Consider boundary value problems for a functional differential equation ( x (n) (t) = (T +x)(t) − (T −x)(t) + f(t), t ∈ [a, b], lx = c, where T +, T − : C[a, b] → L[a, b] are positive linear operators; l: ACn−1 [a, b] → R n is a linear bounded vector-functional, f ∈ L[a, b], c ∈ ℝ n , n ≥ 2. Let the solvability set be the set of all points (T +, T −) ∈ ℝ + 2 such that for all operators T +, T − with kT ±kC→L = T ± the problems have a unique solution for every f and c. A method of finding the solvability sets are proposed. Some new properties of these sets are obtained in various cases. We continue the investigations of the solvability sets started in R. Hakl, A. Lomtatidze, J. Šremr: Some boundary value problems for first order scalar functional differential equations. Folia Mathematica 10, Brno, 2002.
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.