Nonradial positive solutions for a biharmonic critical growth problem

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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

Abstract
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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. 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. Google Scholar
[3]	M. Badiale and E. Serra, Multiplicity results for the supercritical Hénon equation,, Adv. Nonlinear Stud., 4 (2004), 453. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[12]	J. Byeon and Z.-Q. Wang, On the Hénon equation: asymptotic profile of ground states. I,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 23 (2006), 803. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Poincar\'e Anal. Non Lin\'eaire, 25 (2008), 281. Google Scholar
[23]	F. Gazzola, Critical growth problems for polyharmonic operators,, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 251. 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. 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. 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. 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. 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. 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. 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. doi: 10.1002/mana.200610814. Google Scholar
[31]	M. Hénon, Numerical experiments on the stability of spherical stellar systems,, Astronom. Astrophys., 24 (1973), 229. 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. 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. Google Scholar
[34]	E. Mitidieri, A Rellich type identity and applications,, Comm. Partial Differential Equation, 18 (1993), 125. 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. 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. Google Scholar
[37]	S. Peng, Multiple boundary concentrating solutions to Dirichlet problem of Hénon equation,, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 137. 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. doi: 10.1007/s00209-006-0060-9. Google Scholar
[39]	S. I. Pohožaev, On the eigenfunctions of quasilinear elliptic problems,, (Russian), 82 (1970), 192. 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. 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. 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. 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. 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. 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. Google Scholar
[46]	W. C. Troy, Symmetry properties in systems of semilinear elliptic equations,, J. Differential Equations, 42 (1981), 400. 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. 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. 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. 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. Google Scholar
[3]	M. Badiale and E. Serra, Multiplicity results for the supercritical Hénon equation,, Adv. Nonlinear Stud., 4 (2004), 453. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar
[12]	J. Byeon and Z.-Q. Wang, On the Hénon equation: asymptotic profile of ground states. I,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 23 (2006), 803. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Poincar\'e Anal. Non Lin\'eaire, 25 (2008), 281. Google Scholar
[23]	F. Gazzola, Critical growth problems for polyharmonic operators,, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 251. 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. 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. 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. 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. 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. 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. 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. doi: 10.1002/mana.200610814. Google Scholar
[31]	M. Hénon, Numerical experiments on the stability of spherical stellar systems,, Astronom. Astrophys., 24 (1973), 229. 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. 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. Google Scholar
[34]	E. Mitidieri, A Rellich type identity and applications,, Comm. Partial Differential Equation, 18 (1993), 125. 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. 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. Google Scholar
[37]	S. Peng, Multiple boundary concentrating solutions to Dirichlet problem of Hénon equation,, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 137. 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. doi: 10.1007/s00209-006-0060-9. Google Scholar
[39]	S. I. Pohožaev, On the eigenfunctions of quasilinear elliptic problems,, (Russian), 82 (1970), 192. 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. 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. 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. 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. 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. 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. Google Scholar
[46]	W. C. Troy, Symmetry properties in systems of semilinear elliptic equations,, J. Differential Equations, 42 (1981), 400. 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. 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. Google Scholar

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