November  2019, 12(7): 1977-2003. doi: 10.3934/dcdss.2019128

Multiple solutions for a critical quasilinear equation with Hardy potential

Department of Mathematical Science, Tsinghua University, Beijing, China

* Corresponding author

Received  December 2017 Revised  April 2018 Published  December 2018

Fund Project: The authors are supported by NSFC grant 11771235, 11331010, 11571040.

In this paper, we investigate the following quasilinear equation involving a Hardy potential:
$\begin{array}{l}\left\{ {\begin{array}{*{20}{c}}{ - \sum\limits_{i,j = 1}^N {{D_j}} ({a_{ij}}(u){D_i}u) + \frac{1}{2}\sum\limits_{i,j = 1}^N {{{a'}_{ij}}} (u){D_i}u{D_j}u - \frac{\mu }{{|x{|^2}}}u = au + |u{|^{{2^ * } - 2}}u}&{{\rm{in}}\;\Omega ,}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{u = 0}&{{\rm{on}}\;\partial \Omega ,}\end{array}} \right.\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {\rm{P}} \right)\end{array}$
where
$ 2^ * = \frac{2N}{N-2} $
is the Sobolev critical exponent for the embedding of
$ H_0^1(\Omega) $
into
$ L^p(\Omega) $
,
$ a>0 $
is a constant and
$ \Omega\subset \mathbb{R}^N $
is an open bounded domain which contains the origin. We will prove that under some suitable assumptions on
$ a_{ij} $
, when
$ N\geq 7 $
and
$ \mu\in[0,\mu^*) $
for some constant
$ \mu^* $
, problem (P) admits an unbounded sequence of solutions. To achieve this goal, we perform the subcritical approximation and the regularization perturbation.
Citation: Fengshuang Gao, Yuxia Guo. Multiple solutions for a critical quasilinear equation with Hardy potential. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1977-2003. doi: 10.3934/dcdss.2019128
References:
[1]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

A. Ambrosetti and Z. Q. Wang, Positive solutions to a class of quasilinear elliptic equations on $\mathbb R$, Discrete Contin. Dyn. Syst., 9 (2003), 55-68.  doi: 10.3934/dcds.2003.9.55.  Google Scholar

[3]

H. Berestycki and M. Esteban, Existence and bifurcation of solutions for an elliptic degenerate problem, J. Differential Equations, 134 (1997), 1-25.  doi: 10.1006/jdeq.1996.3165.  Google Scholar

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H. Brezis 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.  Google Scholar

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D. Cao and P. Han, Solutions to critical elliptic equations with multi-singular inverse sequare potentials, J. Differential Equations, 224 (2006), 332-372.  doi: 10.1016/j.jde.2005.07.010.  Google Scholar

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D. CaoS. Peng and S. Yan, Multiplicity of solutions for the plasma problem in two dimensions, Adv. Math., 225 (2010), 2741-2785.  doi: 10.1016/j.aim.2010.05.012.  Google Scholar

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D. CaoS. Peng and S. Yan, Infinitely many solutions for $p-$Laplacian equation involving critical Sobolev growth, J. Funct. Anal., 262 (2012), 2861-2902.  doi: 10.1016/j.jfa.2012.01.006.  Google Scholar

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D. Cao and S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var. Part. Diff. Equ., 38 (2010), 471-501.  doi: 10.1007/s00526-009-0295-5.  Google Scholar

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A. CapozziD. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincare and Non Lineaire, 2 (1985), 463-470.  doi: 10.1016/S0294-1449(16)30395-X.  Google Scholar

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G. CeramiS. Solimini and M. Struwe, Some existence results for superlinear elliptic problem involving critical exponents, J. Funct. Anal., 69 (1986), 289-306.  doi: 10.1016/0022-1236(86)90094-7.  Google Scholar

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M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear. Anal. Theor. Meth. App., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.  Google Scholar

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J. M. Coron, Topologie et cas limite des injections de Sobolev (Topology and limit case of Sobolev embeddings), C. R. Acad. Sci. Paris Ser. I Math., 199 (1984), 209-212.   Google Scholar

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A. de BouardN. Hayashi and J. C. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.  doi: 10.1007/s002200050191.  Google Scholar

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Y. Deng, Y. Guo and J. Liu, Existence of solutions for quasilinear elliptic equations with Hardy potential, J. Math. Phys., 57 (2016), 031503, 15 pp. doi: 10.1063/1.4944455.  Google Scholar

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G. Divillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differential Equations, 7 (2002), 1257-1280.   Google Scholar

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J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476.  doi: 10.1006/jdeq.1997.3375.  Google Scholar

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Y. GuoJ. Liu and Z. Wang, On a Brezis- Nirenburg type quasilinear problem, J. Fixed Point Theory Appl., 19 (2017), 719-753.  doi: 10.1007/s11784-016-0371-3.  Google Scholar

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T. Kilpeläinen and J. Malý, The Winer test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161.  doi: 10.1007/BF02392793.  Google Scholar

[19]

A. M. KosevichB. A. Ivanov and A. S. Kovalev, Magnetic solitons in superfluid films, J. Phys. Soc. Japan., 50 (1981), 3262-3267.   Google Scholar

[20]

L. Leblond and J. Marc, Electron capture by polar molecules, Phys. Rev., 153 (1967), 1-4.   Google Scholar

[21]

J. LiuX. Liu and Z.-Q. Wang, Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Comm. PDE, 39 (2014), 2216-2239.  doi: 10.1080/03605302.2014.942738.  Google Scholar

[22]

J. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schr$\ddot{o}$dinger equation Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448.  doi: 10.1090/S0002-9939-02-06783-7.  Google Scholar

[23]

J. LiuY. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schr$\ddot{o}$dinger equation Ⅱ, J. Differential Equations, 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[24]

J. LiuY. Wang and Z.-Q. Wang, Solutions for quasilinear Schr$\ddot{o}$dinger equation via the Nehari method, Comm. PDE, 29 (2004), 879-901.  doi: 10.1081/PDE-120037335.  Google Scholar

[25]

X. LiuJ. Liu and Z.-Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.  doi: 10.1090/S0002-9939-2012-11293-6.  Google Scholar

[26]

X. LiuJ. Liu and Z.-Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations., 254 (2013), 102-124.  doi: 10.1016/j.jde.2012.09.006.  Google Scholar

[27]

X. LiuJ. Liu and Z. Wang, Quasilinear equations via elliptic regularization method, Adv. Non. Stu., 13 (2013), 517-531.  doi: 10.1515/ans-2013-0215.  Google Scholar

[28]

M. Maris, Profile decomposition for sequences of Borel measures, https://arXiv.org/abs/1410.6125. Google Scholar

[29]

M. PoppenbergK. Schmitt and Z.-Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Part. Diff. Equ., 14 (2002), 329-344.  doi: 10.1007/s005260100105.  Google Scholar

[30]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, M. Math Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.  Google Scholar

[31]

C. Tintarev, Concentration analysis and cocompactness, Concentration Analysis and Applications to PDE, 117-141, Trends Math., Birkh$\ddot{a}$user/Springer, Basel, 2013. doi: 10.1007/978-3-0348-0373-1_7.  Google Scholar

[32] C. Tintarev and K. H. Fineseler, Concentration Compactness, Functional Analytic Grounds and Applications, Imperial College Press, London, 2007.  doi: 10.1142/p456.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

A. Ambrosetti and Z. Q. Wang, Positive solutions to a class of quasilinear elliptic equations on $\mathbb R$, Discrete Contin. Dyn. Syst., 9 (2003), 55-68.  doi: 10.3934/dcds.2003.9.55.  Google Scholar

[3]

H. Berestycki and M. Esteban, Existence and bifurcation of solutions for an elliptic degenerate problem, J. Differential Equations, 134 (1997), 1-25.  doi: 10.1006/jdeq.1996.3165.  Google Scholar

[4]

H. Brezis 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.  Google Scholar

[5]

D. Cao and P. Han, Solutions to critical elliptic equations with multi-singular inverse sequare potentials, J. Differential Equations, 224 (2006), 332-372.  doi: 10.1016/j.jde.2005.07.010.  Google Scholar

[6]

D. CaoS. Peng and S. Yan, Multiplicity of solutions for the plasma problem in two dimensions, Adv. Math., 225 (2010), 2741-2785.  doi: 10.1016/j.aim.2010.05.012.  Google Scholar

[7]

D. CaoS. Peng and S. Yan, Infinitely many solutions for $p-$Laplacian equation involving critical Sobolev growth, J. Funct. Anal., 262 (2012), 2861-2902.  doi: 10.1016/j.jfa.2012.01.006.  Google Scholar

[8]

D. Cao and S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var. Part. Diff. Equ., 38 (2010), 471-501.  doi: 10.1007/s00526-009-0295-5.  Google Scholar

[9]

A. CapozziD. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincare and Non Lineaire, 2 (1985), 463-470.  doi: 10.1016/S0294-1449(16)30395-X.  Google Scholar

[10]

G. CeramiS. Solimini and M. Struwe, Some existence results for superlinear elliptic problem involving critical exponents, J. Funct. Anal., 69 (1986), 289-306.  doi: 10.1016/0022-1236(86)90094-7.  Google Scholar

[11]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear. Anal. Theor. Meth. App., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.  Google Scholar

[12]

J. M. Coron, Topologie et cas limite des injections de Sobolev (Topology and limit case of Sobolev embeddings), C. R. Acad. Sci. Paris Ser. I Math., 199 (1984), 209-212.   Google Scholar

[13]

A. de BouardN. Hayashi and J. C. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.  doi: 10.1007/s002200050191.  Google Scholar

[14]

Y. Deng, Y. Guo and J. Liu, Existence of solutions for quasilinear elliptic equations with Hardy potential, J. Math. Phys., 57 (2016), 031503, 15 pp. doi: 10.1063/1.4944455.  Google Scholar

[15]

G. Divillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differential Equations, 7 (2002), 1257-1280.   Google Scholar

[16]

J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476.  doi: 10.1006/jdeq.1997.3375.  Google Scholar

[17]

Y. GuoJ. Liu and Z. Wang, On a Brezis- Nirenburg type quasilinear problem, J. Fixed Point Theory Appl., 19 (2017), 719-753.  doi: 10.1007/s11784-016-0371-3.  Google Scholar

[18]

T. Kilpeläinen and J. Malý, The Winer test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161.  doi: 10.1007/BF02392793.  Google Scholar

[19]

A. M. KosevichB. A. Ivanov and A. S. Kovalev, Magnetic solitons in superfluid films, J. Phys. Soc. Japan., 50 (1981), 3262-3267.   Google Scholar

[20]

L. Leblond and J. Marc, Electron capture by polar molecules, Phys. Rev., 153 (1967), 1-4.   Google Scholar

[21]

J. LiuX. Liu and Z.-Q. Wang, Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Comm. PDE, 39 (2014), 2216-2239.  doi: 10.1080/03605302.2014.942738.  Google Scholar

[22]

J. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schr$\ddot{o}$dinger equation Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448.  doi: 10.1090/S0002-9939-02-06783-7.  Google Scholar

[23]

J. LiuY. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schr$\ddot{o}$dinger equation Ⅱ, J. Differential Equations, 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[24]

J. LiuY. Wang and Z.-Q. Wang, Solutions for quasilinear Schr$\ddot{o}$dinger equation via the Nehari method, Comm. PDE, 29 (2004), 879-901.  doi: 10.1081/PDE-120037335.  Google Scholar

[25]

X. LiuJ. Liu and Z.-Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.  doi: 10.1090/S0002-9939-2012-11293-6.  Google Scholar

[26]

X. LiuJ. Liu and Z.-Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations., 254 (2013), 102-124.  doi: 10.1016/j.jde.2012.09.006.  Google Scholar

[27]

X. LiuJ. Liu and Z. Wang, Quasilinear equations via elliptic regularization method, Adv. Non. Stu., 13 (2013), 517-531.  doi: 10.1515/ans-2013-0215.  Google Scholar

[28]

M. Maris, Profile decomposition for sequences of Borel measures, https://arXiv.org/abs/1410.6125. Google Scholar

[29]

M. PoppenbergK. Schmitt and Z.-Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Part. Diff. Equ., 14 (2002), 329-344.  doi: 10.1007/s005260100105.  Google Scholar

[30]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, M. Math Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.  Google Scholar

[31]

C. Tintarev, Concentration analysis and cocompactness, Concentration Analysis and Applications to PDE, 117-141, Trends Math., Birkh$\ddot{a}$user/Springer, Basel, 2013. doi: 10.1007/978-3-0348-0373-1_7.  Google Scholar

[32] C. Tintarev and K. H. Fineseler, Concentration Compactness, Functional Analytic Grounds and Applications, Imperial College Press, London, 2007.  doi: 10.1142/p456.  Google Scholar
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