## Uniqueness of monotone positive solutions for singular boundary value problems.(English)Zbl 1023.34019

Here, the nonlinear scalar differential equation $\frac 1{p(t)}(p(t)y'(t))'=q(t)f(t,y(t),p(t)y'(t))$ is considered, where $$p$$ and $$q$$ are positive on $$(0,1)$$, “singular” at $$t=0,1$$ and/or $$y=0$$ and $$f\in C((0,1)\times{\mathbb{R}}^+\times{\mathbb{R}}^-)$$, associated to the boundary conditions $\lim\limits_{t\to 0+}p(t)y'(t)=m,\quad m\leq 0,\quad\lim\limits_{t\to 1-}y(t)=0.$
It is proved that, if $f(t,y,v)<0,\quad\text{for all}\quad t\in[0,1],\quad y>0,\quad v\leq 0,\tag{1}$
$\lim\limits_{y\to 0+}\max_{0\leq t\leq 1}f(t,y,v)= -\infty,\quad\lim\limits_{y\to+\infty}\min_{0\leq t\leq 1}f(t,y,v)=0,\quad m_0\leq v\leq m,\tag{2}$
$p\in C[0,1]\cap C^1(0,1),\quad\int\limits_0^1\frac{dt}{p(t)}<\infty,\quad q\in C(0,1),\quad\int\limits_0^1p(t)q(t)dt<\infty,\tag{3}$ and, for each $$\eta>0$$, $$H>\eta$$, there exists a continuous function $$\psi:[0,\infty)\to[0,\infty)$$ such that $\int\limits_{-m}^{+\infty}\frac{ds}{\psi(s)}>\int\limits_0^1p(t)q(t)dt, \quad |f(t,y,v)|\leq\psi(|v|),\quad (t,y,v)\in[0,1]\times[\eta,H]\times{\mathbb{R}},\tag{4}$ then the boundary value problem has a positive decreasing solution.
Furthermore, there exists $$M>0$$, such that for any such solution, $0<y(t)\leq M,\quad 0\leq t<1.$

### MSC:

 34B16 Singular nonlinear boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B05 Linear boundary value problems for ordinary differential equations