Positive solutions for critical singular elliptic equations without Ambrosetti-Rabinowitz type conditions

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

We study the subcritical approximations to Li–Lin’s open problem, proposed by Li and Lin (Arch Ration Mech Anal 203(3): 943-968, 2012). By applying the variational method, we obtain two positive solutions. We establish a nonexistence theorem for positive solutions. Finally, through the combination of the variational method and the sub-supersolution method, we find a global bifurcation phenomenon for positive solutions.

Keywords

Hardy-Sobolev critical exponents / positive solutions / mountain pass lemma / the least action principle / method of sub-supersolutions

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] H. Brézis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. https://doi.org/10.1002/cpa.3160360405

[2] N. Ghoussoub, C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743. https://doi.org/10.1090/S0002-9947-00-02560-5

[3] N. Ghoussoub, X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 21 (2004), 767-793. https://doi.org/10.1016/j.anihpc.2003.07.002

[4] Y.Y. Li, C. S. Lin, A nonlinear elliptic PDE and two Sobolev-Hardy critical exponents, Arch. Ration. Mech. Anal., 203 (2012), 943-968. https://doi.org/10.1007/s00205-011-0467-2

[5] T. Jin, Symmetry and nonexistence of positive solutions of elliptic equations and systems with Hardy terms, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 28 (2011), 965-981. https://doi.org/10.1016/j.anihpc.2011.07.003

[6] S. Yan, J. Yang, Infinitely many solutions for an elliptic problem involving critical Sobolev and Hardy-Sobolev exponents, Calc. Var. Partial Differential Equations, 48 (2013), 587-610. https://doi.org/10.1007/s00526-012-0563-7

[7] G. Cerami, X. Zhong, W. Zou, On some nonlinear elliptic PDEs with Sobolev-Hardy critical exponents and a Li-Lin open problem, Calc. Var. Partial Differential Equations, 54 (2015), 1793-1829. https://doi.org/10.1007/s00526-015-0844-z

[8] X. Zhong, W. Zou, A nonlinear elliptic PDE with multiple Hardy-Sobolev critical exponents in R^N , J. Differential Equations, 292 (2021), 354-387. https://doi.org/10.1016/j.jde.2021.05.027

[9] C. Wang, J. Su, The ground state solutions of Hénon equation with upper weighted critical exponents, J. Differential Equations, 302 (2021), 444-473. https://doi.org/10.1016/j.jde.2021.09.007

[10] Z. Y. Tang, X. H. Tang, On Li-Lin’s open problem, J. Differential Equations, 435 (2025), 113244. https://doi.org/10.1016/j.jde.2025.113244

[11] H. Brezis, J.L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469. https://doi.org/10.5209/rev rema.1997.v10.n2.17459

[12] J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems , volume 74 of Applied Mathematical Sciences. Springer-Verlag, New York, 1989. https://doi.org/10.1007/978-1-4757-2061-7

[13] D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order . Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.

[14] A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7

[15] M. Montenegro, A. C. Ponce, The sub-supersolution method for weak solutions, Proc. Amer. Math. Soc., 136 (2008), 2429-2438. https://doi.org/10.1090/S0002-9939-08-09231-9