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On interior $C^2$-estimates for the Monge-Ampère equation
On the positive solutions for a perturbed negative exponent problem on $\mathbb{R}^3$
Department of Basic Mathematics, Centro de Investigacióne en Mathematicás, Guanajuato, Mexico |
$\begin{align}\left\{\begin{aligned} Δ^2 u&=-\frac{15}{16}(1+ \varepsilon Q)u^{-7} &&\text{ in } \mathbb R^3\\ u &>0 &&\text{ in } \mathbb R^3,\\ u(x) &\sim |x| \text{ as }{|x|\to ∞}. & \end{aligned} \right.\end{align}$ |
$Q$ |
$C^{1}$ |
$\mathbb{R}^3$ |
$\varepsilon >0$ |
References:
[1] |
A. Ambrosetti, A. Garcia and I. Peral,
Perturbation of $\Delta u+ u^{\frac{N+2}{N-2}}=0$, the scalar curvature problem in $\mathbb R^N$ and related topics, J. Funct. Anal, 165 (1999), 117-149.
doi: 10.1006/jfan.1999.3390. |
[2] |
A. Ambrosetti and A. Malchiodi,
Perturbation Methods and Semilinear Elliptic Problems on $\mathbb R^N$, Progress in Mathematics, 240. Birkhäuser Verlag, Basel, 2006. |
[3] |
M. Ben Ayed and K. El Mehdi,
The Paneitz curvature problem on lower-dimensional spheres, Ann. Global Anal. Geom., 31 (2007), 1-36.
|
[4] |
T. Branson,
Differential operators canonically associated to a conformal structure, Math. Scand., 57 (1985), 293-345.
doi: 10.7146/math.scand.a-12120. |
[5] |
R. Cai and S. Santra,
On the $Q$-curvature problem on $\mathbb{S}^3$, Proc. of AMS., 145 (2017), 119-133.
|
[6] |
S. Y. Chang and P. Yang,
Prescribing Gaussian curvature on $\mathbb S^2$, Acta Math., 159 (1987), 215-259.
doi: 10.1007/BF02392560. |
[7] |
S. Y. Chang and P. Yang,
Conformal deformation of metrics on $\mathbb S^2$, J. Differential Geom., 27 (1988), 259-296.
doi: 10.4310/jdg/1214441783. |
[8] |
A. Chang, M. Gursky and P. Yang,
The scalar curvature equation on 2-and 3-spheres, Calc. Var. Partial Differential Equations, 1 (1993), 205-229.
doi: 10.1007/BF01191617. |
[9] |
A. Chang and P. Yang,
Fourth order equations in conformal geometry, Séminairés and Congreé, 4 (2000), 155-165.
|
[10] |
Y. Choi and X. Xu,
Nonlinear biharmonic equations with negative exponents, J. Differential Equations, 246 (2009), 216-234.
doi: 10.1016/j.jde.2008.06.027. |
[11] |
H. Chtioui and A. Rigane,
On the prescribed Q-curvature problem on $\mathbb S^N$, J. Funct. Anal., 261 (2011), 2999-3043.
doi: 10.1016/j.jfa.2011.07.017. |
[12] |
Z. Djadli and A. Malchiodi,
Existence of conformal metrics with constant $Q$-curvature, Annals of Mathematics, 168 (2008), 813-858.
doi: 10.4007/annals.2008.168.813. |
[13] |
Z. Djadli, E. Hebey and M. Ledoux,
Paneitz-type operators and applications, Duke Math. J., 104 (2000), 129-169.
doi: 10.1215/S0012-7094-00-10416-4. |
[14] |
Z. Djadli, A. Malchiodi and M. O. Ahmedou,
Prescribing a fourth order conformal invariant on the standard sphere, Part Ⅰ: A perturbation result, Comm. Contemp. Math., 4 (2002), 375-408.
doi: 10.1142/S0219199702000695. |
[15] |
P. Esposito,
Perturbations of Paneitz-Branson operators on $\mathbb S^N$, Rend. Sem. Mat. Univ. Padova, 107 (2002), 165-184.
|
[16] |
V. Felli,
Existence of conformal metrics on $\mathbb{S}^N$ with prescribed fourth order invariant, Advances in Differential Equations, 7 (2002), 47-76.
|
[17] |
F. Gazzola, H. Grunau and G. Sweers,
Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Mathematics, 1991. Springer-Verlag, Berlin, 2010. |
[18] |
B. Gidas, W. M. Ni and L. Nirenberg,
Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
doi: 10.1007/BF01221125. |
[19] |
M. Gursky,
The Weyl functional, de Rham cohomology, and Kahler-Einstein metrics, Annals of Mathematics, 148 (1998), 315-337.
doi: 10.2307/120996. |
[20] |
F. Hang and P. Yang,
The Sobolev inequality for Paneitz operator on three manifolds, Calc. Var. Partial Differential Equations, 21 (2004), 57-83.
|
[21] |
M. Jiang, L. Wang and J. Wei,
$2π$-periodic self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations, 41 (2011), 535-565.
doi: 10.1007/s00526-010-0375-6. |
[22] |
J. Kazdan and F. Warner,
Curvature functions for compact 2-manifolds, Annals of Mathematics, 99 (1974), 14-47.
doi: 10.2307/1971012. |
[23] |
Y. Y. Li,
Remark on some conformally invariant integral equations: the method of moving spheres, JEMS, 6 (2004), 153-180.
|
[24] |
P. J. McKenna and W. Reichel, Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry,
Electron. J. Differential Equations 2003 (2003), 13 pp. |
[25] |
S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds,
Symmetry, Integrability and Geometry. Methods and Applications 4 (2008), Paper 036, 3 pp. |
[26] |
G. Sweers,
No Gidas-Ni-Nirenberg type result for semilinear biharmonic problems, Math. Nachr., 246/247 (2002), 202-206.
doi: 10.1002/1522-2616(200212)246:1<202::AID-MANA202>3.0.CO;2-G. |
[27] |
J. Wei and X. Xu,
Prescribing Q-curvature problem on $\mathbb{S}^N$, J. Funct. Anal., 257 (2009), 1995-2023.
doi: 10.1016/j.jfa.2009.06.024. |
[28] |
J. Wei and X. Xu,
On conformal deformation of metric of $\mathbb S^N$, J. Funct. Anal., 157 (1998), 292-325.
doi: 10.1006/jfan.1998.3271. |
[29] |
X. Xu,
Exact solutions of nonlinear conformally invariant integral equations in $\mathbb R^3$, Adv. in Math., 194 (2005), 485-503.
doi: 10.1016/j.aim.2004.07.004. |
show all references
References:
[1] |
A. Ambrosetti, A. Garcia and I. Peral,
Perturbation of $\Delta u+ u^{\frac{N+2}{N-2}}=0$, the scalar curvature problem in $\mathbb R^N$ and related topics, J. Funct. Anal, 165 (1999), 117-149.
doi: 10.1006/jfan.1999.3390. |
[2] |
A. Ambrosetti and A. Malchiodi,
Perturbation Methods and Semilinear Elliptic Problems on $\mathbb R^N$, Progress in Mathematics, 240. Birkhäuser Verlag, Basel, 2006. |
[3] |
M. Ben Ayed and K. El Mehdi,
The Paneitz curvature problem on lower-dimensional spheres, Ann. Global Anal. Geom., 31 (2007), 1-36.
|
[4] |
T. Branson,
Differential operators canonically associated to a conformal structure, Math. Scand., 57 (1985), 293-345.
doi: 10.7146/math.scand.a-12120. |
[5] |
R. Cai and S. Santra,
On the $Q$-curvature problem on $\mathbb{S}^3$, Proc. of AMS., 145 (2017), 119-133.
|
[6] |
S. Y. Chang and P. Yang,
Prescribing Gaussian curvature on $\mathbb S^2$, Acta Math., 159 (1987), 215-259.
doi: 10.1007/BF02392560. |
[7] |
S. Y. Chang and P. Yang,
Conformal deformation of metrics on $\mathbb S^2$, J. Differential Geom., 27 (1988), 259-296.
doi: 10.4310/jdg/1214441783. |
[8] |
A. Chang, M. Gursky and P. Yang,
The scalar curvature equation on 2-and 3-spheres, Calc. Var. Partial Differential Equations, 1 (1993), 205-229.
doi: 10.1007/BF01191617. |
[9] |
A. Chang and P. Yang,
Fourth order equations in conformal geometry, Séminairés and Congreé, 4 (2000), 155-165.
|
[10] |
Y. Choi and X. Xu,
Nonlinear biharmonic equations with negative exponents, J. Differential Equations, 246 (2009), 216-234.
doi: 10.1016/j.jde.2008.06.027. |
[11] |
H. Chtioui and A. Rigane,
On the prescribed Q-curvature problem on $\mathbb S^N$, J. Funct. Anal., 261 (2011), 2999-3043.
doi: 10.1016/j.jfa.2011.07.017. |
[12] |
Z. Djadli and A. Malchiodi,
Existence of conformal metrics with constant $Q$-curvature, Annals of Mathematics, 168 (2008), 813-858.
doi: 10.4007/annals.2008.168.813. |
[13] |
Z. Djadli, E. Hebey and M. Ledoux,
Paneitz-type operators and applications, Duke Math. J., 104 (2000), 129-169.
doi: 10.1215/S0012-7094-00-10416-4. |
[14] |
Z. Djadli, A. Malchiodi and M. O. Ahmedou,
Prescribing a fourth order conformal invariant on the standard sphere, Part Ⅰ: A perturbation result, Comm. Contemp. Math., 4 (2002), 375-408.
doi: 10.1142/S0219199702000695. |
[15] |
P. Esposito,
Perturbations of Paneitz-Branson operators on $\mathbb S^N$, Rend. Sem. Mat. Univ. Padova, 107 (2002), 165-184.
|
[16] |
V. Felli,
Existence of conformal metrics on $\mathbb{S}^N$ with prescribed fourth order invariant, Advances in Differential Equations, 7 (2002), 47-76.
|
[17] |
F. Gazzola, H. Grunau and G. Sweers,
Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Mathematics, 1991. Springer-Verlag, Berlin, 2010. |
[18] |
B. Gidas, W. M. Ni and L. Nirenberg,
Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
doi: 10.1007/BF01221125. |
[19] |
M. Gursky,
The Weyl functional, de Rham cohomology, and Kahler-Einstein metrics, Annals of Mathematics, 148 (1998), 315-337.
doi: 10.2307/120996. |
[20] |
F. Hang and P. Yang,
The Sobolev inequality for Paneitz operator on three manifolds, Calc. Var. Partial Differential Equations, 21 (2004), 57-83.
|
[21] |
M. Jiang, L. Wang and J. Wei,
$2π$-periodic self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations, 41 (2011), 535-565.
doi: 10.1007/s00526-010-0375-6. |
[22] |
J. Kazdan and F. Warner,
Curvature functions for compact 2-manifolds, Annals of Mathematics, 99 (1974), 14-47.
doi: 10.2307/1971012. |
[23] |
Y. Y. Li,
Remark on some conformally invariant integral equations: the method of moving spheres, JEMS, 6 (2004), 153-180.
|
[24] |
P. J. McKenna and W. Reichel, Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry,
Electron. J. Differential Equations 2003 (2003), 13 pp. |
[25] |
S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds,
Symmetry, Integrability and Geometry. Methods and Applications 4 (2008), Paper 036, 3 pp. |
[26] |
G. Sweers,
No Gidas-Ni-Nirenberg type result for semilinear biharmonic problems, Math. Nachr., 246/247 (2002), 202-206.
doi: 10.1002/1522-2616(200212)246:1<202::AID-MANA202>3.0.CO;2-G. |
[27] |
J. Wei and X. Xu,
Prescribing Q-curvature problem on $\mathbb{S}^N$, J. Funct. Anal., 257 (2009), 1995-2023.
doi: 10.1016/j.jfa.2009.06.024. |
[28] |
J. Wei and X. Xu,
On conformal deformation of metric of $\mathbb S^N$, J. Funct. Anal., 157 (1998), 292-325.
doi: 10.1006/jfan.1998.3271. |
[29] |
X. Xu,
Exact solutions of nonlinear conformally invariant integral equations in $\mathbb R^3$, Adv. in Math., 194 (2005), 485-503.
doi: 10.1016/j.aim.2004.07.004. |
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