Large and very singular solutions to semilinear elliptic equations

TL;DR

This paper investigates conditions under which very singular solutions exist or are unique for certain semilinear elliptic equations in bounded domains, focusing on the degeneracy of the nonlinearity near the boundary.

Contribution

It establishes a criterion based on degeneracy order for the existence and uniqueness of very singular solutions in semilinear elliptic equations.

Findings

Derived a degeneracy condition for solution existence

Proved sufficiency of the condition for uniqueness

Conjectured the condition's necessity for uniqueness

Abstract

We consider equation $-\Delta u+f(x,u)=0$ in smooth bounded domain $\Omega\in\mathbb{R}^N$, $N\geqslant2$, with $f(x,r)>0$ in $\Omega\times\mathbb{R}^1_+$ and $f(x,r)=0$ on $\partial\Omega$. We find the condition on the order of degeneracy of $f(x,r)$ near $\partial\Omega$, which is a criterion of the existence-nonexistence of a very singular solution with a strong point singularity on $\partial\Omega$. 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.