Blow up boundary solutions of some semilinear fractional equations in the unit ball

Abstract

For γ > 0, we are interested in blow up solutions u ∈ C + (B) of the fractional problem in the unit ball BWe distinguish particularly two orders of singularity at the boundary: solutions exploding at the same rate than δ 1− α 2 (δ denotes the Euclidean distance) and those higher singular than δ 1− α 2 . As a consequence, it will be shown that the classical Keller-Osserman condition can not be readopted in the fractional setting.

Existence and nonexistence of positive solutions to the fractional equation Δ^α/2 u = –u^γ in bounded domains

“…We shall apply Schauder's fixed point theorem to prove the existence of positive solution of problem (2). Before going to the proof, we first give some interesting properties of the potential G α D (δ −λ ) which are technically needed.…”

“…Before going to the proof, we first give some interesting properties of the potential G α D (δ −λ ) which are technically needed. The upcoming lemma has been proved in [2], but for the reader's convenience we include the complete proof.…”

“…2. The question of whether problems (2) or (3) As a consequence, it turns out that the critical power characterizing the existence of a positive solution of problem (1) cannot be…”

Semilinear equations for non-local operators: beyond the fractional Laplacian

“…For the nonlinearity f throughout the paper we assume the condition Semilinear problems for the Laplacian have been studied for at least 40 years and we refer the reader to the monograph [36] for a detailed account. The study of semilinear problems for non-local operators is more recent and is mostly focused on the fractional Laplacian, see [21,14,1,2,5,4,6,12,20]. One of the important differences between the local and nonlocal equations is that in the non-local case the boundary blow-up solutions are possible even for linear equations.…”