June  2021, 14(6): 1801-1818. doi: 10.3934/dcdss.2021065

Multi-bubble nodal solutions to slightly subcritical elliptic problems with Hardy terms in symmetric domains

1. 

Mathematisches Institut, Justus-Liebig-Universit$ \ddot{a} $t Giessen, Arndtstr. 2, 35392 Giessen, Germany

2. 

School of Mathematics and Statistics, Northwestern Polytechnical University, 710129 Xi'an, China

* Corresponding author: Qianqiao Guo

Received  December 2020 Revised  April 2021 Published  June 2021 Early access  May 2021

We consider the slightly subcritical elliptic problem with Hardy term
$ \left\{ \begin{aligned} - \Delta u-\mu\frac{u}{|x|^2} & = |u|^{2^{\ast}-2- \varepsilon}u &&\quad \rm{in } \Omega\subset{\mathbb{R}}^N, \\\ u & = 0&&\quad \rm{on } \partial \Omega, \end{aligned} \right. $
where
$ 0\in \Omega $
and
$ \Omega $
is invariant under the subgroup
$ SO(2)\times\{\pm E_{N-2}\}\subset O(N) $
; here
$ E_n $
denots the
$ n\times n $
identity matrix. If
$ \mu = \mu_0 \varepsilon^ \alpha $
with
$ \mu_0>0 $
fixed and
$ \alpha>\frac{N-4}{N-2} $
the existence of nodal solutions that blow up, as
$ \varepsilon\to0^+ $
, positively at the origin and negatively at a different point in a general bounded domain has been proved in [5]. Solutions with more than two blow-up points have not been found so far. In the present paper we obtain the existence of nodal solutions with a positive blow-up point at the origin and
$ k = 2 $
or
$ k = 3 $
negative blow-up points placed symmetrically in
$ \Omega\cap({\mathbb{R}}^2\times\{0\}) $
around the origin provided a certain function
$ f_k:{\mathbb{R}}^+\times{\mathbb{R}}^+\times I\to{\mathbb{R}} $
has stable critical points; here
$ I = \{t>0:(t,0,\dots,0)\in \Omega\} $
. If
$ \Omega = B(0,1)\subset{\mathbb{R}}^N $
is the unit ball centered at the origin we obtain two solutions for
$ k = 2 $
and
$ N\ge7 $
, or
$ k = 3 $
and
$ N $
large. The result is optimal in the sense that for
$ \Omega = B(0,1) $
there cannot exist solutions with a positive blow-up point at the origin and four negative blow-up points placed on the vertices of a square centered at the origin. Surprisingly there do exist solutions on
$ \Omega = B(0,1) $
with a positive blow-up point at the origin and four blow-up points on the vertices of a square with alternating positive and negative signs. The results of our paper show that the structure of the set of blow-up solutions of the above problem offers fascinating features and is not well understood.
Citation: Thomas Bartsch, Qianqiao Guo. Multi-bubble nodal solutions to slightly subcritical elliptic problems with Hardy terms in symmetric domains. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1801-1818. doi: 10.3934/dcdss.2021065
References:
[1]

T. Aubin, Problémes isopérimétriques et espaces de Sobolev', J. Differential Geometry, 11 (1976), 573-598. 

[2]

A. BahriY. Y. Li and O. Rey, On a variational problem with lack of compactness: The topological effect of the critical points at infinity, Calc. Var. Part. Diff. Equ., 3 (1995), 67-93.  doi: 10.1007/BF01190892.

[3]

T. BartschT. D'Aprile and A. Pistoia, Multi-bubble nodal solutions for slightly subcritical elliptic problems in domains with symmetries, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 1027-1047.  doi: 10.1016/j.anihpc.2013.01.001.

[4]

T. BartschT. D'Aprile and A. Pistoia, On the profile of sign-changing solutions of an almost critical problem in the ball, Bull. London Math. Soc., 45 (2013), 1246-1258.  doi: 10.1112/blms/bdt061.

[5]

T. Bartsch and Q. Guo, Nodal blow-up solutions to slightly subcritical elliptic problems with Hardy-critical term, Adv. Nonlinear Stud., 17 (2017), 55-85.  doi: 10.1515/ans-2016-6008.

[6]

T. Bartsch and Q. Guo, Nodal bubble tower solutions to slightly subcritical elliptic problems with Hardy terms, SN Partial Differ. Equ. Appl., 1 (2020), 26.

[7]

T. BartschA. Micheletti and A. Pistoia, On the existence and the profile of nodal solutions of elliptic equations involving critical growth, Calc. Var. Part. Diff. Equ., 26 (2006), 265-282.  doi: 10.1007/s00526-006-0004-6.

[8]

G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Functional Analysis, 100 (1991), 18-24.  doi: 10.1016/0022-1236(91)90099-Q.

[9]

H. Brezis and L. A. Peletier, Asymptotics for elliptic equations involving critical growth, Partial differential equations and the calculus of variations, Progr. Nonlinear Diff. Equ. Appl. Birkhäuser, Boston, MA, 1 (1989), 149-192. 

[10]

D. Cao and P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Diff. Equ., 205 (2004), 521-537.  doi: 10.1016/j.jde.2004.03.005.

[11]

D. Cao and S. Peng, A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms, J. Diff. Equ., 193 (2003), 424-434.  doi: 10.1016/S0022-0396(03)00118-9.

[12]

D. Cao and S. J. Peng, Asymptotic behavior for elliptic problems with singular coefficient and nearly critical Sobolev growth, Ann. Mat. Pura. Appl., 185 (2006), 189-205.  doi: 10.1007/s10231-005-0150-z.

[13]

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), 229-258. 

[14]

M. del PinoJ. Dolbeault and M. Musso, ``Bubble-tower" radial solutions in the slightly supercritical Brezis-Nirenberg problem, J. Diff. Equ., 193 (2003), 28-306.  doi: 10.1016/S0022-0396(03)00151-7.

[15]

I. Ekeland and N. Ghoussoub, Selected new aspects of the calculus of variations in the large, Bull. Amer. Math. Soc. (N.S.), 39 (2002), 207-265.  doi: 10.1090/S0273-0979-02-00929-1.

[16]

V. Felli and A. Pistoia, Existence of blowing-up solutions for a nonlinear elliptic equation with Hardy potential and critical growth, Comm. Part. Diff. Equ., 31 (2006), 21-56.  doi: 10.1080/03605300500358145.

[17]

V. Felli and S. Terracini, Fountain-like solutions for nonlinear elliptic equations with critical growth and Hardy potential, Comm. Contemp. Math., 7 (2005), 867-904.  doi: 10.1142/S0219199705001994.

[18]

A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Diff. Equ., 177 (2001), 494-522.  doi: 10.1006/jdeq.2000.3999.

[19]

M. Flucher and J. Wei, Semilinear Dirichlet problem with nearly critical exponent, asymptotic location of hot spots, Manuscripta Math., 94 (1997), 337-346.  doi: 10.1007/BF02677858.

[20]

N. Ghoussoub and F. Robert, The Hardy-Schrödinger operator with interior singularity: The remaining cases,, Calc. Var. Part. Diff. Equ., 56 (20017), paper 149, 54 pp. doi: 10.1007/s00526-017-1238-1.

[21]

N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743.  doi: 10.1090/S0002-9947-00-02560-5.

[22]

M. Grossi and F. Takahashi, Nonexistence of multi-bubble solutions to some elliptic equations on convex domains, J. Functional Analysis, 259 (2010), 904-917.  doi: 10.1016/j.jfa.2010.03.008.

[23]

Q. Q. Guo and P. C. Niu, Nodal and positive solutions for singular semilinear elliptic equations with critical exponents in symmetric domains, J. Diff. Equ., 245 (2008), 3974-3985.  doi: 10.1016/j.jde.2008.08.002.

[24]

Z. C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 159-174.  doi: 10.1016/S0294-1449(16)30270-0.

[25]

E. Jannelli, The role played by space dimension in elliptic critical problems, J. Diff. Equ., 156 (1999), 407-426.  doi: 10.1006/jdeq.1998.3589.

[26]

M. Musso and A. Pistoia, Tower of bubbles for almost critical problems in general domains, J. Math. Pures Appl., 93 (2010), 1-40.  doi: 10.1016/j.matpur.2009.08.001.

[27]

M. Musso and J. Wei, Nonradial solutions to critical elliptic equations of Caffarelli-Kohn-Nirenberg type, Int. Math. Res. Not., 2012 (2012), 4120-4162.  doi: 10.1093/imrn/rnr179.

[28]

A. Pistoia and T. Weth, Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 325-340.  doi: 10.1016/j.anihpc.2006.03.002.

[29]

O. Rey, Proof of two conjectures of H. Brézis and L.A. Peletier, Manuscripta Math., 65 (1989), 19-37.  doi: 10.1007/BF01168364.

[30]

O. Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Functional Analysis, 89 (1990), 1-52.  doi: 10.1016/0022-1236(90)90002-3.

[31]

O. Rey, Blow-up points of solutions to elliptic equations with limiting nonlinearity, Differential Integral Equations, 4 (1991), 1155-1167. 

[32]

D. Ruiz and M. Willem, Elliptic problems with critical exponents and Hardy potentials, J. Diff. Equ., 190 (2003), 524-538.  doi: 10.1016/S0022-0396(02)00178-X.

[33]

D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. Amer. Math. Soc., 357 (2005), 2909-2938.  doi: 10.1090/S0002-9947-04-03769-9.

[34]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.

[35]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Diff. Equations, 1 (1996), 241-264. 

show all references

References:
[1]

T. Aubin, Problémes isopérimétriques et espaces de Sobolev', J. Differential Geometry, 11 (1976), 573-598. 

[2]

A. BahriY. Y. Li and O. Rey, On a variational problem with lack of compactness: The topological effect of the critical points at infinity, Calc. Var. Part. Diff. Equ., 3 (1995), 67-93.  doi: 10.1007/BF01190892.

[3]

T. BartschT. D'Aprile and A. Pistoia, Multi-bubble nodal solutions for slightly subcritical elliptic problems in domains with symmetries, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 1027-1047.  doi: 10.1016/j.anihpc.2013.01.001.

[4]

T. BartschT. D'Aprile and A. Pistoia, On the profile of sign-changing solutions of an almost critical problem in the ball, Bull. London Math. Soc., 45 (2013), 1246-1258.  doi: 10.1112/blms/bdt061.

[5]

T. Bartsch and Q. Guo, Nodal blow-up solutions to slightly subcritical elliptic problems with Hardy-critical term, Adv. Nonlinear Stud., 17 (2017), 55-85.  doi: 10.1515/ans-2016-6008.

[6]

T. Bartsch and Q. Guo, Nodal bubble tower solutions to slightly subcritical elliptic problems with Hardy terms, SN Partial Differ. Equ. Appl., 1 (2020), 26.

[7]

T. BartschA. Micheletti and A. Pistoia, On the existence and the profile of nodal solutions of elliptic equations involving critical growth, Calc. Var. Part. Diff. Equ., 26 (2006), 265-282.  doi: 10.1007/s00526-006-0004-6.

[8]

G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Functional Analysis, 100 (1991), 18-24.  doi: 10.1016/0022-1236(91)90099-Q.

[9]

H. Brezis and L. A. Peletier, Asymptotics for elliptic equations involving critical growth, Partial differential equations and the calculus of variations, Progr. Nonlinear Diff. Equ. Appl. Birkhäuser, Boston, MA, 1 (1989), 149-192. 

[10]

D. Cao and P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Diff. Equ., 205 (2004), 521-537.  doi: 10.1016/j.jde.2004.03.005.

[11]

D. Cao and S. Peng, A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms, J. Diff. Equ., 193 (2003), 424-434.  doi: 10.1016/S0022-0396(03)00118-9.

[12]

D. Cao and S. J. Peng, Asymptotic behavior for elliptic problems with singular coefficient and nearly critical Sobolev growth, Ann. Mat. Pura. Appl., 185 (2006), 189-205.  doi: 10.1007/s10231-005-0150-z.

[13]

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), 229-258. 

[14]

M. del PinoJ. Dolbeault and M. Musso, ``Bubble-tower" radial solutions in the slightly supercritical Brezis-Nirenberg problem, J. Diff. Equ., 193 (2003), 28-306.  doi: 10.1016/S0022-0396(03)00151-7.

[15]

I. Ekeland and N. Ghoussoub, Selected new aspects of the calculus of variations in the large, Bull. Amer. Math. Soc. (N.S.), 39 (2002), 207-265.  doi: 10.1090/S0273-0979-02-00929-1.

[16]

V. Felli and A. Pistoia, Existence of blowing-up solutions for a nonlinear elliptic equation with Hardy potential and critical growth, Comm. Part. Diff. Equ., 31 (2006), 21-56.  doi: 10.1080/03605300500358145.

[17]

V. Felli and S. Terracini, Fountain-like solutions for nonlinear elliptic equations with critical growth and Hardy potential, Comm. Contemp. Math., 7 (2005), 867-904.  doi: 10.1142/S0219199705001994.

[18]

A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Diff. Equ., 177 (2001), 494-522.  doi: 10.1006/jdeq.2000.3999.

[19]

M. Flucher and J. Wei, Semilinear Dirichlet problem with nearly critical exponent, asymptotic location of hot spots, Manuscripta Math., 94 (1997), 337-346.  doi: 10.1007/BF02677858.

[20]

N. Ghoussoub and F. Robert, The Hardy-Schrödinger operator with interior singularity: The remaining cases,, Calc. Var. Part. Diff. Equ., 56 (20017), paper 149, 54 pp. doi: 10.1007/s00526-017-1238-1.

[21]

N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743.  doi: 10.1090/S0002-9947-00-02560-5.

[22]

M. Grossi and F. Takahashi, Nonexistence of multi-bubble solutions to some elliptic equations on convex domains, J. Functional Analysis, 259 (2010), 904-917.  doi: 10.1016/j.jfa.2010.03.008.

[23]

Q. Q. Guo and P. C. Niu, Nodal and positive solutions for singular semilinear elliptic equations with critical exponents in symmetric domains, J. Diff. Equ., 245 (2008), 3974-3985.  doi: 10.1016/j.jde.2008.08.002.

[24]

Z. C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 159-174.  doi: 10.1016/S0294-1449(16)30270-0.

[25]

E. Jannelli, The role played by space dimension in elliptic critical problems, J. Diff. Equ., 156 (1999), 407-426.  doi: 10.1006/jdeq.1998.3589.

[26]

M. Musso and A. Pistoia, Tower of bubbles for almost critical problems in general domains, J. Math. Pures Appl., 93 (2010), 1-40.  doi: 10.1016/j.matpur.2009.08.001.

[27]

M. Musso and J. Wei, Nonradial solutions to critical elliptic equations of Caffarelli-Kohn-Nirenberg type, Int. Math. Res. Not., 2012 (2012), 4120-4162.  doi: 10.1093/imrn/rnr179.

[28]

A. Pistoia and T. Weth, Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 325-340.  doi: 10.1016/j.anihpc.2006.03.002.

[29]

O. Rey, Proof of two conjectures of H. Brézis and L.A. Peletier, Manuscripta Math., 65 (1989), 19-37.  doi: 10.1007/BF01168364.

[30]

O. Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Functional Analysis, 89 (1990), 1-52.  doi: 10.1016/0022-1236(90)90002-3.

[31]

O. Rey, Blow-up points of solutions to elliptic equations with limiting nonlinearity, Differential Integral Equations, 4 (1991), 1155-1167. 

[32]

D. Ruiz and M. Willem, Elliptic problems with critical exponents and Hardy potentials, J. Diff. Equ., 190 (2003), 524-538.  doi: 10.1016/S0022-0396(02)00178-X.

[33]

D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. Amer. Math. Soc., 357 (2005), 2909-2938.  doi: 10.1090/S0002-9947-04-03769-9.

[34]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.

[35]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Diff. Equations, 1 (1996), 241-264. 

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