Radial and non radial ground states for a class of dilation invariant fourth order semilinear elliptic equations on R^n

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March 2014, 13(2): 811-821. doi: 10.3934/cpaa.2014.13.811

Radial and non radial ground states for a class of dilation invariant fourth order semilinear elliptic equations on $R^n$

1.	Dipartimento di Matematica, Università di Torino, via Carlo Alberto, 10-10123 Torino, Italy

Received April 2013 Revised July 2013 Published October 2013

Abstract
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We prove existence of extremal functions for some Rellich-Sobolev type inequalities involving the $L^2$ norm of the Laplacian as a leading term and the $L^2$ norm of the gradient, weighted with a Hardy potential. Moreover we exhibit a breaking symmetry phenomenon when the nonlinearity has a growth close to the critical one and the singular potential increases in strength.

Keywords: breaking symmetry., extremal functions, Rellich-Sobolev inequality, Biharmonic operator.

Mathematics Subject Classification: 26D10, 47F0.

Citation: Paolo Caldiroli. Radial and non radial ground states for a class of dilation invariant fourth order semilinear elliptic equations on $R^n$. Communications on Pure & Applied Analysis, 2014, 13 (2) : 811-821. doi: 10.3934/cpaa.2014.13.811

References:

[1]	Adimurthi, M. Grossi and S. Santra, Optimal Hardy-Rellich inequalities, maximum principle and related eigenvalue problem,, J. Funct. Anal., 240 (2006), 36. doi: 10.1016/j.jfa.2006.07.011.
[2]	Adimurthi and S. Santra, Generalized Hardy-Rellich inequalities in critical dimensions and its applications,, Commun. Contemp. Math., 11 (2009), 367. doi: 10.1142/S0219199709003405.
[3]	C. O. Alves and J. M. do Ò, Positive solutions of a fourth-order semilinear problem involving critical growth,, Adv. Nonlinear Stud., 2 (2002), 437.
[4]	M. Bhakta and R. Musina, Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials,, Nonlinear Anal. T.M.A., 75 (2012), 3836. doi: 10.1016/j.na.2012.02.005.
[5]	P. Caldiroli and R. Musina, On Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic operator in cones,, Milan J. Math., 79 (2011), 657. doi: 10.1007/s00032-011-0167-2.
[6]	P. Caldiroli and R. Musina, A class of second order dilation invariant inequalities,, in, ().
[7]	F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions,, Comm. Pure Appl. Math., 54 (2001), 229. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.
[8]	N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities,, Math. Ann., 349 (2011), 1. doi: 10.1007/s00208-010-0510-x.
[9]	N. Ghoussoub and A. Moradifam, "Functional Inequalities: New Perspectives and New Applications,", Mathematical Surveys and Monographs, (2013).
[10]	C.-S. Lin, Interpolation inequalities with weights,, Comm. Part. Diff. Eq., 11 (1986), 1515. doi: 10.1080/03605308608820473.
[11]	P.-L. Lions, The concentration-compactness principle in the calculus of variations. The Limit Case, Part 1,, Rev. Mat. Iberoam., 1 (1985), 145. doi: 10.4171/RMI/6.
[12]	E. Mitidieri, A Rellich type identity and applications,, Comm. Part. Diff. Eq., 18 (1993), 125. doi: 10.1080/03605309308820923.
[13]	E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\R^N$,, Diff. Int. Eq., 9 (1996), 465.
[14]	A. Moradifam, Optimal weighted Hardy-Rellich inequalities on $H^2 \cap H^1_0$,, J. London. Math. Soc., 85 (2011), 22. doi: 10.1112/jlms/jdr045.
[15]	R. Musina, Weighted Sobolev spaces of radially symmetric functions,, Ann. Mat. Pura Appl., (). doi: 10.1007/s10231-013-0348-4.
[16]	E. S. Noussair, C. A. Swanson and J. Yang, Transcritical Biharmonic Equations in $R^N$,, Funkcialaj Ekvacioj, 35 (1992), 533.
[17]	F. Rellich, Halbbeschränkte Differentialoperatoren höherer Ordnung,, in, (1954), 243.
[18]	F. Rellich, "Perturbation Theory of Eigenvalue Problems,", Gordon and Breach, (1969).
[19]	M. Struwe, "Variational Methods,", fourth edition, (2008). doi: PMCid:PMC2582268.
[20]	C. A. Swanson, The best Sobolev constant,, Appl. Anal., 47 (1992), 227. doi: 10.1080/00036819208840142.
[21]	S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent,, Adv. Differential Eq., 1 (1996), 241.
[22]	A. Tertikas and N. B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improvements,, Adv. Math., 209 (2007), 407. doi: 10.1016/j.aim.2006.05.011.

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References:

[1]	Adimurthi, M. Grossi and S. Santra, Optimal Hardy-Rellich inequalities, maximum principle and related eigenvalue problem,, J. Funct. Anal., 240 (2006), 36. doi: 10.1016/j.jfa.2006.07.011.
[2]	Adimurthi and S. Santra, Generalized Hardy-Rellich inequalities in critical dimensions and its applications,, Commun. Contemp. Math., 11 (2009), 367. doi: 10.1142/S0219199709003405.
[3]	C. O. Alves and J. M. do Ò, Positive solutions of a fourth-order semilinear problem involving critical growth,, Adv. Nonlinear Stud., 2 (2002), 437.
[4]	M. Bhakta and R. Musina, Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials,, Nonlinear Anal. T.M.A., 75 (2012), 3836. doi: 10.1016/j.na.2012.02.005.
[5]	P. Caldiroli and R. Musina, On Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic operator in cones,, Milan J. Math., 79 (2011), 657. doi: 10.1007/s00032-011-0167-2.
[6]	P. Caldiroli and R. Musina, A class of second order dilation invariant inequalities,, in, ().
[7]	F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions,, Comm. Pure Appl. Math., 54 (2001), 229. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.
[8]	N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities,, Math. Ann., 349 (2011), 1. doi: 10.1007/s00208-010-0510-x.
[9]	N. Ghoussoub and A. Moradifam, "Functional Inequalities: New Perspectives and New Applications,", Mathematical Surveys and Monographs, (2013).
[10]	C.-S. Lin, Interpolation inequalities with weights,, Comm. Part. Diff. Eq., 11 (1986), 1515. doi: 10.1080/03605308608820473.
[11]	P.-L. Lions, The concentration-compactness principle in the calculus of variations. The Limit Case, Part 1,, Rev. Mat. Iberoam., 1 (1985), 145. doi: 10.4171/RMI/6.
[12]	E. Mitidieri, A Rellich type identity and applications,, Comm. Part. Diff. Eq., 18 (1993), 125. doi: 10.1080/03605309308820923.
[13]	E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\R^N$,, Diff. Int. Eq., 9 (1996), 465.
[14]	A. Moradifam, Optimal weighted Hardy-Rellich inequalities on $H^2 \cap H^1_0$,, J. London. Math. Soc., 85 (2011), 22. doi: 10.1112/jlms/jdr045.
[15]	R. Musina, Weighted Sobolev spaces of radially symmetric functions,, Ann. Mat. Pura Appl., (). doi: 10.1007/s10231-013-0348-4.
[16]	E. S. Noussair, C. A. Swanson and J. Yang, Transcritical Biharmonic Equations in $R^N$,, Funkcialaj Ekvacioj, 35 (1992), 533.
[17]	F. Rellich, Halbbeschränkte Differentialoperatoren höherer Ordnung,, in, (1954), 243.
[18]	F. Rellich, "Perturbation Theory of Eigenvalue Problems,", Gordon and Breach, (1969).
[19]	M. Struwe, "Variational Methods,", fourth edition, (2008). doi: PMCid:PMC2582268.
[20]	C. A. Swanson, The best Sobolev constant,, Appl. Anal., 47 (1992), 227. doi: 10.1080/00036819208840142.
[21]	S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent,, Adv. Differential Eq., 1 (1996), 241.
[22]	A. Tertikas and N. B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improvements,, Adv. Math., 209 (2007), 407. doi: 10.1016/j.aim.2006.05.011.

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