August  2011, 4(4): 809-823. doi: 10.3934/dcdss.2011.4.809

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

Received  September 2009 Revised  December 2009 Published  November 2010

Existence and nonexistence results for positive solutions to a linearly perturbed critical growth biharmonic problem under Steklov boundary conditions, are determined. Furthermore, by investigating the critical dimensions for this problem, a Sobolev inequality with remainder terms, of both interior and boundary type, is deduced.
Citation: Elvise Berchio, Filippo Gazzola. Positive solutions to a linearly perturbed critical growth biharmonic problem. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 809-823. doi: 10.3934/dcdss.2011.4.809
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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