带有变号非线性项Kirchhoff方程基态解的存在性
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摘要
研究了一类带有变号非线性项Kirchhoff方程基态解的存在性。由于非线性项是变号的,相应的Nehari流形不再是一阶连续可微的。因此,利用Nehari流形和单位球面拓扑同胚的性质,将此类方程转化在工作空间的单位球面上来考虑。然后,在此单位球面上利用Ekelend变分原理找到有界极小化序列。最后,利用反证法证明了基态解的存在性。
The existence of ground state solutions for a kind of Kirchhoff equation with sign-changing nonlinearities was studied. Because of the nonlinearities were sign-changing, the corresponding Nehari manifold was not first order continuous and differentiable any more. Therefore,by using of the property that Nehari manifold was topological homeomorphism with unit sphere,this kind of equation was transformed on the unit sphere. Then,the bounded minimizing sequence was found by using of Ekelend variational principle on the unit sphere. Finally,the existence of the ground state solution was proved with reduction to absurdity.
The existence of ground state solutions for a kind of Kirchhoff equation with sign-changing nonlinearities was studied. Because of the nonlinearities were sign-changing, the corresponding Nehari manifold was not first order continuous and differentiable any more. Therefore,by using of the property that Nehari manifold was topological homeomorphism with unit sphere,this kind of equation was transformed on the unit sphere. Then,the bounded minimizing sequence was found by using of Ekelend variational principle on the unit sphere. Finally,the existence of the ground state solution was proved with reduction to absurdity.
引文
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[10]CHEN C Y,KUO Y C,WU T F.The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions[J].Journal of differential equations,2011,250(4):1876-1908.
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[14]郭大钧.非线性泛函分析[M].3版.北京:高等教育出版社,2015.
[15]PANKOV A.On decay of solution to nonlinear Schr9dinger equations[J].Proceedings of the American mathematical society,2008,136(7):2565-2570.
[2]AVCI M,CEKIC B,MASHIYEV R A.Existence and multiplicity of the solutions of the p(x)Kirchhoff type equation via genus theory[J].Mathematical methods in the applied sciences,2011,34(14):1751-1759.
[3]AROSIO A,PANIZZI S.On the well-posedness of the Kirchhoff string[J].Transactions of the American mathematical society,1996,348(1):305-330.
[4]LUO X,WANG Q F.Existence and asymptotic behavior of high energy normalized solutions for the Kirchhoff-type equations in R3[J].Nonlinear analysis(real world applications),2017,33:19-32.
[5]DENG Y B,PENG S J,SHUAI W.Existence and asymptotic behavior of nodal solutions for the Kirchhoff type problems in R2[J].Journal of functiont analysis,2015,269(11):3500-3527.
[6]郝亚文,李宇华.渐近周期的Kirchhoff-Schr9dinger-Poisson系统非平凡解的存在性[J].河南科技大学学报(自然科学版),2017,38(3):95-99.
[7]宋朝霞,张琦.带有奇异项Kirchhoff型方程正解的存在性[J].河南科技大学学报(自然科学版),2017,38(3):91-94.
[8]SZULKIN A,WETH T.Ground state solutions for some indefinite variational problems[J].Journal of functional analysis,2009,257(12):3802-3822.
[9]ZHANG Z T,PERERA K.Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow[J].Journal of mathematical analysis and applications,2006,317(2):456-463.
[10]CHEN C Y,KUO Y C,WU T F.The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions[J].Journal of differential equations,2011,250(4):1876-1908.
[11]BIEGANOWSKI B,MEDERSKI J L.Nonlinear Schr9dinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities[J].Communications on pure and applied analysis,2018,17(1):143-161.
[12]BARTSCH T,WANG Z Q.Existence and multiplicity results for some superlinear elliptic problems on RN[J].Communications in partial differential equations,1995,20(9/10):1725-1741.
[13]LI G B,YE H Y.Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3[J].Journal of differential equations,2014,257(2):566-600.
[14]郭大钧.非线性泛函分析[M].3版.北京:高等教育出版社,2015.
[15]PANKOV A.On decay of solution to nonlinear Schr9dinger equations[J].Proceedings of the American mathematical society,2008,136(7):2565-2570.