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

It is proved that the boundary-value problem
$$ -u''+p(t)u+q(t)u^n=f(t), \quad u(a)=u(b)=0, \quad n\ge 2, $$
has a unique nonnegative solution if the following conditions are fulfilled:
\begin{gather*} 0\le q (t)[(b-t)(t-a)]^{\frac{n+1}{2}}\in L(a,b); \quad 0\le f(t)\sqrt{(b-t)(t-a)}\in L(a,b); \\ 1-\frac1{b-a}\int^{b}_{a}p^-(t)(t-a)(b-t)dt>0. \end{gather*}
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