June  2018, 38(6): 2945-2964. doi: 10.3934/dcds.2018126

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

* Corresponding author: F. Zhou

Received  August 2017 Revised  December 2017 Published  April 2018

Fund Project: H. Chen is supported by NNSF of China, No: 11726614, 11661045, by the Jiangxi Provincial Natural Science Foundation, No: 20161ACB20007, by the Science and Technology Research Project of Jiangxi Provincial Department of Education, No:GJJ160297. F. Zhou is supported by NNSF of China, No: 11726613, 11271133 and 11431005, and STCSM No:13dZ2260400

In this paper, we concern the isolated singular solutions for semi-linear elliptic equations involving Hardy-Leray potential
$- \Delta u + \frac{\mathit{\mu }}{{|x{|^2}}}u = {u^p}\;\;\;{\rm{in }}\;\;\;\Omega \setminus \{ 0\} ,\;\;\;u = 0\;\;\;{\rm{on}}\;\;\;\partial \Omega .\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right)$
We classify the isolated singularities and obtain the existence and stability of positive solutions of (1). Our results are based on the study of nonhomogeneous Hardy problem in a new distributional sense.
Citation: Huyuan Chen, Feng Zhou. Isolated singularities for elliptic equations with hardy operator and source nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2945-2964. doi: 10.3934/dcds.2018126
References:
[1]

O. AdimurthiN. 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. Google Scholar

[2]

P. Aviles, Local behaviour of the solutions of some elliptic equations, Comm. Math. Phys., 108 (1987), 177-192. doi: 10.1007/BF01210610. Google Scholar

[3]

L. BoccardoL. 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. Google Scholar

[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. Google Scholar

[5]

H. Brezis and M. Marcus, Hardy's inequalities revisited, Ann. Sc. Norm. Super. Pisa Cl. Sci., 25 (1997), 217-237. Google Scholar

[6]

H. Brezis and L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469. Google Scholar

[7]

L. CaffarelliB. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

H. Chen, A. Quaas and F. Zhou, On nonhomogeneous elliptic equations with the Hardy-Leray potentials, arXiv: 1705.08047.Google Scholar

[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. Google Scholar

[12]

H. Chen and F. Zhou, Isolated singularities of positive solutions for Choquard equations in sublinear case, Comm. Cont. Math., (2017).Google Scholar

[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. Google Scholar

[14]

J. Davila and L. Dupaigne, Hardy-type inequalities, J. Eur. Math. Soc., 6 (2004), 335-365. Google Scholar

[15]

L. Dupaigne, A nonlinear elliptic PDE with the inverse square potential, J. d'Analyse Mathématique, 86 (2002), 359-398. doi: 10.1007/BF02786656. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Func. Anal., 192 (2002), 186-233. doi: 10.1006/jfan.2001.3900. Google Scholar

[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. Google Scholar

[20]

M. Ghergu and S. Taliaferro, Isolated Singularities in Partial Differential Inequalities, Cambridge University Press, 2016. Google Scholar

[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. Google Scholar

[22]

Q. Han and F. Lin, Elliptic Partial Differential Equations, American Mathematical Soc., 2000. doi: 10.1090/cln/001. Google Scholar

[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. Google Scholar

[24]

P. Lions, Isolated singularities in semilinear problems, J. Diff. Eq., 38 (1980), 441-450. doi: 10.1016/0022-0396(80)90018-2. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., 65 American Mathematical Society, 1986. Google Scholar

[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. Google Scholar

[31]

L. Véron, Singularities of Solutions of Second-Order Quasilinear Equations, Pitman Research Notes in Mathematics Series, 353. Longman, Harlow, 1996. Google Scholar

show all references

References:
[1]

O. AdimurthiN. 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. Google Scholar

[2]

P. Aviles, Local behaviour of the solutions of some elliptic equations, Comm. Math. Phys., 108 (1987), 177-192. doi: 10.1007/BF01210610. Google Scholar

[3]

L. BoccardoL. 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. Google Scholar

[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. Google Scholar

[5]

H. Brezis and M. Marcus, Hardy's inequalities revisited, Ann. Sc. Norm. Super. Pisa Cl. Sci., 25 (1997), 217-237. Google Scholar

[6]

H. Brezis and L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469. Google Scholar

[7]

L. CaffarelliB. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

H. Chen, A. Quaas and F. Zhou, On nonhomogeneous elliptic equations with the Hardy-Leray potentials, arXiv: 1705.08047.Google Scholar

[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. Google Scholar

[12]

H. Chen and F. Zhou, Isolated singularities of positive solutions for Choquard equations in sublinear case, Comm. Cont. Math., (2017).Google Scholar

[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. Google Scholar

[14]

J. Davila and L. Dupaigne, Hardy-type inequalities, J. Eur. Math. Soc., 6 (2004), 335-365. Google Scholar

[15]

L. Dupaigne, A nonlinear elliptic PDE with the inverse square potential, J. d'Analyse Mathématique, 86 (2002), 359-398. doi: 10.1007/BF02786656. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Func. Anal., 192 (2002), 186-233. doi: 10.1006/jfan.2001.3900. Google Scholar

[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. Google Scholar

[20]

M. Ghergu and S. Taliaferro, Isolated Singularities in Partial Differential Inequalities, Cambridge University Press, 2016. Google Scholar

[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. Google Scholar

[22]

Q. Han and F. Lin, Elliptic Partial Differential Equations, American Mathematical Soc., 2000. doi: 10.1090/cln/001. Google Scholar

[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. Google Scholar

[24]

P. Lions, Isolated singularities in semilinear problems, J. Diff. Eq., 38 (1980), 441-450. doi: 10.1016/0022-0396(80)90018-2. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., 65 American Mathematical Society, 1986. Google Scholar

[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. Google Scholar

[31]

L. Véron, Singularities of Solutions of Second-Order Quasilinear Equations, Pitman Research Notes in Mathematics Series, 353. Longman, Harlow, 1996. Google Scholar

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