Large and very singular solutions to semilinear elliptic equations

Abstract

We consider equation $$-\Delta u+f(x,u)=0$$ in smooth bounded domain $$\Omega \in \mathbb {R}^N$$ , $$N\geqslant 2$$ , 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.