Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 997-1007

EXISTENCE AND NONEXISTENCE OF POSITIVE SOLUTIONS TO THE FRACTIONAL EQUATION Δ^α/2u = –u^γ IN BOUNDED DOMAINS

Mohamed Ben Chrouda

Institut Supérieur d'Informatique et de Mathématiques
5000, Monastir, Tunisie; mohamed.benchrouda 'at' isimm.rnu.tn

Abstract. This paper deals with the question of the existence of a positive solution to the boundary value problem involving the fractional Laplacian

Δ^α/2u = –u^γ on D,
u = 0 on D^c,
lim_x→z∈∂Dδ(x)^{1
– α/2}u(x) = f(z),

for γ > 1, where δ(x) denotes the Euclidean distance from x to the boundary ∂D. We distinguish two cases of nonnegative data f: trivial and nontrivial.

2010 Mathematics Subject Classification: Primary 31B05, 31C35, 34B27, 47H10.

Key words: Fractional Laplacian, Martin kernel, Green function.

Reference to this article: M. Ben Chrouda: Existence and nonexistence of positive solutions to the fractional equation Δ^α/2u = –u^γ in bounded domains. Ann. Acad. Sci. Fenn. Math. 42 (2017), 997-1007.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4262

EXISTENCE AND NONEXISTENCE OF POSITIVE SOLUTIONS TO THE FRACTIONAL EQUATION Δα/2u = –uγ IN BOUNDED DOMAINS

Mohamed Ben Chrouda

EXISTENCE AND NONEXISTENCE OF POSITIVE SOLUTIONS TO THE FRACTIONAL EQUATION Δ^α/2u = –u^γ IN BOUNDED DOMAINS