On the positive solutions for a perturbed negative exponent problem on <inline-formula><tex-math id="M1">\begin{document} \mathbb{R}^3 \end{document}</tex-math></inline-formula>

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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 and 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.
[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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