×

zbMATH — the first resource for mathematics

A boundary value problem on the half-line for superlinear differential equations with changing sign weight. (English) Zbl 1269.34027
The paper deals with the existence of positive solutions \(x\) for the superlinear differential equation of the form \[ (r(t)\Phi(x'))'=q(t)f(x) \] satisfying the boundary conditions \[ x(0)=\lim_{t\to+\infty}x(t) = 0. \] Here, \(\Phi(u)=|u|^p\operatorname{sgn}{u}\), and \(p>0\), \(f\) is continuous on \(\mathbb{R}\) such that \(uf(u)>0\), \(u\neq 0\), \(\lim_{u\to 0^+}f(u)/{\Phi(u)}=0\) and \(\lim_{u\to\infty}f(u)/{\Phi(u)}=\infty\). The functions \(r, q\) are continuous, \(r(t)>0\) for \(t\geq 0\) and \(q\) satisfies \(q(t)\leq 0\), \(q(t)\not\equiv 0\) for \(t\in[0,1]\) and \(q(t)\geq 0\) for \(t>1\), \(q(t)\not\equiv 0\) for large \(t\). Let \(R(t):=\int_0^tr^{-1/p}(s)\,ds\) and \(J:=\lim_{T\to\infty}\int_1^T\Big(r^{-1}(t)\int_t^Tq(s)ds\Big)^{1/p}dt\).
The main result of the paper reads as follows: Theorem 1.1. Assume either \(R(\infty)=\infty\) and \(J=\infty\), or \(R(\infty)<\infty\). Then the boundary value problem has a solution. Further, in the remaining case \(J=\infty\) and \(R(\infty)=\infty\), the boundary value problem has no solution.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite