Abstract
In this paper, we establish the existence of multiple solutions for nonhomogeneous singular elliptic equations with cylindrical weights, by using Ekeland’s variational principle and mountain pass theorem without Palais–Smale condition.
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Badiale M., Guida M., Rolando S.: Elliptic equations with decaying cylindrical potentials and power-type nonlinearities. Adv. Differ. Equ. 12, 1321–1362 (2007)
Badiale M., Tarantello G.: A Sobolev–Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch. Ration. Mech. Anal. 163, 252–293 (2002)
Bouchekif M., Matallah A.: On singular nonhomogeneous elliptic equations involving critical Caffarelli–Kohn–Nirenberg exponent. Ric. Mat. 58, 207–218 (2009)
Brézis H., Lieb E.: A relation between point convergence of functions and convergence of functional. Proc. Am. Math. Soc. 88, 486–490 (1983)
Ekeland I.: On the variational principle. J. Math. Anal. Appl. 47, 323–353 (1974)
Gazzini M., Musina R.: On the Hardy–Sobolev–Maz’ja inequalities: symmetry and breaking symmetry of extremal functions. Commun. Contemp. Math. 11, 993–1007 (2009)
Kang D., Deng Y.: Multiple solutions for inhomogeneous elliptic problems involving critical Sobolev–Hardy exponents. Nonlinear Anal. 60, 729–753 (2005)
Mancini G., Sandeep K.: Cylindrical symmetry of extremals of a Hardy–Sobolev inequality. Ann. Mat. Pura Appl. 183, 165–172 (2004)
Maz’ja V.G.: Sobolev Spaces. Springer-Verlag, Berlin (1980)
Musina R.: Ground state solutions of a critical problem involving cylindrical weights. Nonlinear Anal. 68, 3972–3986 (2008)
Secchi S., Smets D., Willem M.: Remarks on a Hardy–Sobolev inequality. C. R. Acad. Sci. Paris, Ser. I 336, 811–815 (2003)
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Bouchekif, M., El Mokhtar Ould El Mokhtar, M. On nonhomogeneous singular elliptic equations with cylindrical weights. Ricerche mat. 61, 147–156 (2012). https://doi.org/10.1007/s11587-011-0121-1
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DOI: https://doi.org/10.1007/s11587-011-0121-1
Keywords
- Cylindrical weight
- Ekeland’s variational principle
- Palais–Smale condition
- Weighted Hardy–Sobolev inequalities