Large and very singular solutions to semilinear elliptic equations

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.