We study the singular periodic boundary value problem of the form (φ(u ' ))' + h(u)u ' = g(u) + e(t), u(0) = u(T), u ' (0) = u ' (T), where φ: R→R is an increasing and odd homeomorphism such that φ(R ) = R, h ∈ C[0, ∞), e ∈ L1[0, T] and g ∈ C(0, ∞) can have a space singularity at x = 0, i.e. lim sup x→0+ |g(x)| = ∞ may hold. We prove new existence results both for the case of an attractive singularity, when lim inf x→0+ g(x) = −∞, and for the case of a strong repulsive singularity, when lim x→0+ R 1 x g(ξ)dξ = ∞. In the latter case we assume that φ(y) = φp(y) = |y| p−2 y, p > 1, is the well-known p-Laplacian. Our results extend and complete those obtained recently by Jebelean and Mawhin and by Liu Bing.
We study singular boundary value problems with mixed boundary conditions of the form (p(t)u ' ) ' + p(t)f(t, u, p(t)u ' ) = 0, lim t→0+ p(t)u ' (t) = 0, u(T) = 0, where [0, T] ⊂ . We assume that D ⊂ R 2 , f satisfies the Carathéodory conditions on (0, T) × D, p ∈ C[0, T] and 1/p need not be integrable on [0, T]. Here f can have time singularities at t = 0 and/or t = T and a space singularity at x = 0. Moreover, f can change its sign. Provided f is nonnegative it can have even a space singularity at y = 0. We present conditions for the existence of solutions positive on [0, T).