Abstract In this note, we study isolated singular positive solutions of Kirchhoff equation Mθ(u)(−Δ)u=upinΩ∖{0},u=0on∂Ω, $$\begin{array}{} \displaystyle M_\theta(u)(-{\it\Delta}) u =u^p \quad{\rm in}\quad {\it\Omega}\setminus \{0\},\qquad u=0\quad {\rm on}\quad \partial {\it\Omega}, \end{array}$$ where p > 1, θ ∈ ℝ, Mθ(u) = θ + ∫Ω |∇ u| dx, Ω is a bounded smooth domain containing the origin in ℝN with N ≥ 2. In the subcritical case: 1 < p < NN−2 $\begin{array}{} \displaystyle \frac{N}{N-2} \end{array}$ if N ≥ 3, 1 < p < + ∞ if N = 2, we employee the Schauder fixed point theorem to derive a sequence of positive isolated singular solutions for the above equation such that Mθ(u) > 0. To estimate Mθ(u), we make use of the rearrangement argument. Furthermore, we obtain a sequence of isolated singular solutions such that Mθ(u) < 0, by analyzing relationship between the parameter λ and the unique solution uλ of −Δu+λup=kδ0inB1(0),u=0on∂B1(0). $$\begin{array}{} \displaystyle -{\it\Delta} u+\lambda u^p=k\delta_0\quad{\rm in}\quad B_1(0),\qquad u=0\quad {\rm on}\quad \partial B_1(0). \end{array}$$ In the supercritical case: NN−2 $\begin{array}{} \displaystyle \frac{N}{N-2} \end{array}$ ≤ p < N+2N−2 $\begin{array}{} \displaystyle \frac{N+2}{N-2} \end{array}$ with N ≥ 3, we obtain two isolated singular solutions ui with i = 1, 2 such that Mθ(ui) > 0 under other assumptions.

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关于基尔霍夫方程的孤立奇点

摘要 在这篇笔记中，我们研究了基尔霍夫方程 Mθ(u)(−Δ)u=upinΩ∖{0},u=0on∂Ω, $$\begin{array}{} \displaystyle M_\theta 的孤立奇异正解(u)(-{\it\Delta}) u =u^p \quad{\rm in}\quad {\it\Omega}\setminus \{0\},\qquad u=0\quad {\rm on}\quad \partial {\it\Omega}, \end{array}$$ 其中 p > 1, θ ∈ ℝ, Mθ(u) = θ + ∫Ω |∇ u| dx, Ω 是一个有界光滑域，包含 ℝN 中的原点，N ≥ 2。在亚临界情况下： 1 < p < NN−2 $\begin{array}{} \displaystyle \frac{N}{N-2} \end{array}$ 如果 N ≥ 3, 1 < p < + ∞ 如果 N = 2，我们使用 Schauder 不动点定理推导出上述方程的一系列正孤立奇异解，使得 Mθ(u) > 0 . 为了估计 Mθ(u)，我们使用重排参数。此外，我们获得了一系列孤立的奇异解，使得 Mθ(u) < 0，通过分析参数λ与−Δu+λup=kδ0inB1(0),u=0on∂B1(0)的唯一解uλ之间的关系。$$\begin{array}{} \displaystyle -{\it\Delta} u+\lambda u^p=k\delta_0\quad{\rm in}\quad B_1(0),\qquad u=0\quad { \rm on}\quad \partial B_1(0)。\end{array}$$ 在超临界情况下：NN−2 $\begin{array}{} \displaystyle \frac{N}{N-2} \end{array}$ ≤ p < N+2N−2 $ \begin{array}{} \displaystyle \frac{N+2}{N-2} \end{array}$ 当 N ≥ 3，我们得到两个孤立的奇异解 ui，i = 1, 2 使得 Mθ(ui ) > 0 在其他假设下。