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March  2012, 11(2): 517-545. doi: 10.3934/cpaa.2012.11.517

Nonradial positive solutions for a biharmonic critical growth problem

1. 

Department of Mathematics, East China Normal University, Shanghai, 200241, China

Received  September 2010 Revised  August 2011 Published  October 2011

We investigate the existence of nonradial positive solutions for a critical semilinear biharmonic problem defined on a unit ball. The solution is obtained as a minimizer of the quotient functional associated to the problem restricted to appropriate subspaces of $H^2\cap H^1_0$ invariant for the action of a subgroup of $O(N)$. By making use of more careful estimates and some new arguments, we extend Serra's result in [41] to the biharmonic case.
Citation: Zhongliang Wang. Nonradial positive solutions for a biharmonic critical growth problem. Communications on Pure and Applied Analysis, 2012, 11 (2) : 517-545. doi: 10.3934/cpaa.2012.11.517
References:
[1]

C. O. Alves, J. M. do Ó and O. H. Miyagaki, Nontrivial solutions for a class of semilinear biharmonic problems involving critical exponents, Nonlinear Anal., 46 (2001), 121-133. doi: 10.1016/S0362-546X(99)00449-6.

[2]

C. O. Alves and J. M. do Ó, Positive solutions of a fourth-order semilinear problem involving critical growth, Adv. Nonlinear Stud., 2 (2002), 437-458.

[3]

M. Badiale and E. Serra, Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Stud., 4 (2004), 453-467.

[4]

A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294. doi: 10.1002/cpa.3160410302.

[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]

V. Barutello, S. Secchi and E. Serra, A note on the radial solutions for the supercritical Hénon equation, J. Math. Anal. Appl., 341 (2008), 720-728. arXiv:0707.1790. doi: 10.1016/j.jmaa.2007.10.052.

[7]

E. Berchio, F. Gazzola and T. Weth, Critical growth biharmonic elliptic problems under Steklov-type boundary conditions, Adv. Differential Equations, 12 (2007), 381-406.

[8]

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: 10.1515/CRELLE.2008.052.

[9]

F. Bernis and H.-Ch. Grunau, Critical exponents and multiple critical dimensions for polyharmonic operators, J. Differential Equations, 117 (1995), 469-486. doi: 10.1006/jdeq.1995.1062.

[10]

F. Bernis, J. García Azorero and I. Peral, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Differential Equations, 1 (1996), 219-240.

[11]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.

[12]

J. Byeon and Z.-Q. Wang, On the Hénon equation: asymptotic profile of ground states. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 803-828. doi: 10.1016/j.anihpc.2006.04.001.

[13]

J. Byeon and Z.-Q. Wang, On the Hénon equation: asymptotic profile of ground states. II, J. Differential Equations, 216 (2005), 78-108. doi: 10.1016/j.jde.2005.02.018.

[14]

M. Calanchi, S. Secchi and E. Terraneo, Multiple solutions for a Hénon-like equation on the annulus, J. Differential Equations, 245 (2008), 1507-1525. arXiv:0705.1492. doi: 10.1016/j.jde.2008.06.018.

[15]

D. Cao and S. Peng, The asymptotic behaviour of the ground state solutions for Hénon equation, J. Math. Anal. Appl., 278 (2003), 1-17. doi: 10.1016/S0022-247X(02)00292-5.

[16]

D. Cao, S. Peng and S. Yan, Asymptotic behaviour of ground state solutions for the Hénon equation, IMA J. Appl. Math., 74 (2009), 468-480. doi: 10.1093/imamat/hxn035.

[17]

G. Chen, W.-M. Ni and J. Zhou, Algorithms and visualization for solutions of nonlinear elliptic equations, Int. J. Bifurcat. Chaos, 10 (2000), 1565-1612. doi: 10.1142/S0218127400001006.

[18]

F. Ebobisse and M. O. Ahmedou, On a nonlinear fourth order elliptic equation involving the critical Sobolev exponent, Nonlinear Anal., 52 (2003), 1535-1552. arXiv:math/0112240. doi: 10.1016/S0362-546X(02)00273-0.

[19]

D. E. Edmunds, D. Fortunato and E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator, Arch. Rational Mech. Anal., 112 (1990), 269-289. doi: 10.1007/BF00381236.

[20]

P. Esposito, A. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in $R^2$, J. Anal. Math., 100 (2006), 249-280. doi: 10.1007/BF02916763.

[21]

J. García Azorero and I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895. doi: 10.2307/2001562.

[22]

M. Gazzini and E. Serra, The Neumann problem for the Hénon equation, trace inequalities and Steklov eigenvalues, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 281-302.

[23]

F. Gazzola, Critical growth problems for polyharmonic operators, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 251-263.

[24]

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: 10.1007/s00526-002-0182-9.

[25]

F. Gazzola and D. Pierotti, Positive solutions to critical growth biharmonic elliptic problems under Steklov boundary conditions, Nonlinear Anal., 71 (2009), 232-238. doi: 10.1016/j.na.2008.10.052.

[26]

Y. Ge, Positive solutions in semilinear critical problems for polyharmonic operators, J. Math. Pures Appl., 84 (2005), 199-245. doi: 10.1016/j.matpur.2004.10.002.

[27]

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.

[28]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.

[29]

H.-Ch. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 3 (1995), 243-252. doi: 10.1007/BF01205006.

[30]

Y. Guo and J. Wei, Supercritical biharmonic elliptic problems in domains with small holes, Math. Nachr., 282 (2009), 1724-1739. doi: 10.1002/mana.200610814.

[31]

M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astronom. Astrophys., 24 (1973), 229-238.

[32]

C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$, Comment. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052.

[33]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1 (1985), 145-201.

[34]

E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equation, 18 (1993), 125-151. doi: 10.1080/03605309308820923.

[35]

W.-M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801-807. doi: 10.1512/iumj.1982.31.31056.

[36]

E. S. Noussair, C. A. Swanson and J. Yang, Critical semilinear biharmonic equations in $R^N$, Proc. Roy. Soc. Edinburgh Sect. A, 121 (1992), 139-148.

[37]

S. Peng, Multiple boundary concentrating solutions to Dirichlet problem of Hénon equation, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 137-162. doi: 10.1007/s10255-005-0293-0.

[38]

A. Pistoia and E. Serra, Multi-peak solutions for the Hénon equation with slightly subcritical growth, Math. Z., 256 (2007), 75-97. doi: 10.1007/s00209-006-0060-9.

[39]

S. I. Pohožaev, On the eigenfunctions of quasilinear elliptic problems, (Russian), Mat. Sb. (N.S.), 82 (1970), 192-212.

[40]

O. Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52. doi: 10.1016/0022-1236(90)90002-3.

[41]

E. Serra, Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differential Equations, 23 (2005), 301-326. doi: 10.1007/s00526-004-0302-9.

[42]

D. Smets, J. Su and M. Willem, Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4 (2002), 467-480. doi: 10.1142/S0219199702000725.

[43]

D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differential Equations, 18 (2003), 57-75. doi: 10.1007/s00526-002-0180-y.

[44]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517. doi: 10.1007/BF01174186.

[45]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996), 241-264.

[46]

W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations, 42 (1981), 400-413. doi: 10.1016/0022-0396(81)90113-3.

[47]

R. C. A. M. Van der Vorst, Variational identities and applications to differential systems, Arch. Rational Mech. Anal., 116 (1992), 375-398. doi: 10.1007/BF00375674.

[48]

R. C. A. M. Van der Vorst, Best constant for the embedding of the space $H^2\cap H^1_0(\Omega)$ into $L^{(2N)/(N-4)(\Omega)$, Differential Integral Equations, 6 (1993), 259-276.

show all references

References:
[1]

C. O. Alves, J. M. do Ó and O. H. Miyagaki, Nontrivial solutions for a class of semilinear biharmonic problems involving critical exponents, Nonlinear Anal., 46 (2001), 121-133. doi: 10.1016/S0362-546X(99)00449-6.

[2]

C. O. Alves and J. M. do Ó, Positive solutions of a fourth-order semilinear problem involving critical growth, Adv. Nonlinear Stud., 2 (2002), 437-458.

[3]

M. Badiale and E. Serra, Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Stud., 4 (2004), 453-467.

[4]

A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294. doi: 10.1002/cpa.3160410302.

[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]

V. Barutello, S. Secchi and E. Serra, A note on the radial solutions for the supercritical Hénon equation, J. Math. Anal. Appl., 341 (2008), 720-728. arXiv:0707.1790. doi: 10.1016/j.jmaa.2007.10.052.

[7]

E. Berchio, F. Gazzola and T. Weth, Critical growth biharmonic elliptic problems under Steklov-type boundary conditions, Adv. Differential Equations, 12 (2007), 381-406.

[8]

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: 10.1515/CRELLE.2008.052.

[9]

F. Bernis and H.-Ch. Grunau, Critical exponents and multiple critical dimensions for polyharmonic operators, J. Differential Equations, 117 (1995), 469-486. doi: 10.1006/jdeq.1995.1062.

[10]

F. Bernis, J. García Azorero and I. Peral, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Differential Equations, 1 (1996), 219-240.

[11]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.

[12]

J. Byeon and Z.-Q. Wang, On the Hénon equation: asymptotic profile of ground states. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 803-828. doi: 10.1016/j.anihpc.2006.04.001.

[13]

J. Byeon and Z.-Q. Wang, On the Hénon equation: asymptotic profile of ground states. II, J. Differential Equations, 216 (2005), 78-108. doi: 10.1016/j.jde.2005.02.018.

[14]

M. Calanchi, S. Secchi and E. Terraneo, Multiple solutions for a Hénon-like equation on the annulus, J. Differential Equations, 245 (2008), 1507-1525. arXiv:0705.1492. doi: 10.1016/j.jde.2008.06.018.

[15]

D. Cao and S. Peng, The asymptotic behaviour of the ground state solutions for Hénon equation, J. Math. Anal. Appl., 278 (2003), 1-17. doi: 10.1016/S0022-247X(02)00292-5.

[16]

D. Cao, S. Peng and S. Yan, Asymptotic behaviour of ground state solutions for the Hénon equation, IMA J. Appl. Math., 74 (2009), 468-480. doi: 10.1093/imamat/hxn035.

[17]

G. Chen, W.-M. Ni and J. Zhou, Algorithms and visualization for solutions of nonlinear elliptic equations, Int. J. Bifurcat. Chaos, 10 (2000), 1565-1612. doi: 10.1142/S0218127400001006.

[18]

F. Ebobisse and M. O. Ahmedou, On a nonlinear fourth order elliptic equation involving the critical Sobolev exponent, Nonlinear Anal., 52 (2003), 1535-1552. arXiv:math/0112240. doi: 10.1016/S0362-546X(02)00273-0.

[19]

D. E. Edmunds, D. Fortunato and E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator, Arch. Rational Mech. Anal., 112 (1990), 269-289. doi: 10.1007/BF00381236.

[20]

P. Esposito, A. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in $R^2$, J. Anal. Math., 100 (2006), 249-280. doi: 10.1007/BF02916763.

[21]

J. García Azorero and I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895. doi: 10.2307/2001562.

[22]

M. Gazzini and E. Serra, The Neumann problem for the Hénon equation, trace inequalities and Steklov eigenvalues, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 281-302.

[23]

F. Gazzola, Critical growth problems for polyharmonic operators, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 251-263.

[24]

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: 10.1007/s00526-002-0182-9.

[25]

F. Gazzola and D. Pierotti, Positive solutions to critical growth biharmonic elliptic problems under Steklov boundary conditions, Nonlinear Anal., 71 (2009), 232-238. doi: 10.1016/j.na.2008.10.052.

[26]

Y. Ge, Positive solutions in semilinear critical problems for polyharmonic operators, J. Math. Pures Appl., 84 (2005), 199-245. doi: 10.1016/j.matpur.2004.10.002.

[27]

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.

[28]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.

[29]

H.-Ch. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 3 (1995), 243-252. doi: 10.1007/BF01205006.

[30]

Y. Guo and J. Wei, Supercritical biharmonic elliptic problems in domains with small holes, Math. Nachr., 282 (2009), 1724-1739. doi: 10.1002/mana.200610814.

[31]

M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astronom. Astrophys., 24 (1973), 229-238.

[32]

C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in $R^n$, Comment. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052.

[33]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1 (1985), 145-201.

[34]

E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equation, 18 (1993), 125-151. doi: 10.1080/03605309308820923.

[35]

W.-M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801-807. doi: 10.1512/iumj.1982.31.31056.

[36]

E. S. Noussair, C. A. Swanson and J. Yang, Critical semilinear biharmonic equations in $R^N$, Proc. Roy. Soc. Edinburgh Sect. A, 121 (1992), 139-148.

[37]

S. Peng, Multiple boundary concentrating solutions to Dirichlet problem of Hénon equation, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 137-162. doi: 10.1007/s10255-005-0293-0.

[38]

A. Pistoia and E. Serra, Multi-peak solutions for the Hénon equation with slightly subcritical growth, Math. Z., 256 (2007), 75-97. doi: 10.1007/s00209-006-0060-9.

[39]

S. I. Pohožaev, On the eigenfunctions of quasilinear elliptic problems, (Russian), Mat. Sb. (N.S.), 82 (1970), 192-212.

[40]

O. Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52. doi: 10.1016/0022-1236(90)90002-3.

[41]

E. Serra, Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differential Equations, 23 (2005), 301-326. doi: 10.1007/s00526-004-0302-9.

[42]

D. Smets, J. Su and M. Willem, Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4 (2002), 467-480. doi: 10.1142/S0219199702000725.

[43]

D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differential Equations, 18 (2003), 57-75. doi: 10.1007/s00526-002-0180-y.

[44]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517. doi: 10.1007/BF01174186.

[45]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996), 241-264.

[46]

W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations, 42 (1981), 400-413. doi: 10.1016/0022-0396(81)90113-3.

[47]

R. C. A. M. Van der Vorst, Variational identities and applications to differential systems, Arch. Rational Mech. Anal., 116 (1992), 375-398. doi: 10.1007/BF00375674.

[48]

R. C. A. M. Van der Vorst, Best constant for the embedding of the space $H^2\cap H^1_0(\Omega)$ into $L^{(2N)/(N-4)(\Omega)$, Differential Integral Equations, 6 (1993), 259-276.

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