# American Institute of Mathematical Sciences

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

 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 $\mathbb{R}^{n}$.

Citation: Saikat Mazumdar. Struwe's decomposition for a polyharmonic operator on a compact Riemannian manifold with or without boundary. Communications on Pure and 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. [2] S. Almaraz, The asymptotic behavior of Palais-Smale sequences on manifolds with boundary, Pacific J. Math, 269 (2014), 1-17. doi: 10.2140/pjm.2014.269.1. [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. [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. [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. [6] K. H. Fieseler and K. Tintarev, Concentration Compactness, Functional-analytic grounds and applications, Imperial College Press, London, 2007. [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. [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. [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. [10] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511551703. [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. [12] E. Hebey, Introduction à l'analyse non linéaire sur les Variétés, Diderot, Paris, 1997. [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. [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, 45-121. doi: 10.4171/RMI/12. [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. [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. [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. [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. [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. [20] C. A. Swanson, The best Sobolev constant, Appl. Anal., 47 (1992), 227-239. doi: 10.1080/00036819208840142.

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##### 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. [2] S. Almaraz, The asymptotic behavior of Palais-Smale sequences on manifolds with boundary, Pacific J. Math, 269 (2014), 1-17. doi: 10.2140/pjm.2014.269.1. [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. [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. [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. [6] K. H. Fieseler and K. Tintarev, Concentration Compactness, Functional-analytic grounds and applications, Imperial College Press, London, 2007. [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. [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. [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. [10] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511551703. [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. [12] E. Hebey, Introduction à l'analyse non linéaire sur les Variétés, Diderot, Paris, 1997. [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. [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, 45-121. doi: 10.4171/RMI/12. [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. [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. [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. [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. [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. [20] C. A. Swanson, The best Sobolev constant, Appl. Anal., 47 (1992), 227-239. doi: 10.1080/00036819208840142.
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