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.

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.

[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.

[1]

Mónica Clapp, Marco Squassina. Nonhomogeneous polyharmonic elliptic problems at critical growth with symmetric data. Communications on Pure and 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 and 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 and Continuous Dynamical Systems, 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 and 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[10]

Diego D. Felix, Marcelo F. Furtado, Everaldo S. Medeiros. Semilinear elliptic problems involving exponential critical growth in the half-space. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4937-4953. doi: 10.3934/cpaa.2020219

[11]

R.M. Brown, L.D. Gauthier. Inverse boundary value problems for polyharmonic operators with non-smooth coefficients. Inverse Problems and Imaging, 2022, 16 (4) : 943-966. doi: 10.3934/ipi.2022006

[12]

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

[13]

Xiaoli Chen, Jianfu Yang. Improved Sobolev inequalities and critical problems. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3673-3695. doi: 10.3934/cpaa.2020162

[14]

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

[15]

Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033

[16]

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

[17]

Claudianor O. Alves, César T. Ledesma. Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2065-2100. doi: 10.3934/cpaa.2021058

[18]

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

[19]

Mónica Clapp, Juan Carlos Fernández, Alberto Saldaña. Critical polyharmonic systems and optimal partitions. Communications on Pure and Applied Analysis, 2021, 20 (11) : 4007-4023. doi: 10.3934/cpaa.2021141

[20]

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

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (145)
  • HTML views (118)
  • Cited by (0)

Other articles
by authors

[Back to Top]