Math 6410 • ODE

\begin{enumerate}
\item State the theorem of Rellich-Kondrachov for bounded open sets \(G\subset\R^N\).
\item Let
\[
  f:\overline{G}\times\R\to\R
\]
be a continuous bounded function such that
\[
  \lvert f(x,u)\rvert\le M,\qquad (x,u)\in\overline{G}\times\R
\]
where \(M\) is a constant. Show that the boundary value problem:
\[
  -\Delta u=f(x,u),\qquad u\in H^1_0(G),
  \tag{8}
\]
has a solution by converting (8) into an equivalent fixed point problem in \(L^2(G)\) and
by applying Schauder's fixed point theorem (give reasons why its hypotheses hold!).
\end{enumerate}