A Dirichlet problem on the half-line for nonlinear equations with indefinite weight

Abstract

We study the existence of positive solutions on the half-line [0, ∞) for the nonlinear second order differential equationsatisfying Dirichlet type conditions, say x(0) = 0, lim t→∞ x(t) = 0. The function b is allowed to change sign and the nonlinearity F is assumed to be asymptotically linear in a neighborhood of zero and infinity. Our results cover also the cases in which b is a periodic function for large t or it is unbounded from below.Keywords. Second order nonlinear differential equation, boundary value problem on the half line, Dirichlet conditions, globally positive solution, disconjugacy, principal solution.MSC 2010: Primary 34B40, Secondary 34B18.

Global Kneser solutions to nonlinear equations with indefinite weight

“…The investigated problem can be also viewed as an extension to the half-line of recent results on nonlinear BVPs on a compact interval, see, e.g., [2] or [26] and references therein, when the weight has indefinite sign or definite sign, respectively. The paper is motivated also by [5,11] and completes some results there. More precisely, in [5] some asymptotic BVPs are studied for (1) when F (u) = |u| β sgn u, β > 0 and b(t) ≤ 0 and in [11] equations with Sturm-Liouville operator, that is when α = 1, are considered.…”

“…The paper is motivated also by [5,11] and completes some results there. More precisely, in [5] some asymptotic BVPs are studied for (1) when F (u) = |u| β sgn u, β > 0 and b(t) ≤ 0 and in [11] equations with Sturm-Liouville operator, that is when α = 1, are considered.…”

“…The BVP (1)-(2) arises in the investigation of radial solutions in a fixed exterior domain for elliptic equations, see, e.g., [7]. Boundary value problems (BVPs), associated to equations of type (1), have attracted considerable attention in the last years, especially when they are examined on unbounded domains, see [1,8,9,10,14,15]. Positive decreasing solutions of (1) are usually called Kneser solutions and have been investigated by many authors, see, e.g.…”

“…When b changes sign, the structure of nonoscillatory solutions is more complicated, due to the possible presence of the so-called weakly oscillatory solutions, that is nonoscillatory solutions with changing-sign derivatives, see, e.g., [4, page 1248]. As far as we know, in this case very few results deal with the existence of Kneser solutions and with their decay at infinity, see, e.g., [9,10,11,22,23]. The investigated problem can be also viewed as an extension to the half-line of recent results on nonlinear BVPs on a compact interval, see, e.g., [2] or [26] and references therein, when the weight has indefinite sign or definite sign, respectively.…”

“…[13, Theorem 4.2.2] Let (7) be nonoscillatory. Assume (9) and denote by y 0 , z 0 the principal solutions of (7) and (8), respectively, such that y 0 (t) > 0, z 0 (t) > 0 for t ≥ T 1 ≥ T and z 0 (T 1 ) = y 0 (T 1 ). Then we have for t ≥ T 1…”