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

Positive solutions to a linearly perturbed critical growth biharmonic problem

Elvise Berchio 1, and  Filippo Gazzola 1,

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.
Keywords: critical exponent, Biharmonic equations, Steklov boundary conditions..
Mathematics Subject Classification: Primary: 35J35, 35J40; Secondary: 35G3.
Citation: Elvise Berchio, Filippo Gazzola. Positive solutions to a linearly perturbed critical growth biharmonic problem. Discrete & 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, (1964).   Google Scholar

[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.  doi: doi:10.1007/s00526-003-0198-9.  Google Scholar

[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.   Google Scholar

[4]

E. Berchio, F. Gazzola and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems,, J. Diff. Eq., 229 (2006), 1.  doi: doi:10.1016/j.jde.2006.04.003.  Google Scholar

[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.  doi: doi:10.3934/cpaa.2009.8.533.  Google Scholar

[6]

E. Berchio, F. Gazzola and T. Weth, Critical growth biharmonic elliptic problems under Steklov-type boundary conditions,, Adv. Diff. Eq., 12 (2007), 381.   Google Scholar

[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.  doi: doi:10.1515/CRELLE.2008.052.  Google Scholar

[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.   Google Scholar

[9]

H. Brezis, "Analyse Fonctionnelle. Théorie et Applications,", Masson, (1983).   Google Scholar

[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.  doi: doi:10.1016/S0362-546X(02)00273-0.  Google Scholar

[11]

D. E. Edmunds, D. Fortunato and E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator,, Arch. Rat. Mech. Anal., 112 (1990), 269.  doi: doi:10.1007/BF00381236.  Google Scholar

[12]

F. Gazzola, Critical growth problems for polyharmonic operators,, Proc. Royal Soc. Edinburgh Sect. A, 128 (1998), 251.   Google Scholar

[13]

F. Gazzola and H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations,, Math. Annalen, 334 (2006), 905.  doi: doi:10.1007/s00208-005-0748-x.  Google Scholar

[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.  doi: doi:10.1007/s00526-002-0182-9.  Google Scholar

[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.  doi: doi:10.1007/s10231-009-0118-5.  Google Scholar

[16]

F. Gazzola, H.-Ch. Grunau and G. Sweers, "Polyharmonic Boundary Value Problems,", Springer, (2010).  doi: 10.1007/978-3-642-12245-3.  Google Scholar

[17]

F. Gazzola and D. Pierotti, Positive solutions to critical growth biharmonic elliptic problems under Steklov boundary conditions,, Nonlinear Analysis, 71 (2009), 232.  doi: doi:10.1016/j.na.2008.10.052.  Google Scholar

[18]

F. Gazzola and G. Sweers, On positivity for the biharmonic operator under Steklov boundary conditions,, Arch. Rat. Mech. Anal., 188 (2008), 399.  doi: doi:10.1007/s00205-007-0090-4.  Google Scholar

[19]

Y. Ge, Positive solutions in semilinear critical problems for polyharmonic operators,, J. Math. Pures Appl., 84 (2005), 199.  doi: doi:10.1016/j.matpur.2004.10.002.  Google Scholar

[20]

H.-Ch. Grunau, Critical exponents and multiple critical dimensions for polyharmonic operators II,, Boll. Unione Mat. Ital., 7 (1995), 815.   Google Scholar

[21]

H.-Ch. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents,, Calc. Var. Partial Diff. Eq., 3 (1995), 243.  doi: doi:10.1007/BF01205006.  Google Scholar

[22]

H.-Ch. Grunau, On a conjecture of P. Pucci and J. Serrin,, Analysis, 16 (1996), 399.   Google Scholar

[23]

E. Jannelli, The role played by space dimension in elliptic critical problems,, J. Diff. Eq, 156 (2000), 407.  doi: doi:10.1006/jdeq.1998.3589.  Google Scholar

[24]

E. Mitidieri, A Rellich type identity and applications,, Comm. Partial Diff. Eq., 18 (1993), 125.  doi: doi:10.1080/03605309308820923.  Google Scholar

[25]

E. Mitidieri, On the definition of critical dimension,, copy available from the author, (1993).   Google Scholar

[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.   Google Scholar

[27]

S. J. Pohozaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$,, Soviet Math. Doklady, 6 (1965), 1408.   Google Scholar

[28]

S. J. Pohozaev, The eigenfunctions of quasilinear elliptic problems,, Math. Sbornik, 82 (1970), 171.   Google Scholar

[29]

P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators,, J. Math. Pures Appl., 69 (1990), 55.   Google Scholar

[30]

R. Soranzo, A priori estimates and existence of positive solutions of a superlinear polyharmonic equation,, Dyn. Syst. Appl., 3 (1994), 465.   Google Scholar

[31]

M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Springer, (1990).   Google Scholar

[32]

C. A. Swanson, The best Sobolev constant,, Appl. Anal., 47 (1992), 227.  doi: doi:10.1080/00036819208840142.  Google Scholar

[33]

W. C. Troy, Symmetry properties in systems of semilinear elliptic equations,, J. Diff. Eq., 42 (1981), 400.  doi: doi:10.1016/0022-0396(81)90113-3.  Google Scholar

[34]

R. C. A. M. van der Vorst, Variational identities and applications to differential systems,, Arch. Rat. Mech. Anal., 116 (1991), 375.  doi: doi:10.1007/BF00375674.  Google Scholar

[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.   Google Scholar

[36]

R. C. A. M. van der Vorst, Fourth order elliptic equations with critical growth,, C. R. Acad. Sci. Paris S\'erie I, 320 (1995), 295.   Google Scholar

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, (1964).   Google Scholar

[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.  doi: doi:10.1007/s00526-003-0198-9.  Google Scholar

[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.   Google Scholar

[4]

E. Berchio, F. Gazzola and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problems,, J. Diff. Eq., 229 (2006), 1.  doi: doi:10.1016/j.jde.2006.04.003.  Google Scholar

[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.  doi: doi:10.3934/cpaa.2009.8.533.  Google Scholar

[6]

E. Berchio, F. Gazzola and T. Weth, Critical growth biharmonic elliptic problems under Steklov-type boundary conditions,, Adv. Diff. Eq., 12 (2007), 381.   Google Scholar

[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.  doi: doi:10.1515/CRELLE.2008.052.  Google Scholar

[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.   Google Scholar

[9]

H. Brezis, "Analyse Fonctionnelle. Théorie et Applications,", Masson, (1983).   Google Scholar

[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.  doi: doi:10.1016/S0362-546X(02)00273-0.  Google Scholar

[11]

D. E. Edmunds, D. Fortunato and E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator,, Arch. Rat. Mech. Anal., 112 (1990), 269.  doi: doi:10.1007/BF00381236.  Google Scholar

[12]

F. Gazzola, Critical growth problems for polyharmonic operators,, Proc. Royal Soc. Edinburgh Sect. A, 128 (1998), 251.   Google Scholar

[13]

F. Gazzola and H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations,, Math. Annalen, 334 (2006), 905.  doi: doi:10.1007/s00208-005-0748-x.  Google Scholar

[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.  doi: doi:10.1007/s00526-002-0182-9.  Google Scholar

[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.  doi: doi:10.1007/s10231-009-0118-5.  Google Scholar

[16]

F. Gazzola, H.-Ch. Grunau and G. Sweers, "Polyharmonic Boundary Value Problems,", Springer, (2010).  doi: 10.1007/978-3-642-12245-3.  Google Scholar

[17]

F. Gazzola and D. Pierotti, Positive solutions to critical growth biharmonic elliptic problems under Steklov boundary conditions,, Nonlinear Analysis, 71 (2009), 232.  doi: doi:10.1016/j.na.2008.10.052.  Google Scholar

[18]

F. Gazzola and G. Sweers, On positivity for the biharmonic operator under Steklov boundary conditions,, Arch. Rat. Mech. Anal., 188 (2008), 399.  doi: doi:10.1007/s00205-007-0090-4.  Google Scholar

[19]

Y. Ge, Positive solutions in semilinear critical problems for polyharmonic operators,, J. Math. Pures Appl., 84 (2005), 199.  doi: doi:10.1016/j.matpur.2004.10.002.  Google Scholar

[20]

H.-Ch. Grunau, Critical exponents and multiple critical dimensions for polyharmonic operators II,, Boll. Unione Mat. Ital., 7 (1995), 815.   Google Scholar

[21]

H.-Ch. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents,, Calc. Var. Partial Diff. Eq., 3 (1995), 243.  doi: doi:10.1007/BF01205006.  Google Scholar

[22]

H.-Ch. Grunau, On a conjecture of P. Pucci and J. Serrin,, Analysis, 16 (1996), 399.   Google Scholar

[23]

E. Jannelli, The role played by space dimension in elliptic critical problems,, J. Diff. Eq, 156 (2000), 407.  doi: doi:10.1006/jdeq.1998.3589.  Google Scholar

[24]

E. Mitidieri, A Rellich type identity and applications,, Comm. Partial Diff. Eq., 18 (1993), 125.  doi: doi:10.1080/03605309308820923.  Google Scholar

[25]

E. Mitidieri, On the definition of critical dimension,, copy available from the author, (1993).   Google Scholar

[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.   Google Scholar

[27]

S. J. Pohozaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$,, Soviet Math. Doklady, 6 (1965), 1408.   Google Scholar

[28]

S. J. Pohozaev, The eigenfunctions of quasilinear elliptic problems,, Math. Sbornik, 82 (1970), 171.   Google Scholar

[29]

P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators,, J. Math. Pures Appl., 69 (1990), 55.   Google Scholar

[30]

R. Soranzo, A priori estimates and existence of positive solutions of a superlinear polyharmonic equation,, Dyn. Syst. Appl., 3 (1994), 465.   Google Scholar

[31]

M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Springer, (1990).   Google Scholar

[32]

C. A. Swanson, The best Sobolev constant,, Appl. Anal., 47 (1992), 227.  doi: doi:10.1080/00036819208840142.  Google Scholar

[33]

W. C. Troy, Symmetry properties in systems of semilinear elliptic equations,, J. Diff. Eq., 42 (1981), 400.  doi: doi:10.1016/0022-0396(81)90113-3.  Google Scholar

[34]

R. C. A. M. van der Vorst, Variational identities and applications to differential systems,, Arch. Rat. Mech. Anal., 116 (1991), 375.  doi: doi:10.1007/BF00375674.  Google Scholar

[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.   Google Scholar

[36]

R. C. A. M. van der Vorst, Fourth order elliptic equations with critical growth,, C. R. Acad. Sci. Paris S\'erie I, 320 (1995), 295.   Google Scholar

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