MULTIPLE SYMMETRIC RESULTS FOR A CLASS OF BIHARMONIC ELLIPTIC SYSTEMS WITH CRITICAL HOMOGENEOUS NONLINEARITY IN RN

doi:10.1016/S0252-9602(17)30099-1

Abstract

Abstract:

This paper is dedicated to studying the existence and multiplicity of symmetric solutions for a class of biharmonic elliptic systems with critical homogeneous nonlinearity in R^N. By virtue of variational methods and the symmetric criticality principle of Palais, we establish several existence and multiplicity results of G-symmetric solutions under certain appropriate hypotheses on the parameters and the weighted functions.

Key words: G-symmetric solution, symmetric criticality principle, critical Sobolev exponent, biharmonic elliptic systems

Zhiying DENG, Yisheng HUANG. MULTIPLE SYMMETRIC RESULTS FOR A CLASS OF BIHARMONIC ELLIPTIC SYSTEMS WITH CRITICAL HOMOGENEOUS NONLINEARITY IN R^N[J].Acta mathematica scientia,Series B, 2017, 37(6): 1665-1684.

Trendmd

References

[1] Brezis H, Nirenberg L. Postive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm Pure Appl Math, 1983, 36(4):437-477[2] Azorero J G, Alonso I P. Hardy inequalities and some critical elliptic and parabolic problems. J Differ Equ, 1998, 144(2):441-476[3] Ghoussoub N, Yuan C. Multiple solutions for quasilinear PDEs involving critical Sobolev and Hardy exponents. Trans Amer Math Soc, 2000, 352(12):5703-5743[4] Chen Z J, Zou W M. On an elliptic problem with critical exponent and Hardy potential. J Differ Equ, 2012, 252(2):969-987[5] Kang D S. Solutions for the quasilinear elliptic problems involving critical Hardy-Sobolev exponents. Acta Math Sci, 2010, 30B(5):1529-1540[6] Yang F. Singular positive radial solutions for a general semilinear elliptic equation. Acta Math Sci, 2012, 32B(6):2377-2387[7] Sun X M. p-Laplace equations with multiple critical exponents and singular cylindrical potential. Acta Math Sci, 2013, 33B(4):1099-1112[8] Hsu T S. Multiple positive solutions for quasilinear elliptic problems involving concave-convex nonlinearities and multiple Hardy-type terms. Acta Math Sci, 2013, 33B(5):1314-1328[9] Lan Y Y, Tand C L. Perturbation methods in semilinear elliptic problems involving critical Hardy-Sobolev exponent. Acta Math Sci, 2014, 34B(3):703-712[10] Deng Y B, Jin L Y. On symmetric solutions of a singular elliptic equation with critical Sobolev-Hardy exponent. J Math Anal Appl, 2007, 329(1):603-616[11] Waliullah S. Minimizers and symmetric minimizers for problems with critical Sobolev exponent. Topol Methods Nonlinear Anal, 2009, 34(2):291-326[12] Lions P L. The concentration-compactness principle in the calculus of variations, The limit case. Rev Mat Iberoamericana, 1985, 1(1) (part I):145-201; 1(2) (part Ⅱ):45-121[13] Deng Z Y, Huang Y S. Existence and multiplicity of symmetric solutions for semilinear elliptic equations with singular potentials and critical Hardy-Sobolev exponents. J Math Anal Appl, 2012, 393(1):273-284[14] Deng Z Y, Huang Y S. Existence and multiplicity of symmetric solutions for the weighted critical quasilinear problems. Appl Math Comput, 2013, 219(9):4836-4846[15] Deng Z Y, Huang Y S. On positive G-symmetric solutions of a weighted quasilinear elliptic equation with critical Hardy-Sobolev exponent. Acta Math Sci, 2014, 34B(5):1619-1633[16] Bianchi G, Chabrowski J, Szulkin A. On symmetric solutions of an elliptic equations with a nonlinearity involving critical Sobolev exponent. Nonlinear Anal, 1995, 25(1):41-59[17] Bartsch T, Willem M. Infinitely many Non-Radial Solutions of an Euclidean Scalar Field Equation. Heidelberg:Mathematisches Institut/Universitat Heidelberg, 1992[18] Chabrowski J. On the existence of G-symmetric entire solutions for semilinear elliptic equations. Rend Circ Mat Palermo, 1992, 41(3):413-440[19] De Morais Filho D C, Souto M A S. Systems of p-Laplacean equations involving homogeneous nonlinearities with critical Sobolev exponent degrees. Comm Partial Differ Equ, 1999, 24(7/8):1537-1553[20] Alves C O, De Morais Filho D C, Miyagaki O H. Multiple solutions for an elliptic system on bounded and unbounded domains. Nonlinear Anal, 2004, 56(4):555-568[21] Furtado M F, Da Silva J P P. Multiplicity of solutions for homogeneous elliptic systems with critical growth. J Math Anal Appl, 2012, 385(2):770-785[22] Hsu T S, Li H L. Multiplicity of positive solutions for singular elliptic systems with critical Sobolev-Hardy and concave exponents. Acta Math Sci, 2011, 31B(3):791-804[23] Li Y X, Gao W J. Existence of multiple positive solutions for singular quasilinear elliptic system with critical Sobolev-Hardy exponents and concave-convex terms. Acta Math Sci, 2013, 33B(1):107-212[24] Nyamoradi N, Hsu T S. Existence of multiple positive solutions for semilinear elliptic systems involving m critical Hardy-Sobolev exponents and m sign-changing weight function. Acta Math Sci, 2014, 34B(2):483-500[25] Kang D S. Positive minimizers of the best constants and solutions to coupled critical quasilinear systems. J Differ Equ, 2016, 260(1):133-148[26] Nyamoradi N, Hsu T S. Multiple solutions for weighted nonlinear elliptic system involving critical exponents. Sci China Math, 2015, 58(1):161-178[27] Chen Z J, Zou W M. Existence and symmetry of positive ground states for a doubly critical Schrödinger system. Trans Amer Math Soc, 2015, 367(5):3599-3646[28] Deng Z Y, Huang Y S. Existence of symmetric solutions for singular semilinear elliptic systems with critical Hardy-Sobolev exponents. Nonlinear Anal RWA, 2013, 14(1):613-625[29] Kang D S, Yang F. Elliptic systems involving multiple critical nonlinearities and symmetric multi-polar potentials. Sci Sin Math, 2014, 57(5):1011-1024[30] Deng Z Y, Huang Y S. Symmetric solutions for a class of singular biharmonic elliptic systems involving critical exponents. Appl Math Comput, 2015, 264:323-334[31] Huang D W, Li Y Q. Multiplicity of solutions for a noncooperative p-Laplacian elliptic systems in R^N. J Differ Equ, 2005, 215(1):206-223[32] Palais R. The Principle of Symmetric Criticality. Comm Math Phy, 1979, 69(1):19-30[33] Wang Y J, Shen Y T. Multiple and sign-changing solutions for a class of semilinear biharmonic equation. J Differ Equ, 2009, 246(8):3109-3125[34] Edmunds D E, Fortunato D, Jannelli E. Critical exponents, critical dimensions and the biharmonic operator. Arch Rational Mech Anal, 1990, 112(3):269-289[35] Ambrosetti A, Rabinowitz P H. Dual variational methods in critical point theory and applications. J Funct Anal, 1973, 14(4):349-381[36] Rabinowitz P H. Methods in Critical Point Theory with Applications To Differential Equations. Miami:Amer Math Soc, 1986[37] Kang D S, Deng Y B. Existence of solution for a singular critical elliptic equation. J Math Anal Appl, 2003, 284(2):724-732[38] Schneider M. Compact embeddings and indefinite semilinear elliptic problems. Nonlinear Anal, 2002, 51(2):283-303[39] Brezis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc, 1983, 88(3):486-490

MULTIPLE SYMMETRIC RESULTS FOR A CLASS OF BIHARMONIC ELLIPTIC SYSTEMS WITH CRITICAL HOMOGENEOUS NONLINEARITY IN R^N

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 8

Metrics

Comments

Recommended 10

[1]	Siwar AMMAR, Mokhles HAMMAMI. BLOWING UP AND MULTIPLICITY OF SOLUTIONS FOR A FOURTH-ORDER EQUATION WITH CRITICAL NONLINEARITY [J]. Acta mathematica scientia,Series B, 2015, 35(6): 1511-1546.
[2]	DENG Zhi-Ying, HUANG Yi-Sheng. ON POSITIVE G-SYMMETRIC SOLUTIONS OF A WEIGHTED QUASILINEAR ELLIPTIC EQUATION WITH CRITICAL HARDY-SOBOLEV EXPONENT [J]. Acta mathematica scientia,Series B, 2014, 34(5): 1619-1633.
[3]	FAN Hai-Ning, LIU Xiao-Chun. MULTIPLE POSITIVE SOLUTIONS FOR A CLASS OF QUASI-LINEAR ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT [J]. Acta mathematica scientia,Series B, 2014, 34(4): 1111-1126.
[4]	Wael ABDELHEDI. CONCENTRATION OF SOLUTIONS FOR THE MEAN CURVATURE PROBLEM [J]. Acta mathematica scientia,Series B, 2013, 33(3): 631-642.
[5]	JIANG Yong-Sheng, ZHOU Huan-Song. BOUND STATES FOR A STATIONARY NONLINEAR SCHRÖDINGER-POISSON SYSTEM WITH SIGN-CHANGING POTENTIAL IN R³ [J]. Acta mathematica scientia,Series B, 2009, 29(4): 1095-1104.
[6]	Han Pigong. NEUMANN PROBLEMS OF A CLASS OF ELLIPTIC EQUATIONS WITH DOUBLY CRITICAL SOBOLEV EXPONENTS [J]. Acta mathematica scientia,Series B, 2004, 24(4): 633-638.
[7]	FU Gong-Zhuo, CHEN Yao-Tian. EXISTENCE OF INFINITELY MANY SOLUTIONS FOR ELLIPTIC PROBLEMS WITH CRITICAL EXPONENT [J]. Acta mathematica scientia,Series B, 2004, 24(3): 395-402.
[8]	TAN Zhong, YAO Zheng-An. THE EXISTENCE OF MULTIPLE SOLUTIONS OF p-LAPLACIAN ELLIPTIC EQUATION [J]. Acta mathematica scientia,Series B, 2001, 21(2): 203-212.