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A remark on Hardy type inequalities with remainder terms
Positive solutions to a linearly perturbed critical growth biharmonic problem
1. | Dipartimento di Matematica del Politecnico, Piazza L. da Vinci 32, Milano, 20133, Italy, Italy |
References:
[1] |
M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables," National Bureau of Standards Applied Mathematics Series, Washington, D.C. 1964. |
[2] |
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 Diff. Eq., 18 (2003), 253-268.
doi: doi:10.1007/s00526-003-0198-9. |
[3] |
E. Berchio and F. Gazzola, Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities, Electronic J. Diff. Eq., 34 (2005), 1-20. |
[4] |
E. Berchio, F. Gazzola and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems, J. Diff. Eq., 229 (2006), 1-23.
doi: doi:10.1016/j.jde.2006.04.003. |
[5] |
E. Berchio, F. Gazzola and D. Pierotti, Nodal solutions to critical growth elliptic problems under Steklov boundary conditions, Comm. Pure Appl. Anal., 8 (2009), 533-557.
doi: doi:10.3934/cpaa.2009.8.533. |
[6] |
E. Berchio, F. Gazzola and T. Weth, Critical growth biharmonic elliptic problems under Steklov-type boundary conditions, Adv. Diff. Eq., 12 (2007), 381-406. |
[7] |
E. Berchio, F. Gazzola and T. Weth, Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems, J. Reine Angew. Math., 620 (2008), 165-183.
doi: doi:10.1515/CRELLE.2008.052. |
[8] |
F. Bernis, J. García Azorero and I. Peral, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Diff. Eq., 1 (1996), 219-240. |
[9] |
H. Brezis, "Analyse Fonctionnelle. Théorie et Applications," Masson, Paris, 1983. |
[10] |
F. Ebobisse and M. O. Ahmedou, On a nonlinear fourth-order elliptic equation involving the critical Sobolev exponent, Nonlin. Anal. TMA, 52 (2003), 1535-1552.
doi: doi:10.1016/S0362-546X(02)00273-0. |
[11] |
D. E. Edmunds, D. Fortunato and E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator, Arch. Rat. Mech. Anal., 112 (1990), 269-289.
doi: doi:10.1007/BF00381236. |
[12] |
F. Gazzola, Critical growth problems for polyharmonic operators, Proc. Royal Soc. Edinburgh Sect. A, 128 (1998), 251-263. |
[13] |
F. Gazzola and H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Annalen, 334 (2006), 905-936.
doi: doi:10.1007/s00208-005-0748-x. |
[14] |
F. Gazzola, H.-Ch. Grunau and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations, 18 (2003), 117-143.
doi: doi:10.1007/s00526-002-0182-9. |
[15] |
F. Gazzola, H.-Ch. Grunau and G. Sweers, Optimal Sobolev and Hardy-Rellich constants under Navier boundary conditions, Ann. Mat. Pura Appl., 189 (2010), 475-486.
doi: doi:10.1007/s10231-009-0118-5. |
[16] |
F. Gazzola, H.-Ch. Grunau and G. Sweers, "Polyharmonic Boundary Value Problems," Springer, 2010.
doi: 10.1007/978-3-642-12245-3. |
[17] |
F. Gazzola and D. Pierotti, Positive solutions to critical growth biharmonic elliptic problems under Steklov boundary conditions, Nonlinear Analysis, 71 (2009), 232-238.
doi: doi:10.1016/j.na.2008.10.052. |
[18] |
F. Gazzola and G. Sweers, On positivity for the biharmonic operator under Steklov boundary conditions, Arch. Rat. Mech. Anal., 188 (2008), 399-427.
doi: doi:10.1007/s00205-007-0090-4. |
[19] |
Y. Ge, Positive solutions in semilinear critical problems for polyharmonic operators, J. Math. Pures Appl., 84 (2005), 199-245.
doi: doi:10.1016/j.matpur.2004.10.002. |
[20] |
H.-Ch. Grunau, Critical exponents and multiple critical dimensions for polyharmonic operators II, Boll. Unione Mat. Ital., 7 (1995), 815-847. |
[21] |
H.-Ch. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents, Calc. Var. Partial Diff. Eq., 3 (1995), 243-252.
doi: doi:10.1007/BF01205006. |
[22] |
H.-Ch. Grunau, On a conjecture of P. Pucci and J. Serrin, Analysis, 16 (1996), 399-403. |
[23] |
E. Jannelli, The role played by space dimension in elliptic critical problems, J. Diff. Eq, 156 (2000), 407-426.
doi: doi:10.1006/jdeq.1998.3589. |
[24] |
E. Mitidieri, A Rellich type identity and applications, Comm. Partial Diff. Eq., 18 (1993), 125-151.
doi: doi:10.1080/03605309308820923. |
[25] |
E. Mitidieri, On the definition of critical dimension, copy available from the author, 1993. |
[26] |
P. Oswald, On a priori estimates for positive solutions of a semilinear biharmonic equation in a ball, Comment. Math. Univ. Carolinae, 26 (1985), 565-577. |
[27] |
S. J. Pohozaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Soviet Math. Doklady, 6 (1965), 1408-1411. |
[28] |
S. J. Pohozaev, The eigenfunctions of quasilinear elliptic problems, Math. Sbornik, 82 (1970), 171-188; first published in Russian on Math. USSR Sbornik, 11 (1970). |
[29] |
P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl., 69 (1990), 55-83. |
[30] |
R. Soranzo, A priori estimates and existence of positive solutions of a superlinear polyharmonic equation, Dyn. Syst. Appl., 3 (1994), 465-487. |
[31] |
M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Springer, Berlin-Heidelberg, 1990. |
[32] |
C. A. Swanson, The best Sobolev constant, Appl. Anal., 47 (1992), 227-239.
doi: doi:10.1080/00036819208840142. |
[33] |
W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Diff. Eq., 42 (1981), 400-413.
doi: doi:10.1016/0022-0396(81)90113-3. |
[34] |
R. C. A. M. van der Vorst, Variational identities and applications to differential systems, Arch. Rat. Mech. Anal., 116 (1991), 375-398.
doi: doi:10.1007/BF00375674. |
[35] |
R. C. A. M. van der Vorst, Best constant for the embedding of the space $H^2\cap H_0^1(\Omega)$ into $L^{\frac{2N}{N-4}}(\Omega)$, Diff. Int. Eq., 6 (1993), 259-276. |
[36] |
R. C. A. M. van der Vorst, Fourth order elliptic equations with critical growth, C. R. Acad. Sci. Paris Série I, 320 (1995), 295-299. |
show all references
References:
[1] |
M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables," National Bureau of Standards Applied Mathematics Series, Washington, D.C. 1964. |
[2] |
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 Diff. Eq., 18 (2003), 253-268.
doi: doi:10.1007/s00526-003-0198-9. |
[3] |
E. Berchio and F. Gazzola, Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities, Electronic J. Diff. Eq., 34 (2005), 1-20. |
[4] |
E. Berchio, F. Gazzola and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems, J. Diff. Eq., 229 (2006), 1-23.
doi: doi:10.1016/j.jde.2006.04.003. |
[5] |
E. Berchio, F. Gazzola and D. Pierotti, Nodal solutions to critical growth elliptic problems under Steklov boundary conditions, Comm. Pure Appl. Anal., 8 (2009), 533-557.
doi: doi:10.3934/cpaa.2009.8.533. |
[6] |
E. Berchio, F. Gazzola and T. Weth, Critical growth biharmonic elliptic problems under Steklov-type boundary conditions, Adv. Diff. Eq., 12 (2007), 381-406. |
[7] |
E. Berchio, F. Gazzola and T. Weth, Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems, J. Reine Angew. Math., 620 (2008), 165-183.
doi: doi:10.1515/CRELLE.2008.052. |
[8] |
F. Bernis, J. García Azorero and I. Peral, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Diff. Eq., 1 (1996), 219-240. |
[9] |
H. Brezis, "Analyse Fonctionnelle. Théorie et Applications," Masson, Paris, 1983. |
[10] |
F. Ebobisse and M. O. Ahmedou, On a nonlinear fourth-order elliptic equation involving the critical Sobolev exponent, Nonlin. Anal. TMA, 52 (2003), 1535-1552.
doi: doi:10.1016/S0362-546X(02)00273-0. |
[11] |
D. E. Edmunds, D. Fortunato and E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator, Arch. Rat. Mech. Anal., 112 (1990), 269-289.
doi: doi:10.1007/BF00381236. |
[12] |
F. Gazzola, Critical growth problems for polyharmonic operators, Proc. Royal Soc. Edinburgh Sect. A, 128 (1998), 251-263. |
[13] |
F. Gazzola and H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Annalen, 334 (2006), 905-936.
doi: doi:10.1007/s00208-005-0748-x. |
[14] |
F. Gazzola, H.-Ch. Grunau and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations, 18 (2003), 117-143.
doi: doi:10.1007/s00526-002-0182-9. |
[15] |
F. Gazzola, H.-Ch. Grunau and G. Sweers, Optimal Sobolev and Hardy-Rellich constants under Navier boundary conditions, Ann. Mat. Pura Appl., 189 (2010), 475-486.
doi: doi:10.1007/s10231-009-0118-5. |
[16] |
F. Gazzola, H.-Ch. Grunau and G. Sweers, "Polyharmonic Boundary Value Problems," Springer, 2010.
doi: 10.1007/978-3-642-12245-3. |
[17] |
F. Gazzola and D. Pierotti, Positive solutions to critical growth biharmonic elliptic problems under Steklov boundary conditions, Nonlinear Analysis, 71 (2009), 232-238.
doi: doi:10.1016/j.na.2008.10.052. |
[18] |
F. Gazzola and G. Sweers, On positivity for the biharmonic operator under Steklov boundary conditions, Arch. Rat. Mech. Anal., 188 (2008), 399-427.
doi: doi:10.1007/s00205-007-0090-4. |
[19] |
Y. Ge, Positive solutions in semilinear critical problems for polyharmonic operators, J. Math. Pures Appl., 84 (2005), 199-245.
doi: doi:10.1016/j.matpur.2004.10.002. |
[20] |
H.-Ch. Grunau, Critical exponents and multiple critical dimensions for polyharmonic operators II, Boll. Unione Mat. Ital., 7 (1995), 815-847. |
[21] |
H.-Ch. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents, Calc. Var. Partial Diff. Eq., 3 (1995), 243-252.
doi: doi:10.1007/BF01205006. |
[22] |
H.-Ch. Grunau, On a conjecture of P. Pucci and J. Serrin, Analysis, 16 (1996), 399-403. |
[23] |
E. Jannelli, The role played by space dimension in elliptic critical problems, J. Diff. Eq, 156 (2000), 407-426.
doi: doi:10.1006/jdeq.1998.3589. |
[24] |
E. Mitidieri, A Rellich type identity and applications, Comm. Partial Diff. Eq., 18 (1993), 125-151.
doi: doi:10.1080/03605309308820923. |
[25] |
E. Mitidieri, On the definition of critical dimension, copy available from the author, 1993. |
[26] |
P. Oswald, On a priori estimates for positive solutions of a semilinear biharmonic equation in a ball, Comment. Math. Univ. Carolinae, 26 (1985), 565-577. |
[27] |
S. J. Pohozaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Soviet Math. Doklady, 6 (1965), 1408-1411. |
[28] |
S. J. Pohozaev, The eigenfunctions of quasilinear elliptic problems, Math. Sbornik, 82 (1970), 171-188; first published in Russian on Math. USSR Sbornik, 11 (1970). |
[29] |
P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl., 69 (1990), 55-83. |
[30] |
R. Soranzo, A priori estimates and existence of positive solutions of a superlinear polyharmonic equation, Dyn. Syst. Appl., 3 (1994), 465-487. |
[31] |
M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Springer, Berlin-Heidelberg, 1990. |
[32] |
C. A. Swanson, The best Sobolev constant, Appl. Anal., 47 (1992), 227-239.
doi: doi:10.1080/00036819208840142. |
[33] |
W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Diff. Eq., 42 (1981), 400-413.
doi: doi:10.1016/0022-0396(81)90113-3. |
[34] |
R. C. A. M. van der Vorst, Variational identities and applications to differential systems, Arch. Rat. Mech. Anal., 116 (1991), 375-398.
doi: doi:10.1007/BF00375674. |
[35] |
R. C. A. M. van der Vorst, Best constant for the embedding of the space $H^2\cap H_0^1(\Omega)$ into $L^{\frac{2N}{N-4}}(\Omega)$, Diff. Int. Eq., 6 (1993), 259-276. |
[36] |
R. C. A. M. van der Vorst, Fourth order elliptic equations with critical growth, C. R. Acad. Sci. Paris Série I, 320 (1995), 295-299. |
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