On the 𝑄-curvature problem on 𝕊³

Cai, Ruilun; Santra, Sanjiban

doi:10.1090/proc/13271

On the $Q$-curvature problem on $\mathbb {S}^3$
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by Ruilun Cai and Sanjiban Santra

Proc. Amer. Math. Soc. 145 (2017), 119-133

DOI: https://doi.org/10.1090/proc/13271

Published electronically: August 29, 2016

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Abstract:

Let $P_{\mathbb {S}^3}=\Delta _0^2+ \frac {1}{2}\Delta _0- \frac {15}{16}$ denote the Panietz operator on the standard sphere $\mathbb {S}^3$. In this paper, we study the following fourth order elliptic equation with a nonlinear term of negative power type: \[ P_{\mathbb {S}^3} u = -\frac {1}{2}Qu^{-7} \mbox { on } \mathbb {S}^3. \] Here $Q$ is a prescribed smooth function on $\mathbb {S}^3$ which is assumed to be a smooth bounded positive function. We prove the existence of positive solutions to the equation under a non-degeneracy assumption on $Q$.

References

Thierry Aubin, Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1636569, DOI 10.1007/978-3-662-13006-3
Jun Ai, Kai-Seng Chou, and Juncheng Wei, Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations 13 (2001), no. 3, 311–337. MR 1865001, DOI 10.1007/s005260000075
Thomas P. Branson, Differential operators canonically associated to a conformal structure, Math. Scand. 57 (1985), no. 2, 293–345. MR 832360, DOI 10.7146/math.scand.a-12120
Sun-Yung Alice Chang and Paul C. Yang, Prescribing Gaussian curvature on $S^2$, Acta Math. 159 (1987), no. 3-4, 215–259. MR 908146, DOI 10.1007/BF02392560
Sun-Yung A. Chang and Paul C. Yang, Conformal deformation of metrics on $S^2$, J. Differential Geom. 27 (1988), no. 2, 259–296. MR 925123
Sun-Yung A. Chang and Paul C. Yang, Fourth order equations in conformal geometry, Global analysis and harmonic analysis (Marseille-Luminy, 1999) Sémin. Congr., vol. 4, Soc. Math. France, Paris, 2000, pp. 155–165 (English, with English and French summaries). MR 1822359
Sun-Yung A. Chang, Matthew J. Gursky, and Paul C. Yang, The scalar curvature equation on $2$- and $3$-spheres, Calc. Var. Partial Differential Equations 1 (1993), no. 2, 205–229. MR 1261723, DOI 10.1007/BF01191617
Sun-Yung A. Chang and Paul C. Yang, A perturbation result in prescribing scalar curvature on $S^n$, Duke Math. J. 64 (1991), no. 1, 27–69. MR 1131392, DOI 10.1215/S0012-7094-91-06402-1
Y. Choi, X. Xu, Classification of solutions of some nonlinear elliptic equations. J. Differential Equations 246 (2009), no. 3, 216–234.
Zindine Djadli and Andrea Malchiodi, Existence of conformal metrics with constant $Q$-curvature, Ann. of Math. (2) 168 (2008), no. 3, 813–858. MR 2456884, DOI 10.4007/annals.2008.168.813
Zindine Djadli, Andrea Malchiodi, and Mohameden Ould Ahmedou, Prescribing a fourth order conformal invariant on the standard sphere. I. A perturbation result, Commun. Contemp. Math. 4 (2002), no. 3, 375–408. MR 1918751, DOI 10.1142/S0219199702000695
Zindine Djadli, Emmanuel Hebey, and Michel Ledoux, Paneitz-type operators and applications, Duke Math. J. 104 (2000), no. 1, 129–169. MR 1769728, DOI 10.1215/S0012-7094-00-10416-4
Filippo Gazzola, Hans-Christoph Grunau, and Guido Sweers, Polyharmonic boundary value problems, Lecture Notes in Mathematics, vol. 1991, Springer-Verlag, Berlin, 2010. Positivity preserving and nonlinear higher order elliptic equations in bounded domains. MR 2667016, DOI 10.1007/978-3-642-12245-3
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
Fengbo Hang and Paul C. Yang, The Sobolev inequality for Paneitz operator on three manifolds, Calc. Var. Partial Differential Equations 21 (2004), no. 1, 57–83. MR 2078747, DOI 10.1007/s00526-003-0247-4
Emmanuel Hebey and Frédéric Robert, Coercivity and Struwe’s compactness for Paneitz type operators with constant coefficients, Calc. Var. Partial Differential Equations 13 (2001), no. 4, 491–517. MR 1867939, DOI 10.1007/s005260100084
B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
Jerry L. Kazdan and F. W. Warner, Curvature functions for compact $2$-manifolds, Ann. of Math. (2) 99 (1974), 14–47. MR 343205, DOI 10.2307/1971012
Yan Yan Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc. (JEMS) 6 (2004), no. 2, 153–180. MR 2055032, DOI 10.4171/jems/6
Stephen M. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary), SIGMA Symmetry Integrability Geom. Methods Appl. 4 (2008), Paper 036, 3. MR 2393291, DOI 10.3842/SIGMA.2008.036
Juncheng Wei and Xingwang Xu, Prescribing $Q$-curvature problem on $\mathbf S^n$, J. Funct. Anal. 257 (2009), no. 7, 1995–2023. MR 2548028, DOI 10.1016/j.jfa.2009.06.024
Juncheng Wei and Xingwang Xu, On conformal deformations of metrics on $S^n$, J. Funct. Anal. 157 (1998), no. 1, 292–325. MR 1637945, DOI 10.1006/jfan.1998.3271
Xingwang Xu and Paul C. Yang, On a fourth order equation in 3-D, ESAIM Control Optim. Calc. Var. 8 (2002), 1029–1042. A tribute to J. L. Lions. MR 1932985, DOI 10.1051/cocv:2002023
Xingwang Xu, Exact solutions of nonlinear conformally invariant integral equations in $\mathbf R^3$, Adv. Math. 194 (2005), no. 2, 485–503. MR 2139922, DOI 10.1016/j.aim.2004.07.004
Paul Yang and Meijun Zhu, On the Paneitz energy on standard three sphere, ESAIM Control Optim. Calc. Var. 10 (2004), no. 2, 211–223. MR 2083484, DOI 10.1051/cocv:2004002