American Institute of Mathematical Sciences

Advanced
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 - A, 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

[1]

Zongming Guo, Long Wei. A perturbed fourth order elliptic equation with negative exponent. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4187-4205. doi: 10.3934/dcdsb.2018132
[2]

V. V. Motreanu. Uniqueness results for a Dirichlet problem with variable exponent. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1399-1410. doi: 10.3934/cpaa.2010.9.1399
[3]

Tiexiang Li, Tsung-Ming Huang, Wen-Wei Lin, Jenn-Nan Wang. On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method. Inverse Problems & Imaging, 2018, 12 (4) : 1033-1054. doi: 10.3934/ipi.2018043
[4]

Nicola Soave, Susanna Terracini. Symbolic dynamics for the $N$-centre problem at negative energies. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3245-3301. doi: 10.3934/dcds.2012.32.3245
[5]

Zongming Guo, Xiaohong Guan, Yonggang Zhao. Uniqueness and asymptotic behavior of solutions of a biharmonic equation with supercritical exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2613-2636. doi: 10.3934/dcds.2019109
[6]

Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012
[7]

Liping Wang, Dong Ye. Concentrating solutions for an anisotropic elliptic problem with large exponent. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3771-3797. doi: 10.3934/dcds.2015.35.3771
[8]

Nicola Soave, Susanna Terracini. Addendum to: Symbolic dynamics for the $N$-centre problem at negative energies. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3825-3829. doi: 10.3934/dcds.2013.33.3825
[9]

Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033
[10]

H. Gajewski, I. V. Skrypnik. To the uniqueness problem for nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 315-336. doi: 10.3934/dcds.2004.10.315
[11]

Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496
[12]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 0 (0) : 0-0. doi: 10.3934/era.2020076
[13]

Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773
[14]

Huiling Li, Xiaoliu Wang, Xueyan Lu. A nonlinear Stefan problem with variable exponent and different moving parameters. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1671-1698. doi: 10.3934/dcdsb.2019246
[15]

V. V. Motreanu. Multiplicity of solutions for variable exponent Dirichlet problem with concave term. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 845-855. doi: 10.3934/dcdss.2012.5.845
[16]

Xudong Shang, Jihui Zhang, Yang Yang. Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Communications on Pure & Applied Analysis, 2014, 13 (2) : 567-584. doi: 10.3934/cpaa.2014.13.567
[17]

Xin Guo, Qiang Fu, Yue Wang, Kenneth C. Land. A numerical method to compute Fisher information for a special case of heterogeneous negative binomial regression. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4179-4189. doi: 10.3934/cpaa.2020187
[18]

Pedro Caro. On an inverse problem in electromagnetism with local data: stability and uniqueness. Inverse Problems & Imaging, 2011, 5 (2) : 297-322. doi: 10.3934/ipi.2011.5.297
[19]

Victor Isakov. On uniqueness in the inverse conductivity problem with local data. Inverse Problems & Imaging, 2007, 1 (1) : 95-105. doi: 10.3934/ipi.2007.1.95
[20]

Germain Gendron. Uniqueness results in the inverse spectral Steklov problem. Inverse Problems & Imaging, 2020, 14 (4) : 631-664. doi: 10.3934/ipi.2020029

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (87)
  • HTML views (176)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences