# American Institute of Mathematical Sciences

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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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