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The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term
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 |
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 [
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. |
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