Positive solutions to a linearly perturbed critical growth biharmonic problem

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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

Abstract
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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 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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