The Paneitz equation in hyperbolic space

Grunau, Hans-Christoph; Ould Ahmedou, Mohameden; Reichel, Wolfgang

Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008) no. 5, pp. 847-864.

[1] Caffarelli L., Gidas B., Spruck J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math. 42 (1989) 271-297. | MR | Zbl

[2] Chang S.-Y.A., On a fourth-order partial differential equation in conformal geometry, in: Christ M., (Eds.), Harmonic Analysis and Partial Differential Equations. Essays in honor of Alberto P. Calderón's 75th birthday, Proceedings of a Conference, University of Chicago, February 1996, Chicago Lectures in Mathematics, The University of Chicago Press, 1999, pp. 127-150. | MR | Zbl

[3] Chang S.-Y.A., Non-Linear Elliptic Equations in Conformal Geometry, Zurich Lectures in Advanced Mathematics, European Math. Soc. (EMS), Zürich, 2004. | MR | Zbl

[4] Chang S.-Y.A., Chen W., A note on a class of higher order conformally covariant equations, Discrete Contin. Dyn. Syst. 7 (2001) 275-281. | MR | Zbl

[5] Chang S.-Y.A., Yang P.C., Extremal metrics of zeta function determinants on 4-manifolds, Ann. Math. 142 (1995) 171-212. | MR | Zbl

[6] Chang S.-Y.A., Yang P.C., On uniqueness of solutions of n-th order differential equations in conformal geometry, Math. Res. Lett. 4 (1997) 91-102. | MR | Zbl

[7] Chang S.-Y.A., Yang P.C., On a fourth order curvature invariant, in: Branson T. (Ed.), Spectral Problems in Geometry and Arithmetic, NSF-CBMS Conference on Spectral Problems in Geometry and Arithmetic, Iowa City, August 18-22, 1997, Contemp. Math., vol. 237, American Mathematical Society, Providence, RI, 1999, pp. 9-28. | MR | Zbl

[8] Chen W., Li C., Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991) 615-622. | MR | Zbl

[9] Y.S. Choi, X. Xu, Nonlinear biharmonic equations with negative exponents, preprint, 1999. | MR

[10] Diaz J.I., Lazzo M., Schmidt P.G., Large radial solutions of a polyharmonic equation with superlinear growth, Electronic J. Differential Equations, Conference 16 (2007) 103-128. | EuDML | MR | Zbl

[11] J.I. Diaz, M. Lazzo, P.G. Schmidt, Asymptotic behavior of large radial solutions of a polyharmonic equation with superlinear growth, in preparation.

[12] Djadli Z., Hebey E., Ledoux M., Paneitz-type operators and applications, Duke Math. J. 104 (2000) 129-169. | MR | Zbl

[13] Z. Djadli, A. Malchiodi, Existence of conformal metrics with constant Q-curvature, Ann. Math., in press. | Zbl

[14] Djadli Z., Malchiodi A., Ould Ahmedou M., Prescribing a fourth order conformal invariant on the standard sphere. I: A perturbation result, Commun. Contemp. Math. 4 (2002) 375-408. | MR | Zbl

[15] Djadli Z., Malchiodi A., Ould Ahmedou M., Prescribing a fourth order conformal invariant on the standard sphere. II: Blow up analysis and applications, Ann. Sc. Norm. Super Pisa, Cl. Sci. (5) 1 (2002) 387-434. | EuDML | Numdam | MR | Zbl

[16] Felli V., Existence of conformal metrics on $S^n$ with prescribed fourth-order invariant, Adv. Differential Equations 7 (2002) 47-76. | MR | Zbl

[17] Ferrero A., Grunau H.-Ch., The Dirichlet problem for supercritical biharmonic equations with power-type nonlinearity, J. Differential Equations 234 (2007) 582-606. | MR | Zbl

[18] Gazzola F., Grunau H.-Ch., Radial entire solutions for supercritical biharmonic equations, Math. Ann. 334 (2006) 905-936. | MR | Zbl

[19] Gursky M., The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE, Commun. Math. Phys. 207 (1999) 131-143. | MR | Zbl

[20] Hartman P., Ordinary Differential Equation, John Wiley and Sons, New York, 1964. | MR | Zbl

[21] Lin C.-S., A classification of solutions of a conformally invariant fourth order equation in $R^n$, Comment. Math. Helv. 73 (1998) 206-231. | MR | Zbl

[22] Loewner C., Nirenberg L., Partial differential equations invariant under conformal or projective transformations, in: Ahlfors L. (Ed.), Contributions to Analysis, Academic Press, New York, 1974, pp. 245-272. | MR | Zbl

[23] Mallet-Paret J., Smith H.L., The Poincaré-Bendixson theorem for monotone cyclic feedback systems, J. Dynamics Differential Equations 2 (1990) 367-421. | MR | Zbl

[24] Mazzeo R., Pacard F., Poincaré-Einstein metrics and the Schouten tensor, Pacific J. Math. 212 (2003) 169-185. | MR | Zbl

[25] Mckenna P.J., Reichel W., Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry, Electronic J. Differential Equations 2003 (37) (2003) 1-13. | EuDML | MR | Zbl

[26] Obata M., Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962) 333-340. | MR | Zbl

[27] Pohožaev S.I., Eigenfunctions of the equation $\Delta u+\lambda f\left(u\right)=0$, Soviet Math. Dokl. 6 (1965) 1408-1411, English translation from, Dokl. Akad. Nauk SSSR 165 (1965) 36-39. | MR | Zbl

[28] Pucci P., Serrin J., A general variational identity, Indiana Univ. Math. J. 35 (1986) 681-703. | MR | Zbl

[29] Robert F., Positive solutions for a fourth order equation invariant under isometries, Proc. Amer. Math. Soc. 131 (2003) 1423-1431. | MR | Zbl

[30] Wei J., Xu X., On conformal deformations of metrics on $S^n$, J. Funct. Anal. 157 (1998) 292-325. | MR | Zbl

[31] Wei J., Xu X., Classification of solutions of higher order conformally invariant equations, Math. Ann. 313 (1999) 207-228. | MR | Zbl