- Previous Article
- DCDS Home
- This Issue
-
Next Article
Linearization of a nonautonomous unbounded system with nonuniform contraction: A spectral approach
Infinitely many solutions for quasilinear equations with critical exponent and Hardy potential in $ \mathbb{R}^N $
Department of Mathematical Science, Tsinghua University, Beijing, China |
$ \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} $ |
$ a_{ij}(u)\!\in \!C^1\!(\mathbb{R},\mathbb{R}) $ |
$ \nu\!>\!0 $ |
$ 0\!\leq\!\mu\!<\!\alpha\bar{\mu} $ |
$ \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} $ |
$ a(x) $ |
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. 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. |
[1] |
Fengshuang Gao, Yuxia Guo. Multiple solutions for a critical quasilinear equation with Hardy potential. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 1977-2003. doi: 10.3934/dcdss.2019128 |
[2] |
Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231 |
[3] |
Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033 |
[4] |
Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 |
[5] |
Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179 |
[6] |
Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singular term. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1363-1380. doi: 10.3934/cpaa.2013.12.1363 |
[7] |
Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095 |
[8] |
Lingwei Ma, Zhong Bo Fang. A new second critical exponent and life span for a quasilinear degenerate parabolic equation with weighted nonlocal sources. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1697-1706. doi: 10.3934/cpaa.2017081 |
[9] |
Yu Su, Zhaosheng Feng. Ground state solutions and decay estimation of Choquard equation with critical exponent and Dipole potential. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022112 |
[10] |
Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361 |
[11] |
Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Bounded solutions of the Boltzmann equation in the whole space. Kinetic and Related Models, 2011, 4 (1) : 17-40. doi: 10.3934/krm.2011.4.17 |
[12] |
Xinkuan Chai. The Boltzmann equation near Maxwellian in the whole space. Communications on Pure and Applied Analysis, 2011, 10 (2) : 435-458. doi: 10.3934/cpaa.2011.10.435 |
[13] |
Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure and Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527 |
[14] |
Shoichi Hasegawa. A critical exponent of Joseph-Lundgren type for an Hénon equation on the hyperbolic space. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1189-1198. doi: 10.3934/cpaa.2017058 |
[15] |
Jiguang Bao, Nguyen Lam, Guozhen Lu. Polyharmonic equations with critical exponential growth in the whole space $ \mathbb{R}^{n}$. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 577-600. doi: 10.3934/dcds.2016.36.577 |
[16] |
Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 |
[17] |
Zifei Shen, Fashun Gao, Minbo Yang. On critical Choquard equation with potential well. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3567-3593. doi: 10.3934/dcds.2018151 |
[18] |
Yingzhe Fan, Yuanjie Lei. The Boltzmann equation with frictional force for very soft potentials in the whole space. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4303-4329. doi: 10.3934/dcds.2019174 |
[19] |
Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351 |
[20] |
Suna Ma, Huiyuan Li, Zhimin Zhang. Novel spectral methods for Schrödinger equations with an inverse square potential on the whole space. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1589-1615. doi: 10.3934/dcdsb.2018221 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]