On isolated singularities of Kirchhoff--type Laplacian problems

2017/08/10 by Chen, Huyuan, Fall, Mouhamed Moustapha, Zhang, Binlin
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1708.03041

Abstract

In this paper, we study isolated singular positive solutions for the following Kirchhoff--type Laplacian problem: -(θ+∫_Ω |∇ u| dx)Δu =u^p \rm in Ω∖ \0\, u=0 \rm on ∂ Ω, where p>1, θ∈ \R, Ω is a bounded smooth domain containing the origin in \R^N with N≥ 2. In the subcritical case: 10. To estimate M_θ(u), we make use of the rearrangement argument. Furthermore, we obtain a sequence of isolated singular solutions such that M_θ(u)<0, by analyzing relationship between the parameter λ and the unique solution u_λ of -Δu+λu^p=kδ₀ \rm in B₁(0), u=0 \rm on ∂ B₁(0). In the supercritical case: N/(N-2)≤ p0 under some appropriate assumptions.

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