June  2020, 19(6): 3159-3187. doi: 10.3934/cpaa.2020137

Radial solutions for a class of Hénon type systems with partial interference with the spectrum

1. 

Departamento de Matemática, Universidade Federal Rural de Pernambuco, 50740-560, Recife - PE, Brazil

2. 

Departamento de Matemática - Centro de Ciências Exatas e de Tecnologia, Universidade Federal de São Carlos, 13565-905, São Carlos - SP, Brazil

3. 

Departamento de Matemática - Instituto de Ciências Exatas, Universidade Federal de Juiz de Fora, 36036-900, Juiz de Fora - MG, Brazil

4. 

Departamento de Ciências Exatas e Tecnológicas, Universidade Estadual de Santa Cruz, 45662-900, Ilhéus - BA, Brazil

* Corresponding author

Received  June 2019 Revised  November 2019 Published  March 2020

Fund Project: O. H. Miyagaki by CNPq/Brazil (Processo 307061/2018-3) and Fapemig/Brazil (CEX APQ-00063/15), F. R. Pereira by INCT-mat

We investigate the existence of radial solutions for a class of Hénon type systems with nonlinearities reaching the critical growth and interacting with the spectrum of the operator with the possibility of double resonance. The proof is made using variational methods, combining Brézis and Nirenberg arguments with Ni compactness result and Rabinowitz linking theorem.

Citation: Eudes. M. Barboza, Olimpio H. Miyagaki, Fábio R. Pereira, Cláudia R. Santana. Radial solutions for a class of Hénon type systems with partial interference with the spectrum. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3159-3187. doi: 10.3934/cpaa.2020137
References:
[1]

C. O. AlvesD. C. de Morais Filho and O. H. Miyagaki, Multiple solutions for an elliptic system on bounded and unbounded domains, Nonlinear Anal., 56 (2004), 555-568.  doi: 10.1016/j.na.2003.10.004.

[2]

C. O. AlvesD. C. de Morais Filho and M. A. S. Souto, On systems of equations involving subcritical or critical sobolev exponents, Nonlinear Anal. Theory Meth. Appl., 42 (2000), 771-787.  doi: 10.1016/S0362-546X(99)00121-2.

[3]

M. Badiale and E. Serra, Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Stud., 4 (2004), 453-467.  doi: 10.1515/ans-2004-0406.

[4]

S. BaeH. O. Choi and D. H. Pahk, Existence of nodal solutions of nonlinear elliptic equations, Proc. R. Soc. Edinb., 137A (2007), 1135-1155.  doi: 10.1017/S0308210505000727.

[5]

E. M. Barboza, O. H. Miyagaki, F. R. Pereira and C. R. Santana, Hénon equation with nonlinearities involving Sobolev critical growth in $H^1_{0, {\rm{rad}}}(B_1)$, to appear.

[6]

E. M. BarbozaJ. M. do Ó and B. Ribeiro, Hénon type equations with jumping nonlinearities involving critical growth, Adv. Differ. Equ., 24 (2019), 713-744. 

[7]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.

[8]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[9]

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

[10]

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

[11]

D. M. Cao and S. G. 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.

[12]

D. M. CaoS. G. 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.

[13]

A. CapozziD. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. Henri Poincare - Anal. Non Lineaire, 2 (1985), 463-470. 

[14]

P. C. CarriãoD. G. de Figueiredo and O. H. Miyagaki, Quasilinear elliptic equations of the Hénon-type: existence of non-radial solution, Commun. Contemp. Math., 11 (2009), 783-798.  doi: 10.1142/S0219199709003594.

[15]

P. ClementD. G. de Figueiredo and E. Mitidieri, Quasilinear elliptic equations with critical exponents, Topol. Meth. Nonlinear Anal., 7 (1966), 133-170.  doi: 10.12775/TMNA.1996.006.

[16]

D. C. de Morais Filho and M. A. S. Souto, Systems of p-laplacean equations involving homogeneous nonlinearities with critical sobolev exponent degrees, Commun. Partial Differ. Equ., 24 (1999), 1537-1553.  doi: 10.1080/03605309908821473.

[17]

L. F. O. FariaO. H. MiyagakiF. R. PereiraM. Squassina and C. Zhang, The Brézis-Nirenberg problem for nonlocal systems, Adv. Nonlinear Anal., 5 (2015), 85-103.  doi: 10.1515/anona-2015-0114.

[18]

F. Gladiali and M. Grossi, Linear perturbations for the critical Hénon problem, Differ. Integral Equ., 28 (2015), 733-752. 

[19]

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

[20]

N. Hirano, Existence of positive solutions for the Hénon equation involving critical Sobolev terms, J. Differ. Equ., 247 (2009), 1311-1333.  doi: 10.1016/j.jde.2009.06.008.

[21]

S. Li and S. Peng, Asymptotic behavior on the Hénon equation with supercritical exponent, Sci. China Ser. A., 52 (2009), 2185-2194.  doi: 10.1007/s11425-009-0094-7.

[22]

W. Long and J. Yang, Existence for critical Hénon-type equations, Differ. Integral Equ., 25 (2012), 567-578. 

[23]

O. H. Miyagaki and F. R. Pereira, Existence results for non-local elliptic systems with nonlinearities interacting with the spectrum, Adv. Differ. Equ., 23 (2018), 555-580. 

[24]

W. 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.

[25]

R. S. Palais, The Principle of Symmetric Criticality, Commun. Math. Phys., 69 (1979), 19-30. 

[26]

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

[27]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Vol. 65, CBMS, American Mathematical Society, 1986. doi: 10.1090/cbms/065.

[28]

S. Secchi, The Brézis-Nirenberg problem for the Hénon equation: ground state solutions, Adv. Nonlinear Stud., 12 (2012), 1-15.  doi: 10.1515/ans-2012-0209.

[29]

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

[30]

R. Servadei, The Yamabe equation in a non-local setting, Adv. Nonlinear Anal., 2 (2013), 235-270.  doi: 10.1515/anona-2013-0008.

[31]

R. Servadei, A critical fractional Laplace equation in the resonant case, Topol. Meth. Nonlinear Anal., 43 (2014), 251-267.  doi: 10.12775/TMNA.2014.015.

[32]

R. Servadei and E. Valdinoci, A Brézis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Math., 12 (2013), 2445-2464.  doi: 10.3934/cpaa.2013.12.2445.

[33]

R. Servadei and E. Valdinoci, The Brézis-Nirenberg result for the fractional laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.

[34]

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

show all references

References:
[1]

C. O. AlvesD. C. de Morais Filho and O. H. Miyagaki, Multiple solutions for an elliptic system on bounded and unbounded domains, Nonlinear Anal., 56 (2004), 555-568.  doi: 10.1016/j.na.2003.10.004.

[2]

C. O. AlvesD. C. de Morais Filho and M. A. S. Souto, On systems of equations involving subcritical or critical sobolev exponents, Nonlinear Anal. Theory Meth. Appl., 42 (2000), 771-787.  doi: 10.1016/S0362-546X(99)00121-2.

[3]

M. Badiale and E. Serra, Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Stud., 4 (2004), 453-467.  doi: 10.1515/ans-2004-0406.

[4]

S. BaeH. O. Choi and D. H. Pahk, Existence of nodal solutions of nonlinear elliptic equations, Proc. R. Soc. Edinb., 137A (2007), 1135-1155.  doi: 10.1017/S0308210505000727.

[5]

E. M. Barboza, O. H. Miyagaki, F. R. Pereira and C. R. Santana, Hénon equation with nonlinearities involving Sobolev critical growth in $H^1_{0, {\rm{rad}}}(B_1)$, to appear.

[6]

E. M. BarbozaJ. M. do Ó and B. Ribeiro, Hénon type equations with jumping nonlinearities involving critical growth, Adv. Differ. Equ., 24 (2019), 713-744. 

[7]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.

[8]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[9]

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

[10]

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

[11]

D. M. Cao and S. G. 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.

[12]

D. M. CaoS. G. 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.

[13]

A. CapozziD. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. Henri Poincare - Anal. Non Lineaire, 2 (1985), 463-470. 

[14]

P. C. CarriãoD. G. de Figueiredo and O. H. Miyagaki, Quasilinear elliptic equations of the Hénon-type: existence of non-radial solution, Commun. Contemp. Math., 11 (2009), 783-798.  doi: 10.1142/S0219199709003594.

[15]

P. ClementD. G. de Figueiredo and E. Mitidieri, Quasilinear elliptic equations with critical exponents, Topol. Meth. Nonlinear Anal., 7 (1966), 133-170.  doi: 10.12775/TMNA.1996.006.

[16]

D. C. de Morais Filho and M. A. S. Souto, Systems of p-laplacean equations involving homogeneous nonlinearities with critical sobolev exponent degrees, Commun. Partial Differ. Equ., 24 (1999), 1537-1553.  doi: 10.1080/03605309908821473.

[17]

L. F. O. FariaO. H. MiyagakiF. R. PereiraM. Squassina and C. Zhang, The Brézis-Nirenberg problem for nonlocal systems, Adv. Nonlinear Anal., 5 (2015), 85-103.  doi: 10.1515/anona-2015-0114.

[18]

F. Gladiali and M. Grossi, Linear perturbations for the critical Hénon problem, Differ. Integral Equ., 28 (2015), 733-752. 

[19]

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

[20]

N. Hirano, Existence of positive solutions for the Hénon equation involving critical Sobolev terms, J. Differ. Equ., 247 (2009), 1311-1333.  doi: 10.1016/j.jde.2009.06.008.

[21]

S. Li and S. Peng, Asymptotic behavior on the Hénon equation with supercritical exponent, Sci. China Ser. A., 52 (2009), 2185-2194.  doi: 10.1007/s11425-009-0094-7.

[22]

W. Long and J. Yang, Existence for critical Hénon-type equations, Differ. Integral Equ., 25 (2012), 567-578. 

[23]

O. H. Miyagaki and F. R. Pereira, Existence results for non-local elliptic systems with nonlinearities interacting with the spectrum, Adv. Differ. Equ., 23 (2018), 555-580. 

[24]

W. 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.

[25]

R. S. Palais, The Principle of Symmetric Criticality, Commun. Math. Phys., 69 (1979), 19-30. 

[26]

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

[27]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Vol. 65, CBMS, American Mathematical Society, 1986. doi: 10.1090/cbms/065.

[28]

S. Secchi, The Brézis-Nirenberg problem for the Hénon equation: ground state solutions, Adv. Nonlinear Stud., 12 (2012), 1-15.  doi: 10.1515/ans-2012-0209.

[29]

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

[30]

R. Servadei, The Yamabe equation in a non-local setting, Adv. Nonlinear Anal., 2 (2013), 235-270.  doi: 10.1515/anona-2013-0008.

[31]

R. Servadei, A critical fractional Laplace equation in the resonant case, Topol. Meth. Nonlinear Anal., 43 (2014), 251-267.  doi: 10.12775/TMNA.2014.015.

[32]

R. Servadei and E. Valdinoci, A Brézis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Math., 12 (2013), 2445-2464.  doi: 10.3934/cpaa.2013.12.2445.

[33]

R. Servadei and E. Valdinoci, The Brézis-Nirenberg result for the fractional laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.

[34]

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

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