# American Institute of Mathematical Sciences

March  2018, 17(2): 333-346. doi: 10.3934/cpaa.2018019

## Biharmonic systems involving multiple Rellich-type potentials and critical Rellich-Sobolev nonlinearities

 School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China

* Corresponding author

Received  February 2017 Revised  June 2017 Published  March 2018

In this paper, the minimizers of a Rellich-Sobolev constant are firstly investigated. Secondly, a system of biharmonic equations is investigated, which involves multiple Rellich-type terms and strongly coupled critical Rellich-Sobolev terms. The existence of nontrivial solutions to the system is established by variational arguments.

Citation: Dongsheng Kang, Liangshun Xu. Biharmonic systems involving multiple Rellich-type potentials and critical Rellich-Sobolev nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (2) : 333-346. doi: 10.3934/cpaa.2018019
##### References:

show all references

##### References:
 [1] Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231 [2] M. Ben Ayed, Abdelbaki Selmi. Asymptotic behavior and existence results for a biharmonic equation involving the critical Sobolev exponent in a five-dimensional domain. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1705-1722. doi: 10.3934/cpaa.2010.9.1705 [3] Zongming Guo, Xiaohong Guan, Yonggang Zhao. Uniqueness and asymptotic behavior of solutions of a biharmonic equation with supercritical exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2613-2636. doi: 10.3934/dcds.2019109 [4] Zifei Shen, Fashun Gao, Minbo Yang. On critical Choquard equation with potential well. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3567-3593. doi: 10.3934/dcds.2018151 [5] 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 [6] Federica Sani. A biharmonic equation in $\mathbb{R}^4$ involving nonlinearities with critical exponential growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 405-428. doi: 10.3934/cpaa.2013.12.405 [7] 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 [8] Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025 [9] T. Ogawa. The degenerate drift-diffusion system with the Sobolev critical exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 875-886. doi: 10.3934/dcdss.2011.4.875 [10] Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033 [11] Maoding Zhen, Jinchun He, Haoyuan Xu, Meihua Yang. Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6523-6539. doi: 10.3934/dcds.2019283 [12] Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 [13] Sergey Zelik. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Communications on Pure & Applied Analysis, 2004, 3 (4) : 921-934. doi: 10.3934/cpaa.2004.3.921 [14] Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307 [15] Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179 [16] Yinbin Deng, Shuangjie Peng, Li Wang. Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795 [17] Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4353-4390. doi: 10.3934/dcds.2018190 [18] Yavdat Il'yasov. On critical exponent for an elliptic equation with non-Lipschitz nonlinearity. Conference Publications, 2011, 2011 (Special) : 698-706. doi: 10.3934/proc.2011.2011.698 [19] Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143 [20] Ting Guo, Xianhua Tang, Qi Zhang, Zu Gao. Nontrivial solutions for the choquard equation with indefinite linear part and upper critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1563-1579. doi: 10.3934/cpaa.2020078

2018 Impact Factor: 0.925