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This paper deals with existence, uniqueness and global behaviour of a positive solution for the fractional boundary value problem D β (ψ(x) p (D α u)) = a(x)u σ in (0, 1) with the condition lim x→0 D β−1 (ψ(x) p (D α u(x))) = lim x→1 ψ(x) p (D α u(x)) = 0 and lim x→0 D α−1 u(x) = u(1) = 0, where β, α ∈ (1, 2], p (t) = t|t| p−2 , p > 1, σ ∈ (1 − p, p − 1), the differential operator is taken in the Riemann-Liouville sense and ψ, a : (0, 1) −→ R are non-negative and continuous functions that may are singular at x = 0 or x = 1 and satisfies some appropriate conditions.

Abstract. In this paper, we establish existence and asymptotic behavior of a positive classical solution to the following semilinear boundary value problem:Here Ω is an annulus in R n , n ≥ 3, σ < 1 and q is a positive function in C γ loc (Ω), 0 < γ < 1, satisfying some appropriate assumptions related to Karamata regular variation theory. Our arguments combine a method of sub-and supersolutions with Karamata regular variation theory.

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In this article, we study the existence and the boundary behavior of positive blow-up solutions for the semilinear elliptic coupled system. Our arguments are based on the subsupersolution method and the Karamata regular variation theory.