描述了一类带权的有狄立克莱边界条件的非线性椭圆方程:-div(|x|~(-2α)▽u)-μ/(|x|~(2(a+1)))u=|x|~(-bp)|u|~(p-2)u+λ|u|~(q-2)u在零点附近正解的存在性问题,其中0∈Ω是R~N(N≥3)中具有光滑边界的有界区域,并在临界的加权Sobolev-Hardy指数情况下得到两个正解.

Let 0 E∈ΩR~N(N≥3)be a bounded domain with smooth boundary.We characterize an exact growth order near zero for positive solutions of a class of weighted nonlinear elliptic equations -div(|x|~(-2a)▽u)-u/(|x|~(2(a+1)))u=|x|~(-bp)|u|~(p-2)u+λ|u|~(q-2)u with Dirichlet boundary condition.Then we obtain multiple positive solutions for equations involving critical weighted Sobolev-Hardy exponents.