We consider the radial symmetric stationary solutions of $$u_{t}=\varDelta u-|x|^{q}u^{-p}$$ . We first give a result on the existence of the negative value functions that satisfy the radial symmetric stationary problem on a finite interval for $$p \in 2{\mathbb{N}}$$ , $$q\in{\mathbb{R}}$$ . Moreover, we give the asymptotic behavior of u(r) and $$u'(r)$$ at both ends of the finite interval. Second, we obtain the existence of the positive radial symmetric stationary solutions with the singularity at $$r=0$$ for $$p\in{\mathbb{N}}$$ and $$q\ge -2$$ . In fact, the behavior of solutions for $$q>-2$$ and $$q=-2$$ are different. In each case of $$q=-2$$ and $$q>-2$$ , we derive the asymptotic behavior for $$r \rightarrow 0$$ and $$r \rightarrow \infty $$ . These facts are studied by applying the Poincare compactification and basic theory of dynamical systems.

中文翻译：

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具有空间相关非线性的 MEMS 型反应扩散方程的径向对称平稳解

我们考虑 $$u_{t}=\varDelta u-|x|^{q}u^{-p}$$ 的径向对称平稳解。我们首先给出在有限区间上满足径向对称平稳问题的负值函数的结果，对于 $$p \in 2{\mathbb{N}}$$ , $$q\in{\mathbb{ R}}$$ 。此外，我们给出了 u(r) 和 $$u'(r)$$ 在有限区间两端的渐近行为。其次，对于 $$p\in{\mathbb{N}}$$ 和 $$q\ge -2$$ ，我们获得了奇点为 $$r=0$$ 的正径向对称平稳解的存在性。实际上，$$q>-2$$ 和 $$q=-2$$ 的解的行为是不同的。在 $$q=-2$$ 和 $$q>-2$$ 的每种情况下，我们推导出 $$r \rightarrow 0$$ 和 $$r \rightarrow \infty $$ 的渐近行为。