We consider the problem $\{\begin{array}{ll}-\mathrm{\Delta }u=\lambda K(x)f(u)& \text{in }{B}_1^c,\\ u(x)=0& \text{on }\mathrm{\partial }{B}_1,\\ u(x)\to 0& \text{as }|x|\to \mathrm{\infty },\end{array}$ where $B_1^c=\{x\in {\mathbb{R}}^n:|x|>1\}$, n>2, λ is a positive parameter, $K:B_1^c\to {\mathbb{R}}^{+}$ belongs to a class of continuous functions which satisfy certain decay assumptions, and $f:[0,\mathrm{\infty })\to \mathbb{R}$ belongs to a class of continuous functions which are superlinear at ∞ with $f(0)<0$. Recently, several authors have studied positive radial solutions to this problem assuming that the weight function is radial. We allow non-radial weights and study the existence and non-existence of positive solutions. We prove the existence of a positive solution for small values of λ using variational methods. A non-existence result is established for large values of λ.

我们考虑问题$\{\begin{array}{ll}-\mathrm{\Delta }你=\lambda ķ(X)F(你)& \text{在 }{乙}_1^C,\\ 你(X)=0& \text{上 }\mathrm{\partial }{乙}_1,\\ 你(X)\to 0& \text{作为 }|X|\to \mathrm{\infty },\end{array}$在哪里${乙}_1^C=\{X\in {\mathbb{R}}^n：|X|>1\}$, n >2, λ为正参数,$ķ：{乙}_1^C\to {\mathbb{R}}^{+}$属于满足某些衰减假设的一类连续函数，并且$F：[0,\mathrm{\infty })\to \mathbb{R}$属于一类在 ∞ 处是超线性的连续函数$F(0)<0$. 最近，一些作者研究了这个问题的正径向解决方案，假设权重函数是径向的。我们允许非径向权重并研究正解的存在和不存在。我们使用变分方法证明了对于较小的λ值存在正解。对于较大的λ值，建立一个不存在的结果。