Our purpose of this paper is to study positive solutions of Lane-Emden equation − Δ u=VupinRN∖ {0} $$\begin{array}{} -{\it\Delta} u = V u^p\quad {\rm in}\quad \mathbb{R}^N\setminus\{0\} \end{array}$$(0.1) perturbed by a non-homogeneous potential V when p∈ [pc,N+2N− 2), $\begin{array}{} p\in [p_c, \frac{N+2}{N-2}), \end{array}$ where p c is the Joseph-Ludgren exponent. When p∈ (NN− 2,pc), $\begin{array}{} p\in (\frac{N}{N-2}, p_c), \end{array}$ the fast decaying solution could be approached by super and sub solutions, which are constructed by the stability of the k -fast decaying solution w k of − Δ u = u p in ℝ N ∖ {0} by authors in [9]. While the fast decaying solution w k is unstable for p∈ (pc,N+2N− 2), $\begin{array}{} p\in (p_c, \frac{N+2}{N-2}), \end{array}$ so these fast decaying solutions seem not able to disturbed like (0.1) by non-homogeneous potential V . A surprising observation that there exists a bounded sub solution of (0.1) from the extremal solution of − Δ u=uN+2N− 2 $\begin{array}{} -{\it\Delta} u = u^{\frac{N+2}{N-2}} \end{array}$ in ℝ N and then a sequence of fast decaying solutions and slow decaying solutions could be derived under appropriated restrictions for V .

中文翻译：

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超临界情况下受非齐次势扰动的Lane-Emden方程

我们这篇论文的目的是研究 Lane-Emden 方程的正解 − Δ u=VupinRN∖ {0} $$\begin{array}{} -{\it\Delta} u = V u^p\quad {\ rm in}\quad \mathbb{R}^N\setminus\{0\} \end{array}$$(0.1) 当 p∈ [pc,N+2N− 2) 时被非齐次势 V 扰动， $\begin{array}{} p\in [p_c, \frac{N+2}{N-2}), \end{array}$ 其中 pc 是 Joseph-Ludgren 指数。当 p∈ (NN− 2,pc), $\begin{array}{} p\in (\frac{N}{N-2}, p_c), \end{array}$ 可以接近快速衰减解由超解和子解构成，这些解是由 [9] 中的作者在 ℝ N ∖ {0} 中 − Δ u = up in ℝ N ∖ {0} 的 k 快速衰减解 wk 的稳定性构建的。而快速衰减解 wk 对于 p∈ (pc,N+2N− 2), $\begin{array}{} p\in (p_c, \frac{N+2}{N-2}), \ end{array}$ 所以这些快速衰减的解决方案似乎不能像 (0. 1) 由非均匀电位 V 。一个令人惊讶的观察结果，即从 − Δ u=uN+2N− 2 $\begin{array}{} -{\it\Delta} u = u^{\frac 的极值解中存在 (0.1) 的有界子解{N+2}{N-2}} \end{array}$ in ℝ N 然后可以在 V 的适当限制下推导出一系列快速衰减解和慢衰减解。