LI Wenmin,ZHANG Jiafeng.The solvability of critical Schr?dinger - Poisson systems in volving vanishing potential[J].Journal of Yanbian University,2023,(03):195-202.
具有消失位势的Schr?dinger - Poisson系统的可解性
- Title:
- The solvability of critical Schr?dinger - Poisson systems in volving vanishing potential
- 文章编号:
- 1004-4353(2023)03-0195-08
- 关键词:
- Schr?dinger - Poisson系统; 变分方法; 截断函数; 消失位势
- Keywords:
- Schr?dinger - Poisson system; variational methods; cut - off function; vanishing potential
- 分类号:
- O177.91
- 文献标志码:
- A
- 摘要:
- 利用山路引理、H?lder不等式、嵌入不等式和估值等,研究了一类含临界非局部项且位势在无穷远处消失的Schr?dinger - Poisson系统的非平凡解的存在性.首先利用嵌入不等式证明了泛函具有紧性,然后利用截断函数对泛函进行了估值,最后用变分方法证明了该系统至少有一个非平凡解.所得结果丰富了椭圆型方程解的相关理论.
- Abstract:
- The existence of nontrivial solutions for a class of Schr?dinger - Poisson systems containing critical nonlocal terms and vanishing potential at infinity was investigated by using the Mountain Pass Lemma, H?lder’s inequality, embedding inequality and valuation.Firstly, we prove that the functional has tightness by using the embedding inequality, then the valuation of the functional is performed using the cut - off function, and finally we prove that there is at least one nontrivial solution of the system by variational method.The results of this paper enrich the theory of solutions of elliptic type equations.
参考文献/References:
[1] SUN J T, CHEN H B, YANG L.Positive solutions of asymptotically linear Schr?dinger - Poisson systems with a radial potential vanishing at infinity [J].Nonlinear Analysis: Theory, Methods and Applications, 2011,74(2):413 - 423.
[2] LIU H D.Positive solutions of an asymptotically periodic Schr?dinger - Poisson system with critical exponent [J].Nonlinear Analysis: Real World Applications, 2016,32:198 - 212.
[3] SHAO L Y.Non - trivial solutions for Schr?dinger - Poisson systems involving critical nonlocal term and potential vanishing at infinity [J].Open Mathematics, 2019,17(1):1156 - 1167.
[4] ZHANG J F, GUO W, CHU C M, et al.Existence of solutions for a Schr?dinger - Poisson system with critical nonlocal term and general nonlinearity [J].Journal of Function Spaces, 2020,2020:2197207.
[5] QIAN X T.Multiplicity of positive solutions for a class of nonlocal problem involving critical exponent [J].Electronic Journal of Qualitative Theory of Differential Equations, 2021,2021(57):1 - 14.
[6] ALVES C O, SOUTO M A S.Existence of solutions for a class of nonlinear Schr?dinger equations with potential vanishing at infinity [J].Journal of Differential Equations, 2013,254(4):1977 - 1991.
[7] SUN J B, WANG Z Q, WILLEM M.Weighted Sobolev embedding with unbounded and decaying radial potentials [J].Journal of Differential Equations, 2007,238(1):201 - 219.
[8] AMBROSETTI A, MALCHIODI A, FELLI V.Ground states of nonlinear Schr?dinger equations with potentials vanishing at infinity [J].Journal of the European Mathematical Society, 2005,7(1):117 - 144.
[9] KHOUTIR S, CHEN H B.Positive ground state solutions for a class of Schr?dinger - Poisson systems in R4 involving critical Sobolev exponent [J].Asymptotic Analysis, 2018,109(1/2):91 - 109.
[10] BREZIS H, LIEB E.A relation between pointwise convergence of functions and convergence of functionals [J].Proceedings of the American Mathematical Society, 1983,88(3):486 - 490.
相似文献/References:
[1]钱晓涛,石志高.一类Kirchhoff型方程解的存在性和多重性[J].延边大学学报(自然科学版),2018,44(04):302.
QIAN Xiaotao,SHI Zhigao.Existence and multiplicity of solutions for a class of
Kirchhoff type equation[J].Journal of Yanbian University,2018,44(03):302.
[2]吴燕林,钱晓涛.一类次临界增长非局部问题的无穷多解[J].延边大学学报(自然科学版),2019,45(04):299.
WU Yanlin,QIAN Xiaotao.Infinitely many solutions for a class of nonlocalproblem with subcritical growth[J].Journal of Yanbian University,2019,45(03):299.
[3]吴燕林,钱晓涛.全空间上一类Kirchhoff型问题正基态解的存在性[J].延边大学学报(自然科学版),2021,47(01):17.
WU Yanlin,QIAN Xiaotao.Existence of ground state positive solution for a class of Kirchhoff type problem on unbounded domain[J].Journal of Yanbian University,2021,47(03):17.
[4]魏其萍,王跃*.一类负模量Kirchhoff - Carrier方程的解[J].延边大学学报(自然科学版),2021,47(02):111.
WEI Qiping,WANG Yue*.The solution to a Kirchhoff - Carrier equation with Negative Modulus[J].Journal of Yanbian University,2021,47(03):111.
备注/Memo
收稿日期: 2023-06-01
基金项目: 国家自然科学基金(11861021); 贵州民族大学自然科学研究项目(GZMUZK[2022]YB23)
第一作者: 李文敏(1999—),女,硕士研究生,研究方向为非线性泛函分析及应用.
通信作者: 张家锋(1981—),男,博士,教授,研究方向为非线性泛函分析及应用.