American Institute of Mathematical Sciences

Advanced
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

[1]

Mónica Clapp, Marco Squassina. Nonhomogeneous polyharmonic elliptic problems at critical growth with symmetric data. Communications on Pure & Applied Analysis, 2003, 2 (2) : 171-186. doi: 10.3934/cpaa.2003.2.171
[2]

Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 581-593. doi: 10.3934/dcdss.2011.4.581
[3]

Yuxia Guo, Ting Liu. Lazer-McKenna conjecture for higher order elliptic problem with critical growth. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1159-1189. doi: 10.3934/dcds.2020074
[4]

Jingbo Dou, Qianqiao Guo. Solutions for polyharmonic elliptic problems with critical nonlinearities in symmetric domains. Communications on Pure & Applied Analysis, 2012, 11 (2) : 453-464. doi: 10.3934/cpaa.2012.11.453
[5]

Olivier Druet, Emmanuel Hebey and Frederic Robert. A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth. Electronic Research Announcements, 2003, 9: 19-25.

[6]

Feliz Minhós, A. I. Santos. Higher order two-point boundary value problems with asymmetric growth. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 127-137. doi: 10.3934/dcdss.2008.1.127
[7]

Nguyen Lam, Guozhen Lu. Existence of nontrivial solutions to Polyharmonic equations with subcritical and critical exponential growth. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2187-2205. doi: 10.3934/dcds.2012.32.2187
[8]

Jiguang Bao, Nguyen Lam, Guozhen Lu. Polyharmonic equations with critical exponential growth in the whole space $ \mathbb{R}^{n}$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 577-600. doi: 10.3934/dcds.2016.36.577
[9]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533
[10]

Yanfang Peng. On elliptic systems with Sobolev critical exponent. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3357-3373. doi: 10.3934/dcds.2016.36.3357
[11]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597
[12]

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
[13]

Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439
[14]

Petr Hasil, Petr Zemánek. Critical second order operators on time scales. Conference Publications, 2011, 2011 (Special) : 653-659. doi: 10.3934/proc.2011.2011.653
[15]

Alexandru Kristály. Asymptotically critical problems on higher-dimensional spheres. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 919-935. doi: 10.3934/dcds.2009.23.919
[16]

Paulo Rabelo. Elliptic systems involving critical growth in dimension two. Communications on Pure & Applied Analysis, 2009, 8 (6) : 2013-2035. doi: 10.3934/cpaa.2009.8.2013
[17]

Shuangjie Peng. Remarks on singular critical growth elliptic equations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 707-719. doi: 10.3934/dcds.2006.14.707
[18]

Angelo Favini, Georgy A. Sviridyuk, Alyona A. Zamyshlyaeva. One Class of Sobolev Type Equations of Higher Order with Additive "White Noise". Communications on Pure & Applied Analysis, 2016, 15 (1) : 185-196. doi: 10.3934/cpaa.2016.15.185
[19]

François Hamel, Emmanuel Russ, Nikolai Nadirashvili. Comparisons of eigenvalues of second order elliptic operators. Conference Publications, 2007, 2007 (Special) : 477-486. doi: 10.3934/proc.2007.2007.477
[20]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (19)
  • HTML views (77)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences