# American Institute of Mathematical Sciences

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

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

 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:
 $$$\!\!\!\!\!\left\{ \begin{array}{ll} \!\!\!\!\!-\!\!\sum\limits_{i,j = 1}^N\!\!\!D_j(a_{ij}(u)D_iu)\!+\!\frac{1}{2}\!\!\!\sum\limits_{i,j = 1}^N\!\!\!\!a_{ij}'(u)D_iuD_ju\!+\!a(x)u\! = \!\nu |u|^{q-2}u\!+\!\frac{\mu u}{|x|^2}\!+\!|u|^{2^*-2}u,\hbox{in } \mathbb{R}^N,\\ u(x)\to0\,\,\hbox{as } |x|\,\,\to\infty, \end{array}\right. \;\;\;\;\;(1)$$$
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\} \!\! , $ \alpha, \gamma>0 $, $ \bar{\mu} = \frac{(N-2)^2}{4} $, $ 2^\ast = \frac{2N}{N-2} $is the Sobolev critical exponent. And $ a(x) $is a finite, positive potential function satisfying suitable decay assumptions. By using truncation method combining with the regularization approximation approach and compactness arguments, we prove the existence of infinitely many solutions for this equation. Citation: Fengshuang Gao, Yuxia Guo. Infinitely many solutions for quasilinear equations with critical exponent and Hardy potential in$ \mathbb{R}^N $. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5591-5616. doi: 10.3934/dcds.2020239 ##### References:  [1] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. [2] D. Cao, S. Peng and S. Yan, Infinitely many solutions for$p-$Laplacian equation involving critical Sobolev growth, J. Funct. Anal., 262 (2012), 2861-2902. doi: 10.1016/j.jfa.2012.01.006. [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. [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. [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. [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. [7] J.-M. Coron, Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 209-212. [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. [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. [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. [11] G. Divillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differential Equations, 7 (2002), 1257-1280. [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. [13] F. Gao and Y. Guo, Solutions for critical quasilinear elliptic equations in$\mathbb{R}^N$, J. Math. Phys., 60 (2019), 26pp. doi: 10.1063/1.5083169. [14] Y. Guo, J. Liu and Z.-Q. Wang, On a Brezis-Nirenburg type quasilinear problem, J. Fixed Point Theory Appl., 19 (2017), 719-753. doi: 10.1007/s11784-016-0371-3. [15] T. Kilpeläinen and J. Malý, The Winer test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793. [16] J. Liu, X. Liu and Z.-Q. Wang, Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth, J. Differential Equations, 261 (2016), 7194-7236. doi: 10.1016/j.jde.2016.09.018. [17] X.-Q. Liu, J.-Q. Liu and Z.-Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263. doi: 10.1090/S0002-9939-2012-11293-6. [18] X.-Q. Liu, J.-Q. Liu and Z.-Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations., 254 (2013), 102-124. doi: 10.1016/j.jde.2012.09.006. [19] J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations. II, J. Differential Equations, 187 (2003), 473-493. doi: 10.1016/S0022-0396(02)00064-5. [20] J. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations. I, Proc. Amer. Math. Soc., 131 (2003), 441-448. doi: 10.1090/S0002-9939-02-06783-7. [21] J.-Q. Liu and Z.-Q. Wang, Multiple solutions for quasilinear elliptic equations with a finite potential well, J. Differential Equations, 257 (2014), 2874-2899. doi: 10.1016/j.jde.2014.06.002. [22] X. Liu and J. Zhao,$p$-Laplacian equations in$\mathbb{R}^N $with finite potential via truncation method, Adv. Nonlinear Stud., 17 (2017), 595-610. doi: 10.1515/ans-2015-5059. [23] M. Poppenberg, K. Schmitt and Z.-Q. Wang, On the existence of solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344. doi: 10.1007/s005260100105. [24] M. Struwe, A global compactness result for elliptic boundary value problem involving limiting nonlinearities, Math Z., 187 (1984), 511-517. doi: 10.1007/BF01174186. [25] C. Tintarev, Concentration analysis and cocompactness, in Concentration Analysis and Applications to PDE, Trends Math., Birkhäuser/Springer, Basel, 2013,117–141. doi: 10.1007/978-3-0348-0373-1_7. [26] K. Tintarev and K.-H. Fineseler, Concentration Compactness. Functional Analytic Grounds and Applications, Imperial College Press, London, 2007. doi: 10.1142/p456. [27] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. [28] C.-L. Xiang, Asymptotic behaviors of solutions to quasilinear elliptic equations with critical Sobolev growth and Hardy potential, J. Differential. Equations, 259 (2015), 3929-3954. doi: 10.1016/j.jde.2015.05.007. [29] J. Zhao, X. Liu and J. Liu,$p$-Laplacian equations in$\mathbb{R}^N$with finite potential via truncation method, the critical case, J. Math. Anal. Appl., 455 (2017), 58-88. doi: 10.1016/j.jmaa.2017.03.085. show all references ##### References:  [1] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. [2] D. Cao, S. Peng and S. Yan, Infinitely many solutions for$p-$Laplacian equation involving critical Sobolev growth, J. Funct. Anal., 262 (2012), 2861-2902. doi: 10.1016/j.jfa.2012.01.006. [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. [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. [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. [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. [7] J.-M. Coron, Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 209-212. [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. [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. [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. [11] G. Divillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differential Equations, 7 (2002), 1257-1280. [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. [13] F. Gao and Y. Guo, Solutions for critical quasilinear elliptic equations in$\mathbb{R}^N$, J. Math. Phys., 60 (2019), 26pp. doi: 10.1063/1.5083169. [14] Y. Guo, J. Liu and Z.-Q. Wang, On a Brezis-Nirenburg type quasilinear problem, J. Fixed Point Theory Appl., 19 (2017), 719-753. doi: 10.1007/s11784-016-0371-3. [15] T. Kilpeläinen and J. Malý, The Winer test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793. [16] J. Liu, X. Liu and Z.-Q. Wang, Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth, J. Differential Equations, 261 (2016), 7194-7236. doi: 10.1016/j.jde.2016.09.018. [17] X.-Q. Liu, J.-Q. Liu and Z.-Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263. doi: 10.1090/S0002-9939-2012-11293-6. [18] X.-Q. Liu, J.-Q. Liu and Z.-Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations., 254 (2013), 102-124. doi: 10.1016/j.jde.2012.09.006. [19] J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations. II, J. Differential Equations, 187 (2003), 473-493. doi: 10.1016/S0022-0396(02)00064-5. [20] J. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations. I, Proc. Amer. Math. Soc., 131 (2003), 441-448. doi: 10.1090/S0002-9939-02-06783-7. [21] J.-Q. Liu and Z.-Q. Wang, Multiple solutions for quasilinear elliptic equations with a finite potential well, J. Differential Equations, 257 (2014), 2874-2899. doi: 10.1016/j.jde.2014.06.002. [22] X. Liu and J. Zhao,$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. [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. [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. [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. [26] K. Tintarev and K.-H. Fineseler, Concentration Compactness. Functional Analytic Grounds and Applications, Imperial College Press, London, 2007. doi: 10.1142/p456. [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. [28] C.-L. 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