Large and very singular solutions to semilinear elliptic equations

We consider equation - Δ u+ f(x, u) = 0 in smooth bounded domain Ω ∈ RN, N⩾ 2 , with f(x, r) > 0 in Ω×R+1 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. © 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.