Large and very singular solutions to semilinear elliptic equations

2021/08/20 by Shishkov, Andrey
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2108.09089

Abstract

We consider equation -Δu+f(x,u)=0 in smooth bounded domain Ω∈ℝ^N, N\geqslant2, with f(x,r)>0 in Ω×ℝ¹₊ and f(x,r)=0 on ∂Ω. We find the condition on the order of degeneracy of f(x,r) near ∂Ω, which is a criterion of the existence-nonexistence of a very singular solution with a strong point singularity on ∂Ω. Moreover, we prove that the mentioned condition is a sufficient condition for the uniqueness of a large solution and conjecture that this condition is also a necessary condition of the uniqueness.

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