# 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 and Applied Analysis, 2018, 17 (2) : 333-346. doi: 10.3934/cpaa.2018019
##### References:
 [1] A. Ambrosetti and H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. [2] M. Bhakta and R. Musina, Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials, Nonlinear Anal., 75 (2012), 3836-3848. [3] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. [4] P. Caldiroli and R. Musina, Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic operator in cones, Milan J. Math., 79 (2011), 657-687. [5] L. D'Ambrosio and E. Jannelli, Nonlinear critical problems for the biharmonic operator with Hardy potential, Calc. Var. Partial Differential Equations, 54 (2015), 365-396. [6] V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations, 31 (2006), 469-495. [7] N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743. [8] E. Jannelli, Critical behavior for the polyharmonic operator with Hardy potential, Nonlinear Anal., 119 (2015), 443-456. [9] E. Jannelli and A. Loiudice, Critical polyharmonic problems with singular nonlinearities, Nonlinear Anal., 110 (2014), 77-96. [10] D. Kang, Concentration compactness principles for the systems of elliptic equations, Differ. Equ. Appl., 4 (2012), 435-444. [11] D. Kang, Elliptic systems involving critical nonlinearities and different Hardy-type terms, J. Math. Anal. Appl., 420 (2014), 930-941. [12] D. Kang and L. Xu, Asymptotic behavior and existence results for the biharmonic problems involving Rellich potentials, J. Math. Anal. Appl., 455 (2017), 1365-1382. [13] D. Kang and J. Yu, Systems of critical elliptic equations involving Hardy-type terms and large ranges of parameters, Appl. Math. Lett., 46 (2015), 77-82. [14] D. Kang and J. Yu, Minimizers to Rayleigh quotients of critical elliptic systems involving different Hardy-type terms, Appl. Math. Lett., 57 (2016), 97-103. [15] E. Lieb and M. Loss, Analysis, 2nd edition, American Mathematical Society, Providence RI, 2001. [16] P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case(Ⅰ), Revista Mathematica Iberoamericana, 1 (1985), 145-201. [17] P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case(Ⅱ), Revista Mathematica Iberoamericana, 1 (1985), 45-121. [18] F. Rellich, Perturbation Theory of Eigenvalue Problems, Courant Institute of Mathematical Sciences, New York University, New York, 1954. [19] S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996), 241-264. [20] M. Willem, Analyse Fonctionnelle Élémentaire, Cassini Éditeurs, Paris, 2003.

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##### References:
 [1] A. Ambrosetti and H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. [2] M. Bhakta and R. Musina, Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials, Nonlinear Anal., 75 (2012), 3836-3848. [3] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. [4] P. Caldiroli and R. Musina, Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic operator in cones, Milan J. Math., 79 (2011), 657-687. [5] L. D'Ambrosio and E. Jannelli, Nonlinear critical problems for the biharmonic operator with Hardy potential, Calc. Var. Partial Differential Equations, 54 (2015), 365-396. [6] V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations, 31 (2006), 469-495. [7] N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743. [8] E. Jannelli, Critical behavior for the polyharmonic operator with Hardy potential, Nonlinear Anal., 119 (2015), 443-456. [9] E. Jannelli and A. Loiudice, Critical polyharmonic problems with singular nonlinearities, Nonlinear Anal., 110 (2014), 77-96. [10] D. Kang, Concentration compactness principles for the systems of elliptic equations, Differ. Equ. Appl., 4 (2012), 435-444. [11] D. Kang, Elliptic systems involving critical nonlinearities and different Hardy-type terms, J. Math. Anal. Appl., 420 (2014), 930-941. [12] D. Kang and L. Xu, Asymptotic behavior and existence results for the biharmonic problems involving Rellich potentials, J. Math. Anal. Appl., 455 (2017), 1365-1382. [13] D. Kang and J. Yu, Systems of critical elliptic equations involving Hardy-type terms and large ranges of parameters, Appl. Math. Lett., 46 (2015), 77-82. [14] D. Kang and J. Yu, Minimizers to Rayleigh quotients of critical elliptic systems involving different Hardy-type terms, Appl. Math. Lett., 57 (2016), 97-103. [15] E. Lieb and M. Loss, Analysis, 2nd edition, American Mathematical Society, Providence RI, 2001. [16] P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case(Ⅰ), Revista Mathematica Iberoamericana, 1 (1985), 145-201. [17] P. L. Lions, The concentration compactness principle in the calculus of variations, the limit case(Ⅱ), Revista Mathematica Iberoamericana, 1 (1985), 45-121. [18] F. Rellich, Perturbation Theory of Eigenvalue Problems, Courant Institute of Mathematical Sciences, New York University, New York, 1954. [19] S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996), 241-264. [20] M. Willem, Analyse Fonctionnelle Élémentaire, Cassini Éditeurs, Paris, 2003.
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