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March  2018, 38(3): 1441-1460. doi: 10.3934/dcds.2018059

On the positive solutions for a perturbed negative exponent problem on $\mathbb{R}^3$

Sanjiban Santra

Department of Basic Mathematics, Centro de Investigacióne en Mathematicás, Guanajuato, Mexico

Received  May 2017 Revised  October 2017 Published  December 2017

Fund Project: The author was supported by CAPES/Brazil (Proc 88881.068018/ 2014-01).

In this paper, we study the following fourth order elliptic problem with a negative nonlinearity :
$\begin{align}\left\{\begin{aligned} Δ^2 u&=-\frac{15}{16}(1+ \varepsilon Q)u^{-7} &&\text{ in } \mathbb R^3\\ u &>0 &&\text{ in } \mathbb R^3,\\ u(x) &\sim |x| \text{ as }{|x|\to ∞}. & \end{aligned} \right.\end{align}$
Here
$Q$
is a
$C^{1}$
bounded function on
$\mathbb{R}^3$
and
$\varepsilon >0$
is a small parameter. We prove the existence, uniqueness of positive solutions for the above perturbed fourth order problem.
Keywords: Perturbed problem, negative exponent, uniqueness, Lypaunov Schmidts method.
Mathematics Subject Classification: Primary: 35G20; Secondary: 35B09, 35B20.
Citation: Sanjiban Santra. On the positive solutions for a perturbed negative exponent problem on $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1441-1460. doi: 10.3934/dcds.2018059
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.  Google Scholar

[2]

A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbb R^N$, Progress in Mathematics, 240. Birkhäuser Verlag, Basel, 2006. Google Scholar

[3]

M. Ben Ayed and K. El Mehdi, The Paneitz curvature problem on lower-dimensional spheres, Ann. Global Anal. Geom., 31 (2007), 1-36.   Google Scholar

[4]

T. Branson, Differential operators canonically associated to a conformal structure, Math. Scand., 57 (1985), 293-345.  doi: 10.7146/math.scand.a-12120.  Google Scholar

[5]

R. Cai and S. Santra, On the $Q$-curvature problem on $\mathbb{S}^3$, Proc. of AMS., 145 (2017), 119-133.   Google Scholar

[6]

S. Y. Chang and P. Yang, Prescribing Gaussian curvature on $\mathbb S^2$, Acta Math., 159 (1987), 215-259.  doi: 10.1007/BF02392560.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. Chang and P. Yang, Fourth order equations in conformal geometry, Séminairés and Congreé, 4 (2000), 155-165.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. Esposito, Perturbations of Paneitz-Branson operators on $\mathbb S^N$, Rend. Sem. Mat. Univ. Padova, 107 (2002), 165-184.   Google Scholar

[16]

V. Felli, Existence of conformal metrics on $\mathbb{S}^N$ with prescribed fourth order invariant, Advances in Differential Equations, 7 (2002), 47-76.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

M. Gursky, The Weyl functional, de Rham cohomology, and Kahler-Einstein metrics, Annals of Mathematics, 148 (1998), 315-337.  doi: 10.2307/120996.  Google Scholar

[20]

F. Hang and P. Yang, The Sobolev inequality for Paneitz operator on three manifolds, Calc. Var. Partial Differential Equations, 21 (2004), 57-83.   Google Scholar

[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.  Google Scholar

[22]

J. Kazdan and F. Warner, Curvature functions for compact 2-manifolds, Annals of Mathematics, 99 (1974), 14-47.  doi: 10.2307/1971012.  Google Scholar

[23]

Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, JEMS, 6 (2004), 153-180.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbb R^N$, Progress in Mathematics, 240. Birkhäuser Verlag, Basel, 2006. Google Scholar

[3]

M. Ben Ayed and K. El Mehdi, The Paneitz curvature problem on lower-dimensional spheres, Ann. Global Anal. Geom., 31 (2007), 1-36.   Google Scholar

[4]

T. Branson, Differential operators canonically associated to a conformal structure, Math. Scand., 57 (1985), 293-345.  doi: 10.7146/math.scand.a-12120.  Google Scholar

[5]

R. Cai and S. Santra, On the $Q$-curvature problem on $\mathbb{S}^3$, Proc. of AMS., 145 (2017), 119-133.   Google Scholar

[6]

S. Y. Chang and P. Yang, Prescribing Gaussian curvature on $\mathbb S^2$, Acta Math., 159 (1987), 215-259.  doi: 10.1007/BF02392560.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. Chang and P. Yang, Fourth order equations in conformal geometry, Séminairés and Congreé, 4 (2000), 155-165.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. Esposito, Perturbations of Paneitz-Branson operators on $\mathbb S^N$, Rend. Sem. Mat. Univ. Padova, 107 (2002), 165-184.   Google Scholar

[16]

V. Felli, Existence of conformal metrics on $\mathbb{S}^N$ with prescribed fourth order invariant, Advances in Differential Equations, 7 (2002), 47-76.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

M. Gursky, The Weyl functional, de Rham cohomology, and Kahler-Einstein metrics, Annals of Mathematics, 148 (1998), 315-337.  doi: 10.2307/120996.  Google Scholar

[20]

F. Hang and P. Yang, The Sobolev inequality for Paneitz operator on three manifolds, Calc. Var. Partial Differential Equations, 21 (2004), 57-83.   Google Scholar

[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.  Google Scholar

[22]

J. Kazdan and F. Warner, Curvature functions for compact 2-manifolds, Annals of Mathematics, 99 (1974), 14-47.  doi: 10.2307/1971012.  Google Scholar

[23]

Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, JEMS, 6 (2004), 153-180.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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