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September  2020, 40(9): 5591-5616. doi: 10.3934/dcds.2020239

Infinitely many solutions for quasilinear equations with critical exponent and Hardy potential in $ \mathbb{R}^N $

Fengshuang Gao and  Yuxia Guo ,

Department of Mathematical Science, Tsinghua University, Beijing, China

* Corresponding author: yguo@mail.tsinghua.edu.cn

Received  February 2020 Published  June 2020

Fund Project: The second author is supported by NSFC grant 11771235.

In this paper, we consider the following critical quasilinear equation with Hardy potential:
$ \begin{equation} \!\!\!\!\!\left\{ \begin{array}{ll} \!\!\!\!\!-\!\!\sum\limits_{i,j = 1}^N\!\!\!D_j(a_{ij}(u)D_iu)\!+\!\frac{1}{2}\!\!\!\sum\limits_{i,j = 1}^N\!\!\!\!a_{ij}'(u)D_iuD_ju\!+\!a(x)u\! = \!\nu |u|^{q-2}u\!+\!\frac{\mu u}{|x|^2}\!+\!|u|^{2^*-2}u,\hbox{in } \mathbb{R}^N,\\ u(x)\to0\,\,\hbox{as } |x|\,\,\to\infty, \end{array}\right. \;\;\;\;\;(1)\end{equation} $
where
$ a_{ij}(u)\!\in \!C^1\!(\mathbb{R},\mathbb{R}) $
,
$ \nu\!>\!0 $
,
$ 0\!\leq\!\mu\!<\!\alpha\bar{\mu} $
, and
$ \max\!\left\{\!\frac{\alpha\bar{\mu}\gamma}{\alpha\bar{\mu}-\mu}\!+\!2,2^*\!\!-\!\frac{2}{N-2}\sqrt{\!\bar{\mu}\!-\!\frac{\mu}{\alpha}}\right\} \!\!<q<2^* $
,
$ \alpha, \gamma>0 $
,
$ \bar{\mu} = \frac{(N-2)^2}{4} $
,
$ 2^\ast = \frac{2N}{N-2} $
 is the Sobolev critical exponent. And
$ a(x) $
 is a finite, positive potential function satisfying suitable decay assumptions. By using truncation method combining with the regularization approximation approach and compactness arguments, we prove the existence of infinitely many solutions for this equation.
Keywords: Quasilinear equation, Hardy potential, critical exponent, whole space, regularization approach.
Mathematics Subject Classification: 35A01, 35J20, 35J62.
Citation: Fengshuang Gao, Yuxia Guo. Infinitely many solutions for quasilinear equations with critical exponent and Hardy potential in $ \mathbb{R}^N $. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5591-5616. doi: 10.3934/dcds.2020239
References:
[1]

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.  Google Scholar

[2]

D. Cao, S. 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

[3]

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

[4]

A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 463-470.  doi: 10.1016/S0294-1449(16)30395-X.  Google Scholar

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G. Cerami, G. Devillanova and S. Solimini, Infinitely many bound states for some nonlinear scalar field equations, Calc. Var. Partial Differential Equations, 23 (2005), 139-168.  doi: 10.1007/s00526-004-0293-6.  Google Scholar

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

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A. de Bouard, N. 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), 15pp. doi: 10.1063/1.4944455.  Google Scholar

[10]

Y. Deng, Y. Guo and S. Yan, Multiple solutions for critical quasilinear elliptic equations, Calc. Var. Partial Differential Equations, 58 (2019), 26pp. doi: 10.1007/s00526-018-1459-y.  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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F. Gao and Y. Guo, Multiple solutions for a critical quasilinear equation with Hardy potential, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 1977-2003.  doi: 10.3934/dcdss.2019128.  Google Scholar

[13]

F. Gao and Y. Guo, Solutions for critical quasilinear elliptic equations in $\mathbb{R}^N$, J. Math. Phys., 60 (2019), 26pp. doi: 10.1063/1.5083169.  Google Scholar

[14]

Y. Guo, J. Liu and Z.-Q. 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

[16]

J. Liu, X. Liu and Z.-Q. Wang, Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth, J. Differential Equations, 261 (2016), 7194-7236.  doi: 10.1016/j.jde.2016.09.018.  Google Scholar

[17]

X.-Q. Liu, J.-Q. 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

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X.-Q. Liu, J.-Q. 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

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J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations. II, J. Differential Equations, 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[20]

J. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations. I, Proc. Amer. Math. Soc., 131 (2003), 441-448.  doi: 10.1090/S0002-9939-02-06783-7.  Google Scholar

[21]

J.-Q. Liu and Z.-Q. Wang, Multiple solutions for quasilinear elliptic equations with a finite potential well, J. Differential Equations, 257 (2014), 2874-2899.  doi: 10.1016/j.jde.2014.06.002.  Google Scholar

[22]

X. Liu and J. Zhao, $p$-Laplacian equations in $\mathbb{R}^N $ with finite potential via truncation method, Adv. Nonlinear Stud., 17 (2017), 595-610.  doi: 10.1515/ans-2015-5059.  Google Scholar

[23]

M. Poppenberg, K. Schmitt and Z.-Q. Wang, On the existence of solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344.  doi: 10.1007/s005260100105.  Google Scholar

[24]

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

[25]

C. Tintarev, Concentration analysis and cocompactness, in Concentration Analysis and Applications to PDE, Trends Math., Birkhäuser/Springer, Basel, 2013,117–141. doi: 10.1007/978-3-0348-0373-1_7.  Google Scholar

[26] K. Tintarev and K.-H. Fineseler, Concentration Compactness. Functional Analytic Grounds and Applications, Imperial College Press, London, 2007.  doi: 10.1142/p456.  Google Scholar
[27]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[28]

C.-L. Xiang, Asymptotic behaviors of solutions to quasilinear elliptic equations with critical Sobolev growth and Hardy potential, J. Differential. Equations, 259 (2015), 3929-3954.  doi: 10.1016/j.jde.2015.05.007.  Google Scholar

[29]

J. Zhao, X. Liu and J. Liu, $p$-Laplacian equations in $\mathbb{R}^N$ with finite potential via truncation method, the critical case, J. Math. Anal. Appl., 455 (2017), 58-88.  doi: 10.1016/j.jmaa.2017.03.085.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

D. Cao, S. 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

[3]

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

[4]

A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 463-470.  doi: 10.1016/S0294-1449(16)30395-X.  Google Scholar

[5]

G. Cerami, G. Devillanova and S. Solimini, Infinitely many bound states for some nonlinear scalar field equations, Calc. Var. Partial Differential Equations, 23 (2005), 139-168.  doi: 10.1007/s00526-004-0293-6.  Google Scholar

[6]

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

[7]

J.-M. Coron, Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 209-212.   Google Scholar

[8]

A. de Bouard, N. 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

[9]

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

[10]

Y. Deng, Y. Guo and S. Yan, Multiple solutions for critical quasilinear elliptic equations, Calc. Var. Partial Differential Equations, 58 (2019), 26pp. doi: 10.1007/s00526-018-1459-y.  Google Scholar

[11]

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

[12]

F. Gao and Y. Guo, Multiple solutions for a critical quasilinear equation with Hardy potential, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 1977-2003.  doi: 10.3934/dcdss.2019128.  Google Scholar

[13]

F. Gao and Y. Guo, Solutions for critical quasilinear elliptic equations in $\mathbb{R}^N$, J. Math. Phys., 60 (2019), 26pp. doi: 10.1063/1.5083169.  Google Scholar

[14]

Y. Guo, J. Liu and Z.-Q. 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

[15]

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

[16]

J. Liu, X. Liu and Z.-Q. Wang, Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth, J. Differential Equations, 261 (2016), 7194-7236.  doi: 10.1016/j.jde.2016.09.018.  Google Scholar

[17]

X.-Q. Liu, J.-Q. 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

[18]

X.-Q. Liu, J.-Q. 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

[19]

J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations. II, J. Differential Equations, 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[20]

J. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations. I, Proc. Amer. Math. Soc., 131 (2003), 441-448.  doi: 10.1090/S0002-9939-02-06783-7.  Google Scholar

[21]

J.-Q. Liu and Z.-Q. Wang, Multiple solutions for quasilinear elliptic equations with a finite potential well, J. Differential Equations, 257 (2014), 2874-2899.  doi: 10.1016/j.jde.2014.06.002.  Google Scholar

[22]

X. Liu and J. Zhao, $p$-Laplacian equations in $\mathbb{R}^N $ with finite potential via truncation method, Adv. Nonlinear Stud., 17 (2017), 595-610.  doi: 10.1515/ans-2015-5059.  Google Scholar

[23]

M. Poppenberg, K. Schmitt and Z.-Q. Wang, On the existence of solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344.  doi: 10.1007/s005260100105.  Google Scholar

[24]

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

[25]

C. Tintarev, Concentration analysis and cocompactness, in Concentration Analysis and Applications to PDE, Trends Math., Birkhäuser/Springer, Basel, 2013,117–141. doi: 10.1007/978-3-0348-0373-1_7.  Google Scholar

[26] K. Tintarev and K.-H. Fineseler, Concentration Compactness. Functional Analytic Grounds and Applications, Imperial College Press, London, 2007.  doi: 10.1142/p456.  Google Scholar
[27]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[28]

C.-L. Xiang, Asymptotic behaviors of solutions to quasilinear elliptic equations with critical Sobolev growth and Hardy potential, J. Differential. Equations, 259 (2015), 3929-3954.  doi: 10.1016/j.jde.2015.05.007.  Google Scholar

[29]

J. Zhao, X. Liu and J. Liu, $p$-Laplacian equations in $\mathbb{R}^N$ with finite potential via truncation method, the critical case, J. Math. Anal. Appl., 455 (2017), 58-88.  doi: 10.1016/j.jmaa.2017.03.085.  Google Scholar

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