Abstract
We consider the following prescribed curvature problem for polyharmonic operator:
where \({m^*=\frac{2N}{N-2m}, N\geq 2m+1,m \in \mathbb{N}_{+}, \tilde{K}}\) is positive and rationally symmetric, \({\mathbb{S}^N}\) is the unit sphere with the induced Riemannian metric \({g=g_{\mathbb{S}^N},}\) and D m is the elliptic differential operator of 2m order given by
where Δ g is the Laplace-Beltrami operator on \({\mathbb{S}^N}\) . We will show that problem (P) has infinitely many non-radial positive solutions, whose energy can be arbitrary large.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ambrosetti A., Azorero G., Peral I.: 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, 117–149 (1999)
Bartsch T., Weth T.: Multiple solutions of a critical polyharmonic equation. J. Reine Angew. Math. 571, 131–143 (2004)
Bartsch T., Weth T., Willem M.: A sobolev inequality with remainder term and critical equations on domains with topology of the domain. Calc. Var. Partial Differ. Equ. 18, 253–268 (2003)
Bahri A., Coron J.: The scalar curvature problem on the standard three dimentional sphere. J. Funct. Anal. 95, 106–172 (1991)
Beckner W.: Sharp Sobolev inequalities on the sphere and the Morse Trudinger inequality. J. Funct. Anal. 187, 197–291 (1993)
Bianchi G.: Non-existence and symmetry of solutions to the scalar curvature equation. Commun. Partial Differ. Equ. 21, 229–234 (1996)
Bianchi G., Egnell H.: An ODE approach to the equation \({-\Delta u+Ku^{\frac{N+2}{N-2}}=0}\) in \({\mathbb{R}^N}\) . Math. Z. 210, 137–166 (1992)
Branson T.: Group representations arising from Lorentz conformal geomtry. J. Funct. Anal. 187, 199–291 (1993)
Cao D., Noussair E., Yan S.: On the scalar curvature equation \({-\Delta u=(1+\epsilon K)u^{\frac{N+2}{N-2}}}\) in \({\mathbb{R}^N}\) . Valc. Var. Partial Differ. Equ. 15, 403–419 (2002)
Chang, S.Y.A., Yang, P.C.: Partial differential equations related to the Gauss-Bonnet-Chern integrand on 4−manifolds, Proc. Conformal, Riemannian and Lagrangian Geometry, Univ. Lecture Ser., vol. 27, pp. 1–30. American Mathematical Society, Providence (2002)
Chang S.Y.A., Yang P.C.: A perturbation result in prescribing scalar curvature on S n. Duke Math. J. 64, 27–69 (1991)
Chen C. C., Lin C. S.: Estimate of the conformal scalar curvature equation via the method of moving planes, II. J. Differ. Geom. 49, 115–178 (1998)
Chen C. C., Lin C. S.: Prescribing scalar curvature on S N, I. A priori estimates. J. Differ. Geom. 57, 67–171 (2001)
del Pino M., Felmer P., Musso M.: Two-bubble solutions in the super-critical Bahri-Coron’s problem. Calc. Var. Partial Differ. Equ. 16, 113–145 (2003)
Ding W. Y., Ni W. M.: On the elliptic equation \({\Delta u+Ku^{\frac{N+2}{N-2}}=0}\) and related topics. Duck Math. J. 52, 485–506 (1985)
Edmunds D. E., Fortunato D., Jannelli E.: Critical exponents, critical dimensions and biharmonic operator. Arch. Rat. Mech. Anal. 112, 269–289 (1990)
Gazzola, F., Grunau, H., Squassina, M.: Existence and non-existence results for critical growth biharmonic elliptic equations (Preprint)
Grunau H.: Positive solutions to semilinear polyharmonic Dirichlet problem operators involving critical Sobolev exponents. Calc. Var. Partial Differ. Equ. 3, 243–252 (1995)
Grunau H., Sweers G.: The maximum principle and positive principle eigenfunctions for polyharmonic equations. Lect. Notes Pure Appl. Math. 194, 163–182 (1998)
Grunau H., Sweers G.: Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions. Math. Ann. 307, 588–626 (1997)
Li Y.: On −Δu = K(x)u 5 in \({\mathbb{R}^3}\) . Commun. Pure Appl. Math. 46, 303–340 (1993)
Li Y.: Prescribed scalar curvature on S n and related problems II, Existence and comapctness. Commun. Pure Appl. Math. 49, 541–597 (1996)
Li Y., Ni W. M.: On the conformal scalar curvature equation in \({\mathbb{R}^N}\) . Duck Math. J. 57, 859–924 (1988)
Lin C. S., Lin S. S.: Positive radial solutions for \({\Delta u+K(x)u^{\frac{N+2}{N-2}}=0}\) in \({\mathbb{R}^N}\) and related topics. Appl. Anal. 38, 121–159 (1990)
Mohamed B.A., Khalil E.M., Mokhless H.: Some existence results for a Paneitz type problem via the theory of critical points at infinity. J. Math. Pures Appl. 84, 247–278 (2005)
Ni W. M.: On the elliptic equation \({\Delta u+K(x)u^{\frac{N+2}{N-2}}=0,}\) its generalizations and applications in geometry. Indiana Univ. Math. J. 31, 493–529 (1982)
Noussair E., Yan S.: The scalar curvature equation on \({\mathbb{R}^N}\) . Nonlinear Anal. 45, 483–514 (2001)
Paneitz, S.: A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (Preprint)
Pucci P., Serrin J.: A general variational identity. Indiana Univ. Math. J. 35, 681–703 (1986)
Pucci P., Serrin J.: Critical exponents and critical dimensions for polyharmonic operators. J. Math. Pures Appl. 69, 55–83 (1990)
Schoen R., Zhang D.: Prescribed calar curvature problem on the n− sphere. Calc. Var. Partial Differ. Equ. 4, 1–25 (1996)
Swanson C.: The best Sobolev constant. Appl. Ana. 47, 227–239 (1992)
Wei J., Xu X.: Classification of solutions of higher order conformally invariant equation. Math. Ann. 313, 207–228 (1999)
Wei J., Yan S.: Infinitely many solutions for the prescribed scalar curcature problem on S N. J. Funct. Anal. 258, 3048–3081 (2010)
Yan S.: Concentration of solutions for the scalar curvature equation on \({\mathbb{R}^N}\) . J. Differ. Equ. 163, 239–264 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Rabinowitz.
Rights and permissions
About this article
Cite this article
Guo, Y., Li, B. Infinitely many solutions for the prescribed curvature problem of polyharmonic operator. Calc. Var. 46, 809–836 (2013). https://doi.org/10.1007/s00526-012-0504-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-012-0504-5