On isolated singularities of Kirchhoff equations

Abstract

Abstract In this note, we study isolated singular positive solutions of Kirchhoff equation M θ ( u ) ( − Δ ) u = u p i n Ω ∖ { 0 } , u = 0 o n ∂ 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 ℝ N with N ≥ 2. In the subcritical case: 1 < p < N N − 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 − Δ u + λ u p = k δ 0 i n B 1 ( 0 ) , u = 0 o n ∂ B 1 ( 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: N N − 2 $\begin{array}{} \displaystyle \frac{N}{N-2} \end{array}$ ≤ p < N + 2 N − 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.