476.
doi: 10.1006/jdeq.1997.3375. |
[20] |
M. Ghergu and S. Taliaferro,
Isolated Singularities in Partial Differential Inequalities, Cambridge University Press, 2016. |
[21] |
B. Gidas and J. Spruck,
Global and local behaviour of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.
doi: 10.1002/cpa.3160340406. |
[22] |
Q. Han and F. Lin,
Elliptic Partial Differential Equations, American Mathematical Soc., 2000.
doi: 10.1090/cln/001. |
[23] |
W. Jeong and Y. Lee,
Stable solutions and finite Morse index solutions of nonlinear elliptic equations with Hardy potential, Nonl. Anal. T.M.A., 87 (2013), 126-145.
doi: 10.1016/j.na.2013.04.007. |
[24] |
P. Lions,
Isolated singularities in semilinear problems, J. Diff. Eq., 38 (1980), 441-450.
doi: 10.1016/0022-0396(80)90018-2. |
[25] |
R. Mazzeo and F. Pacard,
A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis, J. Diff. Geometry., 44 (1996), 331-370.
doi: 10.4310/jdg/1214458975. |
[26] |
Y. Naito and T. Sato,
Positive solutions for semilinear elliptic equations with singular forcing terms, J. Diff. Eq., 235 (2007), 439-483.
doi: 10.1016/j.jde.2007.01.006. |
[27] |
F. Pacard,
Existence and convergence of positive weak solutions of $ -Δ u = u^{\frac{N}{N-2\ }}$ in bounded domains of $ {\mathbb{R}}^N$, Calc. Var. and PDEs., 1 (1993), 243-265.
doi: 10.1007/BF01191296. |
[28] |
Y. Pinchover and K. Tintarev,
Existence of minimizers for Schrödinger operators under domain perturbations with application to Hardy's inequality, Indiana Univ. Math. J., 54 (2005), 1061-1074.
doi: 10.1512/iumj.2005.54.2705. |
[29] |
P. Rabinowitz,
Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., 65 American Mathematical Society, 1986. |
[30] |
M. Struwe,
Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian systems, Ergebnisse der Mathematik und ihrer Grenzgebiete, 34. Springer-Verlag, Berlin, 1996. |
[31] |
L. Véron,
Singularities of Solutions of Second-Order Quasilinear Equations, Pitman Research Notes in Mathematics Series, 353. Longman, Harlow, 1996. |
show all references
References:
[1] |
O. Adimurthi, N. Chaudhuri and M. Ramaswamy,
An improved Hardy-Sobolev inequality and its application, Proc. Amer. Math. Soc., 130 (2002), 489-505.
doi: 10.1090/S0002-9939-01-06132-9. |
[2] |
P. Aviles,
Local behaviour of the solutions of some elliptic equations, Comm. Math. Phys., 108 (1987), 177-192.
doi: 10.1007/BF01210610. |
[3] |
L. Boccardo, L. Orsina and I. Peral,
A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential, Discrete Contin. Dyn. Syst. A, 16 (2006), 513-523.
doi: 10.3934/dcds.2006.16.513. |
[4] |
H. Brezis and P. Lions,
A note on isolated singularities for linear elliptic equations, in Mathematical Analysis and Applications, Acad. Press, 7 (1981), 263-266.
|
[5] |
H. Brezis and M. Marcus,
Hardy's inequalities revisited, Ann. Sc. Norm. Super. Pisa Cl. Sci., 25 (1997), 217-237.
|
[6] |
H. Brezis and L. Vázquez,
Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
|
[7] |
L. Caffarelli, B. Gidas and J. Spruck,
Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.
doi: 10.1002/cpa.3160420304. |
[8] |
D. Cao and Y. Li,
Results on positive solutions of elliptic equations with a critical Hardy-Sobolev operator, Methods Appl. Anal., 15 (2008), 81-95.
doi: 10.4310/MAA.2008.v15.n1.a8. |
[9] |
N. Chaudhuri and F. Cîrstea,
On trichotomy of positive singular solutions associated with the Hardy-Sobolev operator, C. R. Math. Acad. Sci. Paris, 347 (2009), 153-158.
doi: 10.1016/j.crma.2008.12.018. |
[10] |
H. Chen, A. Quaas and F. Zhou, On nonhomogeneous elliptic equations with the Hardy-Leray potentials, arXiv: 1705.08047. |
[11] |
H. Chen and F. Zhou,
Classification of isolated singularities of positive solutions for Choquard equations, J. Diff. Eq., 261 (2016), 6668-6698.
doi: 10.1016/j.jde.2016.08.047. |
[12] |
H. Chen and F. Zhou, Isolated singularities of positive solutions for Choquard equations in sublinear case, Comm. Cont. Math., (2017). |
[13] |
F. Cîrstea, A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials,
Mem. Amer. Math. Soc., 227 (2014), ⅵ+85 pp. |
[14] |
J. Davila and L. Dupaigne,
Hardy-type inequalities, J. Eur. Math. Soc., 6 (2004), 335-365.
|
[15] |
L. Dupaigne,
A nonlinear elliptic PDE with the inverse square potential, J. d'Analyse Mathématique, 86 (2002), 359-398.
doi: 10.1007/BF02786656. |
[16] |
M. Fall and R. Musina, Sharp nonexistence results for a linear elliptic inequality involving Hardy and Leray potentials,
J. Inequal. Appl., (2011), Art. ID 917201, 21 pp. |
[17] |
V. Felli and A. Ferrero,
On semilinear elliptic equations with borderline Hardy potentials, J. d'Analyse Mathématique, 123 (2014), 303-340.
doi: 10.1007/s11854-014-0022-9. |
[18] |
S. Filippas and A. Tertikas,
Optimizing improved Hardy inequalities, J. Func. Anal., 192 (2002), 186-233.
doi: 10.1006/jfan.2001.3900. |
[19] |
A. García and G. Peral,
Hardy inequalities and some critical elliptic and parabolic problems, J. Diff. Eq., 144 (1998), 441-476.
doi: 10.1006/jdeq.1997.3375. |
[20] |
M. Ghergu and S. Taliaferro,
Isolated Singularities in Partial Differential Inequalities, Cambridge University Press, 2016. |
[21] |
B. Gidas and J. Spruck,
Global and local behaviour of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.
doi: 10.1002/cpa.3160340406. |
[22] |
Q. Han and F. Lin,
Elliptic Partial Differential Equations, American Mathematical Soc., 2000.
doi: 10.1090/cln/001. |
[23] |
W. Jeong and Y. Lee,
Stable solutions and finite Morse index solutions of nonlinear elliptic equations with Hardy potential, Nonl. Anal. T.M.A., 87 (2013), 126-145.
doi: 10.1016/j.na.2013.04.007. |
[24] |
P. Lions,
Isolated singularities in semilinear problems, J. Diff. Eq., 38 (1980), 441-450.
doi: 10.1016/0022-0396(80)90018-2. |
[25] |
R. Mazzeo and F. Pacard,
A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis, J. Diff. Geometry., 44 (1996), 331-370.
doi: 10.4310/jdg/1214458975. |
[26] |
Y. Naito and T. Sato,
Positive solutions for semilinear elliptic equations with singular forcing terms, J. Diff. Eq., 235 (2007), 439-483.
doi: 10.1016/j.jde.2007.01.006. |
[27] |
F. Pacard,
Existence and convergence of positive weak solutions of $ -Δ u = u^{\frac{N}{N-2\ }}$ in bounded domains of $ {\mathbb{R}}^N$, Calc. Var. and PDEs., 1 (1993), 243-265.
doi: 10.1007/BF01191296. |
[28] |
Y. Pinchover and K. Tintarev,
Existence of minimizers for Schrödinger operators under domain perturbations with application to Hardy's inequality, Indiana Univ. Math. J., 54 (2005), 1061-1074.
doi: 10.1512/iumj.2005.54.2705. |
[29] |
P. Rabinowitz,
Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., 65 American Mathematical Society, 1986. |
[30] |
M. Struwe,
Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian systems, Ergebnisse der Mathematik und ihrer Grenzgebiete, 34. Springer-Verlag, Berlin, 1996. |
[31] |
L. Véron,
Singularities of Solutions of Second-Order Quasilinear Equations, Pitman Research Notes in Mathematics Series, 353. Longman, Harlow, 1996. |
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