具有井位势的非线性分阶SchrO¨dinger-Poisson 方程组的基态解的存在性和渐近性
Existence and Asymptotic Behavior of Ground State Solutions for Nonlinear Fractional SchrO¨dinger-Poisson Systemswith Steep Potential Well
摘要: 在本文中,  我们研究了如下分 阶 SchrO¨dinger-Poisson 方程组 其中(−∆)α是阶数为α∈(0, 1)的分数阶Laplace算子，λ >是一个参数，是分数阶临街指数. 在 V, f满足适当条件下, 利用变分方法我们证明了基态解存在性. 及当λ → +∞ 基态解的渐近行为.
Abstract: In  this  paper,  we  study  the  following  fractional  SchrO¨dinger-Poisson  system: where (−∆)α denotes the fractional Laplacian of order α∈(0, 1), λ > 0 is a parameter, is the fractional critical exponent. Under appropriate assumptions on V and f ,  we prove the existence of ground state solutions using variational methods. Furthermore, we also study the asymptotic behavior of ground state solutions as λ→+∞.
文章引用：王亚军. 具有井位势的非线性分阶SchrO¨dinger-Poisson 方程组的基态解的存在性和渐近性[J]. 应用数学进展, 2021, 10(5): 1804-1824. https://doi.org/10.12677/AAM.2021.105191

参考文献

[1] Di Nezza, E., Palatucci, G. and Valdinoci, E. (2012) Hitchhiker’s Guide to the Fractional Sobolev Spaces. Bulletin des Sciences Math´ematiques, 136, 521-573. [Google Scholar] [CrossRef] 
[2] Bartsch, T. and Wang, Z. (1995) Existence and Multiplicity Results for Some Superlinear Elliptic Problems on RN . Communications in Partial Differential Equations, 20, 1725-1741. [Google Scholar] [CrossRef] 
[3] Bartsch, T., Pankov, A. and Wang, Z. (2001) Nonlinear Schr¨odinger Equations with Steep Potential Well. Communications in Contemporary Mathematics, 3, 549-569. [Google Scholar] [CrossRef] 
[4] Ding, Y. and Szulkin, A. (2007) Bound States for Semilinear Schr¨odinger Equations with Sign- Changing Potential. Calculus of Variations and Partial Differential Equations, 29, 397-419. [Google Scholar] [CrossRef] 
[5] Jiang, Y. and Zhou, H. (2011) Schr¨odinger-Poisson System with Steep Potential Well. Journal of Differential Equations, 251, 582-608. [Google Scholar] [CrossRef] 
[6] Sun, J. and Wu, T. (2014) Ground State Solutions for an Indefinite Kirchhoff Type Problem with Steep Potential Well. Journal of Differential Equations, 256, 1771-1792. [Google Scholar] [CrossRef] 
[7] Ye, Y. and Tang, C. (2015) Existence and Multiplicity of Solutions for Schr¨odinger-Poisson Equations with Sign-Changing Potential. Calculus of Variations and Partial Differential E- quations, 53, 383-411. [Google Scholar] [CrossRef] 
[8] Zhao, L., Liu, H. and Zhao, F. (2013) Existence and Concentration of Solutions for the Schr¨odinger-Poisson Equations with Steep Well Potential. Journal of Differential Equation- s, 255, 1-23. [Google Scholar] [CrossRef] 
[9] Secchi, S. (2013) Ground State Solutions for Nonlinear Fractional Schr¨odinger Equations in RN .Journal of Mathematical Physics, 54, Article ID: 031501. [Google Scholar] [CrossRef] 
[10] Palatucci, G. and Pisante, A. (2014) Improved Sobolev Embeddings, Profile Decomposition and Concentration-Compactness for Fractional Sobolev Spaces. Calculus of Variations and Partial Differential Equations, 50, 799-829. [Google Scholar] [CrossRef] 
[11] Benci, V. and Fortunato, D. (1998) An Eigenvalue Problem for the Schr¨odinger-Maxwell E- quations. Topological Methods in Nonlinear Analysis, 11, 283-293. [Google Scholar] [CrossRef] 
[12] Landkof, N. (1972) Foundations of Modern Potential Theory. Springer-Verlag, New York-Heidelberg. [Google Scholar] [CrossRef] 
[13] Teng, K. (2016) Existence of Ground State Solutions for the Nonlinear Fractional Schr¨odinger- Poisson System with Critical Sobolev Exponent. Journal of Differential Equations, 261, 3061- 3106. [Google Scholar] [CrossRef] 
[14] Br´ezis, H. and Lieb, E. (1983) A Relation Between Pointwise Convergence of Functions and Convergence of Functionals. Proceedings of the AMS, 88, 486-490. [Google Scholar] [CrossRef] 
[15] Szulkin, A. and Weth, T. (2009) Ground State Solutions for Some Indefinite Variational Prob- lems. Journal of Functional Analysis, 257, 3802-3822. [Google Scholar] [CrossRef] 
[16] Szulkin, A. and Weth, T. (2010) The Method of Nehari Manifold. In: Gao, D.Y. and Motreanu, D., Eds., Handbook of Nonconvex Analysis and Applications, International Press, Somerville, MA, 597-632.
[17] Willem, M. (1996) Minimax Theorems. Birkh¨auser Boston, Inc., Boston, MA. [Google Scholar] [CrossRef] 
[18] Bisci, G.M., Radulescu, D. and Servadei, R. (2016) Variational Methods for Nonlocal Fractional Problems. Vol. 162, Cambridge University Press, Cambridge.

为你推荐