|
|
一类分数阶薛定谔－泊松系统非平凡解的存在性
|
Abstract:
本文研究一类具有变号权的分数阶薛定谔－泊松系统![\"\"](\"Edit_574a5d0b-b91d-4efc-9139-6bc8d2621dd2.png\")
非平凡解的存在性, 其中![\"\"](\"Edit_11c5b562-1cb4-4499-953e-941f3157374a.png\")
, s, t∈(0, 1) 且 4s + 2t > 3, a(x)∈C(R3) 变号且lim|x|→∞ a(x) = a∞ < 0, ![\"\"](\"Edit_b31db360-1a10-4e36-960e-b712ea8a0f0a.png\")
. 应用山路引理, 本文得到该系统至少存在一个非平凡解.
In this paper, we are concerned with the existence of nontrivial solution for a class of fractional Schro¨dinger-Poisson system: ![\"\"](\"Edit_9b068d0b-97d9-4458-af87-27b9de048a0c.png\")
where ![\"\"](\"Edit_07eac06d-e75f-456b-844f-e26ac28bd1d6.png\")
, s, t ∈ (0, 1) and 4s + 2t > 3, a(x) ∈ C(R3) is a sign-changing function with lim|x|→∞ a(x) = a∞ < 0, ![\"\"](\"Edit_78110fa6-509b-4ee3-9e25-d6b324e44740.png\")
. By using mountain pass theorem, we obtain that this system has at least one nontrivial solution.
| [1] | Laskin, N. (2000) Fractional Quantum Mechanics and L′evy Path Integrals. Physics Letters A, 268, 298-305. https://doi.org/10.1016/S0375-9601(00)00201-2 |
| [2] | Laskin, N. (2002) Fractional Schro¨dinger Equations. Physical Review, 66, 56-108. https://doi.org/10.1103/PhysRevE.66.056108 |
| [3] | Molica Bisci, G., R？adulescu, V.D. and Servadei, R. (2016) Variational Methods for Nonlocal Fractional Problems. In: Encyclopedia of Mathematics and Its Applications, 162. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9781316282397 |
| [4] | Chen, M., Li, Q. and Peng, S.J. (2021) Bound States for Fractional Schro¨dinger-Poisson System with Critical Exponent. Discrete and Continuous Dynamical Systems-Series S, 14, 1819-1835. https://doi.org/10.3934/dcdss.2021038 |
| [5] | Guo, L. (2018) Sign-Changing Solutions for Fractional Schro¨dinger-Poisson System in R3. Applicable Analysis, 98, 2085-2104. |
| [6] | Ianni, I. (2013) Sign-Changing Radial Solutions for the Schro¨dinger-Poisson-Slater Problem. Topological Methods in Nonlinear Analysis, 41, 365-385. |
| [7] | Liu, Z. and Zhang, J. (2017) Multiplicity and Concentration of Positive Solutions for the Fractional Schr¨odinger-Poisson Systems with Critical Growth. ESAIM: Control, Optimisation and Calculus of Variations, 23, 1515-1542. https://doi.org/10.1051/cocv/2016063 |
| [8] | Luo, H. and Tang, X. (2018) Ground State and Multiple Solutions for the Fractional Schro¨dinger-Poisson System with Critical Sobolev Exponent. Nonlinear Analysis: Real World Applications, 42, 24-52. https://doi.org/10.1016/j.nonrwa.2017.12.003 |
| [9] | Shen, L. and Yao, X. (2018) Least Energy Solutions for a Class of Fractional Schro¨dinger- Poisson Systems. Journal of Mathematical Physics, 59, Article ID: 081501. https://doi.org/10.1063/1.5047663 |
| [10] | Sun, X. and Teng, K.M. (2020) Positive Bound States for Fractional Schro¨dinger-Poisson System with Critical Exponent. Communications on Pure and Applied Analysis, 19, 3735- 3768. https://doi.org/10.3934/cpaa.2020165 |
| [11] | Teng, K.M. (2016) Existence of Ground State Solutions for the Nonlinear Fractional Schro¨dinger-Poisson Systems with Critical Sobolev Exponent. Journal of Differential Equa- tions, 261, 3061-3106. https://doi.org/10.1016/j.jde.2016.05.022 |
| [12] | Wang, D.B., Zhang, H., Ma, Y. and Guan, W. (2019) Ground State Sign-Changing Solutions for a Class of Nonlinear Fractional Schro¨dinger-Poisson System with Potential Vanishing at Infinity. Journal of Applied Mathematics and Computing, 61, 611-634. https://doi.org/10.1007/s12190-019-01265-y |
| [13] | Yu, Y., Zhao, F. and Zhao, L. (2017) The Concentration Behavior of Ground State Solutions for a Fractional Schro¨dinger-Poisson System. Calculus of Variations and Partial Differential Equations, 56, Article No.116. https://doi.org/10.1007/s00526-017-1199-4 |
| [14] | Yu, Y., Zhao, F. and Zhao, L. (2018) Positive and Sign-Changing Least Energy Solutions for a Fractional Schro¨dinger-Poisson System with Critical Exponent. Applicable Analysis, 99, 2229-2257. |
| [15] | Zhang, J., do O′ , J.M. and Squassina, M. (2016) Fractional Schro¨dinger-Poisson Systems with a General Subcritical or Critical Nonlinearity. Advanced Nonlinear Studies, 16, 15-30. https://doi.org/10.1515/ans-2015-5024 |
| [16] | 余晓辉. 一类薛定诗-泊松方程解的存在性[J]. 应用数学, 2010, 23(3): 648-652. [17] Willem, M. (1996) Minimax Theorems. Birkha¨user, Bosten. |
