The positive solution is studied for a (k, n - k) conjugate boundary value problem. The nonlinear term is allowed to be singular with respect to both the time and space variables. By applying the approximation theorem for completely continuous operators and the Guo-Krasnosel’skii fixed point theorem of cone expansion-compression type, an existence theorem for a positive solution is established.
We study a third order singular boundary value problem with multi-point boundary conditions. Sufficient conditions are obtained for the existence of positive solutions of the problem. Recent results in the literature are significantly extended and improved. Our analysis is mainly based on a nonlinear alternative of Leray-Schauder.
We study the existence of positive solutions for the p-Laplace Emden-Fowler equation. Let H and G be closed subgroups of the orthogonal group O(N) such that H G ⊂ O(N). We denote the orbit of G through x ∈ R N by G(x), i.e., G(x) := {gx: g ∈ G}. We prove that if H(x) G(x) for all x ∈ Ω and the first eigenvalue of the p-Laplacian is large enough, then no H invariant least energy solution is G invariant. Here an H invariant least energy solution means a solution which achieves the minimum of the Rayleigh quotient among all H invariant functions. Therefore there exists an H invariant G non-invariant positive solution.
We study solutions tending to nonzero constants for the third order differential equation with the damping term (a1(t)(a2(t)x ′ (t))′ ) ′ + q(t)x ′ (t) + r(t)f(x(ϕ(t))) = 0 in the case when the corresponding second order differential equation is oscillatory.
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).