A Nonlocal Operator Breaking the Keller-Osserman Condition

Abstract

This work is concerned about the existence of solutions to the nonlocal semilinear problem - N J (x - y) (u (y) - u (x)) y + h (u (x)) = f (x) x ω u = g x N ω, (-) R N J(x-y)(u(y)-u(x%)), dy+h (u(x)) = f(x),& ω u=g, x R N ω. verifying that lim x → ω x ω u (x) = + ∞ known in the literature as large solutions. We find out that the relation between the diffusion and the absorption term is not enough to ensure such existence, not even assuming that the boundary datum g blows up close to ω. On the contrary, the role to obtain large solutions is played only by the interior source f, which gives rise to large solutions even without the presence of the absorption. We determine necessary and sufficient conditions on f providing large solutions and compute the blow-up rates of such solutions in terms of h and f. Finally, we also study the uniqueness of large solutions.