带有卷积非线性项的Kirchhoff方程解的多重性
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摘要
针对带有卷积非线性项的Kirchhoff方程解的多重性研究成果较少、困难较多的情形,通过给出适当的条件来克服非局部项与非线性项之间的相互干扰,进一步再利用Fountain定理来获得方程解的多重性。针对卷积非线性项的出现,采用不同于通常多项式非线性项的处理手法,为解决类似问题提供了参考。
There are few results and more difficulties for the multiplicity of solutions to Kirchhoff equation with convolution nonlinearities. Therefore,on the one hand,overcoming the interference between non-local term and non-linear term was done by giving the proper assumption of one of the nonlinearities,on the other hand,the Fountain theorem was used to obtain the multiplicity of solutions. Due to the appearance of convolution nonlinearities,the solving method is different from the usual nonlinearities.
There are few results and more difficulties for the multiplicity of solutions to Kirchhoff equation with convolution nonlinearities. Therefore,on the one hand,overcoming the interference between non-local term and non-linear term was done by giving the proper assumption of one of the nonlinearities,on the other hand,the Fountain theorem was used to obtain the multiplicity of solutions. Due to the appearance of convolution nonlinearities,the solving method is different from the usual nonlinearities.
引文
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[2] DISI L. Gravitation and quantum-mechanical localizatio of macro-objects[J]. Physics Letters A,1984,105(4):199-202.
[3] WANG Tao. Existence and nonexistence of nontrivial solutions for Choquard type equations[J]. Electron J Differential Equations,2016(2):3-17.
[4] GAO Fashun,YANG Minbo. On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents[J]. J Math Anal Appl,2017,448(2):1006-1041.
[5] LIEB E H. Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation[J]. Studies in Appl Math,1976,57(2):93-105.
[6] LIONS P L. The Choquard equation and related questions[J]. Nonlinear Anal,1980,4(6):1063-1072.
[7] LIONS P L. The concentration-compactness principle in the calculus of variations. The locally compact case. II.Ann. Inst. H. PoincaréAnal[J]. Non Linéaire,1984,1(4):223-283.
[8] MOROZ V,SCHAFTINGEN J V. Groundstates of nonlinear Choquard equations:existence,qualitative properties and decay asymptotics[J]. J Funct Anal,2013,265(2):153-184.
[9] MOROZ V,SCHAFTINGEN J V. Existence of groundstates for a class of nonlinear Choquard equations[J].Trans Amer Math Soc,2015,367(9):6557-6579.
[10] MOROZ V,SCHAFTINGEN J V. Groundstates of nonlinear Choquard equations:Hardy-Littlewood-Sobolev critical exponent[J]. Commun Contemp Math,2015,17(5):1550005.
[11] BARTSCH T,WANG Zhiqiang. Existence and multiplicityresults for some superlinear elliptic problems onRN[J].Comm Partial Differential Equations,1995,20(9/10):1725-1741.
[12] LIEB E H,LOSS M. Analysis[M]. American:American Mathematical Society,2001.
[13] WILLEM M. Minimax Theorems[M]. Boston:Birkhuser,1996.
[14] STEIN E M,WEISS G. Fractional integrals on n-dimensional Euclidean space[J]. J Math Mech,1958(7):503-514.
[15] SCHAFTINGEN J V,XIA Jiankang. Choquard equations under confining external potentials[J]. No DEA Nonlinear Differential Equations Appl,2017,24(1):1-24.