【作者】 朱丽君；

【导师】 廖家锋；

【作者基本信息】 西华师范大学 ， 应用数学， 2022， 硕士

【摘要】 本文主要运用变分法、山路定理和一些分析技巧研究了两类带临界指数的Schr(?)dinger-Poisson系统解的存在性。首先,研究如下带临界指数的Schr(?)dinger-Poisson系统解的存在性(?)(1)其中η∈R\{0},λ>0,系数函数f∈L6/5(R3),l∈L2(R3)∩L∞(R3)且l(x),f(x)均为非零非负函数,可得系统(1)至少存在两个正解。其次,考虑如下带临界指数的Schr(?)dinger-Poisson系统正解的存在性其中系数函数f(x)∈L3/2(R3)且满足存在δ,ρ>0,1<β<2,x0∈R3,使得对(?)x,|x-x0|<ρ,都有f(x)≥δ|x-x0|-β;l∈L2(R3)∩L∞(R3)且l(x),f(x)为非零非负函数;参数λ满足0<λ<λ*=(?){∫R3(|▽u|2+u2)dx:∫R3f(x)u2dx=1}。接下来,考虑如下的带临界指数的Schr(?)dinger-Poisson系统解的存在性(?)(2)其中η∈R\{0},f∈C(R3×R,R),V满足适当的条件,可得系统(2)至少存在一个非平凡解。然后,研究了一类带临界指数的零质量Schr(?)dinger-Poisson系统正基态解的存在性其中a∈L3/2(R3)∩L∞(R3),l∈L2(R3)∩L∞(R3),f(u)满足一定的条件且a(x),l(x),f(u)均为非零非负函数。以上结果推广了近期关于带临界指数的Schr(?)dinger-Poisson系统解的存在性的结果。最后,对本文所研究的两类带临界指数的Schr(?)dinger-Poisson系统解的存在性作简要总结,并给出研究展望。更多还原

【Abstract】 In this paper,we study the existence of solutions for two classes of SchrodingerPoisson systems with critical exponent by using variational method,Mountain pass theorem and some analysis techniques.Firstly,we study the Schrodinger-Poisson system with critical exponent as follows(?)(1)where η∈R\{0},λ>0,the coefficient functions f∈L6/5(R3),l∈L2(R3)∩L∞(R3)and f(x),l(x)are non-zero non-negative functions.We obtain system(1)has at least two positive solutions.Secondly,we consider the existence of positive solutions for the following Schr(?)dinger-Poisson system with critical exponent where the coefficient function f∈L3/2(R3)and satisfies that exist δ,ρ>0,1<β<2,x0∈R3 such that (?)x∈R3,|x-x0|<ρ,we have f(x)≥δ|x-x0|-β;l(x)∈L2(R3)∩L∞(R3)and f(x),l(x)are non-zero non-negative functions.Real parameterλ satisfies 0<λ<λ*=(?){∫R3(|▽u|2+u2)dx:∫R3f(x)u2dx=1}.Next,we consider the existence of solutions for the Schr(?)dinger-Poisson system with critical exponent(?)(2)where η∈R\{0},f∈C(R3×R,R),V satisfy the appropriate conditions.We establish the system(2)has at least one non-trivial solution.Then,we study the existence of positive ground-state solutions for a class of normal zero-mass Schr(?)dinger-Poisson system with critical exponent where a∈L3/2(R3)∩L∞(R3),l∈L2(R3)∩L∞(R3),f(u)satisfy appropriate conditions and f(u),a(x),l(x)are non-zero non-negative functions.The above results generalize recent results on Schr(?)dinger-Poisson systems with critical exponent.In the end,we give a brief summary of the existence of solutions for the two classes of Schr(?)dinger-Poisson systems with critical exponent studied here and draw the research outlook.更多还原