Abstract
In this paper, we consider a nonhomogeneous singular elliptic equation involving a critical Caffarelli–Kohn–Nirenberg exponent. Using Ekeland’s Variational Principle, the Mountain Pass Lemma and the Nehari manifold, we establish the existence of at least two solutions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brezis H., Lieb E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc 88, 486–490 (1983)
Brezis H., Nirenberg L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent. Comm. Pure Appl. Math 36, 437–477 (1983)
Chen Y., Chen J.: Multiple positive solutions for a semilinear equation with critical exponent and prescribed singularity. Nonlinear Anal. 130, 121–137 (2016)
Caffarelli L., Kohn R., Nirenberg L.: First order interpolation inequality with weights. Compos. Math. 53, 259–275 (1984)
Catrina F., Wang Z.: On the Caffarelli–Kohn–Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. Comm. Pure Appl. Math 54, 229–257 (2001)
Chen J., Rocha E.M.: Four solutions of an inhomogeneous elliptic equation with critical exponent and singular term. Nonlinear Anal. 71, 4739–4750 (2009)
Chou K.S., Chu C.W.: On the best constant for a weighted Sobolev–Hardy Inequality. J. Lond. Math. Soc. 2, 137–151 (1993)
Felli V., Schneider M.: Perturbation results of critical elliptic equations of Caffarelli–Kohn–Nirenberg type. J. Differ. Equ. 191, 121–142 (2003)
Jannelli E.: The role played by space dimension in elliptic critical problems. J. Differ. Equ. 156, 407–426 (1999)
Jannelli E., Loiudice A.: Critical polyharmonic problems withsingular nonlinearities. Nonlinear Anal. 110, 77–96 (2014)
Kang D., Deng Y., Peng S.: Multiple solutions for inhomogeneous elliptic problems involving critical Sobolev–Hardy. Nonlinear Anal. 60, 729–753 (2005)
Tarantello G.: On nonhomogeneous elliptic equations involving critical Sobolev exponent. Ann. Inst. Henri Poincaré 9, 281–304 (1992)
Xuan B., Su S., Yan Y.: Existence results for Brezis–Nirenberg problems with Hardy potential and singular coefficients. Nonlinear Anal. 67, 2091–2106 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Benmansour, S., Matallah, A. Multiple Solutions for Nonhomogeneous Elliptic Equations Involving Critical Caffarelli–Kohn–Nirenberg Exponent. Mediterr. J. Math. 13, 4679–4691 (2016). https://doi.org/10.1007/s00009-016-0769-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-016-0769-6