含临界指数的拟线性椭圆系统正对称解的存在性（英文）

Existence of Positive Symmetric Solutions for Singular Quasilinear Elliptic Systems InvolvingCritical Exponents

本文讨论一类拟线性椭圆型系统-Δpu=μ|u|p-2 u|x|p+2αQ(x)(α+β)|x|s|u|α-2 u|v|β+σ1|u|q1-2 u,x∈Ω,-Δpv=μ|v|p-2v|x|p+2βQ(x)(α+β)|x|s|u|α|v|β-2v+σ2|v|q2-2v,x∈Ω,u=v=0,x∈Ω,其中Δpu=div(|▽u|p-2▽u)是p-Laplacian,2≤p<N,ΩRN是一个有界光滑区域,0∈Ω,且Ω关于O(N)的一个闭子群G对称,0≤μ<,=((N-p)/p)p,σ1,σ2≥0,0≤s<p,α,β>1满足α+β=p*(s)=(N-s)p/(N-p),p<q1,q2<p*=Np/(N-p),Q(x)是Ω上的连续G对称函数.应用Palais对称临界原理和变分方法,我们建立了该系统几个全新的正G-对称解的存在性结果.

This paper is concerned with a class of quasilinear elliptic problem of the form -Δpu=μ｜u｜^p-2u/｜x｜p＋2αQ（x）/（α＋β）｜x｜^s｜u｜^α-2u｜v｜^β＋σ1｜u｜^q1-2 u,x∈Ω,-Δpv=μ｜v｜^p-2v/｜x｜p＋2βQ（x）/（α＋β）｜x｜^s｜u｜α｜v｜^β-2v＋σ2｜v｜^q2-2v,x∈Ω,u=v=0,x∈δΩ,whereΔpu=div（｜△↓u｜^p-2△↓u）is the p-Laplacian,2≤p〈N,Ω belong to R^N is a smooth bounded domain,0∈ΩandΩis G-symmetric with respect to a closed subgroup Gof O（N）,0≤μ〈μ^- withμ^-=（（N-p）/p）^p,σ1,σ2 ≥0,0≤s〈pandα,β〉1satisfyα＋β=p^＊（s）=（N-s）p/（N-p）,p〈q1,q2〈p^＊=Np/（N-p）,Q（x）is continuous and G-symmetric on Ω^-.We establish several existence results of positive G-symmetric solutions by using the symmetric criticality principle of Palais and variational methods for this problem.