Existence of nonnegative solutions of singular boundary-value problems for second-order ordinary differential equations

Abstract

It is proved that the boundary-value problem

$$ - u'' + p(t)u + q(t)u^n = f(t),u(a) = u(b) = 0,n \geqslant 2,$$

has a unique nonnegative solution if the following conditions are fulfilled:

$$\begin{gathered} p(t)(b - t)(t - a) \in L(a,b),0 \leqslant q(t)[(b - t)(t - a)]^{\tfrac{{n + 1}}{2}} \in L(a,b),0 \leqslant f(t)\sqrt {(b - t)(t - a)} \in L(a,b), \hfill \\ 1 - \tfrac{1}{{b - a}}\int\limits_a^b {p^ - (t)(t - a)(b - t)dt > 0.} \hfill \\ \end{gathered} $$

. Bibliography: 2 titles.