含有奇异位势和临界指数的拟线性椭圆方程的G-对称解

本文讨论一类奇异拟线性椭圆型方程
-div(|x|^-ap|▽u|^p-2▽u)=μ+h(x)/|x|^(a+1)p|u|^p-2u+k(x)|u|^p-2u/|x|^bq,x∈R^N,
其中1 < p < N, 0 ≤ a < N-p/p, a ≤ b < a + 1, 0 ≤ μ < μ = (N-p/p-a)^p, q=p^*(a, b) = Np/N-(1+a-b)p,h 和k 是R^N上的连续有界函数, 且关于O(N) 的闭子群G满足某些对称性条件. 应用变分方法和Caffarelli-Kohn-Nirenberg 不等式, 在h与k满足适当条件下, 证得了一些G-对称解的存在性和多重性结果.

On G-symmetric solutions of the quasilinear elliptic equations with singular weights and critical exponents

In this work we are concerned with the singular quasilinear elliptic equations of the type -div（｜x｜-ap｜▽u｜p2▽u）=μ＋h（x）/｜x｜（a＋1）p｜u｜p-2u ＋k（x）｜u｜q-2u/｜x｜bq,x ∈RN,where 1pN,0≤aN-p/p,a≤ba＋1,0≤μ μ=（N-p/p-a） p,q=p＊（a,b）=Np/N- （1＋a-b）p,and h and k are continuously bounded functions satisfying some symmetry conditions with respect to a closed subgroup G of O（N）.By variational methods and the Caffarelli-Kohn-Nirenberg inequality,we obtain several existence and multiplicity results of G-symmetric solutions under certain appropriate assumptions on h and k.