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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

Zhongliang Wang 1,

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.
Keywords: Biharmonic operator, nonradial solutions., critical Sobolev exponent.
Mathematics Subject Classification: Primary: 35J35, 35B33; Secondary: 35J6.
Citation: Zhongliang Wang. Nonradial positive solutions for a biharmonic critical growth problem. Communications on Pure & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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