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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