Existence of steady-state solutions for some one-dimensional conduction problems (Q1910031)

Existence of a solution to the problem (1) \((k(x, u)u')'= f(x, u)\), \(0\leq x\leq 1\), \(u(0)=a\), \(u(1)=b\) is proved under the assumption that the functions \(k(x,u)\), \(f(x,u)\) are piecewise continuous in \(x\) and some other hypotheses. The proof uses a priori estimates and the topological transversality theorem. Also a free-boundary problem closely related to (1), is investigated.