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Nehari type ground state solutions for periodic Schrödinger–Poisson systems with variable growth
Complex Variables and Elliptic Equations ( IF 0.9 ) Pub Date : 2020-12-28 , DOI: 10.1080/17476933.2020.1843643
Limin Zhang 1 , Xianhua Tang 1 , Sitong Chen 1
Affiliation  

In this paper, we deal with the variable growth Schrödinger–Poisson Systems in R3: div(|u|p(x)2u)+(V(x)+K(x)ϕ(x))|u|p(x)2u=f(x,u),xR3,Δϕ(x)=K(x)|u|p(x),xR3,uW1,p(x)(R3), where p(x)p(x):=3p(x)3p(x) and V(x), K(x) and f(x,u) are periodic in x. We use the non-Nehari manifold approach to establish the existence of the Nehari type ground state solutions, under the condition: pp+χ(0,1)(t)f(x,τ)|τ|2pˆ1f(x,tτ)|tτ|2pˆ1sign(1t)+pp+χ(0,1)(t)θ0V(x)|tpˆ1|tpˆ|τ|2pˆp(x)0 for all xR3, t>0, τ 0 and the constant θ0(0,1), where pˆ=p+ as t1 and pˆ=p as t<1. In particular, some new inequalities and tricks are used to overcome the difficulties caused by the variable exponent.



中文翻译:

具有可变增长的周期性薛定谔-泊松系统的 Nehari 型基态解

在本文中,我们处理变量增长薛定谔-泊松系统R3-d一世v(||p(X)-2)+((X)+ķ(X)φ(X))||p(X)-2=F(X,),XR3,-Δφ(X)=ķ(X)||p(X),XR3,W1,p(X)(R3),在哪里p(X)p*(X):=3p(X)3-p(X)(X),ķ(X)F(X,)在x中是周期性的。我们使用非 Nehari 流形方法来确定 Nehari 型基态解的存在性,条件是:p-p+χ(0,1)()F(X,τ)|τ|2p^-1-F(X,τ)|τ|2p^-1s一世Gn(1-)+p-p+χ(0,1)()θ0(X)|p^-1|p^|τ|2p^-p(X)0对所有人XR3, t >0,τ 0和常数θ0(0,1), 在哪里p^=p+作为1p^=p-t <1。特别是,一些新的不等式和技巧被用来克服可变指数带来的困难。

更新日期:2020-12-28
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