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Multiple positive solutions for a singular boundary value problem on infinite intervals at resonance. (English) Zbl 1126.34016

Using the Leggett-Williams fixed point theorem the authors establish the existence of three positive solutions to the singular boundary value problem
\[ u''+f(t,u(t))=0,\;t>0, \]
\[ u(0)=0,\;u'(+\infty)=u_{+\infty}, \]
where \(f\in C((0,\infty)^2,\mathbb R^+)\), \(u'(+\infty)=\lim_{t\to+\infty}u'(t)\), \(\mathbb R^+=[0,\infty)\) and \(u'(+\infty)>0.\) The basic assumptions are: There exist \(\lambda\), \(\mu\), \(N\) and \(M\) such that \(-\infty<\lambda<0<\mu<+\infty\), \(0<N\leq1\leq M,\)
\[ \begin{aligned} \delta^\mu f(t,u)&\leq f(t,\delta u)\leq\delta^\lambda f(t,u)\quad\text{for } 0<\delta\leq N,\\ \delta^\lambda f(t,u)&\leq f(t,\delta u)\leq\delta^\mu f(t,u)\quad\text{for }\delta\geq M,\end{aligned} \]
\[ 0<\int_0^1s^{1+\lambda}f(s,1)\,ds<+\infty \quad \text{and}\quad 0<\int_1^\infty(1+s)^\mu f(s,1)\,ds<+\infty. \]

MSC:

34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
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