一类带有不同Rellich项的临界双调和方程组的非平凡解

Abstract

Abstract: In this paper, a system of biharmonic equations is investigated, which involves multiple critical Sobolev nonlinearities and different Rellich-type terms. The minimizers of the related best Soblev constant are found under certain assumptions and the existence of solutions to the system is established by variational arguments.

Key words: Biharmonic system, Solution, Critical exponent, Rellich potential, Variational method

CLC Number:

O175.25

Kang Dongsheng, Xiong Ping. Biharmonic Systems Involving Critical Nonlinearities and Different Rellich-Type Potentials[J].Acta mathematica scientia,Series A, 2017, 37(6): 1105-1118.

Trendmd

References

[1] Davies E, Hinz A. Explicit constants for Rellich ineqaulities in L^p(Ω). Math Z, 1998, 227(2):511-523
[2] Edmunds D, Fortunato D, Jannelli E. Critical exponents, critical dimensions and the biharmonic operator. Arch Ration Mech Anal, 1990, 112(1):269-289
[3] Bhakta M, Musina R. Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials. Nonlinear Anal, 2012, 75(12):3836-3848
[4] D'Ambrosio L, Jennelli E. Nonlinear critical problems for the biharmonic operator with Hardy potential. Calc Var Partial Differential Equations, 2015, 54(1):365-396
[5] Bhakta M. Caffarelli-Kohn-Nirenberg type equations of fourth order with the critical exponent and Rellich potential. J Math Anal Appl, 2016, 433(1):681-700
[6] Hardy G, Littlewood J, Polya G. Inequalities. Cambridge:Cambridge University Press, 1952
[7] Chen Z, Zou W. Existence and symmetry of positive ground states for a doubly critical Schrödinger system. Trans Amer Math Soc, 2015, 367(11):3599-3646
[8] Felli V, Terracini S. Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity. Comm Partial Differential Equations, 2006, 31(2):469-495
[9] Abdellaoui B, Felli V, Peral I. Some remarks on systems of elliptic equations doubly critical in the whole R^N. Calc Var Partial Differential Equations, 2009, 34(1):97-137
[10] Jannelli E. The role played by space dimension in elliptic critcal problems. J Differential Equations, 1999, 156(2):407-426
[11] Kang D. Positive minimizers of the best constants and solutions to coupled critical quasilinear systems. J Differential Equations, 2016, 260(1):133-148
[12] Kang D, Luo J, Shi X L. Solutions to elliptic systems involving doubly critical nonlinearities and Hardy-type potentials. Acta Mathematica Scientia, 2015, 35B(2):423-438
[13] Zhang J, Ma S W. Infinitely many sign-changing solutions for the Brezis-Nirenberg problem involving Hardy-potential. Acta Mathematica Scientia, 2016, 36B(2):527-536
[14] Terracini S. On positive entire solutions to a class of equations with a singular coefficient and critical exponent. Adv Differential Equations, 1996, 1(2):241-264
[15] Kang D. Concentration compactness principles for the systems of critical elliptic equations. Differ Equ Appl, 2012, 4(2):435-444
[16] Lions P L. The concentration compactness principle in the calculus of variations, the limit case (I). Rev Mat Iberoamericana, 1985, 1(1):145-201
[17] Lions P L. The concentration compactness principle in the calculus of variations, the limit case (Ⅱ). Rev Mat Iberoamericana, 1985, 1(2):45-121
[18] Lieb E, Loss M. Analysis. Providence, RI:American Mathematical Society, 2001
[19] Willem M. Analyse Fonctionnelle Élémentaire. Paris:Cassini Éditeurs, 2003
[20] Kang D. Systems of quasilinear elliptic equations involving multiple homogeneous nonlinearities. Appl Math Lett, 2014, 37(1):1-6
[21] Ambrosetti A, Rabinowitz H. Dual variational methods in critical point theory and applications. J Funct Anal, 1973, 14(2):349-381
[22] Brezis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm Pure Appl Math, 1983, 36(2):437-477

Biharmonic Systems Involving Critical Nonlinearities and Different Rellich-Type Potentials

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Wang Jun, Wang Tianlu, Wen Yanhua, Zhou Xianfeng. Global Solutions of IVP for N-Dimensional Nonlinear Fractional Differential Equations with Delay [J]. Acta mathematica scientia,Series A, 2018, 38(5): 903-910.
[2]	Lan Guijie, Fu Yingjie, Wei Chunjin, Zhang Shuwen. Stationary Distribution and Periodic Solution for Stochastic Predator-Prey Systems with Holling-Type Ⅲ Functional Response [J]. Acta mathematica scientia,Series A, 2018, 38(5): 984-1000.
[3]	Cao Jianbing. Perturbation Analysis for Constrained Extremal Solution Problems in Reflexive Strictly Convex Banach Spaces [J]. Acta mathematica scientia,Series A, 2018, 38(4): 649-657.
[4]	Wang Huiling, Gao Jianfang. Oscillation Analysis of Analytical Solutions for a Kind of Nonlinear Neutral Delay Differential Equations with Several Delays [J]. Acta mathematica scientia,Series A, 2018, 38(4): 740-749.
[5]	Shen Wenguo. Interval Bifurcation for the p-Laplacian Equation and Its Applications [J]. Acta mathematica scientia,Series A, 2018, 38(4): 770-778.
[6]	Zou Xia, Wu Shiliang. Traveling Waves in a Nonlocal Dispersal SIR Epidemic Model with Delay and Nonlinear Incidence [J]. Acta mathematica scientia,Series A, 2018, 38(3): 496-513.
[7]	Zhang Shengui. Infinitely Many Solutions for a Bi-Nonlocal Problem Involving p(x)-Laplacian-Like Operator [J]. Acta mathematica scientia,Series A, 2018, 38(3): 514-526.
[8]	Yao Shaowen, Cheng Zhibo. Positive Periodic Solution for Damped Duffing Equation with Singularity [J]. Acta mathematica scientia,Series A, 2018, 38(3): 543-548.
[9]	Lin Jiaoyan. Uniform Estimates in Time for the Solution to a Fluid-Particle Model [J]. Acta mathematica scientia,Series A, 2018, 38(3): 565-570.
[10]	Xing Chao, Ren Mengzhang, Luo Hong. The Existence of Global Weak Solutions to Thermohaline Circulation Equations [J]. Acta mathematica scientia,Series A, 2018, 38(2): 284-290.
[11]	Zhang Shuqin, Hu Lei. Initial Value Problem for Nonlinear Fractional Differential Equations with Variable-order Derivative and a Parameter [J]. Acta mathematica scientia,Series A, 2018, 38(2): 291-303.
[12]	Sun Xiaomei. The Existence of Solutions for Choquard Type Equation [J]. Acta mathematica scientia,Series A, 2018, 38(1): 54-60.
[13]	Ye Yaojun, Hu Yue. Global Existence and Exponential Decay of Solution for Some Higher-Order Wave Equation [J]. Acta mathematica scientia,Series A, 2018, 38(1): 110-121.
[14]	Liang Zhiqing, Zeng Xiaping, Zhou Zewen, Pang Guoping, Huang Junhua. Existence of Periodic Solution of Holling-Tanner System with State Feedback Impulsive Control [J]. Acta mathematica scientia,Series A, 2018, 38(1): 174-189.
[15]	Ouyang Cheng, Mo Jiaqi. Study of Nonlinear Duffing Disturbed Oscillator for Resonance Mechanism [J]. Acta mathematica scientia,Series A, 2018, 38(1): 190-196.