Mixed boundary-value problems for singular second-order ordinary differential equations

It is proved that the boundary-value problem
\begin{gather*} -u''+p_0(t)u(t)+\sum^m_{k=2}q_k(t) u^{2k+1}(t)+f_0(t)\varphi(u(t))=f(t), \quad 0<t<1, \\ u(a)=0, \quad u'(b)=0, \end{gather*}
has a solution, provided that the following conditions are fulfilled:
\begin{gather*} |p_0(t)|(t-a)\in L(a,b), \quad f(t)\sqrt{t-a}\in L(a,b), \\ 0\le f_0(t)\sqrt{t-a}\in L(a,b), \quad 0\le q_k(t)(t-a)^{k+1}\in L(a,b), \\ -c|u|\le\varphi(u)u, \quad c>0, \\ 1-\int^b_a p^-_0(t)(t-a)\,dt>0, \end{gather*}
and, for $\varphi(u)\equiv 0$, the Galerkin method converges in the norm of the space $H^1(a,b;a)$. Several theorems of a similar kind are presented.