Sign-changing Solutions for a Fractional Schrödinger-Poisson System with Concave-convex Nonlinearities and a Steep Potential Well  

一类带陡峭位势和凹凸非线性项的分数阶Schrödinger-Poisson系统的变号解

作　　者：FU Jiao LI Hongying LIAO Jiafeng 付娇;李红英;廖家锋(西华师范大学数学与信息学院,南充637009;2.西华师范大学公共数学学院,南充637009)

机构地区：[1]School of Mathematics and Information,China West Normal University,Nanchong 637009,China [2]College of Mathematics Education,China West Normal University,Nanchong 637009,China

出　　处：《数学理论与应用》2025年第1期25-44,共20页Mathematical Theory and Applications

基　　金：supported by the Natural Science Foundation of Sichuan(No.2023NSFSC0073)。

摘　　要：In this paper,we investigate the following fractional Schrödinger-Poisson system with concave-convex nonlinearities and a steep potential well{(-Δ)^(s)u+V_(λ)(x)u+ϕu=f(x)|u|^(q-2)u+|u|^(p-2)u,in R^(3),(-Δ)^(t)ϕ=u^(2),in R^(3),where s∈(3/4,1),t∈(0,1),q∈(1,2),p∈(4,2_(s)^(*)),2_(s)^(*):=6/3-2s is the fractional critical exponent in dimension 3,V_(λ)(x)=λV(x)+1 withλ>0.Under the case of steep potential well,we obtain the existence of the sign-changing solutions for the above system by using the constraint variational method and the quantitative deformation lemma.Furthermore,we prove that the energy of ground state sign-changing solution is strictly more than twice of the energy of the ground state solution.Our results improve the recent results in the literature.本文研究如下一类带陡峭位势和凹凸非线性项分数阶的Schrödinger-Poisson系统{(-Δ)^(s)u+V_(λ)(x)u+ϕu=f(x)|u|^(q-2)u+|u|^(p-2)u,in R^(3),(-Δ)^(t)ϕ=u^(2),in R^(3),其中s∈(3/4,1),t∈(0,1),q∈(1,2),p∈(4,2_(s)^(*)),2_(s)^(*):=6/3-2s 是三维空间中的分数阶临界指数,V_(λ)(x)=λV(x)+1(λ>0).在陡峭位势下,利用约束变分法和形变引理,我们证明以上系统变号解的存在性,同时证明基态变号解能量严格大于基态解能量的两倍.我们的结果改进了近期相关文献中的结果.

关 键 词：Fractional Schrödinger-Poisson system Concave-convex nonlinearity Sign-changing solution Steep potential well 

分 类 号：O175.2[理学—数学]