Some existence and nonexistence principles for a class of singular boundary value problems. (English) Zbl 0860.34010

The author presents existence and nonexistence results to the following singular boundary value problem \[ \begin{aligned} (p(t)x'(t))'+ p(t)f(t,x(t)) &=0, \qquad 0<t<1,\\ \alpha x(0)- \beta \lim_{t\to 0}p(t)x'(t) &=0, \tag{1}\\ \gamma x(1)+ \delta \lim_{t\to1} p(t)x'(t) &=0, \end{aligned} \] where \(f:(0,1)\times (0,\infty)\to (0,\infty)\) is continuous, \(p\in C^1[0,1]\), and \(p(t)>0\) for \(t\in(0,1)\), and \(\alpha,\beta, \gamma,\delta\geq 0\), \(\beta\gamma+ \alpha\delta+ \alpha\gamma>0\). The nonlinear term \(f(t,u)\) may be singular at \(u=0\) and \(t=0,1\).
An example is \(f(t,u)= t^{-m} (1-t)^{-m} u^{-n}\), where \(m,n>0\). The author shows that, when \(m<2\), \(n>0\), problem (1) has a solution for general boundary conditions. When \(m>2\), no solutions exist.


34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


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