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January  2017, 16(1): 311-330. doi: 10.3934/cpaa.2017015

Struwe's decomposition for a polyharmonic operator on a compact Riemannian manifold with or without boundary

Saikat Mazumdar

Institut Elie Cartan de Lorraine, Université de Lorraine, BP 70239,54506 Vandœuvre-lès-Nancy, France

Received  May 2016 Revised  July 2016 Published  November 2016

Fund Project: This work is part of the PhD thesis of the author, funded by "Fédération Charles Hermite" (FR3198 du CNRS) and "Région Lorraine". The author acknowledges these two institutions for their supports.

Given a high-order elliptic operator on a compact manifold with or without boundary, we perform the decomposition of Palais-Smale sequences for a nonlinear problem as a sum of bubbles. This is a generalization of the celebrated 1984 result of Struwe [19]. Unlike the case of second-order operators, bubbles close to the boundary might appear. Our result includes the case of a smooth bounded domain of $\mathbb{R}^{n}$.

Keywords: Struwe decomposition, polyharmonic operators on manifolds, higher order elliptic problems with critical Sobolev growth.
Mathematics Subject Classification: 35J35, 58J60.
Citation: Saikat Mazumdar. Struwe's decomposition for a polyharmonic operator on a compact Riemannian manifold with or without boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 311-330. doi: 10.3934/cpaa.2017015
References:
[1]

E. H. Abdallah and J. Vétois, Sharp Sobolev asymptotics for critical anisotropic equations, Arch. Ration. Mech. Anal., 192 (2009), 1-36.  doi: 10.1007/s00205-008-0122-8.  Google Scholar

[2]

S. Almaraz, The asymptotic behavior of Palais-Smale sequences on manifolds with boundary, Pacific Math J., 269 (2014), 1-17.  doi: 10.2140/pjm.2014.269.1.  Google Scholar

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.   Google Scholar

[4]

T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-13006-3.  Google Scholar

[5]

T. Bartsch, T. Weth and M. Willem, A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator, Calc. Var. Partial Differential Equations, 18 (2003), 253-268.  doi: 10.1007/s00526-003-0198-9.  Google Scholar

[6] K. H. Fieseler and K. Tintarev, Concentration Compactness, Functional-analytic grounds and applications, Imperial College Press, London, 2007.   Google Scholar
[7]

F. Gazzola, H. C. Grunau and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations, 18 (2003), 117-143.  doi: 10.1007/s00526-002-0182-9.  Google Scholar

[8]

F. Gazzola, H. C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Lecture Notes in Mathematics 1991, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12245-3.  Google Scholar

[9]

Y. Ge, J. Wei and F. Zhou, A critical elliptic problem for polyharmonic operators, J. Funct. Anal., 260 (2011), 2247-2282.  doi: 10.1016/j.jfa.2011.01.005.  Google Scholar

[10] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511551703.  Google Scholar
[11]

E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes in Mathematics 5, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999.  Google Scholar

[12]

E. Hebey, Introduction à l'analyse non linéaire sur les Variétés, Diderot, Paris, 1997. Google Scholar

[13]

E. Hebey and F. Robert, Coercivity and Struwe's compactness for Paneitz type operators with constant coefficients, Calc. Var. Partial Differential Equations, 13 (2001), 491-517.  doi: 10.1007/s005260100084.  Google Scholar

[14]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. Ⅰ, Ⅱ, Rev. Mat. Iberoamericana, 1 (1985), 145-201.  doi: 10.4171/RMI/12.  Google Scholar

[15]

S. Mazumdar, GJMS-type Operators on a compact Riemannian manifold: Best constants and Coron-type solutions, J. Differential Equations, 261 (2016), 4997-5034.  doi: 10.1016/j.jde.2016.07.017.  Google Scholar

[16]

W. Reichel and T. Weth, A priori bounds and a Liouville theorem on a half-space for higherorder elliptic Dirichlet problems, Math. Z., 261 (2009), 805-827.  doi: 10.1007/s00209-008-0352-3.  Google Scholar

[17]

F. Robert, Admissible Q-curvatures under isometries for the conformal GJMS operators, in Nonlinear Elliptic Partial Differential Equations, Contemp. Math, 540, Amer. Math. Soc. , Providence, RI (2011), 241-259. doi: 10.1090/conm/540/10668.  Google Scholar

[18]

N. Saintier, Asymptotic estimates and blow-up theory for critical equations involving the p-Laplacian, Calc. Var. Partial Differential Equations, 25 (2006), 299-331.  doi: 10.1007/s00526-005-0344-7.  Google Scholar

[19]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.  Google Scholar

[20]

C. A. Swanson, The best Sobolev constant, Appl. Anal., 47 (1992), 227-239.  doi: 10.1080/00036819208840142.  Google Scholar

show all references

References:
[1]

E. H. Abdallah and J. Vétois, Sharp Sobolev asymptotics for critical anisotropic equations, Arch. Ration. Mech. Anal., 192 (2009), 1-36.  doi: 10.1007/s00205-008-0122-8.  Google Scholar

[2]

S. Almaraz, The asymptotic behavior of Palais-Smale sequences on manifolds with boundary, Pacific Math J., 269 (2014), 1-17.  doi: 10.2140/pjm.2014.269.1.  Google Scholar

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.   Google Scholar

[4]

T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-13006-3.  Google Scholar

[5]

T. Bartsch, T. Weth and M. Willem, A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator, Calc. Var. Partial Differential Equations, 18 (2003), 253-268.  doi: 10.1007/s00526-003-0198-9.  Google Scholar

[6] K. H. Fieseler and K. Tintarev, Concentration Compactness, Functional-analytic grounds and applications, Imperial College Press, London, 2007.   Google Scholar
[7]

F. Gazzola, H. C. Grunau and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations, 18 (2003), 117-143.  doi: 10.1007/s00526-002-0182-9.  Google Scholar

[8]

F. Gazzola, H. C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Lecture Notes in Mathematics 1991, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12245-3.  Google Scholar

[9]

Y. Ge, J. Wei and F. Zhou, A critical elliptic problem for polyharmonic operators, J. Funct. Anal., 260 (2011), 2247-2282.  doi: 10.1016/j.jfa.2011.01.005.  Google Scholar

[10] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511551703.  Google Scholar
[11]

E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes in Mathematics 5, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999.  Google Scholar

[12]

E. Hebey, Introduction à l'analyse non linéaire sur les Variétés, Diderot, Paris, 1997. Google Scholar

[13]

E. Hebey and F. Robert, Coercivity and Struwe's compactness for Paneitz type operators with constant coefficients, Calc. Var. Partial Differential Equations, 13 (2001), 491-517.  doi: 10.1007/s005260100084.  Google Scholar

[14]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. Ⅰ, Ⅱ, Rev. Mat. Iberoamericana, 1 (1985), 145-201.  doi: 10.4171/RMI/12.  Google Scholar

[15]

S. Mazumdar, GJMS-type Operators on a compact Riemannian manifold: Best constants and Coron-type solutions, J. Differential Equations, 261 (2016), 4997-5034.  doi: 10.1016/j.jde.2016.07.017.  Google Scholar

[16]

W. Reichel and T. Weth, A priori bounds and a Liouville theorem on a half-space for higherorder elliptic Dirichlet problems, Math. Z., 261 (2009), 805-827.  doi: 10.1007/s00209-008-0352-3.  Google Scholar

[17]

F. Robert, Admissible Q-curvatures under isometries for the conformal GJMS operators, in Nonlinear Elliptic Partial Differential Equations, Contemp. Math, 540, Amer. Math. Soc. , Providence, RI (2011), 241-259. doi: 10.1090/conm/540/10668.  Google Scholar

[18]

N. Saintier, Asymptotic estimates and blow-up theory for critical equations involving the p-Laplacian, Calc. Var. Partial Differential Equations, 25 (2006), 299-331.  doi: 10.1007/s00526-005-0344-7.  Google Scholar

[19]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.  Google Scholar

[20]

C. A. Swanson, The best Sobolev constant, Appl. Anal., 47 (1992), 227-239.  doi: 10.1080/00036819208840142.  Google Scholar

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