Multiplicity results for nonhomogenous elliptic equation involving the generalized Paneitz-Branson operator
DOI:
https://doi.org/10.24193/subbmath.2023.4.19Keywords:
Riemannian manifold, multiplicity result, nonhomogenous, Paneitz- Branson operator, critical points theoryAbstract
Let \((M,g)\) be a compact Riemannian manifold of dimension \(n\geq 3\), we consider the multiplicity results of solutions of the following nonhomogenous fourth order elliptic equation involving the generalized Paneitz-Branson operator:\[ P_{g}(u)=f(x)|u|^{2^{♯}-2}u+h(x).\]
Under some conditions and using critical points theory, we prove the existence of two solutions of the elliptic equation. At the end, we give a geometric example when the equation has negative and positive solutions.
References
Ambrosetti, A., Rabinowitz, P., Dual variational methods in critical point theory and
applications, J. Funct. Anal., 11(1973), 349-381.
Benalili, M., Tahri, K., Nonlinear elliptic fourth order equations existence and multiplic-
ity results, NoDEA Nonlinear Di erential Equations Appl., 18(2011), 539-556.
Bernis, F., Garcia-Azorero, J., Peral, I., Existence and multiplicity of non trivial solutions
in semilinear critical problems of fourth order, Adv. Partial Di er. Equ. (Basel), (1996),
-240.
Branson, T.P., rsted, B., Explicit functional determinants in four dimensions, Proc.
Amer. Math. Soc., 113(1991), no. 3, 669-682.
Br ezis, H., Lieb, E.A., A relation between pointwise convergence of functions and con-
vergence of functionals, Proc. Amer. Math. Soc., 88(1983), 486-490.
Cara a, D., Equations elliptiques du quatri eme ordre avec exposants critiques sur les
vari et es riemanniennes compactes, J. Math. Pures Appl., 80(2001), no. 9, 941-960.
Djadli, Z., Hebey, E., Ledoux, M., Paneitz-type operators and applications, Duke Math.
J., 104(2000), no. 1, 129-169.
Ekeland, I., On the variational principle, J. Math. Anal. Appl., 47(1974), 324-353.
Lions, P.L., The concentration-compactness principle in the calculus of variations. The
limit case. I, Rev. Mat. Iberoam., 1(1985), 145-201.
Paneitz, S.M., A quartic conformally covariant di erential operator for arbitrary pseudo-
Riemannian manifolds, SIGMA Symmetry Integrability Geom. Methods Appl., 4(2008),
Paper 036, 3.
Tahri, K., Les Equations Elliptiques du Quatri eme Ordre sur une Vari et e Rie-
mannienne, Abou Bekr Belkaid University, Tlemcen, Algeria, 1-209, (Uni Website:
http://dspace.univ-tlemcen.dz), June 2015.
Downloads
Published
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Transfer of copyright agreement: When the article is accepted for publication, the authors and the representative of the coauthors, hereby agree to transfer to Studia Universitatis Babeș-Bolyai Mathematica all rights, including those pertaining to electronic forms and transmissions, under existing copyright laws, except for the following, which the authors specifically retain: the authors can use the material however they want as long as it fits the NC ND terms of the license. The authors have all rights for reuse according to the license.
