Abstract
We investigate Caffarelli–Kohn–Nirenberg-type inequalities for the weighted biharmonic operator in cones, both under Navier and Dirichlet boundary conditions. Moreover, we study existence and qualitative properties of extremal functions. In particular, we show that in some cases extremal functions do change sign; when the domain is the whole space, we prove some breaking symmetry phenomena.
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References
Adams, R. A., Sobolev spaces, Academic Press, 1970.
Bhakta, M., Musina, R., Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials, (in progress).
Brézis H., Nirenberg L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–477 (1983)
Caffarelli L., Kohn R., Nirenberg L.: First Order Interpolation Inequalities with Weight. Compositio Math. 53, 259–275 (1984)
Calanchi M., Ruf B.: Radial and non radial solutions for Hardy-Hénon type elliptic systems. Calc. Var. PDE 38, 111–133 (2010)
Caldiroli P., Musina R.: On the existence of extremal functions for a weighted Sobolev embedding with critical exponent. Calc. Var. PDE 8, 365–387 (1999)
Caldiroli, P., Musina, R., Rellich inequalities with weights, Calc. Var. PDE (to appear)
Catrina F., Wang Z.-Q.: On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions. Comm. Pure Appl. Math. 54, 229–258 (2001)
de Figueiredo D.G., Peral I., Rossi J.D.: The critical hyperbola for a Hamiltonian system with weights. Ann. Mat. Pura Appl. (4) 187, 531–545 (2008)
Dolbeault J., Esteban M., Loss M., Tarantello G.: On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities Adv. Nonlinear Stud. 9, 713–726 (2009)
Felli V., Schneider M.: Perturbation results of critical elliptic equations of Caffarelli- Kohn-Nirenberg type. J. Diff. Eq. 191, 121–142 (2003)
Gazzola F.: Critical growth problems for polyharmonic operators. Proc. Roy. Soc. Edinburgh Sect. A 128, 251–263 (1998)
Gazzola, F., Grunau, H.-C., Sweers, G., Polyharmonic boundary value problems. Positivity preserving and nonlinear higher order elliptic equations in bounded domains, Lecture Notes in Mathematics 1991. Berlin: Springer, 2010.
Gazzola F., Grunau H.-C., Sweers G.: Optimal Sobolev and Hardy–Rellich constants under Navier boundary conditions. Ann. Mat. Pura Appl. (4) 189, 475–486 (2010)
Lin C.-S.: Interpolation inequalities with weights. Comm. Part. Diff. Eq. 11, 1515–1538 (1986)
Lions, P.-L., The concentration-compactness principle in the Calculus of Variations. The locally compact case, parts 1 and 2. Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984) no. 2, 109–145, no. 4, 223–283.
Liu F., Yang J.: Nontrivial solutions of Hardy-Henon type elliptic systems. Acta Math. Sci. Ser. B Engl. Ed. 27, 673–688 (2007)
Pucci P., Serrin J.: Critical exponents and critical dimensions for polyharmonic operators. J. Math. Pures Appl. 69, 53–83 (1990)
Rellich, F., Halbbeschränkte Differentialoperatoren höherer Ordnung. In: J.C.H. Gerretsen, J. de Groot (Eds.): Proceedings of the International Congress of Mathematicians 1954, Volume III (pp. 243–250) Groningen: Noordhoff 1956.
Rellich F.: Perturbation theory of eigenvalue problems. Gordon and Breach, New York (1969)
Struwe, M., Variational Methods (fourth edition), Springer, 2008.
Swanson C.A.: The best Sobolev constant. Appl. Anal. 47, 227–239 (1992)
Szulkin, A., Waliullah, S., Sign-changing and symmetry-breaking solutions to singular problems, Complex Variables and Elliptic Eq., First published on 02 February 2011.
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Caldiroli, P., Musina, R. On Caffarelli–Kohn–Nirenberg-type Inequalities for the Weighted Biharmonic Operator in Cones. Milan J. Math. 79, 657–687 (2011). https://doi.org/10.1007/s00032-011-0167-2
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DOI: https://doi.org/10.1007/s00032-011-0167-2