American Institute of Mathematical Sciences

Advanced
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 $R^n$

Paolo Caldiroli 1,

1. 

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

Received  April 2013 Revised  July 2013 Published  October 2013

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

P. Caldiroli and R. Musina, A class of second order dilation invariant inequalities,, in, (). Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

N. Ghoussoub and A. Moradifam, "Functional Inequalities: New Perspectives and New Applications,", Mathematical Surveys and Monographs, (2013). Google Scholar

[10]

C.-S. Lin, Interpolation inequalities with weights,, Comm. Part. Diff. Eq., 11 (1986), 1515. doi: 10.1080/03605308608820473. Google Scholar

[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. Google Scholar

[12]

E. Mitidieri, A Rellich type identity and applications,, Comm. Part. Diff. Eq., 18 (1993), 125. doi: 10.1080/03605309308820923. Google Scholar

[13]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\R^N$,, Diff. Int. Eq., 9 (1996), 465. Google Scholar

[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. Google Scholar

[15]

R. Musina, Weighted Sobolev spaces of radially symmetric functions,, Ann. Mat. Pura Appl., (). doi: 10.1007/s10231-013-0348-4. Google Scholar

[16]

E. S. Noussair, C. A. Swanson and J. Yang, Transcritical Biharmonic Equations in $R^N$,, Funkcialaj Ekvacioj, 35 (1992), 533. Google Scholar

[17]

F. Rellich, Halbbeschränkte Differentialoperatoren höherer Ordnung,, in, (1954), 243. Google Scholar

[18]

F. Rellich, "Perturbation Theory of Eigenvalue Problems,", Gordon and Breach, (1969). Google Scholar

[19]

M. Struwe, "Variational Methods,", fourth edition, (2008). doi: PMCid:PMC2582268. Google Scholar

[20]

C. A. Swanson, The best Sobolev constant,, Appl. Anal., 47 (1992), 227. doi: 10.1080/00036819208840142. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

P. Caldiroli and R. Musina, A class of second order dilation invariant inequalities,, in, (). Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

N. Ghoussoub and A. Moradifam, "Functional Inequalities: New Perspectives and New Applications,", Mathematical Surveys and Monographs, (2013). Google Scholar

[10]

C.-S. Lin, Interpolation inequalities with weights,, Comm. Part. Diff. Eq., 11 (1986), 1515. doi: 10.1080/03605308608820473. Google Scholar

[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. Google Scholar

[12]

E. Mitidieri, A Rellich type identity and applications,, Comm. Part. Diff. Eq., 18 (1993), 125. doi: 10.1080/03605309308820923. Google Scholar

[13]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\R^N$,, Diff. Int. Eq., 9 (1996), 465. Google Scholar

[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. Google Scholar

[15]

R. Musina, Weighted Sobolev spaces of radially symmetric functions,, Ann. Mat. Pura Appl., (). doi: 10.1007/s10231-013-0348-4. Google Scholar

[16]

E. S. Noussair, C. A. Swanson and J. Yang, Transcritical Biharmonic Equations in $R^N$,, Funkcialaj Ekvacioj, 35 (1992), 533. Google Scholar

[17]

F. Rellich, Halbbeschränkte Differentialoperatoren höherer Ordnung,, in, (1954), 243. Google Scholar

[18]

F. Rellich, "Perturbation Theory of Eigenvalue Problems,", Gordon and Breach, (1969). Google Scholar

[19]

M. Struwe, "Variational Methods,", fourth edition, (2008). doi: PMCid:PMC2582268. Google Scholar

[20]

C. A. Swanson, The best Sobolev constant,, Appl. Anal., 47 (1992), 227. doi: 10.1080/00036819208840142. Google Scholar

[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. Google Scholar

[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. Google Scholar

[1]

Dongsheng Kang, Liangshun Xu. Biharmonic systems involving multiple Rellich-type potentials and critical Rellich-Sobolev nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (2) : 333-346. doi: 10.3934/cpaa.2018019
[2]

Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171
[3]

José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138
[4]

Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110
[5]

Bernhard Kawohl. Symmetry results for functions yielding best constants in Sobolev-type inequalities. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 683-690. doi: 10.3934/dcds.2000.6.683
[6]

Guillaume Warnault. Regularity of the extremal solution for a biharmonic problem with general nonlinearity. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1709-1723. doi: 10.3934/cpaa.2009.8.1709
[7]

Fanni M. Sélley. Symmetry breaking in a globally coupled map of four sites. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3707-3734. doi: 10.3934/dcds.2018161
[8]

Lucio Cadeddu, Giovanni Porru. Symmetry breaking in problems involving semilinear equations. Conference Publications, 2011, 2011 (Special) : 219-228. doi: 10.3934/proc.2011.2011.219
[9]

Hwai-Chiuan Wang. Stability and symmetry breaking of solutions of semilinear elliptic equations. Conference Publications, 2005, 2005 (Special) : 886-894. doi: 10.3934/proc.2005.2005.886
[10]

Claudia Anedda, Giovanni Porru. Symmetry breaking and other features for Eigenvalue problems. Conference Publications, 2011, 2011 (Special) : 61-70. doi: 10.3934/proc.2011.2011.61
[11]

Sanjay Dharmavaram, Timothy J. Healey. Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1669-1684. doi: 10.3934/dcdss.2019112
[12]

Teemu Tyni, Valery Serov. Scattering problems for perturbations of the multidimensional biharmonic operator. Inverse Problems & Imaging, 2018, 12 (1) : 205-227. doi: 10.3934/ipi.2018008
[13]

Zongming Guo, Juncheng Wei. Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2561-2580. doi: 10.3934/dcds.2014.34.2561
[14]

YanYan Li, Tonia Ricciardi. A sharp Sobolev inequality on Riemannian manifolds. Communications on Pure & Applied Analysis, 2003, 2 (1) : 1-31. doi: 10.3934/cpaa.2003.2.1
[15]

Igor E. Verbitsky. The Hessian Sobolev inequality and its extensions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6165-6179. doi: 10.3934/dcds.2015.35.6165
[16]

Boumediene Abdellaoui, Fethi Mahmoudi. An improved Hardy inequality for a nonlocal operator. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1143-1157. doi: 10.3934/dcds.2016.36.1143
[17]

Linfeng Mei, Zongming Guo. Morse indices and symmetry breaking for the Gelfand equation in expanding annuli. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1509-1523. doi: 10.3934/dcdsb.2017072
[18]

Anna Goƚȩbiewska, Norimichi Hirano, Sƚawomir Rybicki. Global symmetry-breaking bifurcations of critical orbits of invariant functionals. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2005-2017. doi: 10.3934/dcdss.2019129
[19]

Ismail Kombe, Abdullah Yener. A general approach to weighted $L^{p}$ Rellich type inequalities related to Greiner operator. Communications on Pure & Applied Analysis, 2019, 18 (2) : 869-886. doi: 10.3934/cpaa.2019042
[20]

S. S. Dragomir, C. E. M. Pearce. Jensen's inequality for quasiconvex functions. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 279-291. doi: 10.3934/naco.2012.2.279

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences